∎

11institutetext: Zhou Wei22institutetext: Hebei Key Laboratory of Machine
Learning and Computational Intelligence & College of Mathematics and
Information Science, Hebei University, Baoding, 071002, China
22email<EMAIL_ADDRESS>
Michel Théra 33institutetext: XLIM UMR-CNRS 7252, Université de Limoges,
Limoges, France
and 44institutetext: Federation University Australia, Ballarat
44email<EMAIL_ADDRESS>
Jen-Chih Yao 55institutetext: Department of Applied Mathematics, National Sun
Yat-sen University, Kaohsiung, Taiwan
55email<EMAIL_ADDRESS>
# Primal Characterizations of Error Bounds for Composite-convex
Inequalities††thanks: Research of the first author was supported by the
National Natural Science Foundations of China (Grant Nos. 11971422 and
12171419), and funded by Science and Technology Project of Hebei Education
Department (No. ZD2022037) and the Natural Science Foundation of Hebei
Province (A2022201002). Research of the second author benefited from the
support of the FMJH Program PGMO and from the support of EDF.

Zhou Wei  Michel Théra  Jen-Chih Yao

(Received: date / Accepted: date)

###### Abstract

This paper is devoted to primal conditions of error bounds for a general
function. In terms of Bouligand tangent cones, lower Hadamard directional
derivatives and the Hausdorff-Pompeiu excess of subsets, we provide several
necessary and/or sufficient conditions of error bounds with mild assumptions.
Then we use these primal results to characterize error bounds for composite-
convex functions (i.e. the composition of a convex function with a
continuously differentiable mapping). It is proved that the primal
characterization of error bounds can be established via Bouligand tangent
cones, directional derivatives and the Hausdorff-Pompeiu excess if the mapping
is metrically regular at the given point. The accurate estimate on the error
bound modulus is also obtained.

###### Keywords:

Error bound composite-convex inequality Bouligand tangent conelower Hadamard
directional derivative Hausdorff-Pompeiu excess

###### MSC:

90C3190C2549J5246B20

††dedicatory: Dedicated to Roger J-B Wets on the occasion of his 85th
birthday. Roger’s pioneering work helped stochastic optimization become the
stronghold it is today.

##  1 Introduction

The main goal of this paper is to study error bounds for an inequality defined
by a composite-convex function; i.e. the composition of a convex function with
a continuously differentiable mapping. Error bounds of extended-real-valued
functions have been intensively studied for more than half a century. The
starting point of the theory of error bounds goes back to the fundamental
works by Hoffman MP31  and Lojasiewicz Lojasiewicz . Their results were
extensively studied by many authors (cf. AC1988 ; Jour2000 ; KL1999 ; LP1997 ;
Pang1997 ; Penot ; Rob73 ; abassi-thera1 ; abassi-thera2 ) and there have been
significant developments on error bounds for convex and non-convex functions
in recent years. The readers are invited to consult bibliographies AC ; BK ;
BD1 ; CK2020 ; CM2008 ; FHKO2010 ; ioffe-JAMS-2 ; KLT2018 ; KNT2010 ; TsL92 ;
MP52 ; NT2004 ; NT2008 ; NT2009 ; WZ2021 ; Za  and references therein for
theory and applications of error bounds for more details.

Error bounds have played an important role in various aspects of optimization
and variational analysis including for instance, sensitivity analysis of
linear programming (cf. Rob73 ; Rob77 ), convergence analysis of descent
methods (cf. Gul92 ; HLu ; IuD90 ; TsL92 ; LT ; TsB93 ), the so-called
feasibility problems (cf. BB ; BT ; BK ), the domain of image reconstruction
(cf. Combettes ) and many others. Error bounds are closely related with other
notions well known and used in convex analysis and approximation theory such
as the basic constraint qualification, the strong conical hull intersection
property, the linear regularity, the Abadie constraint qualification and
optimality conditions (cf. DR2 ; LP1997 ; Li ; MP52 ). Also error bounds are
extensively discussed with weak sharp minima of functions, metric
subregularity as well as calmness of multifunctions (cf. AC ; BD1 ; BD2 ; Gf ;
ioffe-JAMS-1 ; ioffe-JAMS-2 ; M1  and references therein for more details).

When dealing with error bounds, a large literature is devoted to provide dual
characterizations and criteria in terms of subdifferentials or normal cones.
To the best of our knowledge, Io  was among of the first papers of such kind
to state sufficient conditions for error bounds of a constraint system in
terms of the Clarke subdifferential. In 1997 Lewis and Pang LP1997  studied
error bounds for convex inequality systems and provided necessary conditions
via subdifferentials and normal cones. In 2003 Ngai and Théra NT2004  provided
an error bound estimate and an implicit multifunction theorem in terms of
smooth subdifferentials and abstract subdifferentials. In 2004 Zheng and Ng
ZN2004  proved dual characterizations of error bounds for convex inequalities
in terms of subdifferentials and normal cones. In 2010 subdifferential
characterizations of stability of error bounds for convex constraint
inequalities were given in KNT2010 ; NKT2010 . In 2018 Kruger, López and Théra
KLT2018  extended results in KNT2010 ; NKT2010  and provided subdifferential
characterizations of stability of error bounds for convex inequalities in the
Banach space setting. It is noted that a pretty natural idea is to study error
bounds in terms of various primal derivative-like objects such as directional
derivatives, contingent cones or slopes. Several criteria for error bounds
were worked out in CK2020 ; CM2008 ; FHKO2010 ; NT2008 ; NT2009  based on the
primal-type estimate. In terms of contingent cones and directional
derivatives, Wei, Yao and Zheng WZ2014  proved primal characterizations of
error bounds for a convex inequality (see (WZ2014, , Proposition 5.3)).
Recently, the authors WZ2021  further studied error bounds of the convex
inequality in terms of the Hausdorff-Pompeiu excess (of subsets), Bouligand
tangent cones and directional derivatives, and provided the accurate primal
estimate on the error bound modulus (see (WZ2021, , Theorem 5.1)). Based on
the works in WZ2014 ; WZ2021 , a natural issue is to extend primal results on
characterizations of error bounds and on the error bound modulus by dropping
the convexity assumption. Inspired by this issue, our goal in this article is
to discuss error bounds of the inequality defined by a composite-convex
function. Our work is to provide primal characterizations of error bounds and
the accurate estimate on the error bound modulus in terms of the notions of
Hausdorff-Pompeiu excess of a set beyond another one, of the Bouligand tangent
cone and of the lower Hadamard directional derivative.

The paper is organized as follows. In Section 2, we give some definitions and
preliminary results. In Section 3, we consider two concepts of metric
regularity (of multifunctions) and the Shapiro first order contact property
that are used in our analysis. Section 4 is devoted to the study of error
bounds for the inequality defined by a composite-convex function. We first
consider error bounds for a general inequality defined by a proper lower
semicontinuous (not necessarily convex) function with the Shapiro first order
contact property, and provide sufficient and/or necessary primal conditions of
error bounds in terms of the Bouligand tangent cone, the lower Hadamard
directional derivative and the Hausdorff-Pompeiu excess of subsets (see
Theorems 4.1 and 4.2). When these results are applied to error bounds of
composite-convex inequalities, the primal results on error bounds and on the
error bound modulus can be obtained (see Theorems 4.4 and 4.5).

##  2 Preliminaries

Let $\mathbb{X}$ be a Banach space (or Euclidean space). Let
$\mathbf{B}_{\mathbb{X}}$ denote the closed unit ball of $X$. For
$\bar{x}\in\mathbb{X}$ and $\delta>0$, let $\mathbf{B}(\bar{x},\delta)$ denote
the open ball with center $\bar{x}$ and radius $\delta$. For a subset $\Omega$
of $\mathbb{X}$, we denote by ${\rm cl}(\Omega)$, ${\rm int}(\Omega)$ and
${\rm bd}(\Omega)$ the closure, the interior and the boundary of $\Omega$,
respectively.

Let $A$ be a closed subset of $\mathbb{X}$ and $\bar{x}\in A$. We denote by

$\mathbf{T}^{\mathbf{B}}(A,\bar{x}):=\mathop{\rm Limsup}\limits_{t\rightarrow
0^{+}}\frac{A-\bar{x}}{t}$

the Bouligand tangent cone (also called contingent cone) of $A$ at $\bar{x}$.
Thus, $v\in\mathbf{T}^{\mathbf{B}}(A,\bar{x})$ if and only if there exist a
sequence $\\{v_{n}\\}$ in $X$ converging to $v$ and a sequence $\\{t_{n}\\}$
in $(0,\;+\infty)$ decreasing to 0 such that $\bar{x}+t_{n}v_{n}\in A$ for all
$n\in\mathbb{N}$, where $\mathbb{N}$ denotes the set of all natural numbers.

For any subsets $C$ and $D$ of $\mathbb{X}$, the excess of $C$ beyond $D$ is
defined as:

$\mathbf{e}(C,D):=\sup_{x\in C}\mathbf{d}(x,D),$ (2.1)

where $\mathbf{d}(x,D):=\inf\\{\|x-y\|:y\in D\\}$ and the convention is used
that

$\mathbf{e}(\emptyset,D):=\left\\{\begin{array}[]{cl}0,&{\rm if}\
D\not=\emptyset,\\\ \infty,&{\rm otherwise}.\end{array}\right.$ (2.2)

The following proposition provides a characterization for the Hausdorff-
Pompeiu excess. We refer the reader to (DR2, , Page 138) (or (WZ2022, ,
Proposition 3.1)) for the proof in details.

###### Proposition 1

Let $C$ and $D$ be subsets of $\mathbb{X}$. Then

$\mathbf{e}(C,D)=\inf\\{\tau\geq 0:C\subseteq
D+\tau\mathbf{B}_{\mathbb{X}}\\}.$ (2.3)

Given an extended-real-valued lower semicontinuous function
$\varphi:\mathbb{X}\rightarrow\mathbb{R}\cup\\{+\infty\\}$, we denote by ${\rm
dom}(\varphi):=\\{u\in\mathbb{X}:\varphi(u)<+\infty\\}$ its domain and by

${\rm
epi}(\varphi):=\\{(x,\alpha)\in\mathbb{X}\times\mathbb{R}:\varphi(x)\leq\alpha\\}$

its epigraph. $\varphi$ is said to be proper if its domain is nonempty.

The lower Hadamard directional derivative of $\varphi$ at $x$ along direction
$h$ is defined as

$\varphi_{H}^{\prime}(x;h):=\liminf_{t\rightarrow 0^{+},h^{\prime}\rightarrow
h}\frac{\varphi(x+th^{\prime})-\varphi(x)}{t}.$ (2.4)

We close this section with the following lemma cited from (SYZ, , Theorem
4.1).

###### Lemma 1

Let $\mathbb{X}$ be a Banach space and $\Omega$ be a nonempty closed subset of
$\mathbb{X}$. Let $\gamma\in(0,\;1)$. Then for any $x\not\in\Omega$ there
exists $z\in\Omega$ such that

$\gamma\|x-z\|<\min\\{\mathbf{d}(x,\Omega),\mathbf{d}(x-z,\mathbf{T}^{\mathbf{B}}(\Omega,z))\\}.$

In particular, if $\mathbb{X}$ is of finite dimension, $z$ can be chosen as
the projection of $x$ onto $\Omega$.

