PARABOLIC REGULARITY OF SPECTRAL FUNCTIONS. PART I: THEORY
ASHKAN MOHAMMADI,111Department of Mathematics, Georgetown University,
Washington, D.C. 20057, USA (ashkan.mohammadi@georgetown.edu). and M. EBRAHIM
SARABI222Department of Mathematics, Miami University, Oxford, OH 45065, USA
(sarabim@miamioh.edu). Research of this author is partially supported by the
U.S. National Science Foundation under the grant DMS 2108546.

Abstract. This paper is devoted to the study of the second-order variational
analysis of spectral functions. It is well-known that spectral functions can
be expressed as a composite function of symmetric functions and eigenvalue
functions. We establish several second-order properties of spectral functions
when their associated symmetric functions enjoy these properties. Our main
attention is given to characterize parabolic regularity for this class of
functions. It was observed recently that parabolic regularity can play a
central rule in ensuring the validity of important second-order variational
properties such as twice epi-differentiability. We demonstrates that for
convex spectral functions, their parabolic regularity amounts to that of their
symmetric functions. As an important consequence, we calculate the second
subderivative of convex spectral functions, which allows us to establish
second-order optimality conditions for a class of matrix optimization
problems.
Key words. spectral functions, parabolic regularity, twice epi-
differentiability, composite optimization, second-order optimality conditions
Mathematics Subject Classification (2000) 49J52, 15A18, 49J53

##  1 Introduction

This paper aims to study second-order variational properties, including
parabolic regularity and twice epi-differentiability, of spectral functions.
These are functions
$g:{\mathbb{S}}^{n}\to\overline{\mathbb{R}}:=[-\infty,\infty]$, where
${\mathbb{S}}^{n}$ stands for the real vector space of $n\times n$ symmetric
matrices, that are orthogonally invariant, namely for any $n\times n$
symmetric matrix $X$ and any $n\times n$ orthogonal matrix $U$, we have

$g(X)=g(U^{\top}XU).$

It is well-known (cf. [15, Proposition 4]) that any spectral function $g$ can
be equivalently expressed in a composite form

$g(X):=(\theta\circ\lambda)(X),\;\;X\in{\mathbb{S}}^{n},$ (1.1)

where $\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$ is a permutation-
invariant function on ${\mathbb{R}}^{n}$, called symmetric, and $\lambda$ is a
function, which assigns to each matrix $X\in{\mathbb{S}}^{n}$ its eigenvalue
vector $\big{(}\lambda_{1}(X),\ldots,\lambda_{n}(X)\big{)}$ arranged in
nonincreasing order.

Davis [10] showed that convexity of the permutation-invariant function
$\theta$ in (1.1) is inherited by the spectral function $g$. Similar
observation was made by Lewis [14] about differentiability and strict
differentiability and by Lewis and Sendov [16] about twice differentiability.
It was shown in [14, 15] and [8], respectively, that the calculation of
different notions of subdifferentials of spectral functions and prox-
regularity, which plays an important role in second-order variational
analysis, enjoy this striking pattern as well.

The main question that we are trying to answer in this paper is whether such a
striking pattern can be extended for other important second-order variational
properties, including parabolic regularity (see Definition 5.1) and twice epi-
differentiability. While the former was first introduced two decades ago in
[23, Definition 13.69], its persuasive role in second-order variational
analysis was revealed quite recently in [18, 20]. In particular, it was
demonstrated in [20] that important second-order variational properties of a
composite function $\psi\circ F$, where
$\psi:{\mathbb{E}}_{2}\to(-\infty,\infty]$ is convex and
$F:{\mathbb{E}}_{1}\to{\mathbb{E}}_{2}$ is twice differentiable with
${\mathbb{E}}_{1}$ and ${\mathbb{E}}_{2}$ being finite-dimensional Hilbert
spaces, can be established at any $\bar{x}\in\mbox{\rm dom}\,(\psi\circ F)$
provided that $\psi$ is parabolically regular and that the following metric
subregularity constraint qualification is satisfied (cf. [20, Definition
4.2]): There exists a constant $\kappa\geq 0$ such that the estimate

${\rm dist}\big{(}x,\mbox{\rm dom}\,(\psi\circ F)\big{)}\leq\kappa\,{\rm
dist}(F(x),\mbox{\rm dom}\,\psi)$ (1.2)

holds for all $x$ sufficiently close to $\bar{x}$. Thus, it is natural to ask
whether a similar approach can be utilized for the composite representation in
(1.1) of spectral functions. To do so, two major obstacles seem to hinder
proceeding with the approach in [20]: 1) the lack of twice differentiability
of the inner function $\lambda(\cdot)$ in (1.1); 2) the validity of a
constraint qualification similar to the aforementioned condition for the
composite form (1.1). Given the composite representation (1.1), it follows
from [8, Proposition 2.3] that for any $X\in{\mathbb{S}}^{n}$, the equality

${\rm dist}(X,\mbox{\rm dom}\,g)={\rm dist}(\lambda(X),\mbox{\rm
dom}\,\theta)$ (1.3)

always holds. This simple but important observation from [8] tells us that the
required constraint qualification for dealing with the composite form (1.1) is
automatically satisfied. Moreover, looking closer into the established theory
in [18, 20] tells us that twice differentiability of the inner function was
not required. Indeed, a quadratic expansion will suffice to proceed in both
these publications. Such a quadratic expansion, which is of a parabolic type,
is already achieved in [24] for eigenvalue functions. These open the door for
using the approach from [18, 20] to study second-order variational properties
of spectral functions.

The outline of the paper is as follows. Section 2 recalls important notation
and concepts related to the eigenvalue function. In Section 3, we begin with
establishing a chain rule for subderivative of the spectral function in (1.1).
Section 4 is devoted to the study of the parabolic drivability of spectral
sets. We will obtain a chain rule for the parabolic subderivative, which plays
a central role in the study of parabolic regularity of spectral functions. It
is also shown that the parabolic subderivative is a symmetric function with
respect to a subset of the space of orthogonal matrices. In Section 5, we
demonstrate that the spectral function $g$ in (1.1) is parabolically regular
if and only if the symmetric function $\theta$ in (1.1) enjoys this property.
As a consequence, we are going to calculate the second subderivative of
spectral functions when the symmetric functions associated with them are
convex. This allows us to find second-order optimality conditions for a class
of matrix optimization problems.

##  2 Notation

In what follows, ${\mathbb{E}}$ is a finite-dimensional Hilbert space. By
$\mathbb{B}$ we denote the closed unit ball in the space in question and by
$\mathbb{B}_{r}(x):=x+r\mathbb{B}$ the closed ball centered at $x$ with radius
$r>0$. For any set $C\subset{\mathbb{E}}$, its indicator function is defined
by $\delta_{C}(x)=0$ for $x\in C$ and $\delta_{C}(x)=\infty$ otherwise. We
denote by ${\rm dist}(x,C)$ the distance between $x\in{\mathbb{E}}$ and a set
$C$. For $v\in{\mathbb{E}}$, the subspace $\\{w\in{\mathbb{E}}|\,\langle
w,v\rangle=0\\}$ is denoted by $[v]^{\bot}$. We denote by ${\mathbb{R}}_{+}$
(respectively, ${\mathbb{R}}_{-}$) the set of non-negative (respectively, non-
positive) real numbers. Given an $n\times n$ matrix $Z$ and index sets
$I,J\subseteq\\{1,\ldots,n\\}$, denote by $Z_{IJ}$ the submatrix of $Z$
obtained by removing all the rows of $Z$ not in $I$ and all the columns of $Z$
not in $J$ . The matrix $Z_{I}$ is the submatrix of $Z$ with columns specified
by $I$. Particularly, $Z_{i}$ is the $i$-th column of $Z$ and $Z_{ij}$ is the
entry of $Z$ at $(i,j)$ position. Denote by $Z^{\dagger}$ the Moore–Penrose
generalized inverse of $Z$. Finally, the cardinality of the set
$I\subset{\mathbb{N}}$, where ${\mathbb{N}}$ stands for the set of natural
numbers, is denoted by $|I|$.

Throughout this paper, we denote by ${\mathbb{R}}^{n\times m}$ the space of
all real $n\times m$ matrices and by ${\mathbb{S}}^{n}$ the space of all real
$n\times n$ symmetric matrices equipped with the inner product

$\langle X,Y\rangle=\mbox{\rm tr}\,(XY),\quad X,Y\in{\mathbb{S}}^{n}.$

The induced Frobenius norm of $X\in{\mathbb{S}}^{n}$ is defined via the trace
inner product by $\|X\|=\sqrt{\mbox{\rm tr}\,(X^{2})}$. Given
$X\in{\mathbb{S}}^{n}$, its eigenvalues, in nonincreasing order, are denoted
by

$\lambda_{1}(X)\geq\lambda_{2}(X)\geq\cdots\geq\lambda_{n}(X).$

For any vector $x=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$, denote by
$\mbox{\rm Diag}\,(x)$ the diagonal matrix whose $i$-th diagonal entry is
$x_{i}$ for any $i=1,\ldots,n$. The set of all real $n\times n$ orthogonal
matrices is denoted by ${\mathbb{O}}^{n}$. It is known that for any
$X\in{\mathbb{S}}^{n}$, there exists an orthogonal matrix $U$ for which we
have

$X=U\mbox{\rm
Diag}\,(\lambda(X))U^{\top}\quad\mbox{with}\;\;\lambda(X):=\big{(}\lambda_{1}(X),\ldots,\lambda_{n}(X)\big{)}.$
(2.1)

For a given matrix $X\in{\mathbb{S}}^{n}$, the set of such orthogonal matrices
$U$ is denoted by ${\mathbb{O}}^{n}(X)$. We say that two matrices
$X,Y\in{\mathbb{S}}^{n}$ admit a simultaneous spectral decomposition if there
exists $U\in{\mathbb{O}}^{n}$ such that $U^{\top}XU$ and $U^{\top}YU$ are
diagonal matrices. The matrices $X$ and $Y$ are said to have a simultaneous
ordered spectral decomposition if there exists $U\in{\mathbb{O}}^{n}$ such
that $U^{\top}XU=\mbox{\rm Diag}\,(\lambda(X))$ and $U^{\top}YU=\mbox{\rm
Diag}\,(\lambda(Y))$. It is well-known that for any two matrices
$X,Y\in{\mathbb{S}}^{n}$, the estimate

$\|\lambda(X)-\lambda(Y)\|\leq\|X-Y\|$ (2.2)

always holds. Moreover, equality in this estimate amounts to $X$ and $Y$
admitting a simultaneous ordered spectral decomposition. It is not hard to see
that the estimate in (2.2) amounts to the trace inequality, known as Fan’s
inequality,

$\langle X,Y\rangle\leq\langle\lambda(X),\lambda(Y)\rangle,\quad
X,Y\in{\mathbb{S}}^{n}.$ (2.3)

Assume that $\mu_{1}(X)>\cdots>\mu_{r}(X)$ are distinct eigenvalues of
$X\in{\mathbb{S}}^{n}$ and define then the index sets

$\alpha_{m}:=\big{\\{}i\in\\{1,\ldots,n\\}|\;\lambda_{i}(X)=\mu_{m}(X)\big{\\}}\quad\mbox{for
all}\;\;m=1,\ldots,r.$ (2.4)

Moreover, define ${\ell}_{i}(X)$ for any $i\in\\{1,\ldots,n\\}$ to be the
number of eigenvalues of $X$ that are equal to $\lambda_{i}(X)$ but are ranked
before $\lambda_{i}(X)$ including $\lambda_{i}(X)$. This integer allows us to
locate $\lambda_{i}(X)$ in the group of the eigenvalues of $X$ as follows:

$\lambda_{1}(X)\geq\cdots\geq\lambda_{i-\ell_{i}(X)}>\lambda_{i-\ell_{i}(X)+1}(X)=\cdots=\lambda_{i}(X)\geq\cdots\geq\lambda_{n}(X).$
(2.5)

Note that the index sets $\alpha_{m}$ present a partition of
$\\{1,\ldots,n\\}$, meaning that $\\{1,\ldots,n\\}=\cup_{m=1}^{r}\alpha_{m}$.
In what follows, we often drop $X$ from $\ell_{i}(X)$ when the dependence of
${\ell}_{i}$ on $X$ can be seen clearly from the context. Given an $n\times n$
matrix $W$, it is not hard to see that for any $U\in{\mathbb{O}}^{n}$ and any
$m=1,\ldots,r$, we always have

$U_{\alpha_{m}}^{\top}UWU^{\top}U_{\alpha_{m}}=W_{\alpha_{m}\alpha_{m}}.$
(2.6)

This simple observation will often be utilized in Section 5.

The following estimates are an easy consequence of [24, Proposition 1.4] (cf.
see the proof of [24, Theorem 1.5]) and play a major role in our second-order
variational analysis of eigenvalue functions in this paper.

######  Proposition 2.1 (first-order expansion of eigenvalue functions).

Assume that $X\in{\mathbb{S}}^{n}$ has the eigenvalue decomposition (2.1) for
some $U\in{\mathbb{O}}^{n}(X)$. Let $\mu_{1}>\cdots>\mu_{r}$ be distinct
eigenvalues of $X$. Then for any $H\in{\mathbb{S}}^{n}$ that $H\to 0$ and any
$i\in\\{1,\ldots,n\\}$ the estimates

$\lambda_{i}(X+H)=\lambda_{i}(X)+\lambda_{{\ell}_{i}}\big{(}U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}+U_{\alpha_{m}}^{\top}XH(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{)}+O(\|H\|^{3})$
(2.7)

and

$\lambda_{i}\big{(}X+H\big{)}=\lambda_{i}(X)+\lambda_{{\ell}_{i}}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})+O(\|H\|^{2})$
(2.8)

hold, where $m\in\\{1,\ldots,r\\}$ with $i\in\alpha_{m}$.

Note that the estimate (2.8) clearly tells us that the eigenvalue function
$\lambda_{i}(\cdot)$, $i\in\\{1,\ldots,n\\}$, is directionally differentiable
at $X$ at any direction $H\in{\mathbb{S}}^{n}$ and its directional derivative
$\lambda_{i}^{\prime}(X;H)$ can be calculated by

$\lambda_{i}^{\prime}(X;H):=\lim_{t\downarrow
0}\dfrac{\lambda_{i}(X+tH)-\lambda_{i}(X)}{t}=\lambda_{{\ell}_{i}}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})$
(2.9)

where $m\in\\{1,\ldots,r\\}$ with $i\in\alpha_{m}$. In another words, we have

$\lambda^{{}^{\prime}}(X;H)=\big{(}\lambda(U_{\alpha_{1}}^{\top}HU_{\alpha_{1}}),\ldots,\lambda(U_{\alpha_{r}}^{\top}HU_{\alpha_{r}})\big{)}$

where
$\lambda(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})\in{\mathbb{R}}^{|\alpha_{m}|}$
for any $m=1,\ldots,r$. This observation indicates that both estimates in
(2.7) and (2.8) are, indeed, a first-order estimate of eigenvalue function
$\lambda_{i}(\cdot)$. To obtain a second-order estimate, we need to repeat a
similar argument for each of the symmetric matrices
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$ for any $m\in\\{1,\ldots,r\\}$. To this
end, fix $m\in\\{1,\ldots,r\\}$ and observe that
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}\in{\mathbb{S}}^{|\alpha_{m}|}$. Thus, we
find
$Q_{m}\in{\mathbb{O}}^{|\alpha_{m}|}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})$
such that

$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}=Q_{m}\Lambda(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})Q_{m}^{\top}.$
(2.10)

Denote by $\eta_{1}^{m}>\cdots>\eta_{\rho_{m}}^{m}$ the distinct eigenvalues
of $U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$. Similar to (2.4), define the index
sets

$\beta_{j}^{m}:=\big{\\{}i\in\\{1,\ldots,|\alpha_{m}|\\}\big{|}\;\lambda_{i}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})=\eta_{j}^{m}\big{\\}}\quad\mbox{for
all}\;\;j=1,\ldots,\rho_{m}.$ (2.11)

To state the promised second-order estimate for eigenvalue functions, we need
to clarify some of indices, appeared therein. To do so, pick
$i\in\\{1,\ldots,n\\}$, and observe that there is $m\in\\{1,\dots,r\\}$ such
that $i\in\alpha_{m}$ and that ${\ell}_{i}(X)\in\\{1,\ldots,|\alpha_{m}|\\}$,
where ${\ell}_{i}(X)$ is defined by (2.5). Furthermore, we find
$j\in\\{1,\ldots,\rho_{m}\\}$ such that ${\ell}_{i}(X)\in\beta^{m}_{j}$.
Define now the constant ${\ell}^{\prime}_{i}(X,H)$ by

${\ell}^{\prime}_{i}(X,H)={\ell}_{{\ell}_{i}(X)}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}),$

which , in fact, signifies the number of eigenvalues of
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$ that are equal to
$\lambda_{{\ell}_{i}(X)}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})$ but are ranked
before $\lambda_{{\ell}_{i}(X)}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})$
including $\lambda_{{\ell}_{i}(X)}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})$. As
before, we often drop $X$ and $H$ from ${\ell}^{\prime}_{i}(X,H)$ when the
dependence of ${\ell}^{\prime}_{i}$ on $X$ and $H$ can be seen clearly from
the context. In summary, for any $i\in\\{1,\ldots,n\\}$, there are
$m\in\\{1,\dots,r\\}$ and $j\in\\{1,\dots,\rho_{m}\\}$ for which we have,
respectively,

$i\in\alpha_{m}\quad\mbox{and}\quad{\ell}_{i}(X)\in\beta^{m}_{j}.$ (2.12)

The following second-order estimate of eigenvalue functions was established in
[24, Proposition 2.2] and has important consequences for second-order
variational analysis of eigenvalue functions; see also [26, Proposition 2.1].

######  Proposition 2.2 (second-order expansion of eigenvalue functions).

Assume that $X\in{\mathbb{S}}^{n}$ has the eigenvalue decomposition (2.1) for
some $U\in{\mathbb{O}}^{n}(X)$ and that $H,W\in{\mathbb{S}}^{n}$. Let
$\mu_{1}>\cdots>\mu_{r}$ be distinct eigenvalues of $X$. Then for any $t>0$
sufficiently small and any $i\in\\{1,\ldots,n\\}$ we have

$\lambda_{i}(Y(t))=\lambda_{i}(X)+t\lambda_{{\ell}_{i}}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})+\hbox{${1\over
2}$}t^{2}\lambda_{\scriptsize{\ell}^{\prime}_{i}}\Big{(}{R_{mj}}^{\top}\big{(}U_{\alpha_{m}}^{\top}(W+2H(\mu_{m}I-X)^{\dagger}H)U_{\alpha_{m}}\big{)}R_{mj}\Big{)}+o(t^{2}),$

where $Y(t):=X+tH+\frac{1}{2}t^{2}W+o(t^{2})\in{\mathbb{S}}^{n}$, where
$R_{mj}:=(Q_{m})_{\beta^{m}_{j}}$ with $Q_{m}$ and $\beta^{m}_{j}$ taken from
(2.10) and (2.12), respectively, and where the indices $m$ and $j$ come from
(2.12).

In the framework of Proposition 2.2, we can conclude from (2.9) that for any
$i\in\\{1,\ldots,n\\}$, the parabolic second-order directional derivative of
the eigenvalue function $\lambda_{i}(\cdot)$ at $X$ for $H$ with respect to
$W$, denoted $\lambda_{i}^{{}^{\prime\prime}}(X;H,W)$, exists. Recall that the
latter concept is defined by

$\lambda_{i}^{{}^{\prime\prime}}(X;H,W)=\lim_{t\downarrow
0}\dfrac{\lambda_{i}(X+tH+\frac{1}{2}t^{2}W)-\lambda_{i}(X)-t\lambda^{\prime}_{i}(X;H)}{\frac{1}{2}t^{2}}.$

According to Proposition 2.2, we can conclude further that

$\lambda_{i}^{{}^{\prime\prime}}(X;H,W)=\lambda_{\scriptsize{\ell}^{\prime}_{i}}\Big{(}{R_{mj}}^{\top}\big{(}U_{\alpha_{m}}^{\top}(W+2H(\mu_{m}I-X)^{\dagger}H)U_{\alpha_{m}}\big{)}R_{mj}\Big{)}.$
(2.13)

Combining these with (2.9) brings us to the following estimate for the
eigenvalue function $\lambda(\cdot)$ from (2.1), important for our development
in this paper.