##  3 Metric regularity and the Shapiro first order contact property

This section recalls the two important concepts of metric regularity and of
Shapiro first order contact property that are used in our analysis. It is
known that metric regularity of multifunctions occurs to be closely related to
Lipschtizian properties of inverse mappings. This well-known and significant
property is an extension of surjectivity to nonlinear/set-valued mappings and
goes back to the Banach-Schauder open mapping theorem and to the Lyusternik-
Graves theorem. We first recall the definition of metric regularity.

Definition 3.1. (i) Let $F:\mathbb{X}\rightrightarrows\mathbb{Y}$ be a
multifunction between two Banach spaces and $\bar{x}\in\mathbb{X}$. Recall
that $F$ is said to be metrically regular at $\bar{x}$ for $\bar{y}\in
F(\bar{x})$, if there exists a constant $\kappa\in(0,+\infty)$ along with
neighborhoods $U$ of $\bar{x}$ and $V$ of $\bar{y}$ such that

$\mathbf{d}(x,F^{-1}(y))\leq\kappa\mathbf{d}(y,F(x))\ \ {\it for\ all}\
(x,y)\in U\times V.$ (3.1)

(ii) A single-value mapping $\varPsi:\mathbb{X}\rightarrow\mathbb{Y}$ is said
to be metrically regular at $\bar{x}$ if $\varPsi$ is metrically regular at
$\bar{x}$ for $\varPsi(\bar{x})$.

It is known from Asen-book ; M2 ; Penot2013 ; ioffe-book ; thibault  that
metric regularity of a multifunction is proved to be equivalent to the
covering property. We refer for instance the readers to (M2, , Theorem 1.52)
for more details on this equivalence relationships and the modulus estimates
for metric regularity and the covering property. Further the readers are
invited to consult ioffe-JAMS-1 ; ioffe-JAMS-2  for a survey and the
development on metric regularity and NTVT  for some regular properties of
graphical tangent and normal cones to paraconvex multifunctions.

From (M2, , Lemma 1.56), we always have the automatic closedness of the
derivative image for single-valued metrically regular mappings. The following
proposition, as a result of independent interest, weakens the assumption of
metric regularity therein.

###### Proposition 2

Let $\varphi:\mathbb{X}\rightarrow\mathbb{Y}$ be a mapping between two Banach
spaces and $\bar{x}\in\mathbb{X}$. Suppose that $\varphi$ is Fréchet
differentiable at $\bar{x}$ and that there exist $\kappa,r>0$ such that

$\mathbf{d}(\bar{x},\varphi^{-1}(y))\leq\kappa\|\varphi(\bar{x})-y\|\ \
\forall y\in Y:\|y-\varphi(\bar{x})\|<r.$ (3.2)

Then $\triangledown\varphi(\bar{x})(\mathbb{X})$ is a closed subspace of
$\mathbb{Y}$.

Proof. Let $y_{0}\in{\rm cl}(\triangledown\varphi(\bar{x})(\mathbb{X}))$. Then
we can find a sequence $\\{y_{k}\\}$ in
$\triangledown\varphi(\bar{x})(\mathbb{X})$ such that

$y_{k}\rightarrow y_{0}\ {\rm and}\ \|y_{k+1}-y_{k}\|<\frac{1}{2^{k}}\ {\rm
for\ all}\ k.$ (3.3)

We claim that there exists a sequence $\\{x_{k}\\}$ in $X$ such that

$\|x_{k+1}-x_{k}\|<\frac{3\kappa}{2^{k}}\ {\rm and}\
\|y_{k}-\triangledown\varphi(\bar{x})(x_{k})\|<\frac{1}{2^{k}}\ {\rm for\
all}\ k.$ (3.4)

Granting this, one has $\\{x_{k}\\}$ is a Cauchy sequence in $\mathbb{X}$ that
converges to some $x_{0}\in\mathbb{X}$ and it follows from (3.4) that
$\triangledown\varphi(\bar{x})(x_{k})\rightarrow y_{0}$, which gives
$\triangledown\varphi(\bar{x})(x_{0})=y_{0}$.

We define $x_{k}$ iteratively. Let $x_{1}\in X$ be such that
$\triangledown\varphi(\bar{x})(x_{1})=y_{1}$. Suppose $x_{1},\cdots,x_{k}$
have been given to satisfy (3.4) and construct $x_{k+1}$ as follows.

Choose $u_{k+1}\in\triangledown\varphi(\bar{x})^{-1}(y_{k+1})-x_{k}$. Let
$\varepsilon>0$ be sufficiently small such that

$3\varepsilon\kappa<\frac{1}{2^{2}},\varepsilon<\frac{1}{2^{k+2}}\ \ {\rm
and}\ \ \varepsilon\|u_{k+1}\|<\frac{1}{2^{k+2}}.$ (3.5)

Since $\varphi$ is Fréchet differentiable at $\bar{x}$, then there exists
$\delta\in(0,r)$ such that

$\|\varphi(\bar{x}+w)-\varphi(\bar{x})-\triangledown\varphi(\bar{x})(w)\|<\varepsilon\|w\|\
\ \forall w:\|w\|<\delta.$ (3.6)

Take $t_{k}>0$ sufficiently small such that

$\max\Big{\\{}\frac{3t_{k}}{2^{k}},\frac{3\kappa}{2^{k}}t_{k},t_{k}\|u_{k+1}\|\Big{\\}}<\delta.$

Then by (3.5) and (3.6), one has

$\displaystyle\|\varphi(\bar{x}+t_{k}u_{k+1})-\varphi(\bar{x})\|$
$\displaystyle\leq$
$\displaystyle\|\varphi(\bar{x}+t_{k}u_{k+1})-\varphi(\bar{x})-\triangledown\varphi(\bar{x})(t_{k}u_{k+1})\|+\|t_{k}\triangledown\varphi(\bar{x})(u_{k+1})\|$
$\displaystyle<$ $\displaystyle\varepsilon
t_{k}\|u_{k+1}\|+t_{k}(\|y_{k+1}-y_{k}\|+\|y_{k}-\triangledown\varphi(\bar{x})(x_{k})\|)$
$\displaystyle<$ $\displaystyle
t_{k}\big{(}\frac{1}{2^{k+2}}+\frac{1}{2^{k}}+\frac{1}{2^{k}}\big{)}$
$\displaystyle<$ $\displaystyle\frac{3t_{k}}{2^{k}}<\delta<r.$

This and (3.2) imply that

$\mathbf{d}(\bar{x},\varphi^{-1}(\varphi(\bar{x}+t_{k}u_{k+1})))\leq\kappa\|\varphi(\bar{x}+t_{k}u_{k+1})-\varphi(\bar{x})\|<\frac{3\kappa}{2^{k}}t_{k}$

and thus there is $w_{k}\in\varphi^{-1}(\varphi(\bar{x}+t_{k}u_{k+1}))$ such
that

$\|\bar{x}-w_{k}\|<\frac{3\kappa}{2^{k}}t_{k}.$

Let $v_{k}:=\frac{w_{k}-\bar{x}}{t_{k}}$ and $x_{k+1}:=x_{k}+v_{k}$. Then

$\|x_{k+1}-x_{k}\|=\|v_{k}\|<\frac{3\kappa}{2^{k}}.$

To complete the proof, it remains to show that

$\|y_{k+1}-\triangledown\varphi(\bar{x})(x_{k+1})\|<\frac{1}{2^{k}}.$ (3.7)

Combining (3.5) with (3.6), one has

$\|\varphi(\bar{x}+t_{k}v_{k})-\varphi(\bar{x})-\triangledown\varphi(\bar{x})(t_{k}v_{k})\|<\varepsilon\|t_{k}v_{k}\|<\varepsilon
t_{k}\frac{3\kappa}{2^{k}}<\frac{t_{k}}{2^{k+2}}$ (3.8)

and

$\|\varphi(\bar{x}+t_{k}u_{k+1})-\varphi(\bar{x})-\triangledown\varphi(\bar{x})(t_{k}u_{k+1})\|<\varepsilon\|t_{k}u_{k+1}\|<\frac{t_{k}}{2^{k+2}}.$
(3.9)

Note that $\varphi(\bar{x}+t_{k}v_{k})=\varphi(\bar{x}+t_{k}u_{k+1})$ and it
follows from (3.8) and (3.9) that

$\|\triangledown\varphi(\bar{x})(v_{k})-\triangledown\varphi(\bar{x})(u_{k+1})\|<\frac{1}{2^{k+2}}+\frac{1}{2^{k+2}}=\frac{1}{2^{k+1}}.$

This means that

$\|y_{k+1}-\triangledown\varphi(\bar{x})(x_{k+1})\|=\|\triangledown\varphi(\bar{x})(u_{k+1}+x_{k})-\triangledown\varphi(\bar{x})(x_{k}+v_{k})\|<\frac{1}{2^{k+1}}.$

The proof is complete. $\Box$

As an appropriate substitute of convexity, we consider a tangential concept
introduced in (Sh1, , Definition 2.1) under the name of $o(p)$-convexity and
known in the literature as the Shapiro $p$-order contact property. Definition
3.2. Let $A$ be a closed subset of $\mathbb{X}$ and $p\in\mathbb{N}$. Recall
that $A$ is said to have the $p$-order Shapiro contact property at $a\in A$,
if for any $\varepsilon>0$ there exists $\delta>0$ such that

$\mathbf{d}(x-u,\mathbf{T}^{\mathbf{B}}(A,u))\leq\varepsilon\|x-u\|^{p}\ \
\forall x,u\in A\cap\mathbf{B}(a,\delta).$ (3.10)

In the remainder of this paper, we will use the terminology Shapiro first
order contact property in place of $1$-order Shapiro contact property.

In ADT , Aussel, Daniilids and Thibault established the links between the
Shapiro first order contact property and the notions of subsmoothness and
semismoothness. We refer the reader to (ADT, , Theorem 3.16) and to Thibault’s
book (thibault, , subsection 8.3.2) for more details. In 2019 Shen, Yao and
Zheng SYZ  studied the Shapiro property and the C-Shapiro property, and showed
that the Shapiro property is an extension of convexity and smoothness (see
(SYZ, , Propositions 3.8 and 3.9)). Further, they considered the Shapiro
property of a general multifunction and used it to study calmness for closed
multifunctions. Recently, inspired by the Shapiro property of a multifunction,
the authors WZ2022  consider the epigraphical Shapiro property of a function;
that is,

Definition 3.3. Let $\varphi:\mathbb{X}\rightarrow\mathbb{R}\cup\\{+\infty\\}$
be a proper function and $\bar{x}\in{\rm dom}(\varphi)$. We say that $\varphi$
has the epigraphical Shapiro first order contact property at $\bar{x}$, if the
epigraph ${\rm epi}(\varphi)$ has the Shapiro first order contact property at
$(\bar{x},\varphi(\bar{x}))$.

The following proposition is on the epigraphical Shapiro property which was
proved in (WZ2022, , Propositon 2.1).

###### Proposition 3

Let $\varphi:\mathbb{X}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ be a proper
extended-real-valued lower semicontinuous function and $\bar{x}\in{\rm
dom}(\varphi)$. Consider the following statements:

  * (i)

$\varphi$ has the epigraphical Shapiro first order contact property at
$\bar{x}$;

  * (ii)

for any $\varepsilon>0$ there exists $\delta>0$ such that

$\mathbf{d}((x-u,\varphi(x)-\varphi(u)),\mathbf{T}^{\mathbf{B}}({\rm
epi}(\varphi),(u,\varphi(u))))\leq\varepsilon(\|x-u\|+|\varphi(x)-\varphi(u)|)$
(3.11)

holds for all $x,u\in\mathbf{B}_{\varphi}(\bar{x},\delta)$;

  * (iii)

for any $\varepsilon>0$ there exists $\delta>0$ such that

$\varphi_{H}^{\prime}(u;x-u)\leq\varphi(x)-\varphi(u)+\varepsilon(\|x-u\|+|\varphi(x)-\varphi(u)|)$
(3.12)

holds for all $x,u\in\mathbf{B}_{\varphi}(\bar{x},\delta)$,

where
$\mathbf{B}_{\varphi}(\bar{x},\delta):=\\{x\in\mathbf{B}(\bar{x},\delta):|\varphi(x)-\varphi(\bar{x})|<\delta\\}$.

Then (i)$\Rightarrow$(ii)$\Leftarrow$(iii). Further, assume that $\varphi$ is
continuous around $\bar{x}$. Then (i)$\Leftrightarrow$(ii).

Remark 3.1. It is noted that the implication of (ii)$\Rightarrow$(iii) or
(ii)$\Rightarrow$(i) in Proposition 3 may not be necessarily true. The readers
are invited to consult the counterexample in (WZ2022, , Remark 2.1) for more
details.

The following result was proved by Shapiro and Al-Khayyal in Sh2 .

###### Proposition 4

Let $\varphi:\mathbb{X}\rightarrow\mathbb{Y}$ be a continuously differentiable
mapping and $A$ be a closed convex cone of $\mathbb{Y}$. Suppose that
$\bar{x}\in\varphi^{-1}(A)$ satisfies the following Robinson qualification:

$0\in{\rm int}(\varphi(\bar{x})+\triangledown\varphi(\bar{x})X-A).$

Then $\varphi^{-1}(A)$ has the Shapiro first order contact property at
$\bar{x}$.

The following proposition improves Proposition 4 by weakening the Robinson
qualification. This proposition is a key tool to prove main results in the
paper.

###### Proposition 5

Let $\varphi:\mathbb{X}\rightarrow\mathbb{Y}$ be a continuously differentiable
mapping, $A$ be a closed subset of $\mathbb{Y}$ and
$\bar{x}\in\varphi^{-1}(A)$. Suppose that $A$ has the Shapiro first order
contact property at $\varphi(\bar{x})$ and $\varphi$ is metrically regular
around $\bar{x}$. Then $\varphi^{-1}(A)$ has the Shapiro first order contact
property at $\bar{x}$.

To prove Proposition 5, we need the following two lemmata which are of
independent interest.

###### Lemma 2

Let $\varphi:\mathbb{X}\rightarrow\mathbb{Y}$ and $\bar{x}\in\mathbb{X}$.
Suppose that $\varphi$ is Fréchet differentiable at $\bar{x}$ and image
$\triangledown\varphi(\bar{x})(\mathbb{X})$ is of the second category. Then
there exists $\mu_{0}>0$ such that

$\mathbf{d}(u,\triangledown\varphi(\bar{x})^{-1}(v))\leq\mu_{0}\|\triangledown\varphi(\bar{x})(u)-v\|\
\ \forall(u,v)\in\mathbb{X}\times\mathbb{Y}.$ (3.13)

Further, assume that $\varphi$ is continuously differentiable at $\bar{x}$.
Then there exist $\mu,\delta>0$ such that

$\mathbf{d}(u,\triangledown\varphi(x)^{-1}(v))\leq\mu\|\triangledown\varphi(x)(u)-v\|\
\
\forall(x,u,v)\in\mathbf{B}(\bar{x},\delta)\times\mathbb{X}\times\mathbb{Y}.$
(3.14)

Proof. Since $\triangledown\varphi(\bar{x})(\mathbb{X})$ is of the second
category, it follows from the open mapping theorem that there exists $l>0$
such that

$2l\mathbf{B}_{Y}\subseteq\triangledown\varphi(\bar{x})(\mathbf{B}_{\mathbb{X}}).$
(3.15)

Let $(u,v)\in\mathbb{X}\times\mathbb{Y}$. By virtue of (3.15), one has

$\triangledown\varphi(\bar{x})(u)-v\in\|\triangledown\varphi(\bar{x})(u)-v\|\mathbf{B}_{\mathbb{Y}}\subseteq\triangledown\varphi(\bar{x})\Big{(}\frac{\|\triangledown\varphi(\bar{x})(u)-v\|}{2l}\mathbf{B}_{\mathbb{X}}\Big{)}.$
(3.16)

Note that
$\triangledown\varphi(\bar{x})^{-1}(\triangledown\varphi(\bar{x})(u)-v)=u-\triangledown\varphi(\bar{x})^{-1}(v)$
by the linearity of $\triangledown\varphi(\bar{x})$ and then (3.16) gives that

$\displaystyle\mathbf{d}(u,\triangledown\varphi(\bar{x})^{-1}(v))$
$\displaystyle=$
$\displaystyle\mathbf{d}(0,u-\triangledown\varphi(\bar{x})^{-1}(v))$
$\displaystyle=$
$\displaystyle\mathbf{d}(0,\triangledown\varphi(\bar{x})^{-1}(\triangledown\varphi(\bar{x})(u)-v))$
$\displaystyle\leq$
$\displaystyle\frac{1}{2l}\|\triangledown\varphi(\bar{x})(u)-v\|.$

This means that (3.13) holds with $\mu_{0}:=\frac{1}{2l}$.

To prove (3.14), we next show that there exists $\delta>0$ such that

$l\mathbf{B}_{Y}\subseteq\triangledown\varphi(x)(\mathbf{B}_{\mathbb{X}})\ \
\forall x\in\mathbf{B}(\bar{x},\delta).$ (3.17)

Noting that $\varphi$ is continuously differentiable at $\bar{x}$, then there
exists $\delta>0$ such that $\varphi$ is Fréchet differentiable on
$\mathbf{B}(\bar{x},\delta)$ and

$\|\triangledown\varphi(x)-\triangledown\varphi(\bar{x})\|<\frac{l}{2}\ \
\forall x\in\mathbf{B}(\bar{x},\delta)$ (3.18)

(taking a smaller $\delta$ if necessary). Then by (3.15) and (3.18), one has

$2l\mathbf{B}_{\mathbb{Y}}\subseteq(\triangledown\varphi(x)+\triangledown\varphi(\bar{x})-\triangledown\varphi(x))(\mathbf{B}_{\mathbb{X}})\subseteq\triangledown\varphi(x)(\mathbf{B}_{\mathbb{X}})+\frac{l}{2}\mathbf{B}_{\mathbb{Y}}\
\ \forall x\in\mathbf{B}(\bar{x},\delta).$

By the Rädstrom cancellation lemma (Ra, , Lemma 2.3), this gives that

$\frac{3l}{2}\mathbf{B}_{\mathbb{Y}}\subseteq{\rm
cl}(\triangledown\varphi(x)(\mathbf{B}_{\mathbb{X}}))\ \ \forall
x\in\mathbf{B}(\bar{x},\delta).$ (3.19)

Applying (Jameson, , P.183, Theorem A.1) again gives that
$\triangledown\varphi(x)(\mathbf{B}_{\mathbb{X}})$ and ${\rm
cl}(\triangledown\varphi(x)(\mathbf{B}_{\mathbb{X}}))$ have the same interior
and thus that (3.17) holds by (3.19). The proof is complete. $\Box$

###### Lemma 3

Let $\varphi:\mathbb{X}\rightarrow\mathbb{Y}$ be a mapping and $A$ be a closed
subset of $\mathbb{Y}$. Suppose that $\bar{x}\in\varphi^{-1}(A)$ is such that
$\varphi$ is Fréchet differentiable at $\bar{x}$ and metrically regular around
$\bar{x}$. Then there exists $\delta>0$ such that

$\mathbf{T}^{\mathbf{B}}(\varphi^{-1}(A),x)=\triangledown\varphi(x)^{-1}(\mathbf{T}^{\mathbf{B}}(A,\varphi(x)))\
\ \forall x\in\mathbf{B}(\bar{x},\delta)\cap\varphi^{-1}(A).$ (3.20)

Proof. Since $\varphi$ is metrically regular around $\bar{x}$, there exist
$\kappa>0$ along with neighborhoods $U$ of $\bar{x}$ and $V$ of
$\varphi(\bar{x})$ such that

$\mathbf{d}(x,\varphi^{-1}(y))\leq\kappa\|\varphi(x)-y\|\ \ \forall(x,y)\in
U\times V.$ (3.21)

Take $\delta>0$ sufficiently small such that

$\mathbf{B}(\bar{x},\delta)\subseteq U\ \ {\rm and}\ \
\varphi(\mathbf{B}(\bar{x},\delta))\subseteq V.$ (3.22)

Let $x\in\mathbf{B}(\bar{x},\delta)\cap\varphi^{-1}(A)$. Choose any
$u\in\triangledown\varphi(x)^{-1}(\mathbf{T}^{\mathbf{B}}(A,\varphi(x)))$.
Then $\triangledown\varphi(x)(u)\in\mathbf{T}^{\mathbf{B}}(A,\varphi(x))$ and
thus there exist $t_{k}\rightarrow 0^{+}$ and
$v_{k}\rightarrow\triangledown\varphi(x)(u)$ such that

$\varphi(x)+t_{k}v_{k}\in A\ \ {\rm for\ all}\ k.$

By virtue of (3.21) and (3.22), for any $k$ sufficiently large, one has

$\mathbf{d}(x+t_{k}u,\varphi^{-1}(\varphi(x)+t_{k}v_{k}))\leq\kappa\|\varphi(x+t_{k}u)-\varphi(x)-t_{k}v_{k}\|$

and consequently there is
$x_{k}\in\varphi^{-1}(\varphi(x)+t_{k}v_{k})\subseteq\varphi^{-1}(A)$ such
that

$\|x+t_{k}u-x_{k}\|<2\kappa\|\varphi(x+t_{k}u)-\varphi(x)-t_{k}v_{k}\|.$
(3.23)