######  Corollary 2.3.

Assume that $X\in{\mathbb{S}}^{n}$ has the eigenvalue decomposition (2.1) for
some $U\in{\mathbb{O}}^{n}(X)$ and that $H,W\in{\mathbb{S}}^{n}$. Then for any
$t>0$ sufficiently small we have

$\lambda\big{(}X+tH+\hbox{${1\over
2}$}t^{2}W+o(t^{2})\big{)}=\lambda(X)+t\lambda^{\prime}(X;H)+\hbox{${1\over
2}$}t^{2}\lambda^{{}^{\prime\prime}}(X;H,W)+o(t^{2}).$ (2.14)

We proceed with recalling some concepts, utilized extensively in this paper.
Given a nonempty set $C\subset{\mathbb{E}}$ with $\bar{x}\in C$, the tangent
cone $T_{C}(\bar{x})$ to $C$ at $\bar{x}$ is defined by

$T_{C}(\bar{x})=\big{\\{}w\in{\mathbb{E}}|\;\exists\,t_{k}{\downarrow}0,\;\;w_{k}\to
w\;\;\mbox{ as }\;k\to\infty\;\;\mbox{with}\;\;\bar{x}+t_{k}w_{k}\in
C\big{\\}}.$

We say a tangent vector $w\in T_{C}(\bar{x})$ is derivable if there exist a
constant $\varepsilon>0$ and an arc $\xi:[0,\varepsilon]\to C$ such that
$\xi(0)=\bar{x}$ and $\xi^{\prime}_{+}(0)=w$, where
$\xi^{\prime}_{+}(0):=\lim_{t\downarrow 0}{[\xi(t)-\xi(0)]}/{t}$ signifies the
right derivative of $\xi$ at $0$. The set $C$ is called geometrically
derivable at $\bar{x}$ if every tangent vector $w$ to $C$ at $\bar{x}$ is
derivable. The geometric derivability of $C$ at $\bar{x}$ can be equivalently
described by the sets $[C-\bar{x}]/{t}$ converging to $T_{C}(\bar{x})$ as
$t\downarrow 0$ in the sense of the Painlevé-Kuratowski set convergence (cf.
[23, Definition 4.1]).

Given a function $f:{\mathbb{E}}\to\overline{\mathbb{R}}$, its domain is
defined by $\mbox{\rm
dom}\,f=\big{\\{}x\in{\mathbb{E}}|\;f(x)<\infty\big{\\}}$. The function $f$ is
called locally Lipschitz continuous around $\bar{x}$ relative to
$C\subset\mbox{\rm dom}\,f$ with constant $\ell\geq 0$ if $\bar{x}\in C$ with
$f(\bar{x})$ finite and there exists a neighborhood $U$ of $\bar{x}$ such that

$|f(x)-f(y)|\leq\ell\,\|x-y\|\quad\mbox{for all }\;x,y\in U\cap C.$

Such a function is called locally Lipschitz continuous relative to $C$ if it
is locally Lipschitz continuous around every $\bar{x}\in C$ relative to $C$.
Piecewise linear-quadratic functions (not necessarily convex) and an indicator
function of a nonempty set are important examples of functions that are
locally Lipschitz continuous relative to their domains. The subderivative
function of $f$ at $\bar{x}$, denoted by ${\rm
d}f(\bar{x})\colon{\mathbb{E}}\to\overline{\mathbb{R}}$, is defined by

${\mathrm{d}}f(\bar{x})(\bar{w})=\liminf_{\begin{subarray}{c}t\downarrow 0\\\
w\to\bar{w}\end{subarray}}{\frac{f(\bar{x}+tw)-f(\bar{x})}{t}}.$

When $f$ is convex, its subdifferential at $\bar{x}$ with $f(\bar{x})$ finite,
denoted by $\partial f(\bar{x})$, is understood in the sense of convex
analysis, namely $v\in\partial f(\bar{x})$ if $f(x)\geq f(\bar{x})+\langle
v,x-\bar{x}\rangle$ for any $x\in{\mathbb{E}}$. Given a nonempty convex set
$C\subset{\mathbb{E}}$, its normal cones to $C$ at $\bar{x}\in C$ is defined
by $N_{C}(\bar{x})=\partial\delta_{C}(\bar{x})$. The second-order tangent set
to $C\subset{\mathbb{E}}$ at $\bar{x}\in C$ for a tangent vector $w\in
T_{C}(\bar{x})$ is given by

$T_{C}^{2}(\bar{x},w)=\big{\\{}u\in{\mathbb{E}}|\;\exists\,t_{k}{\downarrow}0,\;\;u_{k}\to
u\;\;\mbox{ as }\;k\to\infty\;\;\mbox{with}\;\;\bar{x}+t_{k}w+\hbox{${1\over
2}$}t_{k}^{2}u_{k}\in C\big{\\}}.$ (2.15)

A set $C$ is called parabolically derivable at $\bar{x}$ for $w$ if
$T_{C}^{2}(\bar{x},w)$ is nonempty and for each $u\in T_{C}^{2}(\bar{x},w)$
there are $\varepsilon>0$ and an are $\xi:[0,\varepsilon]\to C$ with
$\xi(0)=\bar{x}$, $\xi^{\prime}_{+}(0)=w$, and $\xi^{\prime\prime}_{+}(0)=u$,
where $\xi^{\prime\prime}_{+}(0):=\lim_{t\downarrow
0}[{\xi(t)-\xi(0)-t\xi^{\prime}_{+}(0)}]/{\hbox{${1\over 2}$}t^{2}}$. It is
known that if $C$ is convex and parabolically derivable at $\bar{x}$ for $w$,
then the second-order tangent set $T_{C}^{2}(\bar{x},w)$ is a nonempty convex
set in ${\mathbb{E}}$ (cf. [3, page 163]).

##  3 Subderivatives of Spectral Functions

In this section, we present two important results about the spectral
functions, central to our developments in this paper. We begin with an
observation about symmetric functions. Recall that a function
$\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$ is called symmetric if for
every $x\in{\mathbb{R}}^{n}$ and every $n\times n$ permutation matrix $Q$, we
have $\theta(Qx)=\theta(x)$. Recall also that $Q$ is a permutation matrix if
all its components are either $0$ and $1$ and each row and each column has
exactly one nonzero element. We denote by ${\mathbb{P}}^{n}$ the set of all
$n\times n$ permutation matrices. As pointed out before, for any spectral
function $g:{\mathbb{S}}^{n}\to\overline{\mathbb{R}}$, there exists a
symmetric function $\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$
satisfying (1.1). Indeed, $\theta$ can be chosen as the restriction of $g$ to
diagonal matrices, namely

$\theta(x)=g\big{(}\mbox{\rm Diag}\,(x)\big{)}\quad\mbox{for
all}\;\;x\in{\mathbb{R}}^{n}.$ (3.1)

A set $C\subset{\mathbb{S}}^{n}$ is called a spectral set if $\delta_{C}$ is a
spectral function. Likewise, $\Theta\subset{\mathbb{R}}^{n}$ is called a
symmetric set if $\delta_{\Theta}$ is a symmetric function. Similar to (1.1),
it is easy to see that for any spectral set $C\subset{\mathbb{S}}^{n}$, there
exists a symmetric set $\Theta\subset{\mathbb{R}}^{n}$ such that

$C=\big{\\{}X\in{\mathbb{S}}^{n}\big{|}\;\lambda(X)\in\Theta\big{\\}},$ (3.2)

where $\Theta$ can be chosen as

$\Theta=\big{\\{}x\in{\mathbb{R}}^{n}\big{|}\;\mbox{\rm Diag}\,(x)\in
C\big{\\}}.$ (3.3)

The composite forms (1.1) and (3.1) readily imply, respectively, that

$\mbox{\rm
dom}\,g=\big{\\{}X\in{\mathbb{S}}^{n}\big{|}\;\lambda(X)\in\mbox{\rm
dom}\,\theta\big{\\}}\quad\mbox{and}\quad\mbox{\rm
dom}\,\theta=\big{\\{}x\in{\mathbb{R}}^{n}\big{|}\;\mbox{\rm
Diag}\,(x)\in\mbox{\rm dom}\,g\big{\\}}.$ (3.4)

Next, we are going to justify a similar estimate as (1.3) for domains of
symmetric functions, which allows us to show via the established theory for
composite functions in [20] that second-order variational properties of
spectral functions are inherited by symmetric functions.

######  Proposition 3.1.

Let $g:{\mathbb{S}}\to\overline{\mathbb{R}}$ be a spectral function,
represented by (1.1). Then for any $x\in{\mathbb{R}}^{n}$ we have

${\rm dist}(x,\mbox{\rm dom}\,\theta)={\rm dist}\big{(}\mbox{\rm
Diag}\,(x),\mbox{\rm dom}\,g\big{)},$ (3.5)

where $\theta$ is taken by (1.1).

[Proof. ]For any $x\in{\mathbb{R}}^{n}$, we know that there exist a
permutation matrix $P\in{\mathbb{O}}^{n}$ such that $\lambda(\mbox{\rm
Diag}\,(x))=Px$. Since $\theta$ is a symmetric function, $\mbox{\rm
dom}\,\theta$ is a symmetric set. Thus for any $X\in\mbox{\rm dom}\,g$, we
have $P^{\top}\lambda(X)\in\mbox{\rm dom}\,\theta.$ This, coupled with (2.2),
leads us to

$\displaystyle\|\mbox{\rm Diag}\,(x)-X\|\geq\|\lambda(\mbox{\rm
Diag}\,(x))-\lambda(X)\|$ $\displaystyle=$ $\displaystyle\|Px-\lambda(X)\|$
$\displaystyle=$ $\displaystyle\|x-P^{\top}\lambda(X)\|\geq{\rm
dist}(x,\mbox{\rm dom}\,\theta)$

for all $X\in\mbox{\rm dom}\,g$, which in turn brings us to

${\rm dist}(x,\mbox{\rm dom}\,\theta)\leq{\rm dist}\big{(}\mbox{\rm
Diag}\,(x),\mbox{\rm dom}\,g\big{)}.$

To prove the opposite inequality, pick any $y\in\mbox{\rm dom}\,\theta.$ By
(3.4), we get $\mbox{\rm Diag}\,(y)\in\mbox{\rm dom}\,g$, which implies that

$\|x-y\|=\|\mbox{\rm Diag}\,(x)-\mbox{\rm Diag}\,(y)\|\geq{\rm dist}(\mbox{\rm
Diag}\,(x),\mbox{\rm dom}\,g).$

Combining these clearly justifies (3.5).

Note that the identity in (1.3) allows us to show that second-order
variational properties of a symmetric function $\theta$ from (1.1) are
disseminated to the spectral function $g$. Appealing to (3.5), we will show in
the coming sections that those variational properties of the spectral function
$g$ are inherited by the symmetric function $\theta$ from (1.1). This will be
achieved using the second-order variational theory in [20] for the composite
form (3.1). Note also that the results in [20] were proven under a constraint
qualification, which is similar to (1.2). According to (3.5), such a
constraint qualification automatically holds for the composite form (3.1).
Moreover, the inner mapping $x\mapsto\mbox{\rm Diag}\,(x)$ in this composite
form is twice continuously differentiable, which allows us to exploit the
results in [17, 20].

######  Proposition 3.2.

Let $g:{\mathbb{S}}^{n}\to\overline{\mathbb{R}}$ be a spectral function,
represented by (1.1), and let the symmetric function $\theta$, taken from
(1.1), be locally Lipschitz continuous relative to its domain. Then for any
$X\in{\mathbb{S}}^{n}$ with $g(X)$ finite and any $v\in{\mathbb{R}}^{n}$, we
have

${\rm d}\theta(\lambda(X))(v)={\rm d}g\big{(}\mbox{\rm
Diag}\,(\lambda(X))\big{)}(\mbox{\rm Diag}\,(v)).$ (3.6)

In particular, if $g=\delta_{C}$, where $C\subset{\mathbb{S}}^{n}$ is a
spectral set, then we get for any $X\in C$ that

$T_{\Theta}(\lambda(X))=\big{\\{}v\in{\mathbb{R}}^{n}|\;\mbox{\rm
Diag}\,(v)\in T_{C}(\mbox{\rm Diag}\,(\lambda(X))\big{\\}},$ (3.7)

where $\Theta$ is taken from (3.2).

[Proof. ]It follows from (1.1) that the symmetric function $\theta$ satisfies
(3.1), which means that $\theta$ can be represented as a composite function of
$g$ and the linear mapping $x\mapsto\mbox{\rm Diag}\,(x)$ with
$x\in{\mathbb{R}}^{n}$. We also deduce from the imposed assumption on $\theta$
and the inequality in (2.2) that $g$ is locally Lipschitz continuous relative
to its domain. This, together with (3.5) and [17, Theorem 3.4], justifies
(3.6). To justify (3.7), recall the representation $\Theta$ from (3.3), which
can be equivalently expressed as $\delta_{\Theta}(x)=\delta_{C}(\mbox{\rm
Diag}\,(x))$ for any $x\in{\mathbb{R}}^{n}$. The claim equality in (3.7)
results from (3.6) and the fact that ${\rm
d}\theta(\lambda(X))=\delta_{T_{\Theta}(\lambda(X))}$ and ${\rm
d}g\big{(}\mbox{\rm
Diag}\,(\lambda(X))\big{)}=\delta_{T_{C}(\scriptsize{\mbox{\rm
Diag}\,}(\lambda(X)))}$.

######  Remark 3.3 (symmetric property of subderivatives).

If $\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$ is a symmetric function
and $X\in{\mathbb{S}}^{n}$ with $\theta(\lambda(X))$ finite, one may wonder
whether the subderivative ${\rm d}\theta(\lambda(X))$ is a symmetric function.
This can be easily disproven by taking $\theta=\delta_{{\mathbb{R}}_{-}^{n}}$
and $X\in{\mathbb{S}}^{n}$ such that all its eigenvalues are not the same; see
Example 3.7 for more details. While this may seem disappointing, we can show
that ${\rm d}\theta(\lambda(X))$ is a symmetric function with respect to a
subset of ${\mathbb{P}}^{n}$. Indeed, assume that ${\mathbb{P}}^{n}_{X}$ is
the set of all $n\times n$ block diagonal matrices in the form $Q=\mbox{\rm
Diag}\,(P_{1},\ldots,P_{r})$, where
$P_{m}\in{\mathbb{R}}^{|\alpha_{m}|\times|\alpha_{m}|}$ is a permutation
matrix for any $m=1,\ldots,r$ with $\alpha_{m}$ taken from (2.4) and $r$ being
the number of distinct eigenvalues of $X$. It is clear that
${\mathbb{P}}^{n}_{X}\subset{\mathbb{P}}^{n}$ and that if
$Q\in{\mathbb{P}}^{n}_{X}$, then we have $Q\lambda(X)=\lambda(X)$. Moreover,
for any $v\in{\mathbb{R}}^{n}$ and $Q\in{\mathbb{P}}^{n}_{X}$, we get

$\displaystyle{\rm d}\theta(\lambda(X))(v)$ $\displaystyle=$
$\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\ v^{\prime}\to
v\end{subarray}}\frac{\theta\big{(}\lambda(X)+tv^{\prime}\big{)}-\theta\big{(}\lambda(X)\big{)}}{t}$
$\displaystyle=$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\
v^{\prime}\to
v\end{subarray}}\frac{\theta\big{(}\lambda(X)+tQv^{\prime}\big{)}-\theta\big{(}\lambda(X)\big{)}}{t}$
$\displaystyle\geq$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\
w\to
Qv\end{subarray}}\frac{\theta\big{(}\lambda(X)+tw\big{)}-\theta\big{(}\lambda(X)\big{)}}{t}={\rm
d}\theta(\lambda(X))(Qv).$

Since $Q^{-1}=\mbox{\rm Diag}\,(P_{1}^{-1},\ldots,P_{r}^{-1})$, we can show
similarly that ${\rm d}\theta(\lambda(X))(v)\leq{\rm d}\theta(\lambda(X))(Qv)$
for any $v\in{\mathbb{R}}^{n}$ and $Q\in{\mathbb{P}}^{n}_{X}$, which leads us
to

${\rm d}\theta(\lambda(X))(v)={\rm d}\theta(\lambda(X))(Qv)\quad\mbox{for
all}\;\;v\in{\mathbb{R}}^{n},\;Q\in{\mathbb{P}}^{n}_{X},$

demonstrating that ${\rm d}\theta(\lambda(X))$ is a symmetric function with
respect to ${\mathbb{P}}^{n}_{X}$.

We proceed by proving a chain rule for subderivatives of spectral functions.
We begin with recalling a useful characterization of the subdifferential of
the spectral functions.

######  Proposition 3.4.

Assume that $\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$ is a proper,
lower semicontinuous (lsc), convex, and symmetric function. Then the following
properties are equivalent:

  * (a)

$Y\in\partial(\theta\circ\lambda)(X)$;

  * (b)

$\lambda(Y)\in\partial\theta(\lambda(X))$ and the matrices $X$ and $Y$ have
simultaneous ordered spectral decomposition, meaning that there exists
$U\in{\mathbb{O}}^{n}(X)\,\cap\,{\mathbb{O}}^{n}(Y)$ such that

$X=U{\Lambda}(X)U^{\top}\quad\mbox{and}\quad Y=U{\Lambda}(Y)U^{\top},$

where ${\Lambda}(X)=\mbox{\rm Diag}\,(\lambda(X)\big{)}$ and
${\Lambda}(Y)=\mbox{\rm Diag}\,(\lambda(Y)\big{)}$.

[Proof. ]It follows from [4, Corollary 5.2.3] that $\theta\circ\lambda$ is lsc
and convex if and only if $\theta$ is lsc and convex. The claimed equivalence
then results from [4, Theorem 5.2.4].

Given a matrix $X\in{\mathbb{S}}^{n}$ with $r$ distinct eigenvalues and the
index sets $\alpha_{m}$, $m=1,\ldots,r$, from (2.4), recall that
$\cup_{m=1}^{r}\alpha_{m}=\\{1,\ldots,n\\}$. In what follows, we partition a
vector $p\in{\mathbb{R}}^{n}$ into $(p_{\alpha_{1}},\ldots,p_{\alpha_{r}})$,
where $p_{\alpha_{m}}\in{\mathbb{R}}^{|\alpha_{m}|}$ for any $m=1,\ldots,r$.