Denote $u_{k}:=\frac{x_{k}-x}{t_{k}}$ for all $k$. Note that

$\varphi(x+t_{k}u)=\varphi(x)+\triangledown\varphi(x)(t_{k}u)+o(t_{k})$

and then (3.23) gives that

$\|u_{k}-u\|<2\kappa\left\|\triangledown\varphi(x)(u)-v_{k}+\frac{o(t_{k})}{t_{k}}\right\|,$

which implies that $u_{k}\rightarrow u$ as
$v_{k}\rightarrow\triangledown\varphi(x)(u)$. Noting that
$x+t_{k}u_{k}=x_{k}\in\varphi^{-1}(A)$, it follows that
$u\in\mathbf{T}^{\mathbf{B}}(\varphi^{-1}(A),x)$ and thus

$\mathbf{T}^{\mathbf{B}}(\varphi^{-1}(A),x)\supseteq\triangledown\varphi(x)^{-1}(\mathbf{T}^{\mathbf{B}}(A,\varphi(x))).$
(3.24)

On the other hand, let $h\in\mathbf{T}^{\mathbf{B}}(\varphi^{-1}(A),x)$. Then
there exist $t_{k}\rightarrow 0^{+}$ and $h_{k}\rightarrow h$ such that
$x+t_{k}h_{k}\in\varphi^{-1}(A)$ for all $k$. Using the Fréchet
differentiability of $\varphi$ at $x$, one has

$\varphi(x+t_{k}h_{k})=\varphi(x)+\triangledown\varphi(x)(t_{k}h_{k})+o(\|t_{k}h_{k}\|).$

Denote

$v_{k}:=\triangledown\varphi(x)(h_{k})+\frac{o(\|t_{k}h_{k}\|)}{t_{k}}\ \ {\rm
for\ all}\ k.$

Then $v_{k}\rightarrow\triangledown\varphi(x)(h)$ and
$\varphi(x)+t_{k}v_{k}=\varphi(x+t_{k}h_{k})\in A$, which implies that
$\triangledown\varphi(x)(h)\in\mathbf{T}^{\mathbf{B}}(A,\varphi(x))$. Hence

$\mathbf{T}^{\mathbf{B}}(\varphi^{-1}(A),x)\subseteq\triangledown\varphi(x)^{-1}(\mathbf{T}^{\mathbf{B}}(A,\varphi(x)))$

and (3.20) holds by (3.24). The proof is complete. $\Box$

Proof of Proposition 5. Since $\varphi$ is continuously differentiable and
metrically regular around $\bar{x}$, it follows from (M2, , Theorem 1.57) that
$\triangledown\varphi(\bar{x})$ is surjective. By virtue of Lemma 2 and Lemma
3 , there exist $\mu,\delta>0$ such that (3.14) and (3.20) hold.

Let $\varepsilon>0$. Since $\varphi$ is continuously differentiable at
$\bar{x}$, there exists $\delta_{1}\in(0,\delta)$ such that

$\|\varphi(x)-\varphi(u)-\triangledown\varphi(u)(x-u)\|<\frac{\varepsilon}{2\mu}\|x-u\|\
\ \forall x,u\in\mathbf{B}(\bar{x},\delta_{1})$ (3.25)

and

$\|\varphi(x)-\varphi(u)\|\leq K\|x-u\|\ \ \forall
x,u\in\mathbf{B}(\bar{x},\delta_{1})$ (3.26)

where $K:=\|\triangledown\varphi(\bar{x})\|+1$.

Noting $A$ has the Shapiro first order contact property at $\varphi(\bar{x})$,
it follows that there exists $\delta_{2}\in(0,\delta_{1})$ such that

$\mathbf{d}(y-v,\mathbf{T}^{\mathbf{B}}(A,v))\leq\frac{\varepsilon}{2K\mu}\|y-v\|\
\ \forall y,v\in A\cap\mathbf{B}(\varphi(\bar{x}),\delta_{2}).$ (3.27)

Take $\delta_{3}\in(0,\delta_{2})$ such that
$\varphi(\mathbf{B}(\bar{x},\delta_{3}))\subseteq\mathbf{B}(\varphi(\bar{x}),\delta_{2})$.
Let $x,u\in\varphi^{-1}(A)\cap\mathbf{B}(\bar{x},\delta_{3})$. Take a sequence
$\\{v_{n}\\}$ in $\mathbf{T}^{\mathbf{B}}(A,\varphi(u))$ such that

$\|\varphi(x)-\varphi(u)-v_{n}\|\rightarrow\mathbf{d}(\varphi(x)-\varphi(u),\mathbf{T}^{\mathbf{B}}(A,\varphi(u))).$

By virtue of (3.14) and (3.20), one has

$\displaystyle\mathbf{d}(x-u,\mathbf{T}^{\mathbf{B}}(\varphi^{-1}(A),u))$
$\displaystyle=$
$\displaystyle\mathbf{d}(x-u,\triangledown\varphi(u)^{-1}(\mathbf{T}^{\mathbf{B}}(A,\varphi(u))))$
$\displaystyle\leq$
$\displaystyle\mathbf{d}(x-u,\triangledown\varphi(u)^{-1}(v_{n}))$
$\displaystyle\leq$ $\displaystyle\mu\|\triangledown\varphi(u)(x-u)-v_{n}\|$
$\displaystyle\leq$
$\displaystyle\mu\big{(}\|\triangledown\varphi(u)(x-u)+\varphi(u)-\varphi(x)\|+\|\varphi(x)-\varphi(u)-v_{n}\|\big{)}.$

By letting $n\rightarrow\infty$, one gets

$\displaystyle\mathbf{d}(x-u,\mathbf{T}^{\mathbf{B}}(\varphi^{-1}(A),u))$
$\displaystyle\leq$
$\displaystyle\mu\big{(}\|\varphi(x)-\varphi(u)-\triangledown\varphi(u)(x-u)\|+\mathbf{d}(\varphi(x)-\varphi(u),\mathbf{T}^{\mathbf{B}}(A,\varphi(u)))\|\big{)}.$

This together with (3.25), (3.26) and (3.27) gives

$\displaystyle\mathbf{d}(x-u,\mathbf{T}^{\mathbf{B}}(\varphi^{-1}(A),u))$
$\displaystyle\leq$
$\displaystyle\mu\big{(}\|\varphi(x)-\varphi(u)-\triangledown\varphi(u)(x-u)\|+\mathbf{d}(\varphi(x)-\varphi(u),\mathbf{T}^{\mathbf{B}}(A,\varphi(u)))\|\big{)}$
$\displaystyle\leq$
$\displaystyle\mu\big{(}\frac{\varepsilon}{2\mu}\|x-u\|+\frac{\varepsilon}{2K\mu}\|\varphi(x)-\varphi(u)\|\big{)}$
$\displaystyle\leq$
$\displaystyle\mu\big{(}\frac{\varepsilon}{2\mu}\|x-u\|+\frac{\varepsilon}{2K\mu}K\|x-u\|\big{)}$
$\displaystyle=$ $\displaystyle\varepsilon\|x-u\|.$

Hence $\varphi^{-1}(A)$ has the Shapiro first order contact property at
$\bar{x}$. The proof is complete. $\Box$

##  4 Main results

In this section, we study error bounds of the inequality defined by a general
function and aim to provide primal criteria of error bounds via Bouligand
tangent cones, lower Hadamard directional derivatives and the Hausdorff-
Pompeiu excess of subsets. Then we apply these primal results to error bounds
of the composite-convex function, and establish primal characterizations of
error bounds and an accurate estimate of the error bound modulus in terms of
Bouligand tangent cones, directional derivatives of convex functions and the
Hausdorff-Pompeiu excess. We first consider error bounds of the inequality
defined by a general lower semicontinuous function.

Given a proper lower semicontinuous extended-real-valued function
$\varphi:\mathbb{X}\rightarrow\mathbb{R}\cup\\{+\infty\\}$, we consider the
following inequality:

$\varphi(x)\leq 0.$ (4.1)

We denote by $\mathbf{S}_{\varphi}:=\\{x\in\mathbb{X}:\varphi(x)\leq 0\\}$ the
solution set. Recall that inequality (4.1) is said to have a local error bound
at $\bar{x}\in\mathbf{S}_{\varphi}$, if there exist
$\tau,\delta\in(0,+\infty)$ such that

$\mathbf{d}(x,\mathbf{S}_{\varphi})\leq\tau[\varphi(x)]_{+},\ \ \forall
x\in\mathbf{B}(\bar{x},\delta),$ (4.2)

where $[\varphi(x)]_{+}:=\max\\{\varphi(x),0\\}$. We denote by

$\tau(\mathbf{S}_{\varphi},\bar{x}):=\inf\\{\tau>0:\ {\rm there\ exists}\
\delta>0\ {\rm such\ that}\ \eqref{3.2}\ {\rm holds}\\}$ (4.3)

the local error bound modulus of $\mathbf{S}_{\varphi}$ at $\bar{x}$.

The theory of error bounds has been a subject of intense study during many
years due to its numerous applications in optimization and variational
analysis. They include for instance areas like sensitivity analysis of linear
programming, convergence analysis of descent methods, the feasibility problems
and the domain of image reconstruction. This notion has been proved to have
close connections with several notions in convex analysis and approximation
like the basic constraint qualification (BCQ), the Abadie constraint
qualification (ACQ), the strong conical hull intersection property (CHIP), the
linear regularity and normal property, the metric subregularity as well as the
calmness of multifunctions.

The following theorem gives necessary and/or sufficient criteria of local
error bounds for the inequality (4.1) in terms of the Bouligand tangent cone,
the lower Hadamard directional derivative and the Hausdorff-Pompeiu excess of
a set beyond another set.

###### Theorem 4.1

Let $\varphi:X\rightarrow\mathbb{R}\cup\\{+\infty\\}$ be a proper lower
semicontinuous function and $\bar{x}\in\mathbf{S}_{\varphi}$.

  * (i)

Suppose that $\mathbf{S}_{\varphi}$ has the Shapiro first order contact
property around $\bar{x}$ and inequality (4.1) has a local error bound at
$\bar{x}$. Then there exist $\tau,r\in(0,+\infty)$ such that

$\mathbf{e}\big{(}\varphi^{\prime}_{\mathbf{H}}(x,\cdot)^{-1}(-\infty,1],\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)\big{)}\leq\tau$
(4.4)

holds for all $x\in\mathbf{S}_{\varphi}\cap\mathbf{B}(\bar{x},r)$ with
$\varphi(x)=0$.

  * (ii)

Suppose that ${\rm bd}(\mathbf{S}_{\varphi})\subseteq\varphi^{-1}(0)$,
$\varphi$ has the epigraphical Shapiro first order contact property at
$\bar{x}$ and that there exist $\tau,r\in(0,+\infty)$ such that (4.4) holds
for all $x\in{\rm bd}(\mathbf{S}_{\varphi})\cap\mathbf{B}(\bar{x},r)$. Then
inequality (4.1) has a local error bound at $\bar{x}$.

Proof. (i) By the local error bound of inequality (4.1) at $\bar{x}$, there
exist $\tau,\delta\in(0,+\infty)$ such that (4.2) holds. Note that
$\mathbf{S}_{\varphi}$ has the Shapiro first order contact property around
$\bar{x}$ and thus there exists $r\in(0,\delta)$ such that
$\mathbf{S}_{\varphi}$ has the Shapiro first order contact property on
$\mathbf{S}_{\varphi}\cap\mathbf{B}(\bar{x},r)$. Let
$x\in\mathbf{S}_{\varphi}\cap\mathbf{B}(\bar{x},r)$ with $\varphi(x)=0$. Take
any $h\in X$ such that $\varphi^{\prime}_{\mathbf{H}}(x,h)\leq 1$. We need to
show that

$\mathbf{d}(h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))\leq\tau.$ (4.5)

Note that $\varphi^{\prime}_{\mathbf{H}}(x,h)\leq 1$ and then there exist
$t_{n}\rightarrow 0^{+}$ and $h_{n}\rightarrow h$ such that

$\frac{\varphi(x+t_{n}h_{n})-\varphi(x)}{t_{n}}<1+\frac{1}{n}.$ (4.6)

If there exists a subsequence $\\{n_{k}\\}\subseteq\mathbb{N}$ such that
$\varphi(x+t_{n_{k}}h_{n_{k}})\leq 0$ for all $k$, then one has
$h\in\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)$ and thus (4.5) holds by
$\mathbf{d}(h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))=0$.

We next consider the case that $\varphi(x+t_{n}h_{n})>0$ for all $n$. Since
$\mathbf{S}_{\varphi}$ has the Shapiro first order contact property at $x$, by
Proposition 2.1, for any $k\in\mathbb{N}$, there exist $r_{k}\in(0,+\infty)$
such that $\mathbf{B}(x,r_{k})\subseteq\mathbf{B}(\bar{x},\delta)$ and

$\mathbf{d}(u-x,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))\leq\mathbf{d}(u,\mathbf{S}_{\varphi})+\frac{1}{k}\|u-x\|\
\ \forall u\in\mathbf{B}(x,r_{k}).$ (4.7)

Take a subsequence $\\{n_{k}\\}\subseteq\mathbb{N}$ such that
$\|t_{n_{k}}h_{n_{k}}\|<\frac{1}{k}$. Then by virtue of (4.2), (4.6) and
(4.7), one has

$\displaystyle\mathbf{d}(x+t_{n_{k}}h_{n_{k}}-x,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))$
$\displaystyle\leq$
$\displaystyle\mathbf{d}(x+t_{n_{k}}h_{n_{k}},\mathbf{S}_{\varphi})+\frac{1}{k}t_{n_{k}}\|h_{n_{k}}\|$
$\displaystyle\leq$
$\displaystyle\tau\varphi(x+t_{n_{k}}h_{n_{k}})+\frac{1}{k}t_{n_{k}}\|h_{n_{k}}\|$
$\displaystyle<$
$\displaystyle\tau\big{(}1+\frac{1}{n_{k}}\big{)}t_{n_{k}}+\frac{1}{k}t_{n_{k}}\|h_{n_{k}}\|$

thanks to $\varphi(x)=0$. This implies that

$\displaystyle\mathbf{d}(h_{n_{k}},\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))\leq\tau\big{(}1+\frac{1}{n_{k}}\big{)}+\frac{1}{k}\|h_{n_{k}}\|$

and consequently (4.5) holds by letting $k\rightarrow\infty$.

(ii) Let $\varepsilon>0$ such that $(1+\tau)\varepsilon<1$. Since $\varphi$
has the Shapiro first order contact property at $\bar{x}$, then there exists
$\delta_{1}\in(0,r)$ such that

$\mathbf{d}((x-u,\alpha-\beta),\mathbf{T}^{\mathbf{B}}({\rm
epi}(\varphi),(u,\beta)))<\varepsilon(\|x-u\|+\|\alpha-\beta\|)$ (4.8)

holds for all $(x,\alpha),(u,\beta)\in{\rm
epi}(\varphi)\cap(\mathbf{B}(\bar{x},\delta_{1})\times(\varphi(\bar{x})-\delta_{1},\varphi(\bar{x})+\delta_{1}))$.

Take $\delta\in(0,\frac{\delta_{1}}{2})$ and let
$x\in\mathbf{B}(\bar{x},\delta)\backslash\mathbf{S}_{\varphi}$. Then
$\mathbf{d}(x,\mathbf{S}_{\varphi})\leq\|x-\bar{x}\|<\delta$. Choose any
$\gamma\in(0,1)$ such that

$\gamma>\max\Big{\\{}\frac{\mathbf{d}(x,\mathbf{S}_{\varphi})}{\delta},(1+\tau)\varepsilon\Big{\\}}.$
(4.9)

By using Lemma 1, there exists $a\in{\rm bd}(\mathbf{S}_{\varphi})$ such that

$\gamma\|x-a\|\leq\min\\{\mathbf{d}(x-a,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},a)),\mathbf{d}(x,\mathbf{S}_{\varphi})\\}.$
(4.10)

Then

$\|\bar{x}-a\|\leq\|\bar{x}-x\|+\|x-a\|<\delta+\frac{\mathbf{d}(x,\mathbf{S}_{\varphi})}{\gamma}<2\delta<\delta_{1}$

and it follows from (4.8) that

$\mathbf{d}((x-a,\varphi(x)-\varphi(a)),\mathbf{T}^{\mathbf{B}}({\rm
epi}(\varphi),(a,\varphi(a))))<\varepsilon(\|x-a\|+|\varphi(x)-\varphi(a)|).$

Thus, there is $(h,\alpha)\in\mathbf{T}^{\mathbf{B}}({\rm
epi}(\varphi),(a,\varphi(a)))={\rm epi}(\varphi_{H}^{\prime}(a,\cdot))$ such
that

$\varphi_{H}^{\prime}(a,h)\leq\alpha\ \ {\rm and}\ \
\|x-a-h\|+|\varphi(x)-\varphi(a)-\alpha|<\varepsilon(\|x-a\|+|\varphi(x)-\varphi(a)|).$
(4.11)

Case 1: $\alpha=0$. Then for any $\lambda>0$, one has
$\varphi_{H}^{\prime}(a,\lambda h)\leq 0$ and then (4.4) implies that

$\mathbf{d}(\lambda h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},a))\leq 1.$

This means $h\in\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},a)$ by letting
$\lambda\rightarrow+\infty$. By virtue of (4.10) and (4.11), one has

$\displaystyle\gamma\|x-a\|\leq\mathbf{d}(x-a,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},a))\leq\|x-a-h\|$
$\displaystyle<$ $\displaystyle\varepsilon(\|x-a\|+|\varphi(x)-\varphi(a)|)$
$\displaystyle=$ $\displaystyle\varepsilon\|x-a\|+\varepsilon\varphi(x)$

the equality follows by $\varphi(x)>0$ and $\varphi(a)=0$. This implies that

$(\gamma-\varepsilon)\mathbf{d}(x,\mathbf{S}_{\varphi})\leq(\gamma-\varepsilon)\|x-a\|<\varepsilon\varphi(x).$

By letting $\gamma\rightarrow 1^{-}$, one has

$\mathbf{d}(x,\mathbf{S}_{\varphi})\leq\frac{\varepsilon}{1-\varepsilon}\varphi(x).$
(4.12)

Case 2: $\alpha\not=0$. Note that $\varphi_{H}^{\prime}(a,\cdot)$ is
positively homogeneous and then by (4.11), one has

$\varphi_{H}^{\prime}\big{(}a,\frac{h}{|\alpha|}\big{)}\leq 1.$

This and (4.4) imply that

$d\big{(}\frac{h}{|\alpha|},\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},a)\big{)}\leq\tau$

and consequently

$\mathbf{d}(h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},a))\leq\tau|\alpha|.$

By virtue of (4.10) and (4.11), one has

$\displaystyle\gamma\|x-a\|$ $\displaystyle\leq$
$\displaystyle\mathbf{d}(x-a,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},a))$
$\displaystyle\leq$
$\displaystyle\|x-a-h\|+\mathbf{d}(h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},a))$
$\displaystyle\leq$
$\displaystyle\|x-a-h\|+\tau(|\varphi(x)-\varphi(a)-\alpha|+|\varphi(x)-\varphi(a)|)$
$\displaystyle<$
$\displaystyle\varepsilon(\|x-a\|+|\varphi(x)-\varphi(a)|)+\tau(\varepsilon\|x-a\|+\varepsilon|\varphi(x)-\varphi(a)|+|\varphi(x)-\varphi(a)|)$

and thus

$\displaystyle(\gamma-\varepsilon-\tau\varepsilon)\mathbf{d}(x,\mathbf{S}_{\varphi})\leq(\gamma-\varepsilon-\tau\varepsilon)\|x-a\|$
$\displaystyle<$
$\displaystyle(\tau+\varepsilon+\tau\varepsilon)|\varphi(x)-\varphi(a)|$
$\displaystyle=$ $\displaystyle(\tau+\varepsilon+\tau\varepsilon)\varphi(x)$

the equality holds by $\varphi(x)>0$ and $\varphi(a)=0$. By letting
$\gamma\rightarrow 1^{-}$, one has

$\displaystyle(1-\varepsilon-\tau\varepsilon)\mathbf{d}(x,\mathbf{S}_{\varphi})<(\tau+\varepsilon+\tau\varepsilon)\varphi(x)$

and consequently

$\displaystyle\mathbf{d}(x,\mathbf{S}_{\varphi})\leq\frac{\tau+\varepsilon+\tau\varepsilon}{1-\varepsilon-\tau\varepsilon}\varphi(x).$

This and (4.12) imply that inequality (4.1) has the local error bound at
$\bar{x}$ (with the constant
$\max\\{\frac{\varepsilon}{1-\varepsilon},\frac{\tau+\varepsilon+\tau\varepsilon}{1-\varepsilon-\tau\varepsilon}\\}$).
The proof is complete. $\Box$

The following theorem, immediate from (i) and (ii) in Theorem 4.1 provides an
equivalent primal condition for the local error bound under the Shapiro first
order contact property.