######  Theorem 3.5 (subderivatives of spectral functions).

Let $\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$ be a symmetric function
and let $X\in{\mathbb{S}}^{n}$ with $(\theta\circ\lambda)(X)$ finite. If
$\theta$ is either lsc and convex or locally Lipschitz continuous around
$\lambda(X)$ relative to its domain, then for all $H\in{\mathbb{S}}^{n}$ we
have

${\rm d}(\theta\circ\lambda)(X)(H)={\rm
d}\theta(\lambda(X))(\lambda^{{}^{\prime}}(X;H)).$ (3.8)

[Proof. ]Pick any $H\in{\mathbb{S}}^{n}$ and deduce from Lemma 2.1 that
$\lambda^{\prime}(X;.)$ is a Lipschitz continuous and positively homogeneous
function. Moreover, we have
$\lambda^{\prime}(X;E)+{O(t^{2}\|E\|^{2})}/{t}\to\lambda^{\prime}(X;H)$ as
$t\downarrow 0$ and $E\to H$. This and the definition of subderivative give us
the relationships

$\displaystyle\displaystyle{\rm d}(\theta\circ\lambda)(X)(H)$ $\displaystyle=$
$\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\ E\to
H\end{subarray}}\frac{\theta\big{(}\lambda(X+tE)\big{)}-\theta\big{(}\lambda(X)\big{)}}{t}$
(3.9) $\displaystyle=$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow
0\\\ E\to
H\end{subarray}}\frac{\theta\big{(}\lambda(X)+t\lambda^{\prime}(X;E)+O(t^{2}\|E\|^{2})\big{)}-\theta\big{(}\lambda(X)\big{)}}{t}$
$\displaystyle=$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\
E\to
H\end{subarray}}\frac{\theta\big{(}\lambda(X)+t[\lambda^{\prime}(X;E)+{O(t^{2}\|E\|^{2})}/{t}]\big{)}-\theta\big{(}\lambda(X)\big{)}}{t}$
$\displaystyle\geq$ $\displaystyle{\rm
d}\theta\big{(}\lambda(X)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)},\displaystyle$

which verify the inequality “$\geq$” in (3.8). To justify the opposite
inequality in (3.9), observe that if ${\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))=\infty$, the latter inequality
clearly holds. Thus, assume that ${\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))<\infty$. If $\theta$ is lsc and
convex, $\theta\circ\lambda$ is lsc and convex due to [4, Corollary 5.2.3].
Thus, it follows from [23, Theorem 8.30] that ${\rm
d}(\theta\circ\lambda)(X)(H)=\sup_{Y\in\partial(\theta\circ\lambda)(X)}\langle
Y,H\rangle$. Let $\varepsilon>0$ and choose
$Y\in\partial(\theta\circ\lambda)(X)$ such that

${\rm d}(\theta\circ\lambda)(X)(H)\leq\varepsilon+\langle Y,H\rangle.$

Since $Y\in\partial(\theta\circ\lambda)(X)$, it follows from Proposition 3.4
that there is $U\in{\mathbb{O}}^{n}(X)\,\cap\,{\mathbb{O}}^{n}(Y)$ such that
$\lambda(Y)\in\partial\theta(\lambda(X))$. Set ${\Lambda}(Y):=U^{\top}YU$ and
use Fan’s inequality to conclude

$\displaystyle{\rm d}(\theta\circ\lambda)(X)(H)\leq\varepsilon+\langle
Y,H\rangle=\varepsilon+\langle{\Lambda}(Y),U^{T}HU\rangle$ $\displaystyle=$
$\displaystyle\varepsilon+\sum_{m=1}^{r}\langle{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}\rangle$
$\displaystyle\leq$
$\displaystyle\varepsilon+\sum_{m=1}^{r}\langle\lambda(Y)_{\alpha_{m}},\lambda(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})\rangle$
$\displaystyle=$
$\displaystyle\varepsilon+\langle\lambda(Y),\lambda^{{}^{\prime}}(X;H)\rangle$
$\displaystyle\leq$ $\displaystyle\varepsilon+{\rm
d}\theta(\lambda(X))(\lambda^{{}^{\prime}}(X;H)),$

where the last inequality results from the fact that $\theta$ is convex and
$\lambda(Y)\in\partial\theta(\lambda(X))$. Letting $\varepsilon\downarrow 0$,
we get the opposite inequality in (3.9), which proves (3.8) in this case.
Suppose now that $\theta$ is locally Lipschitz continuous around $\lambda(X)$
relative to its domain. To prove the opposite inequality in (3.9), by
definition, there exist sequences $t_{k}\downarrow 0$ and
$v_{k}\to\lambda^{\prime}(X;H)$ such that

${\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))=\lim_{\begin{subarray}{c}k\to\infty\end{subarray}}\frac{\theta\big{(}\lambda(X)+t_{k}v_{k}\big{)}-\theta\big{(}\lambda(X)\big{)}}{t_{k}}.$
(3.10)

Since ${\rm d}\theta(\lambda(X))(\lambda^{\prime}(X;H))<\infty$, we can assume
without loss of generality that $\lambda(X)+t_{k}v_{k}\in\mbox{\rm
dom}\,\theta$ for all $k\in{\mathbb{N}}$. Take the function $g$ from (1.1) and
appeal to (1.3) to get

${\rm dist}(X+t_{k}H,\mbox{\rm dom}\,g)={\rm
dist}\big{(}\lambda(X+t_{k}H),\mbox{\rm dom}\,\theta\big{)},\quad
k\in{\mathbb{N}},$

which in turn brings us to the relationships

$\displaystyle{\rm dist}\Big{(}H,\frac{\mbox{\rm dom}\,g-X}{t_{k}}\Big{)}$
$\displaystyle=$ $\displaystyle\frac{1}{t_{k}}\,{\rm
dist}\big{(}\lambda(X)+t_{k}\lambda^{\prime}(X,H)+O(t^{2}_{k}),\mbox{\rm
dom}\,\theta\big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{t_{k}}\,\big{\|}\lambda(X)+t_{k}\lambda^{\prime}(X,H)+O(t^{2}_{k})-\lambda(X)-t_{k}v_{k}\big{\|}$
$\displaystyle=$
$\displaystyle\big{\|}\lambda^{\prime}(X;H)-v_{k}+\frac{O(t^{2}_{k})}{t_{k}}\big{\|}\;\mbox{
for all }\;k\in{\mathbb{N}}.\displaystyle$

So for each $k\in{\mathbb{N}}$, we find a matrix $E_{k}\in{\mathbb{S}}^{n}$
such that $X+t_{k}E_{k}\in\mbox{\rm dom}\,g$ and

$\|H-E_{k}\|<\big{\|}\lambda^{\prime}(X;H)-v_{k}+\frac{O(t^{2}_{k})}{t_{k}}\big{\|}+\frac{1}{k},$

which in turn yields $E_{k}\to H$ as $k\to\infty$. Combining these with (3.10)
and (2.8), we arrive at

$\displaystyle{\rm
d}\theta\big{(}\lambda(X)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}$
$\displaystyle=$
$\displaystyle\displaystyle\lim_{k\to\infty}\Big{[}\frac{g(X+t_{k}E_{k})-g(X)}{t_{k}}+\frac{\theta\big{(}\lambda(X)+t_{k}v_{k})\big{)}-\theta\big{(}\lambda(X+t_{k}E_{k})\big{)}}{t_{k}}\Big{]}$
$\displaystyle\geq$
$\displaystyle\displaystyle\liminf_{k\to\infty}\frac{g(X+t_{k}E_{k})-g(X)}{t_{k}}-\kappa\lim_{k\to\infty}\big{\|}\frac{\lambda(X+t_{k}E_{k})-\lambda(X)}{t_{k}}-v_{k}\big{\|}$
$\displaystyle\geq$ $\displaystyle{\rm
d}g(X)(H)-\kappa\lim_{k\to\infty}\big{\|}\lambda^{\prime}(X;E_{k})+\frac{O(t^{2}_{k})}{t_{k}}-v_{k}\big{\|}={\rm
d}g(X)(H),$

where $\kappa\geq 0$ is a Lipschitz constant of $\theta$ around $\lambda(X)$
relative to its domain. This verifies the inequality “$\leq$” in (3.8) and
completes the proof of the theorem.

As an immediate conclusion of Theorem 3.5, we obtain a simple representation
of tangent cones to spectral sets.

######  Corollary 3.6 (tangent cone to the spectral sets).

Let $C$ be a spectral set represented by (3.2). Then for any $X\in C$, we have

$T_{C}(X)=\big{\\{}H\in{\mathbb{S}}^{n}\big{|}\;\lambda^{\prime}(X;H)\in
T_{\Theta}(\lambda(X))\big{\\}}.$

[Proof. ]Taking the symmetric set $\Theta$ from (3.2), we can apply Theorem
3.5 for the symmetric function $\delta_{\Theta}$. The claimed representation
of the tangent cone to $C$ at $X$ follows from the facts that ${\rm
d}\delta_{\Theta}(X)=\delta_{T_{C}(X)}$ and ${\rm
d}\delta_{\Theta}(\lambda(X))=\delta_{T_{\Theta}(\lambda(X))}$.

######  Example 3.7 (tangent cone to ${\mathbb{S}}^{n}_{-}$).

Suppose that ${\mathbb{S}}^{n}_{-}$ stands for the cone of all $n\times n$
symmetric and negative semidefinite matrices. This cone is a spectral set and

${\mathbb{S}}^{n}_{-}=\big{\\{}X\in{\mathbb{S}}^{n}\big{|}\;\lambda(X)\in{\mathbb{R}}^{n}_{-}\big{\\}}.$

Take $X\in{\mathbb{S}}^{n}_{-}$ and assume that $\mu_{1}>\cdots>\mu_{r}$ are
its distinct eigenvalues. If $\mu_{1}<0$, then we clearly have
$\lambda(X)\in\mbox{\rm int}\,{\mathbb{R}}^{n}_{-}$ and hence
$T_{{\mathbb{R}}^{n}_{-}}(\lambda(X))={\mathbb{R}}^{n}$. Using this, together
with Corollary 3.6, we get $T_{{\mathbb{S}}^{n}_{-}}(X)={\mathbb{S}}^{n}$. If
$\mu_{1}=0$, then we obtain
$T_{{\mathbb{R}}^{n}_{-}}(\lambda(X))={\mathbb{R}}^{|\alpha_{1}|}_{-}\times{\mathbb{R}}^{n-|\alpha_{1}|}$,
where $\alpha_{1}$ is defined by (2.4). Appealing to Corollary 3.6 tells us
that

$T_{{\mathbb{S}}^{n}_{-}}(X)=\big{\\{}H\in{\mathbb{S}}^{n}\big{|}\;\lambda_{1}(U_{\alpha_{1}}^{\top}HU_{\alpha_{1}})\leq
0\big{\\}},$ (3.11)

where $U$ is taken from (2.1).

The tangent cone description for the set ${\mathbb{S}}^{n}_{-}$ in (3.11) can
be alternatively obtained by [3, Proposition 2.61]. Indeed, it is easy to see
that ${\mathbb{S}}^{n}_{-}=\\{X\in{\mathbb{S}}^{n}\big{|}\;\lambda_{1}(X)\leq
0\\}$ in which $\lambda_{1}$ is known to be a convex function (cf. [23,
Exercise 2.54]). Obviously, we can find ${X}\in{\mathbb{S}}^{n}$ with
$\lambda_{1}({X})<0$, a condition known as the Slater condition and assumed in
[3, Proposition 2.61]. In contrast, our approach relies upon the metric
subregularity, automatically satisfied for spectral sets. This allows to
calculate the tangent cone of spectral sets even if the Slater condition fails
therein as the following example demonstrates.

######  Example 3.8 (failure of the Slater condition in spectral sets).

Assume that $k\in{\mathbb{N}}$ and consider the set

$C=\big{\\{}X\in{\mathbb{S}}^{n}_{+}\big{|}\;\sum^{n}_{i=1}\lambda^{k}_{i}(X)=1\big{\\}}.$
(3.12)

This is clearly a spectral subset of ${\mathbb{S}}^{n}$. When $k=1$, the set
$C$ is called spectahedron. Note that $C$ can be represented in the form of
(3.2) with the symmetric set $\Theta$ defined by

$\Theta=\big{\\{}(z_{1},\ldots,z_{n})\in{\mathbb{R}}^{n}\big{|}\;\sum^{n}_{i=1}z^{k}_{i}=1,\;z_{i}\geq
0\;\;\mbox{for all}\;\;i\in\\{1,\ldots,n\\}\big{\\}}.$ (3.13)

Set $\Phi(z)=(\sum^{n}_{i=1}z^{k}_{i}-1,z_{1},\ldots,z_{n})$ with
$z=(z_{1},\ldots,z_{n})$ and $D=\\{0\\}\times{\mathbb{R}}_{+}^{n}$ and observe
that

$\Theta=\\{z\in{\mathbb{R}}^{n}|\;\Phi(z)\in D\\}.$ (3.14)

We claim now that

$N_{D}(\Phi(z))\cap\mbox{\rm ker}\,\nabla\Phi(z)^{*}=\\{0\\}$ (3.15)

for any $z\in\Theta$. To justify it, pick $z=(z_{1},\ldots,z_{n})\in\Theta$
and assume that $(b_{0},\ldots,b_{n})\in N_{D}(\Phi(z))\cap\mbox{\rm
ker}\,\nabla\Phi(z)^{*}$. This implies that $b_{i}z_{i}=0$ and
$kz_{i}^{k-1}b_{0}+b_{i}=0$ for any $i=1,\ldots,n$. It is not hard to see that
these conditions lead us to $b_{i}=0$ for any $i=0,\ldots,n$, which proves our
claim. Take $X\in C$ and define the active index set
$I(\lambda(X))=\big{\\{}i\in\\{1,\ldots,n\\}|\;\lambda_{i}(X)=0\big{\\}}$. It
follows from [23, Theorem 6.14] and (3.15) that the tangent cone to $\Theta$
at $\lambda(X)$ can be calculated as

$T_{\Theta}(\lambda(X))=\big{\\{}(w_{1},\ldots,w_{n})\in{\mathbb{R}}^{n}\big{|}\;\sum^{n}_{i=1}w_{i}\lambda^{k-1}_{i}(X)=0,\;w_{i}\geq
0\;\;\mbox{for all}\;\;i\in I(\lambda(X))\big{\\}}.$

Appealing to Corollary 3.6 tells us that

$T_{C}(X)=\big{\\{}H\in{\mathbb{S}}^{n}\big{|}\;\sum^{n}_{i=1}\lambda^{\prime}_{i}(X;H)\lambda^{k-1}_{i}(X)=0,\;\
\lambda^{\prime}_{i}(X;H)\geq 0\;\;\mbox{for all}\;\;i\in
I(\lambda(X))\big{\\}}.$ (3.16)

Note that due to the presence of the equality constraint, the Slater condition
fails for $C$ and thus [3, Proposition 2.61] can’t be applied.

##  4  Parabolic Epi-Differentiability of Spectral Functions

The main objective of this section is to provide a systematic study of two
important second-order variational properties of spectral sets and functions:
1) parabolic derivability; and 2) a chain rule for parabolic subderivatives.
To achieve these goals, we begin with justifying that certain second-order
approximations of spectral sets enjoy an outer Lipschitzian property, which is
central to our developments in this section. Suppose that
$C\subset{\mathbb{S}}^{n}$ is a spectral set with the representation in (3.2)
and that $X\in C$ and $H\in T_{C}(X)$. Define the set-valued mapping
$S_{H}:{\mathbb{R}}^{n}\rightrightarrows{\mathbb{S}}^{n}$ via the second-order
tangent set to the symmetric set $\Theta$ in (3.2) by

$S_{H}(p):=\big{\\{}W\in{\mathbb{S}}^{n}\big{|}\;\lambda^{{}^{\prime\prime}}(X;H,W)+p\in
T^{2}_{\Theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}\big{\\}}.$ (4.1)

For any parameter $p\in{\mathbb{R}}^{n}$, the set-valued mapping $S_{H}(p)$
presents a second-order tangential approximation of the spectral set (3.2) at
$X$ for $H$. Note that by Corollary 3.6, the condition $H\in T_{C}(X)$ amounts
to $\lambda^{\prime}(X;H)\in T_{\Theta}(\lambda(X))$, which is required in the
definition of the second-order tangent set to $\Theta$ at $\lambda(X)$ in
(4.1); see (2.15). Note also that reducing the estimate (1.3), which was
stated for the spectral function in (1.1), to the spectral set $C$ in (3.2)
gives us the estimate

${\rm dist}(X,C)={\rm dist}(\lambda(X),\Theta)\quad\mbox{for
all}\;\;X\in{\mathbb{S}}^{n},$ (4.2)

which will be utilized broadly in this section.

######  Proposition 4.1 (uniform outer Lipschitzian property of $S_{H}$).

Assume that $C\subset{\mathbb{S}}^{n}$ is a spectral set with the
representation (3.2) and that $X\in C$ and $H\in T_{C}(X)$. Then the mapping
$S_{H}$ in (4.1) enjoys the following uniform outer Lipschitzian property at
the origin:

$S_{H}(p)\subset S_{H}(0)+\|p\|\mathbb{B}\;\mbox{ for all
}\;p\in{\mathbb{R}}^{n}.$ (4.3)

[Proof. ]Let $p\in{\mathbb{R}}^{n}$ and pick then $W\in S_{H}(p)$. It follows
from (4.1) that $\lambda^{{}^{\prime\prime}}(X;H,W)+p\in
T^{2}_{\Theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}$. By (2.15),
there exists a sequence $t_{k}\downarrow 0$ such that

$\lambda(X)+t_{k}\lambda^{\prime}(X;H)+\hbox{${1\over
2}$}t_{k}^{2}\lambda^{{}^{\prime\prime}}(X;H,W)+\hbox{${1\over
2}$}t_{k}^{2}p+o(t^{2}_{k})\in\Theta\quad\mbox{for all}\;\;k\in{\mathbb{N}}.$

For any $k$ sufficiently large, we conclude from (2.14) that

$\lambda(X+t_{k}H+\hbox{${1\over
2}$}t_{k}^{2}W)=\lambda(X)+t_{k}\lambda^{\prime}(X;H)+\hbox{${1\over
2}$}t_{k}^{2}\lambda^{{}^{\prime\prime}}(X;H,W)+o(t^{2}_{k}),$

which in turn implies via (4.2) that

$\displaystyle{\rm dist}\big{(}X+t_{k}H+\hbox{${1\over
2}$}t_{k}^{2}W,C\big{)}={\rm dist}\big{(}\lambda(X+t_{k}H+\hbox{${1\over
2}$}t_{k}^{2}W),\Theta\big{)}\leq\hbox{${1\over
2}$}t_{k}^{2}\|p\|+o(t_{k}^{2}).$

This ensures the existence of a matrix $Y_{k}\in C$ such that

$\|D_{k}\|\leq\frac{1}{2}\Big{(}\|p\|+\frac{o(t_{k}^{2})}{t_{k}^{2}}\Big{)}\;\mbox{
with }\;D_{k}:=\frac{X+t_{k}H+\hbox{${1\over
2}$}t_{k}^{2}W-Y_{k}}{t_{k}^{2}}.$

Passing to a subsequence, if necessary, ensures the existence of
$D\in{\mathbb{S}}^{n}$ such that $D_{k}\to D$ as $k\to\infty$. This yields the
estimate

$\|D\|\leq\hbox{${1\over 2}$}\|p\|.$ (4.4)

It follows from $X+t_{k}H+\hbox{${1\over
2}$}t_{k}^{2}W-t_{k}^{2}D_{k}=Y_{k}\in C$ and (3.2) that
$\lambda(X+t_{k}H+\hbox{${1\over 2}$}t_{k}^{2}W-t_{k}^{2}D_{k})\in\Theta$.
Taking into account (2.14), we get for any $k\in{\mathbb{N}}$ sufficiently
large that

$\lambda(X+t_{k}H+\hbox{${1\over
2}$}t_{k}^{2}W-t_{k}^{2}D_{k})=\lambda(X)+t_{k}\lambda^{\prime}(X;H)+\hbox{${1\over
2}$}t_{k}^{2}\lambda^{\prime\prime}\big{(}X;H,W-2D\big{)}+o(t_{k}^{2})\in\Theta.$

By the definition of the second-order tangent set, we arrive at

$\lambda^{\prime\prime}\big{(}X;H,W-2D\big{)}\in
T_{\Theta}^{2}(\lambda(X),\lambda(X;H)),$

which yeilds $W-2D\in S_{H}(0)$. This, combined with (4.4), justifies the
claimed inclusion in (4.3) and thus completes the proof.

The outer Lipschitzian property for second-order tangential approximations was
appeared first in [18, Theorem 4.3] for sets $C$ as the one in (3.2) with the
eigenvalue function $\lambda(\cdot)$ replaced with a twice differentiable
function, under an adaptation of the constraint qualification (1.2) for this
setting. Proposition 4.1 demonstrates that the latter result can be achieved
without the assumed twice differentiability in [18] when we still have a
second-order expansion for functions in our settings.

Next, we are going to achieve a chain rule for second-order tangent sets of
spectral sets, which heavily relies upon Proposition 4.1. First, we recall
[18, Theorem 4.5], where a similar result was proven for constraint systems in
finite dimensional Hilbert spaces.

######  Proposition 4.2.

Let $D$ be a closed subset of ${\mathbb{E}}_{2}$ and let
$\Omega=\big{\\{}x\in{\mathbb{E}}_{1}\big{|}\;\Phi(x)\in D\big{\\}}$, where
$\Phi:{\mathbb{E}}_{1}\to{\mathbb{E}}_{2}$ is a twice differentiable function
between two Euclidean spaces, and $\bar{x}\in\Omega$. Suppose further that
there are $\kappa\geq 0$ and $\varepsilon>0$ such that the estimate

${{\rm dist}}(x,\Omega)\leq\kappa\,{\rm dist}\big{(}\Phi(x),D\big{)}\;\mbox{
for all }\;x\in\mathbb{B}_{\varepsilon}(\bar{x})$

holds. Then for all $w\in T_{\Omega}(\bar{x})$, we have

$T^{2}_{\Omega}(\bar{x},w)=\big{\\{}u\in{\mathbb{E}}_{1}|\;\nabla\Phi(\bar{x})u+\nabla^{2}\Phi(\bar{x})(w,w)\in
T^{2}_{D}\big{(}\Phi(\bar{x}),\nabla\Phi(\bar{x})w\big{)}\big{\\}}.$ (4.5)

If furthermore the set $D$ is parabolically derivable at $\Phi(\bar{x})$ for
$\nabla\Phi(\bar{x})w$, then the constraint set $\Omega$ is parabolically
derivable at $\bar{x}$ for $w$.

[Proof. ]Both claims were justified in [18, Theorem 4.5] under an extra
assumption that the set $D$ is regular in the sense of [23, Definition 6.4].
However, it is not hard to see from the proof of this result that this
condition can be dropped with no harm.