###### Theorem 4.2

Let $\bar{x}\in\mathbf{S}_{\varphi}$. Suppose that ${\rm
bd}(\mathbf{S}_{\varphi})\subseteq\varphi^{-1}(0)$, $\varphi$ has the
epigraphical Shapiro first order contact property at $\bar{x}$ and that
$\mathbf{S}_{\varphi}$ has the Shapiro first order contact property around
$\bar{x}$. Then inequality (4.1) has the local error bound at $\bar{x}$ if and
only if

$\limsup_{x\stackrel{{\scriptstyle{\rm
bd}(\mathbf{S}_{\varphi})}}{{\longrightarrow}}\bar{x}}\mathbf{e}\big{(}\varphi^{\prime}_{\mathbf{H}}(x,\cdot)^{-1}(-\infty,1],\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)\big{)}<+\infty.$

Further, one has the following accurate estimate for the local error bound
modulus:

$\tau(\mathbf{S}_{\varphi},\bar{x})=\limsup_{x\stackrel{{\scriptstyle{\rm
bd}(\mathbf{S}_{\varphi})}}{{\longrightarrow}}\bar{x}}\mathbf{e}\big{(}\varphi^{\prime}_{\mathbf{H}}(x,\cdot)^{-1}(-\infty,1],\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)\big{)}.$
(4.13)

Proof. Thanks to (i) and (ii) in Theorem 4.1 ,it suffices to prove (4.13). We
denote

$\alpha:=\limsup_{x\stackrel{{\scriptstyle{\rm
bd}(\mathbf{S}_{\varphi})}}{{\longrightarrow}}\bar{x}}\mathbf{e}\big{(}\varphi^{\prime}_{\mathbf{H}}(x,\cdot)^{-1}(-\infty,1],\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)\big{)}.$

We first consider the case $\tau(\mathbf{S}_{\varphi},\bar{x})<+\infty$. Let
$\tau\in(0,\tau(\mathbf{S}_{\varphi},\bar{x}))$. Then by using the proof of
(i) in Theorem 4.1, there exists $\delta>0$ such that

$\mathbf{e}\big{(}\varphi^{\prime}_{\mathbf{H}}(x,\cdot)^{-1}(-\infty,1],\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)\big{)}\leq\tau\
\ \forall x\in\mathbf{B}(\bar{x},\delta)\cap{\rm bd}(\mathbf{S}_{\varphi}).$

This means that $\alpha\leq\tau$ and thus
$\alpha\leq\tau(\mathbf{S}_{\varphi},\bar{x})$ by letting
$\tau\downarrow\tau(\mathbf{S}_{\varphi},\bar{x})$.

Note that $\alpha<+\infty$ and for any $\nu>0$ there exists $\delta>0$ such
that

$\mathbf{e}\big{(}\varphi^{\prime}_{\mathbf{H}}(x,\cdot)^{-1}(-\infty,1],\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)\big{)}\leq\alpha+\nu\
\ \forall x\in\mathbf{B}(\bar{x},\delta)\cap{\rm bd}(\mathbf{S}_{\varphi}).$

Applying the proof of (ii) in Theorem 4.1, for any $\varepsilon>0$
sufficiently small, one has

$\tau(\mathbf{S}_{\varphi},\bar{x})\leq\frac{(\alpha+\nu)+\varepsilon+(\alpha+\nu)\varepsilon}{1-\varepsilon-(\alpha+\nu)\varepsilon}.$

By letting $\varepsilon\downarrow 0$, one has

$\tau(\mathbf{S}_{\varphi},\bar{x})\leq\alpha+\nu\ \ \forall\nu>0,$

and consequently $\tau(\mathbf{S}_{\varphi},\bar{x})\leq\alpha$ by letting
$\nu\downarrow 0$. Hence (4.13) holds.

We next consider the case $\tau(\mathbf{S}_{\varphi},\bar{x})=+\infty$. We
claim that $\alpha=+\infty$ (otherwise, $\alpha<+\infty$ and by using the
proof of (ii) in Theorem 4.1 again, one can obtain that
$\tau(\mathbf{S}_{\varphi},\bar{x})<+\infty$, a contradiction). The proof is
complete. $\Box$

Remark 4.1. For the case that $\varphi$ is convex, the Shapiro first order
contact property holds automatically and thus Theorem 4.2 reduces to (WZ2021,
, Theorem 5.3), which means that Theorem 4.2 is an extension of (WZ2021, ,
Theorem 5.3) from the convex case to the non-convex one.

It is noted that (4.4) is a key inequality to characterize the local error
bound of inequality (4.1). We are now in a position to give a characterization
of (4.4) via the following proposition.

###### Proposition 6

Let $\tau>0$ and $x\in\mathbf{S}_{\varphi}$. Then (4.4) holds if and only if

$\mathbf{d}(h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))\leq\tau\max\\{\varphi_{H}^{\prime}(x,h),0\\}\
\ \forall h\in X.$ (4.14)

Proof. The necessity part. Suppose that (4.4) holds. Let $h\in X$ be such that
$\varphi_{H}^{\prime}(x,h)\leq 0$. Then for any $\lambda>0$, one has
$\varphi_{H}^{\prime}(x,\lambda h)\leq 0$ and it follows from (4.4) that

$\mathbf{d}(\lambda
h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))\leq\tau$

This implies that

$\mathbf{d}(h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))\leq\frac{\tau}{\lambda}\
\ {\rm for\ any}\ \lambda>0.$

By letting $\lambda\rightarrow+\infty$, one has
$h\in\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)$ as
$\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)$ is closed, which implies
that (4.14) holds.

Let $h\in X$ be such that $\varphi_{H}^{\prime}(x,h)>0$. Note that
$\varphi_{H}^{\prime}(a,\cdot)$ is positively homogeneous and then

$\varphi_{H}^{\prime}\left(\mathbf{T}^{\mathbf{B}}(x,\frac{h}{\varphi_{H}^{\prime}(x,h)}\right)=1.$

By virtue of (4.4), one has

$d\left(\mathbf{T}^{\mathbf{B}}(\frac{h}{\varphi_{H}^{\prime}(x,h)}h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)\right)\leq\tau$

and consequently

$\mathbf{d}(h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))\leq\tau\varphi_{H}^{\prime}(x,h),$

which implies that (4.14) holds.

The sufficiency part. Suppose that (4.14) holds. Let $h\in X$ be such that
$\varphi_{H}^{\prime}(x,h)\leq 1$. Then (4.14) implies that

$\mathbf{d}(h,\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x))\leq\tau\max\\{\varphi_{H}^{\prime}(x,h),0\\}\leq\tau.$

Hence (4.4) holds. The proof is complete. $\Box$

The following theorem follows immediately from Theorem 4.1 and Proposition 6.

###### Theorem 4.3

Let $\bar{x}\in\mathbf{S}_{\varphi}$ and $\tau>0$. Suppose that ${\rm
bd}(\mathbf{S}_{\varphi})\subseteq\varphi^{-1}(0)$, $\varphi$ has the
epigraphical Shapiro first order contact property at $\bar{x}$ and that
$\mathbf{S}_{\varphi}$ has the Shapiro first order contact property around
$\bar{x}$. Then the following statements are equivalent:

  * (i)

Inequality (4.1) has the local error bound at $\bar{x}$ with constant
$\tau>0$;

  * (ii)

There exists $\delta>0$ such that (4.4) holds for all $x\in{\rm
bd}(\mathbf{S}_{\varphi})\cap\mathbf{B}(\bar{x},\delta)$;

  * (ii)

There exists $\delta>0$ such that (4.14) holds for all $x\in{\rm
bd}(\mathbf{S}_{\varphi})\cap\mathbf{B}(\bar{x},\delta)$.

Given $x\in X$, we can consider the inequality
$\varphi_{H}^{\prime}(x,\cdot)\leq 0$ defined by the lower Hadamard
directional derivative $\varphi_{H}^{\prime}(x,\cdot)$ and study global error
bounds of the inequality $\varphi_{H}^{\prime}(x,\cdot)\leq 0$; that is, there
exists $\tau>0$ such that

$\mathbf{d}(h,S_{\varphi_{H}^{\prime}(x,\cdot)})\leq\tau\max\\{\varphi_{H}^{\prime}(x,h),0\\}\
\ {\rm for\ all}\ h\in X,$ (4.15)

where $S_{\varphi_{H}^{\prime}(x,\cdot)}:=\\{u\in
X:\varphi_{H}^{\prime}(x,u)\leq 0\\}$.

The following corollary, immediate from Theorem 4.1, shows that the local
error bound of $\mathbf{S}_{\varphi}$ at $\bar{x}$ is, to some degree,
equivalent to the global error bound of the inequality
$\varphi_{H}^{\prime}(x,\cdot)\leq 0$ for all $x$ close to $\bar{x}$ with the
same constant. This corollary is inspired from (WZ2018, , Theorem 4.4) which
studies the convex inequality defined by the Clarke directional derivative of
a locally Lipschitz function and its global error bound.

###### Corollary 1

Let $\bar{x}\in\mathbf{S}_{\varphi}$ and $\tau>0$. Assume that ${\rm
bd}(\mathbf{S}_{\varphi})\subseteq\varphi^{-1}(0)$, $\varphi$ has the
epigraphical Shapiro first order contact property at $\bar{x}$ and that
$\mathbf{S}_{\varphi}$ has the Shapiro first order contact property around
$\bar{x}$.

  * (i)

Suppose that inequality (4.1) has the local error bound at $\bar{x}$. Then
there exists $\delta>0$ such that for any $x\in{\rm
bd}(\mathbf{S}_{\varphi})\cap\mathbf{B}(\bar{x},\delta)$, the inequality
$\varphi_{H}^{\prime}(x,\cdot)\leq 0$ has the global error bound with the same
constant.

  * (ii)

Suppose that there exists a neighborhood $U$ of $\bar{x}$ such that
$\ker\varphi_{H}^{\prime}(x,\cdot)\subseteq\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)$
for all $x\in U\cap{\rm bd}(\mathbf{S}_{\varphi})$. Then inequality (4.1) has
the local error bound at $\bar{x}$ if and only if there exists $\delta>0$ such
that for any $x\in{\rm
bd}(\mathbf{S}_{\varphi})\cap\mathbf{B}(\bar{x},\delta)$, the inequality
$\varphi_{H}^{\prime}(x,\cdot)\leq 0$ has the global error bound with the same
constant.

Proof. By the definition of the lower Hadamard directional derivative, for any
$x\in{\rm bd}(\mathbf{S}_{\varphi})$, one can verify that

$\\{h\in
X:\varphi_{H}^{\prime}(x,h)<0\\}\subseteq\mathbf{T}^{\mathbf{B}}(\mathbf{S}_{\varphi},x)\subseteq\\{h\in
X:\varphi_{H}^{\prime}(x,h)\leq 0\\}$

and thus the conclusions follow from Theorem 4.3. The proof is complete.
$\Box$

Now, we are in a position to study error bounds of a composite-convex
function. We apply main results obtained above to establish primal
characterizations of error bounds and give an accurate estimate of the error
bound modulus in terms of Bouligand tangent cones, directional derivatives of
convex functions and the Hausdorff-Pompeiu excess.

Throughout the rest of this section, we always assume that that $\mathbb{Y}$
is a Banach space, $g:\mathbb{X}\rightarrow\mathbb{Y}$ is a continuously
differentiable mapping and that
$f:\mathbb{Y}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is a proper lower
semicontinuous and convex function.

We consider the following composite-convex inequality:

$(f\circ g)(x)\leq 0.$ (4.16)

We denote by $\mathcal{S}:=\\{x\in X:(f\circ g)(x)\leq 0\\}$ the solution set
of (4.16). For any given $\bar{x}\in\mathcal{S}$, we denote by
$\tau(\mathcal{S},\bar{x})$, defined as said in (4.3), the local error bound
modulus of $\mathcal{S}$ at $\bar{x}$.

###### Theorem 4.4

Denote $S_{f}:=\\{y\in Y:f(y)\leq 0\\}$ and assume that ${\rm
bd}(S_{f})\subseteq f^{-1}(0)$. Let $\bar{x}\in\mathcal{S}$ be such that
$g(\bar{x})\in{\rm int}({\rm dom}f)$ and $g$ is metrically regular around
$\bar{x}$. Then composite-convex inequality (4.16) has a local error bound at
$\bar{x}$ if and only if

$\limsup_{x\stackrel{{\scriptstyle{\rm
bd}(\mathcal{S})}}{{\longrightarrow}}\bar{x}}\;\mathbf{e}\big{(}\\{h\in
X:\mathbf{d}^{+}f(g(x),\triangledown g(x)(h))\leq 1\\},\triangledown
g(x)^{-1}(\mathbf{T}^{\mathbf{B}}(S_{f},g(x))\big{)}<+\infty.$

Further, one has the following accurate estimate for the local error bound
modulus:

$\tau(\mathcal{S},\bar{x})=\limsup_{x\stackrel{{\scriptstyle{\rm
bd}(\mathcal{S})}}{{\longrightarrow}}\bar{x}}\mathbf{e}\big{(}\\{h\in
X:\mathbf{d}^{+}f(g(x),\triangledown g(x)(h))\leq 1\\},\triangledown
g(x)^{-1}(\mathbf{T}^{\mathbf{B}}(S_{f},g(x))\big{)}.$

To prove Theorem 4.4, we need the following lemma which is of some independent
interest.

###### Lemma 4

Let $\varphi:=f\circ g$ and $\bar{x}\in X$ be such that $g(\bar{x})\in{\rm
int}({\rm dom}f)$. Then there exists $\delta>0$ such that

$\varphi_{H}^{\prime}(x,h)=\mathbf{d}^{+}f(g(x),\triangledown g(x)(h))\ \
\forall h\in X$ (4.17)

holds for all $x\in\mathbf{B}(\bar{x},\delta)$.

Proof. Note that $g(\bar{x})\in{\rm int}({\rm dom}f)$ and then (Ph, ,
Proposition 1.6) implies that $f$ is locally Lipschtizian around $g(\bar{x})$;
that is, there exist $L,r>0$ such that

$|f(y_{1})-f(y_{2})|\leq L\|y_{1}-y_{2}\|\ \ \forall
y_{1},y_{2}\in\mathbf{B}(g(\bar{x}),r).$ (4.18)

Using the continuity of $g$, there is $\delta>0$ such that

$g(\mathbf{B}(\bar{x},\delta))\subseteq\mathbf{B}(g(\bar{x}),r).$ (4.19)

Let $x\in\mathbf{B}(\bar{x},\delta)$ and take any $h\in X$. Then for any $t>0$
sufficiently small and $h^{\prime}$ close to $h$, one has

$\displaystyle\frac{1}{t}[\varphi(x,th^{\prime})-\varphi(x)]=\frac{1}{t}[f(g(x,th^{\prime}))-f(g(x))]$
$\displaystyle=$ $\displaystyle\frac{1}{t}[f(g(x)+\triangledown
g(x)(th^{\prime})+o(t))-f(g(x))]$ $\displaystyle=$
$\displaystyle\frac{1}{t}[f(g(x)+\triangledown
g(x)(th^{\prime})+o(t))-f(g(x)+\triangledown g(x)(th))]$
$\displaystyle+\frac{1}{t}[f(g(x)+\triangledown g(x)(th))-f(g(x))].$

Using (4.18) and (4.19), for any $t>0$ sufficiently small and $h^{\prime}$
close to $h$, one has

$\displaystyle\left\|\frac{1}{t}[f(g(x)+\triangledown
g(x)(th^{\prime})+o(t))-f(g(x)+\triangledown g(x)(th))]\right\|$
$\displaystyle\leq$ $\displaystyle\frac{L}{t}\|t\triangledown
g(x)(h^{\prime}-h)+o(t)\|\rightarrow 0\ (t\rightarrow
0^{+},h^{\prime}\rightarrow h).$

Then

$\displaystyle\liminf_{t\rightarrow 0^{+},h^{\prime}\rightarrow
h}\frac{1}{t}[\varphi(x,th^{\prime})-\varphi(x)]$ $\displaystyle=$
$\displaystyle\liminf_{t\rightarrow 0^{+}}\frac{1}{t}[f(g(x)+t\triangledown
g(x)(h))-f(g(x))]$ $\displaystyle=$
$\displaystyle\mathbf{d}^{+}f(g(x),\triangledown g(x)(h)).$

This means that (4.17) holds. The proof is complete. $\Box$

Proof of Theorem 4.4. Let $\varphi:=f\circ g$. By the metric regularity of $g$
around $\bar{x}$, there exist $\kappa,\delta_{0}>0$ such that

$\mathbf{d}(x,g^{-1}(y))\leq\kappa\|g(x)-y\|\ \
\forall(x,y)\in\mathbf{B}(\bar{x},\delta_{0})\times\mathbf{B}(g(\bar{x}),\delta_{0}).$
(4.20)

Note that ${\rm bd}(S_{f})\subseteq f^{-1}(0)$ and then

${\rm bd}(\mathcal{S})\subseteq g^{-1}({\rm bd}(S_{f}))\subseteq
g^{-1}(f^{-1}(0))=\varphi^{-1}(0).$

By virtue of Lemma 3 and Proposition 5, there is $\delta\in(0,\delta_{0})$
sufficiently small such that $g^{-1}(S_{f})$ has the Shapiro first order
contact property on $\mathbf{B}(\bar{x},\delta)$ and

$\mathbf{T}^{\mathbf{B}}(g^{-1}(S_{f}),x)=\triangledown
g(x)^{-1}(\mathbf{T}^{\mathbf{B}}(S_{f}),g(x))\ \ \forall
x\in\mathbf{B}(\bar{x},\delta)\cap g^{-1}(S_{f}).$ (4.21)

This means that $\mathcal{S}$ has the Shapiro first order contact property
around $\bar{x}$ as $\mathcal{S}=g^{-1}(S_{f})$.

Define $\varPsi:X\times\mathbb{R}\rightarrow Y\times\mathbb{R}$ as

$\varPsi(x,r):=(g(x),r)\ \ \forall(x,r)\in X\times\mathbb{R}.$

Then one can verify that

${\rm epi}(\varphi)=\varPsi^{-1}({\rm epi}(f)).$

For any
$(x,y)\in\mathbf{B}(\bar{x},\delta)\times\mathbf{B}(g(\bar{x}),\delta)$ and
any $r,s\in(\varphi(\bar{x})-\delta,\varphi(\bar{x})+\delta)$, one has

$\varPsi^{-1}(y,s)=g^{-1}(y)\times\\{s\\}$

and it follows from (4.20) that

$\displaystyle\mathbf{d}((x,r),\varPsi^{-1}(y,s))$ $\displaystyle=$
$\displaystyle\mathbf{d}((x,r),g^{-1}(y)\times\\{s\\})$ $\displaystyle=$
$\displaystyle\mathbf{d}(x,g^{-1}(y))+|r-s|$ $\displaystyle\leq$
$\displaystyle\kappa\|g(x)-y\|+|r-s|$ $\displaystyle\leq$
$\displaystyle(\kappa+1)\|\varPsi(x,r)-(y,s)\|,$

which implies that $\varPsi$ is metrically regular around
$(\bar{x},\varphi(\bar{x}))$. Thus, Proposition 5 gives that
$\varPsi^{-1}({\rm epi}(f))$ has the Shapiro first order contact property at
$(\bar{x},\varphi(x))$ and so $\varphi$ has the epigraphical Shapiro first
order contact property at $\bar{x}$ (thanks to ${\rm
epi}(\varphi)=\varPsi^{-1}({\rm epi}(f))$).

Finally, by applying Theorem 4.2 and Lemma 4, one can obtain the proof of
Theorem 4.4. The proof is complete. $\Box$

The following theorem follows immediately from Theorem 4.4.

###### Theorem 4.5

Suppose that $f$ is continuous and $\bar{x}\in\mathcal{S}$ is such that $g$ is
metrically regular around $\bar{x}$. Then composite-convex inequality (4.16)
has a local error bound at $\bar{x}$ if and only if

$\limsup_{x\stackrel{{\scriptstyle{\rm
bd}(\mathcal{S})}}{{\longrightarrow}}\bar{x}}\mathbf{e}\big{(}\\{h\in
X:\mathbf{d}^{+}f(g(x),\triangledown g(x)(h))\leq 1\\},\triangledown
g(x)^{-1}(\mathbf{T}^{\mathbf{B}}(S_{f},g(x))\big{)}<+\infty.$

Further, one has the following accurate estimate for the local error bound
modulus:

$\tau(\mathcal{S},\bar{x})=\limsup_{x\stackrel{{\scriptstyle{\rm
bd}(\mathcal{S})}}{{\longrightarrow}}\bar{x}}\mathbf{e}\big{(}\\{h\in
X:\mathbf{d}^{+}f(g(x),\triangledown g(x)(h))\leq 1\\},\triangledown
g(x)^{-1}(\mathbf{T}^{\mathbf{B}}(S_{f},g(x))\big{)}.$

The following example is to show the application of Theorem 4.5 when one
verifies the error bound modulus of the composite-convex inequality.

Example 4.1. Let $f(y):=y-1$ for all $y\in\mathbb{R}$, $g(x):=x^{3}$ for all
$x\in\mathbb{R}$ and $\bar{x}:=1$. We consider the composite-convex
inequality:

$(f\circ g)(x)\leq 0.$

Then one can verify that $\mathcal{S}=S_{f}=(-\infty,1]$,
$\mathbf{T}^{\mathbf{B}}(S_{f},g(\bar{x}))=(-\infty,0]$ and

$\\{h:\mathbf{d}^{+}f(g(\bar{x}),\triangledown g(\bar{x})(h))\leq
1\\}=(-\infty,\frac{1}{3}].$

This implies that

$\limsup_{x\stackrel{{\scriptstyle{\rm
bd}(\mathcal{S})}}{{\longrightarrow}}\bar{x}}\mathbf{e}\big{(}\\{h\in
X:\mathbf{d}^{+}f(g(x),\triangledown g(x)(h))\leq 1\\},\triangledown
g(x)^{-1}(\mathbf{T}^{\mathbf{B}}(S_{f},g(x))\big{)}=\frac{1}{3}.$

Hence Theorem 4.5 implies that $\mathcal{S}$ has a local error bound at
$\bar{x}$ and moreover $\tau(\mathcal{S},\bar{x})=\frac{1}{3}$. $\Box$

Finally, we give an example to show the local error bound may not be satisfied
if the metric regularity assumption is dropped in Theorem 4.5.

Example 4.2. Let $f(y):=y$ for all $y\in\mathbb{R}$, $g(x):=x^{3}$ for all
$x\in\mathbb{R}$ and $\bar{x}:=0$. We consider the composite-convex
inequality:

$(f\circ g)(x)\leq 0.$

Then $\mathcal{S}=S_{f}=(-\infty,0]$, and one can verify that $g$ is not
metrically regular at $\bar{x}$ since $\triangledown g(\bar{x})=0$ is not
surjective. However, for any $x>0$ sufficiently small, one has

$\frac{d(x,\mathcal{S})}{f(g(x))}=\frac{x}{x^{3}}\rightarrow+\infty,\ \ {\rm
as}\ x\rightarrow 0^{+}.$

This means that $\mathcal{S}$ has no local error bound at $\bar{x}$, and thus
the conclusions in Theorem 4.5. do not hold. $\Box$

Acknowledgements. The authors are very grateful to the anonymous reviewers for
their suggestions and comments that improved the presentation of this paper.