######  Theorem 4.3 (second-order tangent sets of spectral sets).

Assume that $C\subset{\mathbb{S}}^{n}$ is a spectral set with the
representation in (3.2) and that $X\in C$ and $H\in T_{C}(X)$. Then we have

$T^{2}_{C}(X,H)=\big{\\{}W\in{\mathbb{S}}^{n}\big{|}\;\lambda^{\prime\prime}\big{(}X;H,W\big{)}\in
T^{2}_{\Theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}\big{\\}},$ (4.6)

where $\Theta$ is taken from (3.2). Moreover, the following properties are
satisfied.

  * (a)

If the symmetric set $\Theta$ is parabolically derivable at $\lambda(X)$ for
$\lambda^{\prime}(X;H)$, then $C$ is parabolically derivable at $X$ for $H$.

  * (b)

If the symmetric set $C$ is parabolically derivable at $X$ for any $H\in
T_{C}(X)$, then $\Theta$ is parabolically derivable at $\lambda(X)$ for any
$v\in T_{\Theta}(\lambda(X))$.

[Proof. ]First note from Corollary 3.6 that the condition $H\in T_{C}(X)$
amounts to $\lambda^{\prime}(X;H)\in T_{\Theta}(\lambda(X))$. Let
$W\in{\mathbb{S}}^{n}$. Employing (4.2) and (2.14) tells us that for any $t>0$
sufficiently small, we have

$\displaystyle{\rm dist}\big{(}X+tH+\hbox{${1\over 2}$}t^{2}W,C\big{)}$
$\displaystyle=$ $\displaystyle{\rm dist}\big{(}\lambda(X+tH+\hbox{${1\over
2}$}t^{2}W),\Theta\big{)}$ (4.7) $\displaystyle=$ $\displaystyle{\rm
dist}\big{(}\lambda(X)+t\lambda^{\prime}(X;H)+\hbox{${1\over
2}$}t^{2}\lambda^{{}^{\prime\prime}}(X;H,W)+o(t^{2}),\Theta\big{)}.$

Take $W\in T^{2}_{C}(X,H)$. By (2.15), there exists a sequence
$t_{k}\downarrow 0$ such that $X+t_{k}H+\frac{1}{2}t_{k}^{2}W+o(t_{k}^{2})\in
C$. By (4.7), we get $\lambda(X)+t_{k}\lambda^{\prime}(X;H)+\hbox{${1\over
2}$}t_{k}^{2}\lambda^{{}^{\prime\prime}}(X;H,W)+o(t_{k}^{2})\in\Theta$, which
clearly demonstrates that $\lambda^{\prime\prime}\big{(}X;H,W\big{)}\in
T^{2}_{\Theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}$ and thus proves
the inclusion “$\subset$” in (4.6). The opposite inclusion in (4.6) can be
established via a similar argument and (4.7), which proves the claimed
representation of the second-order tangent set to $C$ in (4.6).

To prove (a), suppose that the symmetric set $\Theta$ is parabolically
derivable at $\lambda(X)$ for $\lambda^{\prime}(X;H)$. To justify the same
property for $C$ at $X$ for $H$, pick $W\in T^{2}_{C}(X,H)$. By (4.6), we
obtain $\lambda^{{}^{\prime\prime}}(X;H,W)\in
T^{2}_{\Theta}(\lambda(X),\lambda^{\prime}(X;H))$. Since $\Theta$ is
parabolically derivable at $\lambda(X)$ for $\lambda^{\prime}(X;H)$, we find
$\varepsilon>0$ such that for all $t\in[0,\varepsilon]$, we have

$\lambda(X)+t\lambda^{\prime}(X;H)+\hbox{${1\over
2}$}t^{2}\lambda^{{}^{\prime\prime}}(X;H,W)+o(t^{2})\in\Theta.$

Reducing $\varepsilon>0$ if necessary, pick $t\in[0,\varepsilon]$ and conclude
from (4.7) that $X+tH+\frac{1}{2}t^{2}W+o(t^{2})\in C$. Defining the arc
$\xi:[0,\varepsilon]\to C$ by $\xi(t)=X+tH+\frac{1}{2}t^{2}W+o(t^{2})$ for
$t\in[0,\varepsilon]$, we can readily see that $\xi(0)=X$,
$\xi_{+}^{\prime}(0)=H$, and $\xi^{\prime\prime}_{+}(0)=W$. To finish the
proof of parabilic derivability of $C$ at $X$ for $H$, it remains to show that
$T^{2}_{C}(X,H)\neq\emptyset$. To this end, pick $Z\in{\mathbb{S}}^{n}$ and
$y\in T^{2}_{\Theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}$. In fact,
such $y$ exists since $\Theta$ is parabolic derivable at $\lambda(X)$ for
$\lambda^{\prime}(X;H)$. Therefore, we have

$\lambda^{{}^{\prime\prime}}(X;H,Z)+p\in
T^{2}_{\Theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}\quad\mbox{ with
}\;\;p:=y-\lambda^{{}^{\prime\prime}}(X;H,Z),$

which can be equivalently expressed as $Z\in S_{H}(p)$ via the mapping $S_{H}$
in (4.1). Appealing to Proposition 4.1 and the established outer Lipschitzian
property in (4.3), we find a matrix $W\in S_{H}(0)$ such that
$\|Z-W\|\leq\|p\|$. This tells us that

$\lambda^{{}^{\prime\prime}}(X;H,W)\in
T^{2}_{\Theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}.$

Using the chain rule (4.6) leads us to $W\in T^{2}_{C}(X,H)$, and thus
$T^{2}_{C}(X,H)\neq\emptyset$. This shows that $C$ is parabolically derivable
at $X$ for $H$, and thus proves (a).

Turning into the proof of (b), observe first that $\Theta$ can be represented
as the constraint system (3.3). Adapting the estimate in (3.5) for the latter
constraint system gives us the estimate

${\rm dist}(x,\Theta)={\rm dist}(\mbox{\rm Diag}\,(x),C)\quad\mbox{for
all}\;\;x\in{\mathbb{R}}^{n}.$

This, together with twice differentiability of the mapping $x\mapsto\mbox{\rm
Diag}\,(x)$ with $x\in{\mathbb{R}}^{n}$, allows us to conclude from (4.5) that
for any $v\in T_{\Theta}(\lambda(X))$ we always have

$w\in T^{2}_{\Theta}(\lambda(X),v)\iff\mbox{\rm Diag}\,(w)\in
T^{2}_{C}\big{(}\mbox{\rm Diag}\,(\lambda(X)),\mbox{\rm Diag}\,(v)\big{)}.$

To justify (b), pick $v\in T_{\Theta}(\lambda(X))$. We are going to show that
$\Theta$ is parabolically derivable at $\lambda(X)$ for $v$. According to
Theorem 4.2, this will be ensured provided that $C$ is parabolically derivable
at $\mbox{\rm Diag}\,(\lambda(X))$ for $\mbox{\rm Diag}\,(v)$. Since $C$ is a
spectral set, it is easy to see that

$W\in T^{2}_{C}\big{(}\mbox{\rm Diag}\,(\lambda(X)),\mbox{\rm
Diag}\,(v)\big{)}\iff UWU^{\top}\in T_{C}^{2}\big{(}X,U\mbox{\rm
Diag}\,(v)U^{\top}\big{)},$ (4.8)

where $U$ is taken from (2.1). Moreover, it follows from $v\in
T_{\Theta}(\lambda(X))$ and Proposition 3.2 that $\mbox{\rm Diag}\,(v)\in
T_{C}\big{(}\mbox{\rm Diag}\,(\lambda(X))\big{)}$, which tells us that
$U\mbox{\rm Diag}\,(v)U^{\top}\in T_{C}(X)$. By assumption, we know that $C$
is parabolically derivable at $X$ for $U\mbox{\rm Diag}\,(v)U^{\top}$. This,
combined with (4.8), confirms that $C$ is parabolically derivable at
$\mbox{\rm Diag}\,(\lambda(X))$ for $\mbox{\rm Diag}\,(v)$. Employing now
Theorem 4.2 proves that $\Theta$ is parabolically derivable at $\lambda(X)$
for $v$ and hence completes the proof.

######  Example 4.4 (second-order tangent set to ${\mathbb{S}}^{n}_{-}$).

In the framework of Example 3.7, we are going to calculate the second-order
tangent set to ${\mathbb{S}}^{n}_{-}$ at $X\in{\mathbb{S}}^{n}_{-}$ for any
$H\in T_{{\mathbb{S}}^{n}_{-}}(X)$. To this end, we deduce from Theorem 4.3
that

$T^{2}_{{\mathbb{S}}^{n}_{-}}(X,H)=\big{\\{}W\in{\mathbb{S}}^{n}\big{|}\;\lambda^{\prime\prime}\big{(}X;H,W\big{)}\in
T^{2}_{{\mathbb{R}}^{n}_{-}}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}\big{\\}}.$

It follows from [23, Proposition 13.12] that ${\mathbb{R}}^{n}_{-}$ is
parabolically derivable at $\lambda(X)$ for $\lambda^{\prime}(X;H)$, which
together with Theorem 4.3 implies that ${\mathbb{S}}^{n}_{-}$ enjoys the same
property at $X$ for $H$. Moreover, we deduce from [23, Proposition 13.12] that

$T^{2}_{{\mathbb{R}}^{n}_{-}}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}=T_{T_{{\mathbb{R}}^{n}_{-}}(\lambda(X))}\big{(}\lambda^{\prime}(X;H)\big{)}.$
(4.9)

If $\mu_{1}<0$, we have $\lambda(X)\in\mbox{\rm int}\,{\mathbb{R}}^{n}_{-}$.
This implies that $T_{{\mathbb{R}}^{n}_{-}}(\lambda(X))={\mathbb{R}}^{n}$,
which together with (4.9) yields
$T^{2}_{{\mathbb{R}}^{n}_{-}}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}={\mathbb{R}}^{n}$
and thus $T^{2}_{{\mathbb{S}}^{n}_{-}}(X,H)={\mathbb{S}}^{n}$. Now, assume
that $\mu_{1}=0$. According to Example 3.7, we have
$T_{{\mathbb{R}}^{n}_{-}}(\lambda(X))={\mathbb{R}}^{|\alpha_{1}|}_{-}\times{\mathbb{R}}^{n-|\alpha_{1}|}$,
where $\alpha_{1}$ is defined by (2.4). To proceed, since $H\in
T_{{\mathbb{S}}^{n}_{-}}(X)$, we need by (3.11) to consider two cases: 1)
$\lambda_{1}(U_{\alpha_{1}}^{\top}HU_{\alpha_{1}})<0$; and 2)
$\lambda_{1}(U_{\alpha_{1}}^{\top}HU_{\alpha_{1}})=0$. If the former holds, we
obtain

$T_{T_{{\mathbb{R}}^{n}_{-}}(\lambda(X))}\big{(}\lambda^{\prime}(X;H)\big{)}=T_{{\mathbb{R}}^{|\alpha_{1}|}_{-}\times{\mathbb{R}}^{n-|\alpha_{1}|}}\big{(}\lambda^{\prime}(X;H)\big{)}={\mathbb{R}}^{n},$

which together with (4.9) brings us again to
$T^{2}_{{\mathbb{R}}^{n}_{-}}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}={\mathbb{R}}^{n}$
and thus $T^{2}_{{\mathbb{S}}^{n}_{-}}(X,H)={\mathbb{S}}^{n}$. If the latter
holds, denote by $\eta_{1}^{1}>\cdots>\eta_{\rho_{1}}^{1}$ the distinct
eigenvalues of $U_{\alpha_{1}}^{\top}HU_{\alpha_{1}}$ and take the index set
$\beta_{1}^{1}$ from (2.11). Recall that $|\beta_{1}^{1}|\leq|\alpha_{1}|$.
Using this, we obtain

$T_{T_{{\mathbb{R}}^{n}_{-}}(\lambda(X))}\big{(}\lambda^{\prime}(X;H)\big{)}=T_{{\mathbb{R}}^{|\alpha_{1}|}_{-}\times{\mathbb{R}}^{n-|\alpha_{1}|}}\big{(}\lambda^{\prime}(X;H)\big{)}={\mathbb{R}}^{|\beta_{1}^{1}|}_{-}\times{\mathbb{R}}^{n-|\beta_{1}^{1}|}.$

This, combined with (2.13) and (4.9), leads us to

$\displaystyle T^{2}_{{\mathbb{S}}^{n}_{-}}(X,H)$ $\displaystyle=$
$\displaystyle\big{\\{}W\in{\mathbb{S}}^{n}\big{|}\;\lambda^{\prime\prime}\big{(}X;H,W\big{)}\in{\mathbb{R}}^{|\beta_{1}^{1}|}_{-}\times{\mathbb{R}}^{n-|\beta_{1}^{1}|}\big{\\}}$
$\displaystyle=$
$\displaystyle\big{\\{}W\in{\mathbb{S}}^{n}\big{|}\;\lambda_{1}\Big{(}{R_{11}}^{\top}\big{(}U_{\alpha_{1}}^{\top}(W-2HX^{\dagger}H)U_{\alpha_{1}}\big{)}R_{11}\Big{)}\leq
0\big{\\}},$

where $R_{11}=(Q_{1})_{\beta_{1}^{1}}$ is taken from Proposition 2.2. We
should point out that the second-order tangent set to ${\mathbb{S}}^{n}_{-}$
was calculated by finding the parabolic second-order directional derivative of
the maximum eigenvalue function in [3, page 474]; see also [26, page 583] for
a different derivation of this object.

In the next example, we obtain the second-order tangent set to the spectral
set defined in (3.12). Note again that while obtaining such result by [3,
Proposition 3.92] requires the Slater condition, our approach shows that no
constraint qualification is needed for this propose.

######  Example 4.5.

Let $C$ be the spectral set in (3.12) and $X\in C$. Given $H\in T_{C}(X)$, we
aim to determine $T^{2}_{C}(X,H)$ using Theorem 4.3. We know from Example 3.8
that $C$ has the spectral representation in (3.2) with the symmetric set
$\Theta$ defined by (3.13). Moreover, we showed that $\Theta$ can be
equivalently described as the constraint set in (3.14) with
$\Phi(z)=(\sum^{n}_{i=1}z_{i}^{k}-1,z)$ for all
$z=(z_{1},\ldots,z_{n})\in{\mathbb{R}}^{n}$. We deduce from [23, Proposition
13.13] that $w\in T^{2}_{\Theta}(\lambda(X),\lambda^{\prime}(X;H))$ if and
only if we have

$\nabla\Phi(\lambda(X))w+\nabla^{2}\Phi(\lambda(X))\big{(}\lambda^{\prime}(X;H),\lambda^{\prime}(X;H)\big{)}\in
T^{2}_{\\{0\\}\times{\mathbb{R}}^{n}_{+}}\big{(}\Phi(\lambda(X)),\nabla\Phi(\lambda(X))(\lambda^{\prime}(X;H))\big{)}.$
(4.10)

Using the index set $I(\lambda(X))$ taken from Example 3.8, define the index
set

$I\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}:=\big{\\{}i\in
I(\lambda(X))\big{|}\;\lambda^{\prime}_{i}(X;H)=0\big{\\}}$

and conclude then from [23, Proposition 13.12] that

$\displaystyle
T^{2}_{\\{0\\}\times{\mathbb{R}}^{n}_{+}}\big{(}\Phi(\lambda(X)),\nabla\Phi(\lambda(X))(\lambda^{\prime}(X;H))\big{)}=T_{T_{\\{0\\}\times{\mathbb{R}}^{n}_{+}}(\Phi(\lambda(X)))}\big{(}\nabla\Phi(\lambda(X))(\lambda^{\prime}(X;H))\big{)}$
$\displaystyle=\big{\\{}(w_{0},\ldots,w_{n})\big{|}\>w_{0}=0,\;w_{i}\geq
0\quad\mbox{for all}\;\;i\in
I\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}\big{\\}}.$

This, coupled with (4.10), yields $(w_{1},\ldots,w_{n})\in
T^{2}_{\Theta}(\lambda(X),\lambda^{\prime}(X;H))$ if and only if $w_{i}\geq 0$
for all $i\in I\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}$ and

$\sum^{n}_{i=1}\lambda_{i}(X)^{k-1}w_{i}+(k-1)\sum^{n}_{i=1}\lambda_{i}(X)^{k-2}\lambda^{\prime}_{i}(X;H)^{2}=0.$

Appealing now to Theorem 4.3, we conclude that $C$ is parabolically derivable
at $X$ for $H$ and that $W\in T^{2}(X,H)$ if and only if
$\lambda_{i}^{{}^{\prime\prime}}(X;H,W)\geq 0$ for all $i\in
I\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}$ and

$\sum^{n}_{i=1}\lambda_{i}(X)^{k-1}\lambda_{i}^{{}^{\prime\prime}}(X;H,W)+(k-1)\sum^{n}_{i=1}\lambda_{i}(X)^{k-2}\lambda^{\prime}_{i}(X;H)^{2}=0.$

Note that when $k=1$ for which $C$ reduces to the spectahedron, the above
equation simplifies as
$\sum^{n}_{i=1}\lambda_{i}^{{}^{\prime\prime}}(X;H,W)=0$.

We proceed with characterizing parabolic epi-differentiability of spectral
functions. We begin with recalling the concept of the parabolic subderivative,
introduced by Ben-Tal and Zowe in [2]. Let
$f:{\mathbb{E}}\to\overline{\mathbb{R}}$ and let $\bar{x}\in{\mathbb{E}}$ with
$f(\bar{x})$ finite and $w\in{\mathbb{E}}$ with ${\rm d}f(\bar{x})(w)$ finite.
The parabolic subderivative of $f$ at $\bar{x}$ for $w$ with respect to $z$ is
defined by

${\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;z):=\liminf_{\begin{subarray}{c}t\downarrow
0\\\ z^{\prime}\to
z\end{subarray}}\dfrac{f(\bar{x}+tw+\frac{1}{2}t^{2}z^{\prime})-f(\bar{x})-t{\rm
d}f(\bar{x})(w)}{\frac{1}{2}t^{2}}.$

Recall from [23, Definition 13.59] that $f$ is called parabolically epi-
differentiable at $\bar{x}$ for $w$ if

$\mbox{\rm dom}\,{\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;\cdot)=\big{\\{}z\in{\mathbb{E}}|\,{\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;z)<\infty\big{\\}}\neq\emptyset,$

and for every $z\in{\mathbb{E}}$ and every sequence $t_{k}\downarrow 0$ there
exists a sequences $z_{k}\to z$ such that

${\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;z)=\lim_{k\to\infty}\dfrac{f(\bar{x}+t_{k}w+\frac{1}{2}t_{k}^{2}z_{k})-f(\bar{x})-t_{k}{\rm
d}f(\bar{x})(w)}{\frac{1}{2}t_{k}^{2}}.$ (4.11)

We say that $f$ is parabolically epi-differentiable at $\bar{x}$ if it
satisfies this condition at $\bar{x}$ for any $w\in{\mathbb{E}}$ where ${\rm
d}f(\bar{x})(w)$ is finite. Note that the inclusion $\mbox{\rm dom}\,{\rm
d}f(\bar{x})\subset T_{\scriptsize\mbox{\rm dom}\,f}(\bar{x})$ always holds
and equality happens when, in addition, $f$ is locally Lipschitz continuous
around $\bar{x}$ relative to its domain; see [20, Proposition 2.2]. A list of
important functions, appearing in different classes of constrained and
composite optimization problems, that are parabolically epi-differentiable at
any points of their domains can be found in [20, Example 4.7]. By definition,
it is not hard to see that the inclusion

$\mbox{\rm dom}\,{\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;\cdot)\subset
T_{\scriptsize\mbox{\rm dom}\,f}^{2}(\bar{x},w)$ (4.12)

always holds for any $w\in T_{\scriptsize\mbox{\rm dom}\,f}(\bar{x})$. The
following result, taken from [20, Propositions 2.1 and 4.1], presents
conditions under which we can ensure equality in the latter inclusion.

######  Proposition 4.6 (properties of parabolic subderivatives).

Let $f:{\mathbb{E}}\to\overline{\mathbb{R}}$ be finite at $\bar{x}$, locally
Lipschitz continuous around $\bar{x}$ relative to its domain, and parabolic
epi-differentiable at $\bar{x}$ for $w\in T_{\scriptsize\mbox{\rm
dom}\,f}(\bar{x})$. Then the following properties hold.

  * (a)

$\mbox{\rm dom}\,{\rm d}f(\bar{x})=T_{\scriptsize\mbox{\rm dom}\,f}(\bar{x})$
and $\mbox{\rm dom}\,{\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;.)=T_{\scriptsize\mbox{\rm
dom}\,f}^{2}(\bar{x},w)$.

  * (b)

$\mbox{\rm dom}\,f$ is parabolically derivable at $\bar{x}$ for $w$.

The next result presents sufficient conditions under which spectral functions
are parabolically epi-differentiable. Moreover, it achieves a useful formula
for parabolic subderivatives of this class of functions.

######  Theorem 4.7 (parabolic subderivatives of spectral function).

Let $\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$ be a symmetric function,
which is locally Lipschitz continuous relative to its domain. Let
$X\in{\mathbb{S}}^{n}$ with $(\theta\circ\lambda)(X)$ finite. Then the
following properties hold.

  * (a)

If $H\in T_{\scriptsize\mbox{\rm dom}\,(\theta\circ\lambda)}(X)$ and $\theta$
is parabolically epi-differentiable at $\lambda(X)$ for
$\lambda^{\prime}(X;H)$, then $\theta\circ\lambda$ is parabolically epi-
differentiable at $X$ for $H$ and its parabolic subderivative at $X$ for $H$
and its domain can be calculated, respectively, by

${\rm
d}^{2}(\theta\circ\lambda)(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)={\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;\lambda^{{}^{\prime\prime}}(X;H,W)\big{)}$
(4.13)

and

$\mbox{\rm dom}\,{\rm
d}^{2}(\theta\circ\lambda)(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;\;.\;)=T_{\scriptsize\mbox{\rm
dom}\,(\theta\circ\lambda)}^{2}(X,H).$ (4.14)

Moreover, if $\theta$ is lsc and convex, the parabolic subderivative
$W\mapsto{\rm
d}^{2}(\theta\circ\lambda)(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)$
is a convex function.

  * (b)

If $\theta\circ\lambda$ is parabolically epi-differentiable at $X$, then
$\theta$ is parabolically epi-differentiable at $\lambda(X)$.

[Proof. ]To justify (a), we proceed concurrently to show that
$\theta\circ\lambda$ is parabolically epi-differentiable at $X$ for $H$ and
that (4.13) and (4.11) hold for $\theta\circ\lambda$. To this end, set
$g:=\theta\circ\lambda$ and pick $W\in{\mathbb{S}}^{n}$ and proceed with
considering two cases. Assume first that $W\notin T_{\scriptsize\mbox{\rm
dom}\,g}^{2}(X,H)$. Employing the inclusion in (4.12) for $g$, we get ${\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)=\infty$. On the
other hand, by (3.4) and Theorem 4.3, we obtain

$T^{2}_{\scriptsize\mbox{\rm
dom}\,g}(X,H)=\big{\\{}W\in{\mathbb{S}}^{n}\big{|}\;\lambda^{\prime\prime}\big{(}X;H,W\big{)}\in
T^{2}_{\scriptsize\mbox{\rm
dom}\,\theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}\big{\\}}.$ (4.15)

This, combined with $W\notin T_{\scriptsize\mbox{\rm dom}\,g}^{2}(X,H)$,
yields $\lambda^{{}^{\prime\prime}}(X;H,W)\notin T^{2}_{\scriptsize\mbox{\rm
dom}\,\theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}$. Observe from
Corolllary 3.6 and (3.4) that $H\in T_{\scriptsize\mbox{\rm dom}\,g}(X)$
amounts to $\lambda^{\prime}(X;H)\in T_{\scriptsize\mbox{\rm
dom}\,\theta}(\lambda(X))$. By Theorem 4.6(a), we arrive at

$\mbox{\rm dom}\,{\rm
d}^{2}\theta(\lambda(X))(\lambda(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;\cdot)=T_{\scriptsize\mbox{\rm
dom}\,\theta}^{2}(\lambda(X),\lambda^{\prime}(X;H)).$ (4.16)

Combining these tells us that ${\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;\lambda^{{}^{\prime\prime}}(X;H,W)\big{)}=\infty,$
which in turn justifies (4.13) for every $W\notin T_{\scriptsize\mbox{\rm
dom}\,g}^{2}(X,H)$. To verify (4.11) for $g$ in this case, consider an
arbitrary sequence $t_{k}\downarrow 0$, set $W_{k}:=W$ for all
$k\in{\mathbb{N}}$, and observe that

$\displaystyle\infty={\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)$
$\displaystyle\leq$
$\displaystyle\liminf_{k\to\infty}\dfrac{g(X+t_{k}H+\frac{1}{2}t_{k}^{2}W_{k})-g(X)-t_{k}{\rm
d}g(X)(W)}{\frac{1}{2}t_{k}^{2}}.$

This clearly justifies (4.11) for all $W\notin T_{\scriptsize\mbox{\rm
dom}\,g}^{2}(X,H)$.

Turning now to the case $W\in T_{\scriptsize\mbox{\rm dom}\,g}^{2}(X,H)$, we
observe that since $\theta$ is parabolically epi-differentiable at
$\lambda(X)$ for $\lambda^{\prime}(X;H)$, Theorem 4.6(b) tells us that
$\mbox{\rm dom}\,\theta$ is parabolically derivable at $\lambda(X)$ for
$\lambda^{\prime}(X;H)$. We conclude from Proposition 4.3 that $\mbox{\rm
dom}\,g$ is parabolically derivable at $X$ for $H$. In particular, we have

$T_{\scriptsize\mbox{\rm dom}\,g}^{2}(X,H)\neq\emptyset.$ (4.17)

Pick now $W\in T_{\scriptsize\mbox{\rm dom}\,g}^{2}(X,H)$ and consider then an
arbitrary sequence $t_{k}\downarrow 0$. Thus, by the definition of parabolic
derivability, we find a sequence $W_{k}\to W$ as $k\to\infty$ such that

$X_{k}:=X+t_{k}H+\hbox{${1\over 2}$}t_{k}^{2}W_{k}\in\mbox{\rm
dom}\,g\quad\mbox{for all}\;\;k\in{\mathbb{N}}.$ (4.18)

Moreover, since $\theta$ is parabolically epi-differentiable at $\lambda(X)$
for $\lambda^{\prime}(X;H)$, we find a sequence $w_{k}\to
w:=\lambda^{{}^{\prime\prime}}(X;H,W)$ such that

$\displaystyle{\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)$
$\displaystyle=$
$\displaystyle\lim_{k\to\infty}\dfrac{\theta(\lambda(X)+t_{k}\lambda^{\prime}(X;H)+\frac{1}{2}t_{k}^{2}w_{k})-\theta(\lambda(X))-t_{k}{\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))}{\frac{1}{2}t_{k}^{2}}.$

It follows from (4.15) and $W\in T_{\scriptsize\mbox{\rm dom}\,g}^{2}(X,H)$
that $w\in T^{2}_{\scriptsize\mbox{\rm
dom}\,\theta}\big{(}\lambda(X),\lambda^{\prime}(X;H)\big{)}$. Combining this
with (4.16) tells us that ${\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)<\infty$.
This implies that
$y_{k}:=\lambda(X)+t_{k}\lambda^{\prime}(X;H)+\frac{1}{2}t_{k}^{2}w_{k}\in\mbox{\rm
dom}\,\theta$ for all $k$ sufficiently large. Using this together with (3.8),
(4.18), and (4), we obtain

$\displaystyle{\rm d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)$
$\displaystyle\leq$
$\displaystyle\liminf_{k\to\infty}\dfrac{g(X+t_{k}H+\frac{1}{2}t_{k}^{2}W_{k})-g(X)-t_{k}{\rm
d}g(X)(H)}{\hbox{${1\over 2}$}t_{k}^{2}}$ (4.19) $\displaystyle\leq$
$\displaystyle\limsup_{k\to\infty}\dfrac{\theta(\lambda(X_{k}))-\theta(\lambda(X))-t_{k}{\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))}{\frac{1}{2}t_{k}^{2}}$
$\displaystyle\leq$
$\displaystyle\limsup_{k\to\infty}\dfrac{\theta(y_{k})-\theta(\lambda(X))-t_{k}{\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))}{\frac{1}{2}t_{k}^{2}}+\limsup_{k\to\infty}\dfrac{\theta(\lambda(X_{k}))-\theta(y_{k})}{\frac{1}{2}t_{k}^{2}}$
$\displaystyle\leq$ $\displaystyle{\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)+\limsup_{k\to\infty}\ell\|\lambda{{}^{\prime\prime}}(X;H,W)-w_{k}+\frac{o(t_{k}^{2})}{t_{k}^{2}}\|$
$\displaystyle=$ $\displaystyle{\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w),$

where $\ell\geq 0$ is a Lipschitz constant of $\theta$ around $\lambda(X)$
relative to its domain. On the other hand, for any sequence $t_{k}\downarrow
0$ and any sequence $W_{k}\to W$, we can always conclude from (2.14) and (3.8)
that

$\displaystyle\liminf_{k\to\infty}\frac{g(X+t_{k}H+\frac{1}{2}t_{k}^{2}W_{k})-g(X)-t_{k}{\rm
d}g(X)(H)}{\hbox{${1\over 2}$}t_{k}^{2}}$ $\displaystyle=$
$\displaystyle\liminf_{k\to\infty}\dfrac{\theta\big{(}\lambda(X)+t_{k}\lambda^{\prime}(X;H)+\frac{1}{2}t_{k}^{2}\lambda^{{}^{\prime\prime}}(X;H,W)+o(t_{k}^{2})\big{)}-\theta(\lambda(X))-t_{k}{\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))}{\frac{1}{2}t_{k}^{2}}$
$\displaystyle\geq$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\
w^{\prime}\to
w\end{subarray}}\dfrac{\theta\big{(}\lambda(X)+t\lambda^{\prime}(X;H)+\frac{1}{2}t^{2}w^{\prime}\big{)}-\theta(\lambda(X))-t{\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))}{\frac{1}{2}t^{2}}$
$\displaystyle=$ $\displaystyle{\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w).$

This clearly yields the inequality

${\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)\leq{\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)\quad\mbox{with}\;\;w=\lambda^{{}^{\prime\prime}}(X;H,W).$

Combining this and (4.19) implies that

${\rm d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)={\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)$
(4.20)

and that

${\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)=\lim_{k\to\infty}\dfrac{g(X+t_{k}H+\frac{1}{2}t_{k}^{2}W_{k})-g(X)-t_{k}{\rm
d}g(X)(W)}{\frac{1}{2}t_{k}^{2}},$

which in turn prove both (4.13) and (4.11) for any $W\in
T_{\scriptsize\mbox{\rm dom}\,g}^{2}(X,H)$. As argued above, we also have
${\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)<\infty$,
which together with (4.20) tells us that ${\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)<\infty$. This
brings us to the inclusion

$T_{\scriptsize\mbox{\rm dom}\,g}^{2}(X,H)\subset\mbox{\rm dom}\,{\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;\cdot).$

Since the opposite inclusion always holds (see (4.12)), we arrive at (4.14).
Combining this and (4.17) indicates that $\mbox{\rm dom}\,{\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;\cdot)\neq\emptyset$,
and hence shows that $g$ is parabolically epi-differentiable at $X$ for $H$.
Finally, assume that $\theta$ is lsc and convex. By [4, Corollary 5.2.3], the
spectral function $g=\theta\circ\lambda$ is convex. According to [23, Example
13.62], parabolic epi-differentiability of $g$ at $X$ for $H$ amounts to
parabolic derivability of $\mbox{\rm epi}\,g$ at $(X,g(X))$ for $(H,{\rm
d}g(X)(H))$ and

$\mbox{\rm epi}\,{\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;\;.\;)=T^{2}_{\scriptsize\mbox{\rm
epi}\,g}\big{(}(X,g(X)),(H,{\rm d}g(X)(H))\big{)}.$

Since $g$ is convex, it follows from parabolic derivability of $\mbox{\rm
epi}\,g$ at $(X,g(X))$ for $(H,{\rm d}g(X)(H))$ that
$T^{2}_{\scriptsize\mbox{\rm epi}\,g}\big{(}(X,g(X)),(H,{\rm
d}g(X)(H))\big{)}$ is a convex set. The above equality then confirms that
$W\mapsto{\rm
d}^{2}(\theta\circ\lambda)(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)$
is a convex function and hence completes the proof of (a).

Turning into the proof of (b), we conclude from (1.1) that the symmetric
function $\theta$ satisfies (3.1), which means that $\theta$ can be
represented as a composite function of $g$ and the linear mapping
$x\mapsto\mbox{\rm Diag}\,(x)$ with $x\in{\mathbb{R}}^{n}$. We also deduce
from the imposed assumption on $\theta$ and the inequality in (2.2) that $g$
is locally Lipschitz continuous relative to its domain. Pick $v\in\mbox{\rm
dom}\,{\rm d}\theta(\lambda(X))=T_{\scriptsize\mbox{\rm
dom}\,\theta}(\lambda(X))$ and apply the chain rule in (3.7) to the
representation (3.4) of $\mbox{\rm dom}\,\theta$ to obtain $\mbox{\rm
Diag}\,(v)\in T_{\scriptsize\mbox{\rm dom}\,g}(\mbox{\rm
Diag}\,(\lambda(X)))$. To justify the parabolic epi-differentiability of
$\theta$ at $\lambda(X)$ for $v$, we are going to use [20, Theorem 4.4(iii)]
by showing that $g$ is parabolically epi-differentiable at $\mbox{\rm
Diag}\,(\lambda(X))$ for $\mbox{\rm Diag}\,(v)$. To this end, it is not hard
to see for any $U\in{\mathbb{O}}^{n}(X)$ that

${\rm d}g\big{(}\mbox{\rm Diag}\,(\lambda(X))\big{)}\big{(}\mbox{\rm
Diag}\,(v)\big{)}={\rm d}g(X)(U\mbox{\rm Diag}\,(v)U^{\top})$ (4.21)

Indeed, since $g$ is orthogonally invariant, we get for any
$U\in{\mathbb{O}}^{n}(X)$ that

$\displaystyle{\rm d}g(X)(U\mbox{\rm Diag}\,(v)U^{\top})$ $\displaystyle=$
$\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\ W\to U\mbox{\rm
Diag}\,(v)U^{\top}\end{subarray}}\frac{g\big{(}X+tW\big{)}-\theta\big{(}X)\big{)}}{t}$
$\displaystyle=$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\
U^{\top}WU\to\mbox{\rm Diag}\,(v)\end{subarray}}\frac{g\big{(}\mbox{\rm
Diag}\,(\lambda(X))+tU^{\top}WU\big{)}-\theta\big{(}\mbox{\rm
Diag}\,(\lambda(X))\big{)}}{t}$ $\displaystyle\geq$ $\displaystyle{\rm
d}g\big{(}\mbox{\rm Diag}\,(\lambda(X))\big{)}\big{(}\mbox{\rm
Diag}\,(v)\big{)}.$

A similar argument leads us to ${\rm d}g\big{(}\mbox{\rm
Diag}\,(\lambda(X))\big{)}\big{(}\mbox{\rm Diag}\,(v)\big{)}\leq{\rm
d}g(X)(U\mbox{\rm Diag}\,(v)U^{\top})$ and thus proves (4.21). Similarly, we
can show that

${\rm d}^{2}g\big{(}\mbox{\rm Diag}\,(\lambda(X))\big{)}\big{(}\mbox{\rm
Diag}\,(v)\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W\big{)}={\rm
d}^{2}g(X)(U\mbox{\rm
Diag}\,(v)U^{\top}\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;UWU^{\top}),\;\;W\in{\mathbb{S}}^{n}.$
(4.22)

Since $g$ is a spectral function, $\mbox{\rm dom}\,g$ is a spectral set. This
and $\mbox{\rm Diag}\,(v)\in T_{\scriptsize\mbox{\rm dom}\,g}(\mbox{\rm
Diag}\,(\lambda(X)))$ tell us that $U\mbox{\rm Diag}\,(v)U^{\top}\in
T_{\scriptsize\mbox{\rm dom}\,g}(X)$. Since $g$ is parabolically epi-
differentiable at $X$ for $U\mbox{\rm Diag}\,(v)U^{\top}$, the equality in
(4.22) confirms that $g$ enjoys the same property at $\mbox{\rm
Diag}\,(\lambda(X))$ for $\mbox{\rm Diag}\,(v)$. Combining this, (3.5), and
[20, Theorem 4.4(iii)] shows that $\theta$ is parabolically epi-differentiable
at $\lambda(X)$ for $v$ and hence completes the proof.

We close this section by revealing that the parabolic subderivative of
spectral functions is symmetric with respect to a subset of
${\mathbb{P}}^{n}$, the set of all $n\times n$ permutation matrices. This
plays a central role in next section when we are going to study parabolic
regularity of spectral functions. To this end, recall from Remark 3.3 that if
$\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$ is a symmetric function and
$X\in{\mathbb{S}}^{n}$ with $\theta(\lambda(X))$ finite, the subderivative
function ${\rm d}\theta(\lambda(X))$ is a symmetric function with respect to
${\mathbb{P}}^{n}_{X}$, which is a subset of ${\mathbb{P}}^{n}$ consisting of
all $n\times n$ block diagonal matrices in the form $Q=\mbox{\rm
Diag}\,(P_{1},\ldots,P_{r})$, where
$P_{m}\in{\mathbb{R}}^{|\alpha_{m}|\times|\alpha_{m}|}$, $m=1,\ldots,r$, is a
permutation matrix with $\alpha_{m}$ taken from (2.4) and $r$ being the number
of distinct eigenvalues of $X$. Consider now $H\in{\mathbb{S}}^{n}$ with ${\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))$ finite. Take the orthogonal
matrix $U$ from (2.1) and $m\in\\{1,\ldots,r\\}$. Suppose that $\rho_{m}$ is
the number of distinct eigenvalues of $U^{\top}_{\alpha_{m}}HU_{\alpha_{m}}$
and pick then the index sets $\beta_{j}^{m}$ for $j=1,\ldots,\rho_{m}$ from
(2.11). Denote by ${\mathbb{P}}^{n}_{X,H}$ a subset of ${\mathbb{P}}^{n}_{X}$
consisting of all $n\times n$ matrices with representation $\mbox{\rm
Diag}\,(P_{1},\ldots,P_{r})$ such that for each $m=1,\ldots,r$, the
$|\alpha_{m}|\times|\alpha_{m}|$ permutation matrix $P_{m}$ has a block
diagonal representation $P_{m}=\mbox{\rm
Diag}\,(B^{m}_{1},\ldots,B^{m}_{\rho_{m}})$, where
$B_{j}^{m}\in{\mathbb{R}}^{|\beta_{j}^{m}|\times|\beta_{j}^{m}|}$ is a
permutation matrix for any $j=1,\ldots,\rho_{m}$. It is not hard to see that

$Q\lambda(X)=\lambda(X)\quad\mbox{and}\quad
Q\lambda^{\prime}(X;H)=\lambda^{\prime}(X;H)\quad\mbox{for
any}\;\;Q\in{\mathbb{P}}^{n}_{X,H}.$ (4.23)

######  Proposition 4.8.

Assume that $\theta:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$ is a symmetric
function and $X\in{\mathbb{S}}^{n}$ with $\theta(\lambda(X))$ finite and that
$H\in{\mathbb{S}}^{n}$ with ${\rm d}\theta(\lambda(X))(\lambda^{\prime}(X;H))$
finite. Then for any $w\in{\mathbb{R}}^{n}$ and any permutation matrix
$Q\in{\mathbb{P}}^{n}_{X,H}$, we have

${\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;Qw\big{)}={\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w\big{)},$

which means that the parabolic subderivative $w\mapsto{\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)$
is symmetric with respect to ${\mathbb{P}}^{n}_{X,H}$.

[Proof. ]Pick $w\in{\mathbb{R}}^{n}$ and $Q\in{\mathbb{P}}^{n}_{X,H}$. Since
$\theta$ is symmetric, it follows from (4.23) that

$\displaystyle{\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)$
$\displaystyle=$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\
w^{\prime}\to
w\end{subarray}}\dfrac{\theta(\lambda(X)+t\lambda^{\prime}(X;H)+\frac{1}{2}t^{2}w^{\prime})-\theta(\lambda(X))-t{\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))}{\frac{1}{2}t^{2}}$
$\displaystyle=$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\
w^{\prime}\to
w\end{subarray}}\dfrac{\theta(\lambda(X)+t\lambda^{\prime}(X;H)+\frac{1}{2}t^{2}Qw^{\prime})-\theta(\lambda(X))-t{\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))}{\frac{1}{2}t^{2}}$
$\displaystyle\geq$ $\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\
v\to
Qw\end{subarray}}\dfrac{\theta(\lambda(X)+t\lambda^{\prime}(X;H)+\frac{1}{2}t^{2}v)-\theta(\lambda(X))-t{\rm
d}\theta(\lambda(X))(\lambda^{\prime}(X;H))}{\frac{1}{2}t^{2}}$
$\displaystyle=$ $\displaystyle{\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;Qw).$

Similarly, we can show that ${\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;w)\leq{\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;Qw)$,
which justifies the claimed equality and hence ends the proof.

##  5  Parabolic Regularity of Spectral Functions

This section is devoted to the study of parabolic regularity of spectral
functions whose central role in second-order variational analysis was revealed
recently in [18]. As demonstrated in [18], parabolic regularity can be viewed
as an important second-order regularity with remarkable consequences among
which we should highlight twice epi-differentiability of extended-real-valued
functions. We begin with recalling the concepts of the second subderivative
and parabolic regualrity for functions, respectively. Given a function
$f:{\mathbb{E}}\to\overline{\mathbb{R}}$ and $\bar{x}\in{\mathbb{E}}$ with
$f(\bar{x})$ finite, define the parametric family of second-order difference
quotients for $f$ at $\bar{x}$ for $\bar{v}\in{\mathbb{E}}$ by

$\Delta_{t}^{2}f(\bar{x},\bar{v})(w)=\dfrac{f(\bar{x}+tw)-f(\bar{x})-t\langle\bar{v},\,w\rangle}{\frac{1}{2}t^{2}}\quad\quad\mbox{with}\;\;w\in{\mathbb{E}},\;\;t>0.$

The second subderivative of $f$ at $\bar{x}$ for $\bar{v}$ is defined by

${\rm d}^{2}f(\bar{x},\bar{v})(w)=\liminf_{\begin{subarray}{c}t\downarrow 0\\\
w^{\prime}\to
w\end{subarray}}\Delta_{t}^{2}f(\bar{x},\bar{v})(w^{\prime}),\;\;w\in{\mathbb{E}}.$

The importance of the second subderivative resides in the fact that it can
characterize the quadratic growth condition for optimization problems; see
[23, Theorem 13.24]. So, it is crucial for many applications to calculate it
in terms of the initial data of an optimization problem. This task was carried
out for major classes of functions including the convex piecewise linear-
quadratic functions in the sense of [23, Definition 10.20] in [23, Proposition
13.9], the second-order/ice-cream cone in [18, Example 5.8], the cone of
positive semidefinite symmetric matrices in [20, Example 3.7], and the
augmented Lagrangian of constrained optimization problems in [18, Theorem
8.3]. We are going to calculate it for the spectral function $g$ in (1.1) when
the symmetric function $\theta$ therein is convex.

######  Definition 5.1 (parabolic regularity).

A convex function $f:{\mathbb{E}}\to\overline{\mathbb{R}}$ is parabolically
regular at $\bar{x}$ for $\bar{v}\in\partial f(\bar{x})$ if $f(\bar{x})$ is
finite and if for any $w$ such that ${\rm d}^{2}f(\bar{x},\bar{v})(w)<\infty$,
there exist, among the sequences $t_{k}\downarrow 0$ and $w_{k}\to w$ with
$\Delta_{t_{k}}^{2}f(\bar{x},\bar{v})(w_{k})\to{\rm
d}^{2}f(\bar{x},\bar{v})(w)$, those with the additional property that
$\limsup_{k\to\infty}{\|w_{k}-w\|}/{t_{k}}<\infty.$ We say that $f$ is
parabolically regular at $\bar{x}$ if it is parabolically regular at $\bar{x}$
for every $\bar{v}\in\partial f(\bar{x})$. A nonempty convex set
$C\subset{\mathbb{E}}$ is said to be parabolically regular at $\bar{x}$ if the
indicator function $\delta_{C}$ is parabolically regular at $\bar{x}$.

Parabolic regularity was introduced in [23, Definition 13.65] for extended-
real-valued functions but was not scrutinized therein. It was shown in [18,
20] that polyhedral convex sets, the second-order/ice-cream cone, the cone of
positive semidefinite symmetric matrices are parabolically regular. One can
also find in [23, Corollary 13.68] that convex piecewise linear-quadratic
functions are parabolically regular. Recall that the critical cone of a convex
function $f:{\mathbb{E}}\to\overline{\mathbb{R}}$ at $\bar{x}$ for
$\bar{v}\in\partial f(\bar{x})$ is defined by

$K_{f}(\bar{x},\bar{v})=\\{w\in{\mathbb{E}}\>|\>{\rm
d}f(\bar{x})(w)=\langle\bar{v},w\rangle\\}.$

When $f=\delta_{C}$, where $C$ is a nonempty convex subset of ${\mathbb{E}}$,
the critical cone of $\delta_{C}$ at $\bar{x}$ for $\bar{v}$ is denoted by
$K_{C}(\bar{x},\bar{v})$. In this case, the above definition of the critical
cone of a function boils down to the well known concept of the critical cone
of a set (see [9, page 109]), namely
$K_{C}(\bar{x},\bar{v})=T_{C}(\bar{x})\cap[\bar{v}]^{\perp}$ since ${\rm
d}\delta_{C}(\bar{x})=\delta_{T_{C}(\bar{x})}$.

The following result is a special case of a more general characterization of
parabolic regularity from [20, Proposition 3.6] and will be utilized in our
approach in this section.

######  Proposition 5.2 (characterization of parabolic regularity).

Assume that $f:{\mathbb{E}}\to\overline{\mathbb{R}}$ is convex, finite at
$\bar{x}\in{\mathbb{E}}$, and $\bar{v}\in\partial f(\bar{x})$. Then $f$ is
parabolically regular at $\bar{x}$ for $\bar{v}$ if and only if we have

${\rm d}^{2}f(\bar{x},\bar{v})(w)=\inf_{z\in{\mathbb{E}}}\big{\\{}{\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;z)-\langle
z,\bar{v}\rangle\big{\\}}$

for any $w\in K_{f}(\bar{x},\bar{v})$. Furthermore, for any $w\in\mbox{\rm
dom}\,{\rm d}^{2}f(\bar{x},\bar{v})$, there exists a $\bar{z}\in\mbox{\rm
dom}\,{\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;\cdot)$ such that

${\rm d}^{2}f(\bar{x},\bar{v})(w)={\rm
d}^{2}f(\bar{x})(w\;\rule[-1.13809pt]{0.56905pt}{7.68236pt}\;\bar{z})-\langle\bar{z},\bar{v}\rangle.$

Our results so far required that the symmetric function $\theta$ in (1.1) be
locally Lipschitz continuous with respect to its domain. In what follows, we
need to assume further that $\theta$ is an lsc convex function. This
assumption allows us to use the characterization of the subgradients of the
spectral function $g$ in (1.1), recorded in Proposition 3.4. We begin our
analysis of the second subderivative of spectral functions by finding a lower
estimate for it.

######  Proposition 5.3 (lower estimate for second subderivatives).

Assume that $g:{\mathbb{S}}^{n}\to\overline{\mathbb{R}}$ has the spectral
representation in (1.1) and is lsc and convex and that $Y\in\partial g(X)$.
Let $\mu_{1}>\cdots>\mu_{r}$ be the distinct eigenvalues of $X$ and
$U\in{\mathbb{O}}^{n}(X)\,\cap\,{\mathbb{O}}^{n}(Y)$. Then for any
$H\in{\mathbb{S}}^{n}$ we have

${\rm d}^{2}g(X,Y)(H)\geq{\rm
d}^{2}\theta\big{(}\lambda(X),\lambda(Y)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{\rangle},$
(5.1)

where $\alpha_{m}$, $m=1,\ldots,r$, are defined in (2.4).

[Proof. ]Let $H\in{\mathbb{S}}^{n}$ and pick sequences $H_{k}\to H$ and
$t_{k}\downarrow 0$. Setting
$\Delta_{t_{k}}\lambda(X)(H_{k}):=\big{(}\lambda(X+t_{k}H_{k})-\lambda(X)\big{)}/t_{k}$,
we get

$\displaystyle\Delta^{2}_{t_{k}}g\big{(}X,Y\big{)}(H_{k})$ $\displaystyle=$
$\displaystyle\frac{\theta\big{(}\lambda(X+t_{k}H_{k})\big{)}-\theta\big{(}\lambda(X)\big{)}-t_{k}\big{\langle}Y,H_{k}\big{\rangle}}{\hbox{${1\over
2}$}t_{k}^{2}}$ $\displaystyle=$
$\displaystyle\frac{\theta\big{(}\lambda(X)+t_{k}\Delta_{t_{k}}\lambda(X)(H_{k})\big{)}-\theta\big{(}\lambda(X)\big{)}-t_{k}\big{\langle}\lambda(Y),\Delta_{t_{k}}\lambda(X)(H_{k})\big{\rangle}}{\hbox{${1\over
2}$}t_{k}^{2}}$
$\displaystyle+\frac{\big{\langle}\lambda(Y),\Delta_{t_{k}}\lambda(X)(H_{k})\big{\rangle}-\big{\langle}Y,H_{k}\big{\rangle}}{\hbox{${1\over
2}$}t_{k}}$ $\displaystyle=$
$\displaystyle\Delta^{2}_{t_{k}}\theta\big{(}\lambda(X),\lambda(Y)\big{)}(\Delta_{t_{k}}\lambda(X)(H_{k})\big{)}+\frac{\big{\langle}\lambda(Y),\Delta_{t_{k}}\lambda(X)(H_{k})\big{\rangle}-\big{\langle}Y,H_{k}\big{\rangle}}{\hbox{${1\over
2}$}t_{k}}.$

It follows from $Y=U{\Lambda}(Y)U^{\top}$ that

$\big{\langle}Y,H_{k}\big{\rangle}=\big{\langle}U{\Lambda}(Y)U^{\top},H_{k}\big{\rangle}=\big{\langle}{\Lambda}(Y),U^{\top}H_{k}U\big{\rangle}=\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H_{k}U_{\alpha_{m}}\big{\rangle}.$
(5.2)

On the other hand, it results from (2.7) and Fan’s inequality that

$\displaystyle\big{\langle}\lambda(Y),\Delta_{t_{k}}\lambda(X)(H_{k})\big{\rangle}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{r}\sum_{j\in\alpha_{m}}\frac{\lambda_{j}(Y)\big{(}\lambda_{j}(X+t_{k}H_{k})-\lambda_{j}(X)\big{)}}{t_{k}}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{r}\sum_{j\in\alpha_{m}}\lambda_{j}(Y)\lambda_{l_{j}}\big{(}U_{\alpha_{m}}^{\top}H_{k}U_{\alpha_{m}}+t_{k}U_{\alpha_{m}}^{\top}H_{k}(\mu_{m}I-X)^{\dagger}H_{k}U_{\alpha_{m}}\big{)}+O(t_{k}^{2})$
$\displaystyle\geq$
$\displaystyle\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H_{k}U_{\alpha_{m}}+t_{k}U_{\alpha_{m}}^{\top}H_{k}(\mu_{m}I-X)^{\dagger}H_{k}U_{\alpha_{m}}\big{\rangle}+O(t_{k}^{2}).$

Combining this with (5.2) brings us to

$\displaystyle\frac{\big{\langle}\lambda(Y),\Delta_{t_{k}}\lambda(X)(H_{k})\big{\rangle}-\big{\langle}Y,H_{k}\big{\rangle}}{\hbox{${1\over
2}$}t_{k}}$ $\displaystyle\geq$ $\displaystyle
2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H_{k}(\mu_{m}I-X)^{\dagger}H_{k}U_{\alpha_{m}}\big{\rangle}+O(t_{k}).$

This leads us to the estimate

$\displaystyle\Delta^{2}_{t_{k}}g\big{(}X,Y\big{)}(H_{k})$ $\displaystyle\geq$
$\displaystyle\Delta^{2}_{t_{k}}\theta\big{(}\lambda(X),\lambda(Y)\big{)}(\Delta_{t_{k}}\lambda(X)(H_{k})\big{)}$
$\displaystyle+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H_{k}(\mu_{m}I-X)^{\dagger}H_{k}U_{\alpha_{m}}\big{\rangle}+O(t_{k}),$

which in turn clearly justifies the lower estimate in (5.1) for the second
subderivative of $g$ at $X$ for $Y$ since
$\Delta_{t_{k}}\lambda(X)(H_{k})\to\lambda^{\prime}(X;H)$ as $k\to\infty$.

We proceed with a result about the critical cone of spectral functions.

######  Proposition 5.4 (critical cone of spectral functions).

Assume that $g:{\mathbb{S}}^{n}\to\overline{\mathbb{R}}$ has the spectral
representation in (1.1) and is lsc and convex and that $Y\in\partial g(X)$.
Let $\mu_{1}>\cdots>\mu_{r}$ be the distinct eigenvalues of $X$. Then, we have
$H\in K_{g}(X,Y)$ if and only if $\lambda^{\prime}(X;H)\in
K_{\theta}\big{(}\lambda(X),\lambda(Y)\big{)}$ and the matrices
${\Lambda}(Y)_{\alpha_{m}\alpha_{m}}$ and
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$ have a simultaneous ordered spectral
decomposition for any $m=1,\ldots,r$ with $\alpha_{m}$ taken from (2.4) and
$U\in{\mathbb{O}}^{n}(X)\,\cap\,{\mathbb{O}}^{n}(Y)$.

[Proof. ]By [4, Corollary 5.2.3], the symmetric function $\theta$ in (1.1) is
lsc and convex. Thus, we find
$U\in{\mathbb{O}}^{n}(X)\,\cap\,{\mathbb{O}}^{n}(Y)$ and get
$\lambda(Y)\in\partial\theta\big{(}\lambda(X)\big{)}$ due to Proposition 3.4.
Pick $H\in K_{g}(X,Y)$ and deduce from the definition of the critical cone
that ${\rm d}g(X)(H)=\big{\langle}Y,H\big{\rangle}$. We then conclude from
$Y=U{\Lambda}(Y)U^{\top}$ with ${\Lambda}(Y)=\mbox{\rm
Diag}\,(\lambda(Y)\big{)}$ and Fan’s inequality that

$\displaystyle\big{\langle}Y,H\big{\rangle}$ $\displaystyle=$
$\displaystyle\big{\langle}{\Lambda}(Y),U^{\top}HU\big{\rangle}=\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}\big{\rangle}$
(5.3) $\displaystyle\leq$
$\displaystyle\sum_{m=1}^{r}\big{\langle}\lambda(Y)_{\alpha_{m}},\lambda(U^{\top}_{\alpha_{m}}HU_{\alpha_{m}})\big{\rangle}=\big{\langle}\lambda(Y),\lambda^{\prime}(X;H)\big{\rangle}$
$\displaystyle\leq$ $\displaystyle{\rm
d}\theta\big{(}\lambda(X)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}={\rm
d}g(X)(H),$

where the last inequality results from
$\lambda(Y)\in\partial\theta\big{(}\lambda(X)\big{)}$, [23, Exercise 8.4], and
the convexity of $\theta$ and where the last equality comes from (3.8). These
relationships clearly imply that ${\rm
d}\theta\big{(}\lambda(X)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}=\big{\langle}\lambda(Y),\lambda^{\prime}(X;H)\big{\rangle}$,
meaning that $\lambda^{\prime}(X;H)\in K_{\theta}(\lambda(X),\lambda(Y))$, and
that
$\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}\big{\rangle}=\big{\langle}\lambda(Y)_{\alpha_{m}},\lambda(U^{\top}_{\alpha_{m}}HU_{\alpha_{m}})\big{\rangle}$
for any $m=1,\ldots,r$. By Fan’s inequality, these equalties are equivalent to
saying that the matrices ${\Lambda}(Y)_{\alpha_{m}\alpha_{m}}$ and
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$ have a simultaneous ordered spectral
decomposition for any $m=1,\ldots,r$.

To prove the opposite claim, assume $\lambda^{\prime}(X;H)\in
K_{\theta}\big{(}\lambda(X),\lambda(Y)\big{)}$ and the matrices
${\Lambda}(Y)_{\alpha_{m}\alpha_{m}}$ and
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$ have a simultaneous ordered spectral
decomposition for any $m=1,\ldots,r$. The latter tells us via Fan’s inequality
that
$\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}\big{\rangle}=\big{\langle}\lambda(Y)_{\alpha_{m}},\lambda(U^{\top}_{\alpha_{m}}HU_{\alpha_{m}})\big{\rangle}$
for any $m=1,\ldots,r$. Moreover, the former yields ${\rm
d}\theta\big{(}\lambda(X)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}=\big{\langle}\lambda(Y),\lambda^{\prime}(X;H)\big{\rangle}$.
Taking these into account demonstrates that both inequalities in (5.3) are
indeed equalities. This leads us to ${\rm
d}g(X)(H)=\big{\langle}Y,H\big{\rangle}$, which implies that $H\in
K_{g}(X,Y)$.

The characterization of the critical cone of spectral functions, obtained
above, is a generalization of a similar result, established recently in [6,
Proposition 4] for the spectral function $g$ from (1.1) when the symmetric
function $\theta$ therein is a polyhedral function, meaning a function that
its epigraph is a polyhedral convex set. To obtain a full characterization of
the critical cone of spectral functions, we should know when the matrices
${\Lambda}(Y)_{\alpha_{m}\alpha_{m}}$ and
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$ in Proposition 5.4 have a simultaneous
ordered spectral decomposition since the critical cone
$K_{\theta}\big{(}\lambda(X),\lambda(Y)\big{)}$ can be often calculated rather
easily. This remains an open question for now and will be a subject of our
future research.

As mentioned before, we can partition any vector $p\in{\mathbb{R}}^{n}$ into
$(p_{\alpha_{1}},\ldots,p_{\alpha_{r}})$ with $\alpha_{m}$, $m=1,\ldots,r$,
taken from (2.4). Pick $m\in\\{1,\ldots,r\\}$ and recall from (2.11) that the
index set $\alpha_{m}=\cup_{i=1}^{\rho_{m}}\beta_{i}^{m}$. This allows to
partition further $p_{\alpha_{m}}$ into
$(y_{\beta_{1}^{m}},\ldots,y_{\beta_{\rho_{m}}^{m}})$, where
$y_{\beta_{i}^{m}}\in{\mathbb{R}}^{|\beta_{i}^{m}|}$ for any
$i=1,\ldots,\rho_{m}$. In summary, we can equivalently write $p$ as

$(y_{\beta_{1}^{1}},\ldots,y_{\beta_{\rho_{1}}^{1}},\ldots,y_{\beta_{1}^{r}},\ldots,y_{\beta_{\rho_{r}}^{r}}),$
(5.4)

where $r$, taken from (2.4), and $\rho_{m}$, taken from (2.11), stand for the
number of distinct eigenvalues of $X$ and
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$, respectively. Thus, the representation
of $p$ in (5.4) is associated with the permutation matrices in
${\mathbb{P}}^{n}_{X,H}$, defined prior to Proposition 4.8, as a subset of
${\mathbb{P}}^{n}$. In fact, any permutation matrix
$Q\in{\mathbb{P}}^{n}_{X,H}$ has a representation of the form $\mbox{\rm
Diag}\,(B^{1}_{1},\ldots,B^{1}_{\rho_{1}},\ldots,B^{r}_{1},\ldots,B^{r}_{\rho_{r}})$,
where $B^{m}_{j}\in{\mathbb{R}}^{|\beta_{j}^{m}|\times|\beta_{j}^{m}|}$ is a
permutation matrix for any $j\in\\{1,\ldots,\rho_{m}\\}$ and
$m\in\\{1,\ldots,r\\}$. Denote by ${\mathbb{R}}^{n}_{\downarrow}$ the set of
all vectors $(x_{1},\ldots,x_{n})$ such that $x_{1}\geq\cdots\geq x_{n}$.

######  Proposition 5.5.

Assume that the spectral function $g=\theta\circ\lambda$ in (1.1) is lsc and
convex and that $Y\in\partial g(X)$ and $H\in K_{g}(X,Y)$. If $\theta$ is
parabolically regular at $\lambda(X)$ for $\lambda(Y)$, then the following
properties hold.

  * (a)

There exists $z\in{\mathbb{R}}^{n}$, which has a representation in the form in
(5.4) with $y_{i}^{m}\in{\mathbb{R}}^{|\beta_{i}^{m}|}_{\downarrow}$ for any
$i\in\\{1,\ldots,\rho_{m}\\}$ and $m\in\\{1,\ldots,r\\}$, satisfying

${\rm d}^{2}\theta(\lambda(X),\lambda(Y))(\lambda^{\prime}(X;H))={\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;z)-\langle\lambda(Y),z\rangle.$
(5.5)

  * (b)

There exists a matrix $\widehat{W}\in{\mathbb{S}}^{n}$ such that
$\lambda^{{}^{\prime\prime}}(X;H,\widehat{W})=z$, where $z$ comes from (a).

[Proof. ]We deduce from $Y\in\partial g(X)$ and Proposition 3.4 that
$\lambda(Y)\in\partial\theta(\lambda(X))$. Also it follows from $H\in
K_{g}(X,Y)$ and Proposition 5.4 that $\lambda^{\prime}(X;H)\in
K_{\theta}\big{(}\lambda(X),\lambda(Y)\big{)}$. Employing now Proposition 5.2
ensures the existence of $p\in{\mathbb{R}}^{n}$ satisfying (5.5). As explained
above, any such a vector $p$ has a representation in the form of (5.4) with
$y_{\beta_{i}^{m}}\in{\mathbb{R}}^{|\beta_{i}^{m}|}$ for any
$i\in\\{1,\ldots,\rho_{m}\\}$ and $m\in\\{1,\ldots,r\\}$. We are going to show
that we can find a vector $p$ with the representation in (5.4) such that
$y_{\beta_{i}^{m}}\in{\mathbb{R}}^{|\beta_{i}^{m}|}_{\downarrow}$ for any
$i\in\\{1,\ldots,\rho_{m}\\}$ and $m\in\\{1,\ldots,r\\}$, meaning the
components of each $y_{\beta_{i}^{m}}$ have nonincreasing order. To this end,
pick $m\in\\{1,\ldots,r\\}$ and $i\in\\{1,\ldots,\rho_{m}\\}$ and choose then
a $|\beta_{i}^{m}|\times|\beta_{i}^{m}|$ permutation matrix $B^{m}_{i}$ such
that
$q_{\beta_{i}^{m}}:=B^{m}_{i}y_{\beta_{i}^{m}}\in{\mathbb{R}}^{|\beta_{i}^{m}|}_{\downarrow}$.
Set $Q:=\mbox{\rm
Diag}\,(B_{1}^{1},\ldots,B_{\rho_{1}}^{1},\ldots,B_{1}^{r},\ldots,B_{\rho_{r}}^{r})$
and observe that $Q\in{\mathbb{P}}^{n}_{X,H}$. Moreover, let

$z:=(q_{\beta_{1}^{1}},\ldots,q_{\beta_{\rho_{1}}^{1}},\ldots,q_{\beta_{1}^{r}},\ldots,q_{\beta_{\rho_{r}}^{r}}).$
(5.6)

Clearly, we have $z=Qp$. It follows from Proposition 4.8 that

${\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;z)={\rm
d}^{2}\theta(\lambda(X))\big{(}\lambda^{\prime}(X;H)\;\rule[-1.42262pt]{0.56905pt}{9.38945pt}\;p).$

We claim now that $\langle\lambda(Y),z\rangle\leq\langle\lambda(Y),p\rangle$.
To justify it, suppose that

$(\lambda(Y)_{\beta_{1}^{1}},\ldots,\lambda(Y)_{\beta_{\rho_{1}}^{1}},\ldots,\lambda(Y)_{\beta_{1}^{r}},\ldots,\lambda(Y)_{\beta_{\rho_{r}}^{r}})$

is a partition of the vector $\lambda(Y)$ corresponding to (5.4). Note that
$\lambda(Y)_{\beta_{i}^{m}}\in{\mathbb{R}}^{|\beta_{i}^{m}|}_{\downarrow}$ for
any $i\in\\{1,\ldots,\rho_{m}\\}$ and $m\in\\{1,\ldots,r\\}$. Thus, we get

$\langle\lambda(Y),p\rangle=\sum_{m=1}^{r}\sum_{i=1}^{\rho_{m}}\langle\lambda(Y)_{\beta_{i}^{m}},y_{\beta_{i}^{m}}\rangle\leq\sum_{m=1}^{r}\sum_{i=1}^{\rho_{m}}\langle\lambda(Y)_{\beta_{i}^{m}},q_{\beta_{i}^{m}}\rangle=\langle\lambda(Y),z\rangle,$

where the inequality is a consequence of the Hardy-Littlewood-Pólya inequality
(cf. [4, Proposition 1.2.4]). Set $\varphi(x)={\rm
d}^{2}\theta(\lambda(X))(\lambda^{\prime}(X;H)\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;x)-\langle\lambda(Y),x\rangle$
for any $x\in{\mathbb{R}}^{n}$ and observe from Proposition 5.2 that $p$ is a
minimizer of $\varphi$. But we showed above that $\varphi(z)\leq\varphi(p)$,
which tells us that $z$ is also a minimizer of $\varphi$. Thus, we arrive at
$\varphi(z)=\varphi(p)$, which implies that (5.5) holds for $z$. This proves
(a).

Turning now to the proof of (b), pick the vector $z$ from (5.6). We can
equivalently write via the index sets $\alpha_{m}$, $m=1,\ldots,r$, from (2.4)
that

$z=(z_{\alpha_{1}},\ldots,z_{\alpha_{r}})\quad\mbox{with}\quad
z_{\alpha_{m}}=(q_{\beta_{1}^{m}},\ldots,q_{\beta_{\rho_{m}}^{m}})\in{\mathbb{R}}^{|\alpha_{m}|}\quad\mbox{for
all}\;\;m\in\\{1,\ldots,r\\}.$ (5.7)

Take the $|\alpha_{m}|\times|\alpha_{m}|$ matrix $Q_{m}$, $m=1,\ldots,r$, from
(2.10) and consider the $n\times n$ block diagonal matrix

$A=\mbox{\rm Diag}\,\big{(}Q_{1}\mbox{\rm
Diag}\,(z_{\alpha_{1}})Q_{1}^{\top},\ldots,Q_{r}\mbox{\rm
Diag}\,(z_{\alpha_{r}})Q_{r}^{\top}\big{)}.$ (5.8)

We claim that there exists a matrix $\widehat{W}\in{\mathbb{S}}^{n}$ such that
for any $m=1,\ldots,r$ the relationship

$U_{\alpha_{m}}^{\top}\widehat{W}U_{\alpha_{m}}=U_{\alpha_{m}}^{\top}\big{(}2H(X-\mu_{m}I)^{\dagger}H+UAU^{\top}\big{)}U_{\alpha_{m}}$
(5.9)

holds, where $\mu_{1}>\cdots>\mu_{r}$ are the distinct eigenvalues of $X$ and
$U\in{\mathbb{O}}^{n}(X)$. Indeed, to find such a matrix $\widehat{W}$, let
$W\in{\mathbb{S}}^{n}$ and set $\widehat{W}=UWU^{\top}$ in the above equality.
This, coupled with (2.6), leads us to

$W_{\alpha_{m}\alpha_{m}}=U_{\alpha_{m}}^{\top}UWU^{\top}U_{\alpha_{m}}=U_{\alpha_{m}}^{\top}\widehat{W}U_{\alpha_{m}}=2U_{\alpha_{m}}^{\top}\big{(}H(X-\mu_{m}I)^{\dagger}H+UAU^{\top}\big{)}U_{\alpha_{m}}$

for all $m=1,\ldots,r$. Define the matrix $W$ as the block diagonal matrix
$\mbox{\rm Diag}\,(W_{\alpha_{1}\alpha_{1}},\ldots,W_{\alpha_{r}\alpha_{r}})$
from which we can obtain the claimed matrix $\widehat{W}$. Suppose now that
$i\in\\{1,\ldots,n\\}$. By (2.12), there are $m\in\\{1,\ldots,r\\}$ and
$j\in\\{1,\ldots,\rho_{m}\\}$ such that $i\in\alpha_{m}$ and
$\ell_{i}\in\beta_{j}^{m}$. According to (2.13), we have

$\displaystyle\lambda_{i}^{{}^{\prime\prime}}(X;H,\widehat{W})$
$\displaystyle=$
$\displaystyle\lambda_{\scriptsize{\ell}^{\prime}_{i}}\Big{(}(Q_{m})_{\beta^{m}_{j}}^{\top}\big{(}U_{\alpha_{m}}^{\top}(\widehat{W}+2H(\mu_{m}I-X)^{\dagger}H)U_{\alpha_{m}}\big{)}(Q_{m})_{\beta^{m}_{j}}\Big{)}$
(5.10) $\displaystyle=$
$\displaystyle\lambda_{\scriptsize{\ell}^{\prime}_{i}}\Big{(}(Q_{m})_{\beta^{m}_{j}}^{\top}\big{(}U_{\alpha_{m}}^{\top}UAU^{\top}U_{\alpha_{m}}\big{)}(Q_{m})_{\beta^{m}_{j}}\Big{)}$
$\displaystyle=$
$\displaystyle\lambda_{\scriptsize{\ell}^{\prime}_{i}}\Big{(}(Q_{m})_{\beta^{m}_{j}}^{\top}Q_{m}\mbox{\rm
Diag}\,(z_{\alpha_{m}})Q_{m}^{\top}(Q_{m})_{\beta^{m}_{j}}\Big{)}$
$\displaystyle=$
$\displaystyle\lambda_{\scriptsize{\ell}^{\prime}_{i}}\big{(}\mbox{\rm
Diag}\,(q_{\beta_{j}^{m}})\big{)},$

where the last two equalities result from (2.6). Consider now a partition of
$\lambda^{{}^{\prime\prime}}(X;H,\widehat{W})$ corresponding to (5.6) as

$(\eta_{\beta_{1}^{1}},\ldots,\eta_{\beta_{\rho_{1}}^{1}},\ldots,\eta_{\beta_{1}^{r}},\ldots,\eta_{\beta_{\rho_{r}}^{r}}).$

Thus, it follows from (5.10) and the definition $\ell^{\prime}_{i}$ that

$\eta_{\beta_{j}^{m}}=\lambda\big{(}\mbox{\rm
Diag}\,(q_{\beta_{j}^{m}})\big{)}=q_{\beta_{j}^{m}}$

for any $j\in\\{1,\ldots,\rho_{m}\\}$ and $m\in\\{1,\ldots,r\\}$ since
$q_{\beta_{j}^{m}}\in{\mathbb{R}}^{|\beta_{j}^{m}|}_{\downarrow}$. This
confirms that $\lambda^{{}^{\prime\prime}}(X;H,\widehat{W})=z$ and hence
completes the proof of (b).

We are now ready to characterize parabolic regularity of spectral functions.
We begin with the following result in which we provide a sufficient condition
to calculate the domain of the second subderivative of spectral functions.

######  Proposition 5.6.

Assume that the spectral function $g=\theta\circ\lambda$ in (1.1) is convex
and that $Y\in\partial g(X)$. Then, we have $\mbox{\rm dom}\,{\rm
d}^{2}g(X,Y)\subset K_{g}(X,Y)$. Equality holds if, in addition, $g$ is
parabolically epi-differentiable at $X$ for any $H\in K_{g}(X,Y)$.

[Proof. ]The claimed inclusion results from [20, Proposition 2.1(ii)-(iii)].
To establish the second claim, it follows from parabolic epi-differentiability
of $g$ at $X$ for any $H\in K_{g}(X,Y)$ that $\mbox{\rm dom}\,{\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;\;.\;)\neq\emptyset$.
This, coupled with [20, Proposition 3.4], confirms that $\mbox{\rm dom}\,{\rm
d}^{2}g(X,Y)=K_{g}(X,Y)$ and hence completes the proof.

Note that the assumption of parabolic epi-differentiability of $g$ in the
result above can be ensured via Theorem 4.7(a) by parabolic epi-
differentiability of $\theta$. An important class of functions for which this
assumption automatically is satisfied is polyhedral functions; see [23,
Exercise 13.61]. This class of functions allows us to cover many examples of
spectral functions, which are important for applications. We should add that
polyhedral functions that are symmetric were characterized in [6, Proposition
1].

Our next result presents a characterization of parabolic regularity of
spectral functions.

######  Theorem 5.7 (parabolic regularity of spectral functions).

Assume that the spectral function $g=\theta\circ\lambda$ in (1.1) is locally
Lipschitz continuous with respect to its domain, lsc, and convex. Let
$\mu_{1}>\cdots>\mu_{r}$ be the distinct eigenvalues of $X$. Then the
following properties hold.

  * (a)

If $Y\in\partial g(X)$ and $\theta$ is parabolically regular at $\lambda(X)$
for $\lambda(Y)$ and parabolically epi-differentiable at $\lambda(X)$, then
$g$ is parabolically regular at $X$ for $Y$ and for any $H\in K_{g}(X,Y)$ we
have

${\rm d}^{2}g(X,Y)(H)={\rm
d}^{2}\theta\big{(}\lambda(X),\lambda(Y)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{\rangle},$

where $\alpha_{m}$, $m=1,\ldots,r$, come from (2.4),
$U\in{\mathbb{O}}^{n}(X)\,\cap\,{\mathbb{O}}^{n}(Y)$, and
${\Lambda}(Y)=\mbox{\rm Diag}\,(\lambda(Y)\big{)}$.

  * (b)

If $g$ is parabolically epi-differentiable and parabolically regular at $X$,
then $\theta$ is parabolically regular at $\lambda(X)$.

[Proof. ]We begin with the proof of (a). To justify (a), it suffices by
Proposition 5.2 to show that for any $H\in K_{g}(X,Y)$, we have

${\rm d}^{2}g(X,Y)(H)=\inf_{W\in{\mathbb{S}}^{n}}\big{\\{}{\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)-\big{\langle}Y,W\big{\rangle}\big{\\}}.$
(5.11)

To this end, pick $H\in K_{g}(X,Y)$ and deduce from [23, Proposition 13.64]
and (4.13), respectively, that

$\displaystyle{\rm d}^{2}g(X,Y)(H)$ $\displaystyle\leq$
$\displaystyle\inf_{W\in{\mathbb{S}}^{n}}\big{\\{}{\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)-\big{\langle}Y,W\big{\rangle}\big{\\}}$
(5.12) $\displaystyle=$ $\displaystyle\inf_{W\in{\mathbb{S}}^{n}}\big{\\{}{\rm
d}^{2}\theta\big{(}\lambda(X)\big{)}\big{(}\lambda^{\prime}(X;H)\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;\lambda^{{}^{\prime\prime}}(X;H,W)\big{)}-\big{\langle}Y,W\big{\rangle}\big{\\}}.$

Since $H\in K_{g}(X,Y)$ , it results from Proposition 5.4 that the matrices
${\Lambda}(Y)_{\alpha_{m}\alpha_{m}}$ and
$U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}$ have a simultaneous ordered spectral
decomposition for any $m=1,\ldots,r$. This means that there are matrices
$\widehat{Q}_{m}\in{\mathbb{O}}^{|\alpha_{m}|}({\Lambda}(Y)_{\alpha_{m}\alpha_{m}})\cap{\mathbb{O}}^{|\alpha_{m}|}(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})$,
$m=1,\ldots,r$, such that

${\Lambda}(Y)_{\alpha_{m}\alpha_{m}}=\widehat{Q}_{m}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}}\widehat{Q}_{m}^{\top}\quad\mbox{and}\quad
U_{\alpha_{m}}^{\top}HU_{\alpha_{m}}=\widehat{Q}_{m}\Lambda(U_{\alpha_{m}}^{\top}HU_{\alpha_{m}})\widehat{Q}_{m}^{\top}.$
(5.13)

Replace the matrices $Q_{m}$ in the definition of the matrix $A$ in (5.8) with
$\widehat{Q}_{m}$, $m=1,\ldots,r$, and observe that the same conclusion can be
achieved as the one in Proposition 5.5(b) for the updated matrix $A$. In fact,
the matrices $\widehat{Q}_{m}$ enjoy all the properties of $Q_{m}$ together
with the relationships in (5.13), which are important for our argument below.
Since $\theta$ is parabolically regular at $\lambda(X)$ for $\lambda(Y)$, we
conclude from Proposition 5.5(a) that there is $z\in{\mathbb{R}}^{n}$ with a
representation in the form in (5.6) with
$q_{\beta_{i}^{m}}\in{\mathbb{R}}^{|\beta_{i}^{m}|}_{\downarrow}$ for any
$i\in\\{1,\ldots,\rho_{m}\\}$ and $m\in\\{1,\ldots,r\\}$, satisfying (5.5).
According to Proposition 5.5(b), there exists a matrix
$\widehat{W}\in{\mathbb{S}}^{n}$ such that
$\lambda^{{}^{\prime\prime}}(X;H,\widehat{W})=z$. Employing now (5.12), (5.5),
(5.9), and $Y=U{\Lambda}(Y)U^{\top}$ and using a similar argument as (5.2), we
arrive at

$\displaystyle{\rm d}^{2}g(X,Y)(H)$ $\displaystyle\leq$ $\displaystyle{\rm
d}^{2}\theta\big{(}\lambda({\overline{X}})\big{)}\big{(}\lambda^{\prime}(X;H)\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;\lambda^{{}^{\prime\prime}}(X;H,\widehat{W})\big{)}-\big{\langle}Y,\widehat{W}\big{\rangle}$
(5.14) $\displaystyle=$ $\displaystyle{\rm
d}^{2}\theta\big{(}\lambda(X)\big{)}\big{(}\lambda^{\prime}(X;H)\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;z\big{)}-\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}\widehat{W}U_{\alpha_{m}}\big{\rangle}$
$\displaystyle=$ $\displaystyle{\rm
d}^{2}\theta(\lambda(X),\lambda(Y))(\lambda^{\prime}(X;H))+\langle\lambda(Y),z\rangle$
$\displaystyle-\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}\big{(}2H(X-\mu_{m}I)^{\dagger}H+UAU^{\top}\big{)}U_{\alpha_{m}}\big{\rangle}$
$\displaystyle=$ $\displaystyle{\rm
d}^{2}\theta(\lambda(X),\lambda(Y))(\lambda^{\prime}(X;H))+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}\big{(}H(\mu_{m}I-X)^{\dagger}H\big{)}U_{\alpha_{m}}\big{\rangle}$
$\displaystyle+\langle\lambda(Y),z\rangle-\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}\big{(}UAU^{\top}\big{)}U_{\alpha_{m}}\big{\rangle}.$

By the definition of $A$ in (5.8), the equality in (2.6), and the
representation of the vector $z$ in (5.7), we obtain

$\displaystyle\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}\big{(}UAU^{\top}\big{)}U_{\alpha_{m}}\big{\rangle}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},\widehat{Q}_{m}\mbox{\rm
Diag}\,(z_{\alpha_{m}})\widehat{Q}_{m}^{\top}\big{\rangle}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{r}\big{\langle}\widehat{Q}_{m}^{\top}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}}\widehat{Q}_{m},\mbox{\rm
Diag}\,(z_{\alpha_{m}})\big{\rangle}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},\mbox{\rm
Diag}\,(z_{\alpha_{m}})\big{\rangle}=\langle\lambda(Y),z\rangle,$

where the penultimate equality results from the first relationship in (5.13)
and the last one is a consequence of ${\Lambda}(Y)=\mbox{\rm
Diag}\,(\lambda(Y))$. This, coupled with (5.1), (5.12), and (5.14), brings us
to

$\displaystyle{\rm
d}^{2}\theta\big{(}\lambda(X),\lambda(Y)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{\rangle}$
$\displaystyle\leq$ $\displaystyle{\rm d}^{2}g(X,Y)(H)$ $\displaystyle\leq$
$\displaystyle\inf_{W\in{\mathbb{S}}^{n}}\big{\\{}{\rm
d}^{2}g(X)(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)-\big{\langle}Y,W\big{\rangle}\big{\\}}$
$\displaystyle\leq$ $\displaystyle{\rm
d}^{2}\theta\big{(}\lambda({\overline{X}})\big{)}\big{(}\lambda^{\prime}(X;H)\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;\lambda^{{}^{\prime\prime}}(X;H,\widehat{W})\big{)}-\big{\langle}Y,\widehat{W}\big{\rangle}$
$\displaystyle=$ $\displaystyle{\rm
d}^{2}\theta\big{(}\lambda(X),\lambda(Y)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{\rangle}.$

These relationships clearly justify (5.11) and hence imply that $g$ is
parabolically regular at $X$ for $Y$. Moreover, they confirm the claimed
formula for the second subderivative of $g$ at $X$ for $Y$ for any $H\in
K_{g}(X,Y)$ and hence complete the proof of (a).

Turning into the proof of (b), we need to show that $\theta$ is parabolically
regular at $\lambda(X)$ for any $v\in\partial\theta(\lambda(X))$. To justify
it, pick $v\in\partial\theta(\lambda(X))$ and $U\in{\mathbb{O}}^{n}(X)$ and
deduce from [15, Theorem 6] that $U\mbox{\rm Diag}\,(v)U^{\top}\in\partial
g(X)$. Since $g$ is convex and orthogonally invariant, the latter yields
$\mbox{\rm Diag}\,(v)\in\partial g({\Lambda}(X))$, where
${\Lambda}(X)=\mbox{\rm Diag}\,(\lambda(X))$. We claim that $g$ is
parabolically regular at ${\Lambda}(X)$ for $\mbox{\rm Diag}\,(v)$. To this
end, set $Z:=U\mbox{\rm Diag}\,(v)U^{\top}$ and take $H\in
K_{g}({\Lambda}(X),\mbox{\rm Diag}\,(v))$. We conclude from (4.21) and the
definition of the critical cone that $UHU^{\top}\in K_{g}(X,Z)$. Thus, it also
follows from parabolic regularity of $g$ at $X$ for $Z$ and Proposition 5.2
that there is $W\in{\mathbb{S}}^{n}$ such that

${\rm d}^{2}g(X,Z)(UHU^{\top})={\rm
d}^{2}g(X)(UHU^{\top}\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)-\big{\langle}Z,W\big{\rangle}.$
(5.15)

Since $g$ is orthogonally invariant, we get

$\displaystyle{\rm d}^{2}g(X,Z)(UHU^{\top})$ $\displaystyle=$
$\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\ H^{\prime}\to
UHU^{\top}\end{subarray}}\frac{g(X+tH^{\prime})-g(X)-\langle
Z,H^{\prime}\rangle}{\hbox{${1\over 2}$}t^{2}}$ $\displaystyle=$
$\displaystyle\liminf_{\begin{subarray}{c}t\downarrow 0\\\ H^{\prime}\to
UHU^{\top}\end{subarray}}\frac{g({\Lambda}(X)+tU^{\top}H^{\prime}U)-g({\Lambda}(X))-\langle
U^{\top}ZU,U^{\top}H^{\prime}U\rangle}{\hbox{${1\over 2}$}t^{2}}$
$\displaystyle\geq$ $\displaystyle{\rm d}^{2}g\big{(}{\Lambda}(X),\mbox{\rm
Diag}\,(v)\big{)}(H).$

Similarly, we can show that ${\rm d}^{2}g(X,Z)(UHU^{\top})\leq{\rm
d}^{2}g\big{(}{\Lambda}(X),\mbox{\rm Diag}\,(v)\big{)}(H)$, which leads us to
${\rm d}^{2}g(X,Z)(UHU^{\top})={\rm d}^{2}g\big{(}{\Lambda}(X),\mbox{\rm
Diag}\,(v)\big{)}(H)$. It follows from (4.22) that

${\rm d}^{2}g(X)(UHU^{\top}\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;W)={\rm
d}^{2}g({\Lambda}(X))(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;U^{\top}WU).$

We also have $\big{\langle}Z,W\big{\rangle}=\big{\langle}\mbox{\rm
Diag}\,(v),U^{\top}WU\big{\rangle}$. Combining these with (5.15) brings us to

${\rm d}^{2}g\big{(}{\Lambda}(X),\mbox{\rm Diag}\,(v)\big{)}(H)={\rm
d}^{2}g({\Lambda}(X))(H\;\rule[-1.99168pt]{0.56905pt}{10.52737pt}\;U^{\top}WU)-\big{\langle}\mbox{\rm
Diag}\,(v),U^{\top}WU\big{\rangle}.$

Since $H\in K_{g}({\Lambda}(X),\mbox{\rm Diag}\,(v))$ was taken arbitrarily,
Proposition 5.2 confirms that $g$ is parabolically regular at ${\Lambda}(X)$
for $\mbox{\rm Diag}\,(v)$. Recall that the symmetric function $\theta$
satisfies (3.1), which means that $\theta$ can be represented as
$\theta=g\circ F$ with $F(x):=\mbox{\rm Diag}\,(x)$ for all
$x\in{\mathbb{R}}^{n}$. We also deduce from the imposed assumption on $\theta$
and (2.2) that $g$ is locally Lipschitz continuous relative to its domain.
According to [17, Theorem 3.6], we get $\partial\theta(\lambda(X))=\nabla
F(\lambda(X))^{*}\partial g({\Lambda}(X))$. It is not hard to see that for any
$x\in{\mathbb{R}}^{n}$, $\nabla F(x)$ is a linear operator from
${\mathbb{R}}^{n}$ into ${\mathbb{S}}^{n}$, defined by $\nabla
F(x)(y)=\sum_{i=1}^{n}y_{i}E_{ii}$ for any
$y=(y_{1},\ldots,y_{n})\in{\mathbb{R}}^{n}$, where $E_{ii}$, $i=1,\ldots,n$,
are the $n\times n$ matrix with $(i,i)$ entry equal to $1$ and elsewhere equal
to $0$. This tells us that the adjoint operator $\nabla
F(x)^{*}:{\mathbb{S}}^{n}\to{\mathbb{R}}^{n}$ has a representation in the form
$\nabla F(x)^{*}B=\big{(}\mbox{\rm tr}\,(E_{11}B),\ldots,\mbox{\rm
tr}\,(E_{nn}B)\big{)}$ for any $B\in{\mathbb{S}}^{n}$; see [1, Example 1.8]
for more details. Remember that $v\in\partial\theta(\lambda(X))$ and
$\mbox{\rm Diag}\,(v)\in\partial g({\Lambda}(X))$. Thus, we have $\nabla
F(\lambda(X))^{*}\mbox{\rm Diag}\,(v)=v.$ On the other hand, it follows from
the proof of Theorem 4.7(b) that parabolic epi-differentiability of $g$ at $X$
for $UHU^{\top}\in\mbox{\rm dom}\,{\rm d}g(X)$ implies that of $g$ at
${\Lambda}(X)$ for $H$. Combining these and [20, Theorem 5.4] tells us that
$\theta$ is parabolically regular at $\lambda(X)$ for $v$ and hence completes
the proof of (b).

Given the spectral function $g$ in (1.1), it was shown in [7, Proposition 10]
that if $\theta$ is ${\cal C}^{2}$-cone reducible in the sense of [7,
Definition 6], then $g$ enjoys the same property. Note that ${\cal
C}^{2}$-cone reducibility of $\theta$ is strictly stronger assumption than
parabolic regularity of this function, utilized in Theorem 5.7, as shown in
[18, Theorem 6.2] and [18, Example 6.4]. Note also we showed in Theorem 5.7(b)
that parabolic regularity of $g$ yields that of $\theta$; such a result was
not achieved for ${\cal C}^{2}$-cone reducibility in [7].

In many important applications of the spectral function $g$ in (1.1), the
symmetric function $\theta$ is a polyhedral function. In this case, all the
assumptions in Theorem 5.7 are satisfied automatically. Furthermore, the
second subderivative of $g$ has a simple representation as demonstrated below.

######  Corollary 5.8.

Assume that $g:{\mathbb{S}}^{n}\to\overline{\mathbb{R}}$ has the spectral
representation in (1.1) with the symmetric function $\theta$ being polyhedral.
If $\mu_{1}>\cdots>\mu_{r}$ are the distinct eigenvalues of $X$ and
$Y\in\partial g(X)$, then $g$ is parabolically regular at $X$ for $Y$ and for
any $H\in{\mathbb{S}}^{n}$ we have

${\rm
d}^{2}g(X,Y)(H)=\delta_{K_{g}(X,Y)}(H)+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{\rangle},$

where $\alpha_{m}$, $m=1,\ldots,r$, come from (2.4),
$U\in{\mathbb{O}}^{n}(X)\,\cap\,{\mathbb{O}}^{n}(Y)$, and
${\Lambda}(Y)=\mbox{\rm Diag}\,(\lambda(Y)\big{)}$.

[Proof. ]It follows from [23, Exercise 13.61] and [20, Example 3.2],
respectively, that $\theta$ is parabolically epi-differentiable and
parabolically regular at $\lambda(X)$. By Theorem 5.7(a), the spectral
function $g$ is parabolically regular at $X$ for $Y$. Moreover, we know from
Proposition 5.7 that $\mbox{\rm dom}\,{\rm d}^{2}g(X,Y)=K_{g}(X,Y)$. Take
$H\in K_{g}(X,Y)$ and observe from Proposition 5.4 that
$\lambda^{\prime}(X;H)\in K_{\theta}\big{(}\lambda(X),\lambda(Y)\big{)}$.
Thus, we obtain from [23, Proposition 13.9] that

${\rm
d}^{2}\theta\big{(}\lambda(X),\lambda(Y)\big{)}\big{(}\lambda^{\prime}(X;H)\big{)}=\delta_{K_{\theta}(\lambda(X),\lambda(Y))}\big{(}\lambda^{\prime}(X;H)\big{)}=0.$

Employing now Theorem 5.7(a) tells us that

${\rm
d}^{2}g(X,Y)(H)=2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{\rangle}\quad\mbox{for
all}\;\;H\in K_{g}(X,Y).$

On the other hand, if $H\notin K_{g}(X,Y)$, we have ${\rm
d}^{2}g(X,Y)(H)=\infty$ since $\mbox{\rm dom}\,{\rm d}^{2}g(X,Y)=K_{g}(X,Y)$.
This proves the claimed formula for the second subderivative of $g$ at $X$ for
$Y$.

Note that the conjugate function of the parabolic subderivative of the
spectral function $g$ in (1.1) with $\theta$ therein being polyhedral was
recently calculated in [6, Propositions 6 & 10] by dividing a polyhedral
function into two parts. In general, such a result gives us an upper bound for
the second subderivative (cf. [23, Proposition 13.64]). According to
Proposition 5.2, parabolic regularity is, indeed, equivalent to saying that
the latter conjugate function coincides with the second subderivative. We
should add that parabolic regularity of $g$ was not discussed in [6] and so
(5.8) can’t be derived from the aforementioned results in [6].

We continue to apply the formula of the second subderivative, obtained in
Corollary 5.8, for two important examples of spectral functions and show how
one can simplify the established formula for the second subderivative in these
cases.

######  Example 5.9.

  * (a)

Assume that $g:{\mathbb{S}}^{n}\to{\mathbb{R}}$ is defined by
$g(X)=\lambda_{\max}(X)$, where $\lambda_{\max}$ stands for the maximum
eigenvalue of $X$. $g$ is a spectral function and satisfies the representation
(1.1) and $\theta(x)=\max\\{x_{1},\ldots,x_{n}\\}$ with
$x=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$. Take $Y\in\partial g(X)$ and
observe from Proposition 3.4 that $Y=U\mbox{\rm Diag}\,(\lambda(Y))U^{\top}$,
where $\lambda(Y)\in\partial\theta(\lambda(X))$ and
$U\in{\mathbb{O}}^{n}(X)\cap{\mathbb{O}}^{n}(Y)$. Recall from (2.4) that
$\alpha_{1}=\big{\\{}i\in\\{1,\ldots,n\\}|\;\lambda_{i}(X)=\lambda_{\max}(X)\big{\\}}$.
It follows from [23, Exercise 8.31] that

$\partial\theta(\lambda(X))=\big{\\{}(t_{1},\ldots,t_{n})|\;\sum_{i=1}^{n}t_{i}=1,\;t_{i}\geq
0\;\;\mbox{for all}\;i\in\alpha_{1},\;t_{i}=0\;\;\mbox{otherwise}\big{\\}}.$

Taking into the consideration the formula for the second subderivative from
Corollary 5.8 and the notation therein and the description of $\partial
g(\lambda(X))$, we conclude that ${\Lambda}(Y)_{\alpha_{m}\alpha_{m}}=0$ for
all $m\geq 2$. Moreover, we have

$Y=U\mbox{\rm
Diag}\,(\lambda(Y))U^{\top}=\sum_{i=1}^{n}\lambda_{i}(Y)U_{i}U_{i}^{T}=\sum_{i\in\alpha_{1}}\lambda_{i}(Y)U_{i}U_{i}^{T}=U_{\alpha_{1}}{\Lambda}(Y)_{\alpha_{1}\alpha_{1}}U_{\alpha_{1}}^{\top}.$

Combining this and Corollary 5.8, we obtain for any $H\in{\mathbb{S}}^{n}$
that

$\displaystyle{\rm d}^{2}g(X,Y)(H)$ $\displaystyle=$
$\displaystyle\delta_{K_{g}(X,Y)}(H)+2\big{\langle}{\Lambda}(Y)_{\alpha_{1}\alpha_{1}},U_{\alpha_{1}}^{\top}H(\mu_{1}I-X)^{\dagger}HU_{\alpha_{1}}\big{\rangle}$
$\displaystyle=$
$\displaystyle\delta_{K_{g}(X,Y)}(H)+2\big{\langle}Y,H(\mu_{1}I-X)^{\dagger}H\big{\rangle}.$

This is the same formula, obtained in [25, Theorem 2.1] for the second
subderivative of the first leading eigenvalue of a symmetric matrix. Note that
it was proven in [20, Example 3.3] that all leading eigenvalues of a symmetric
matrix is parabolically regular. Also one can find their second subderivatives
in [25, Theorem 2.1]. Since the leading eigenvalues, except the first one
which is the maximum eigenvalue, are not convex, Theorem 5.7 and Corollary 5.8
can’t be utilized to cover them. That requires to extend the established
theory in this section for subdifferentially regular functions in the sense of
[23, Definition 7.25], a task that we leave for our future research.

  * (b)

Suppose that $g=\delta_{{\mathbb{S}}^{n}_{-}}$. As shown in Example 3.7, $g$
is a spectral function satisfying (1.1) with
$\theta=\delta_{{\mathbb{R}}^{n}_{-}}$. Take $Y\in
N_{{\mathbb{S}}^{n}_{-}}(X)$ and observe from Proposition 3.4 that
$Y=U\mbox{\rm Diag}\,(\lambda(Y))U^{\top}$, where $\lambda(Y)\in
N_{{\mathbb{R}}^{n}_{-}}(\lambda(X))$ and
$U\in{\mathbb{O}}^{n}(X)\cap{\mathbb{O}}^{n}(Y)$. If
$\mu_{1}=\lambda_{1}(X)<0$, it follows from $X\in\mbox{\rm
int}\,{\mathbb{S}}^{n}_{-}$ that $g$ is twice differentiable and ${\rm
d}^{2}g(X,Y)(H)=0$ for any $H\in{\mathbb{S}}^{n}$. Assume now that
$\mu_{1}=\lambda_{1}(X)=0$. Recall from (2.4) that
$\alpha_{1}=\big{\\{}i\in\\{1,\ldots,n\\}|\;\lambda_{i}(X)=\mu_{1}\big{\\}}$.
Thus we obtain

$N_{{\mathbb{R}}^{n}_{-}}(\lambda(X))=\big{\\{}(t_{1},\ldots,t_{n})|\;t_{i}\geq
0\;\;\mbox{for all}\;i\in\alpha_{1},\;t_{i}=0\;\;\mbox{otherwise}\big{\\}}.$

Arguing similar to (a) leads us to

${\rm
d}^{2}\delta_{{\mathbb{S}}^{n}_{-}}(X,Y)(H)=\delta_{K_{{\mathbb{S}}^{n}_{-}}(X,Y)}(H)-2\big{\langle}Y,HX^{\dagger}H\big{\rangle}\quad\mbox{for
all}\;H\in{\mathbb{S}}^{n}.$

This formula was obtained perviously in [20, Example 3.7] via a different
approach. Similarly, we can show that if $Y\in N_{{\mathbb{S}}^{n}_{+}}(X)$,
the second subderivative of $\delta_{{\mathbb{S}}^{n}_{+}}$ at $X$ for $Y$ can
be calculated by

${\rm
d}^{2}\delta_{{\mathbb{S}}^{n}_{+}}(X,Y)(H)=\delta_{K_{{\mathbb{S}}^{n}_{+}}(X,Y)}(H)-2\big{\langle}Y,HX^{\dagger}H\big{\rangle}\quad\mbox{for
all}\;H\in{\mathbb{S}}^{n}.$

The second subderivative can be utilized to establish second-order optimality
conditions for different classes of optimization problems. Doing so often
requires obtaining a chain rule for the second subderivative, a task carried
out in Theorem 5.7 and Corollary 5.8. Given a twice differentiable function
$\varphi:{\mathbb{S}}^{n}\to{\mathbb{R}}$ and a spectral function $g$,
consider the optimization problem

$\mbox{minimize}\;\varphi(X)+g(X)\quad\mbox{subject
to}\;\;X\in{\mathbb{S}}^{n}.$ (5.16)

Below, we present a result in which second-order optimality conditions for
this optimization problem are established. For simplicity, we are going to
assume that $g$ has the assumed representation in Corollary 5.8 but one can
easily extend it for any $g$ satisfying the assumptions in Theorem 5.7.

######  Theorem 5.10.

Assume that $X$ is a feasible solution to (5.16), where the spectral function
$g$ has the representation (1.1) with $\theta$ therein being a polyhedral
function. If $-\nabla\varphi(X)\in\partial g(X)$, then the following second-
order optimality conditions for (5.16) are satisfied.

  * (a)

If $X$ is a local minimizer of (5.16), then the second-order necessary
condition

$\nabla^{2}\varphi(X)(H,H)+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{\rangle}\geq
0$

holds for all $H\in K_{g}(X,-\nabla\varphi(X))$.

  * (b)

The validity of the second-order condition

$\nabla^{2}\varphi(X)(H,H)+2\sum_{m=1}^{r}\big{\langle}{\Lambda}(Y)_{\alpha_{m}\alpha_{m}},U_{\alpha_{m}}^{\top}H(\mu_{m}I-X)^{\dagger}HU_{\alpha_{m}}\big{\rangle}>0$

for all $H\in K_{g}(X,-\nabla\varphi(X))$ amounts to the existence of the
constants $\ell\geq 0$ and $\varepsilon>0$ for which the quadratic growth
condition

$\varphi(X^{\prime})+g(X^{\prime})\geq\varphi(X)+g(X)+\frac{\ell}{2}\|X^{\prime}-X\|^{2}\quad\mbox{for
all}\;\;X^{\prime}\in\mathbb{B}_{\varepsilon}(X)$

is satisfied.

[Proof. ]It follows from [23, Exercise 13.18] that

${\rm d}^{2}(\varphi+g)(X,0)(H)=\nabla^{2}\varphi(X)(H,H)+{\rm
d}^{2}g(X,-\nabla\varphi(X))(H)$

for any $H\in{\mathbb{S}}^{n}$. Both claims in (a) and (b) then result
immediately from [23, Theorem 13.24] and Corollary 5.8.

Our next application is to provide sufficient conditions for twice epi-
differentiability of spectral functions, a concept with important consequences
in second-order variational analysis and parametric optimization. Recall from
[23, Definition 13.6] that a function $f:{\mathbb{E}}\to\overline{\mathbb{R}}$
is said to be twice epi-differentiable at $\bar{x}$ for
$\bar{v}\in{\mathbb{E}}$, with $f(\bar{x})$ finite, if the sets $\mbox{\rm
epi}\,\Delta_{t}^{2}f(\bar{x},\bar{v})$ converge to $\mbox{\rm epi}\,{\rm
d}^{2}f(\bar{x},\bar{v})$ as $t\downarrow 0$ in the sense of set convergence
from [23, Definition 4.1], where ‘epi ’ stands for the epigraph of a function.
This can be equivalently described via [23, Proposition 7.2] that for every
sequence $t_{k}\downarrow 0$ and every $w\in{\mathbb{E}}$, there exists a
sequence $w_{k}\to w$ such that

${\rm
d}^{2}f(\bar{x},\bar{v})(w)=\lim_{k\to\infty}\Delta_{t_{k}}^{2}f(\bar{x},\bar{v})(w_{k}).$

Twice epi-differentiability is a geometric notion of second-order
approximation for extended-real-valued functions and was defined by
Rockafellar in [21]. Its central role in second-order variational analysis,
parametric optimization, and numerical algorithms has been demonstrated in
[23, 18, 20, 13]. It was observed recently in [20, Corollary 5.5] that
parabolic regularity of certain composite functions yields their twice epi-
differentiability. A similar conclusion can be drawn for spectral functions as
demonstrated below.

######  Corollary 5.11 (twice epi-differentiability of spectral functions).

Assume that the spectral function $g=\theta\circ\lambda$ in (1.1) is locally
Lipschitz continuous with respect to its domain, lsc, and convex. If
$Y\in\partial g(X)$ and $\theta$ is parabolically regular at $\lambda(X)$ for
$\lambda(Y)$ and parabolically epi-differentiable at $\lambda(X)$, then $g$ is
twice epi-differentiable at $X$ for $Y$.