##  5 Conclusions

This paper is to establish primal characterizations of error bounds for a
composite-convex inequality (that is defined by a composition of a convex
function with a continuously differentiable mapping). To this aim, several
primal necessary and/or sufficient conditions for a general inequality, under
the assumption of Shapiro contact property, are given in terms of Bouligand
tangent cones, directional derivatives and the Hausdorff-Pompeiu excess. Then
it is proved that the composite-convex inequality satisfies the Shapiro
contact property at the given point where the continuously differentiable
mapping is metrically regular and thus primal characterizations of error
bounds can be obtained. Our works actually extend the existing primal results
on error bounds for the convex inequality to the non-convex case. The future
work would be to investigate a broader class of non-convex inequality for
which these primal results on error bounds are valid.

## References

  * (1) M. Abassi, M. Théra: About error bounds in metrizable topological vector spaces, in revision in Set-Valued and Variational Analysis, (2021).
  * (2) M. Abassi, M. Théra: Strongly regular points of mappings, Fixed Point Theory Algorithms Sci. Eng. p. Paper No. 14 (2021). DOI 10.1186/s13663-021-00699-z. URL https://doi.org/10.1186/ s13663- 021- 00699- z
  * (3) D. Aussel, A. Daniillids, L. Thibault: Subsmooth sets: Functional characterizations and related concepts, Trans. Amer. Math. Soc., 357 (2005) 1275-1301.
  * (4) A. Auslender, J. P. Crouzeix: Global regularity theorems, Math. Oper. Res. 13(2)(1988) 243-253.
  * (5) D. Azé, J. -N. Corvellec: Characterizations of error bounds for lower semicontinuous functions on metric spaces, ESAIM Control Optim. Calc. Var., 10 (2004) 409-425.
  * (6) H. H. Bauschke, J. M. Borwein: On projection algorithms for solving convex feasibility problems, SIAM Rev., 38(3) (1996) 367-426.
  * (7) A. Beck, M. Teboulle: Convergence rate analysis and error bounds for projection algorithms in convex feasibility problems, Optim. Methods Softw., 18/4 (2003) 377-394.
  * (8) E. M. Bednarczuk, A. Y. Kruger: Error bounds for vector-valued functions: necessary and sufficient conditions, Nonlinear Anal., 75(3) (2012) 1124-1140.
  * (9) J. V. Burke, S. Deng: Weak sharp minima revisited. I. Basic theory, Control Cybernet., 31/3 (2002) 439-469. Well-Posedness in Optimization and Related Topics (Warsaw, 2001).
  * (10) J. V. Burke, S. Deng: Weak sharp minima revisited. II. Application to linear regularity and error bounds, Math. Program., 104/2-3 (2005) 235-261.
  * (11) N. D. Cuong, A. Y. Kruger: Error bounds revisited, arXiv e-prints, arXiv 2012.03941 (2020)
  * (12) P. L. Combettes: Hilbertian convex feasibility problem: convergence of projection methods, Appl. Math. Optim., 35/3 (1997) 311-330.
  * (13) J.-N. Corvellec, V. V. Motreanu: Nonlinear error bounds for lower semicontinuous functions on metric spaces, Math. Program., 114/2 (2008) 291-319.
  * (14) A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solutions Mapping, Springer, Berlin (2009).
  * (15) A. L. Dontchev: Lectures on variational analysis, Applied Mathematical Sciences, 205, Springer, Cham (2021).
  * (16) M. J. Fabian, R. Henrion, A. Y. Kruger, J. V. Outrata: Error bounds: necessary and sufficient conditions, Set-Valued Var. Anal., 18/2 (2010) 121-149.
  * (17) H. Gfrerer: First order and second order characterizations of metric subregularity and calmness of constrant set mapping, SIAM J. Optim. 21/4 (2011) 1439-1474.
  * (18) O. Güler, Augmented Lagrangian algorithms for linear programming, J. Optim. Theory Appl., 75/3 (1992) 445-478.
  * (19) R. Hesse, D. R. Luke: Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems, SIAM J. Optim. 23/4 (2013) 2397-2419.
  * (20) A.J. Hoffman: On approximate solutions of systems of linear inequalities, J. Res. Nat. Bur. Stand., 49 (1952) 263-265.
  * (21) A. D. Ioffe: Regular points of Lipschitz functions, Trans. Amer. Math. Soc., 251 (1979) 61-69.
  * (22) A. D. Ioffe: Metric regularity-a survey. Part I. Theory, J. Aust. Math. Soc., 101/2(2016) 188-243.
  * (23) A. D. Ioffe: Metric regularity-a survey. Part II. Applications, J. Aust. Math. Soc., 101/3 (2016) 376-417.
  * (24) A. D. Ioffe: Variational analysis of regular mappings, Theory and applications, Springer Monographs in Mathematics, Springer, Cham (2017).
  * (25) A. Jourani: Hoffman’s error bound, local controllability, and sensitivity analysis, SIAM J. Control Optim., 38/3 (2000) 947-970.
  * (26) A. N. Iusem, A. R. De Pierro: On the convergence properties of Hildreth’s quadratic programming algorithm, Math. Program., 47 (1990) 37-51.
  * (27) G. Jameson: Ordered Linear Spaces, Springer-Verlag, Berlin (1970).
  * (28) D. Klatte, W. Li: Asymptotic constraint qualifications and global error bounds for convex inequalities, Math. Program., 84/1 (1999) 137-160.
  * (29) A. Y. Kruger, M. A. López, M. A. Théra: Perturbation of error bounds, Math. Program., 168 (2018) 533-554.
  * (30) A. Y. Kruger, H. V. Ngai, M. Théra: Stability of error bounds for convex constraint systems in Banach spaces, SIAM J. Optim., 20/6 (2010) 3280-3296.
  * (31) S. Łojasiewicz: Sur le problème de la division, Studia Math., 18 (1959) 87-136.
  * (32) A. S. Lewis, J. S. Pang: Error bounds for convex inequality systems, in Generalized Convexity, Generalized Monotonicity: Recent Results, Proceedings of the Fifth Symposium on Generalized Convexity, Luminy, 1996, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997, pp. 75-110.
  * (33) Z.-Q. Luo, P. Tseng: On a global error bound for a class of monotone affine variational inequality problems, Oper. Res. Lett., 11/3 (1992) 159-165.
  * (34) Z.-Q. Luo, P. Tseng: Error bounds and convergence analysis of feasible descent methods: a general approach, Annals of Operations Research, 46 (1993) 157-178.
  * (35) W. Li: Abadie’s constraint qualification, metric regularity, and error bounds for differentiable convex inequalities, SIAM J. Optim., 7 (1997) 966-978.
  * (36) B. S. Mordukhovich: Complete characterization of openness, metric regularity, and Lipschitzian properties of set-valued mappings, Trans. Amer. Math. Soc., 340(1993) 1-35.
  * (37) B. S. Mordukhovich: Variational Analysis and Generalized Differentiation I, Springer-verlag, Berlin Heidelberg (2006).
  * (38) K. F. Ng, X. Y. Zheng: Error bounds for lower semicontinuous functions in normed spaces, SIAM J. Optim., 12/1 (2001) 1-17.
  * (39) H. V. Ngai, A. Y. Kruger, M. Théra: Stability of error bounds for semi-infinite convex constraint systems, SIAM J. Optim, 20 (2010) 2080-2096.
  * (40) H. V. Ngai, M. Théra: Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal. 12/1-2 (2004) 195-223.
  * (41) H. V. Ngai, M. Théra: Error bounds in metric spaces and applications to the perturbation stability of metric regularity, SIAM J. Optim., 19/1 (2008) 1-20.
  * (42) H. V. Ngai, M. Théra: Error bounds for systems of lower semicontinuous functions in Asplund spaces, Math. Program., 116/1-2 (2009) 397-427.
  * (43) H. V. Ngai, N. H. Tron, N. V. Vu, M. Théra: Variational Analysis of Paraconvex Multifunctions, J. Optim. Theory Appl., (2021) https://doi.org/10.1007/s10957-022-02021-2.
  * (44) J. S. Pang: Error bounds in mathematical programming, Math. Program., 79/1-3 (1997) 299-332.
  * (45) J.-P. Penot: Error bounds, calmness and their applications in nonsmooth analysis, In: Nonlinear analysis and optimization II. Optimization, Contemporary Mathematics, vol. 514, 225-247. American Mathematical Society, Providence (2010).
  * (46) J.-P., Penot: Calculus without derivatives, Graduate Texts in Mathematics, vol. 266. Springer, New York (2013)
  * (47) R. R. Phelps: Convex functions, Monotone operators and Differentiability, Lecture Notes in Math, 1364, Springer, New York (1989)
  * (48) H. Rådström: An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc., 3 (1952) 165-169.
  * (49) S. M. Robinson: Bounds for error in the solution set of a perturbed linear program, Linear Algebra Appl., 6 (1973) 69-81.
  * (50) S. M. Robinson: A characterization of stability in linear programming, Oper. Res., 25 (1977) 435-447.
  * (51) A. Shapiro: Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim., 4 (1994) 130-141.
  * (52) A. Shapiro, F. Al-Khayyal: First order conditions for isolated locally optimal solutions, J. Optim. Theory Appl., 77 (1993) 189-196.
  * (53) Z. Shen, J.-C. Yao, X.Y. Zheng: Calmness and the Abadie CQ for multifunctions and linear regularity for a collection of closed sets, SIAM J. Optim. 29/3 (2019) 2291-2319
  * (54) L. Thibault: Unilateral variational analysis in Banach spaces, World Scientific (2022).
  * (55) P. Tseng, D. P. Bertsekas: On the convergence of the exponential multiplier method for convex programming, Math. Program., 60/1 (1993) 1-19.
  * (56) Z. Wei, C. Tammer, J.-C. Yao: Characterizations for Strong Abadie Constraint Qualification and Applications to Calmness, J. Optim. Theory Appl., 189 (2021) 1-18.
  * (57) Z. Wei, J.-C. Yao: On Constraint qualifications of a nonconvex inequality, Optim. Lett., 12 (2018) 1117-1139.
  * (58) Z. Wei, J.-C. Yao: On applications of the calmness moduli for multifunctions to error bounds, Optimization, https://doi.org/10.1080/02331934.2021.1906873.
  * (59) Z. Wei, J.-C. Yao, X.Y. Zheng: Strong Abadie CQ, ACQ, calmness and linear regularity, Math. Program., 145 (2014) 97-131.
  * (60) C. Zălinescu: Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces, Proc. 12th Baikal Internat. Conf. on Optimization Methods and Their Appl. Irkutsk, Russia, 272-284 (2001).
  * (61) X. Y. Zheng, K. F. Ng: Metric regularity and constraint qualifications for convex inequalities on Banach spaces, SIAM J. Optim., 14 (2004) 757-772.


# Orca: Ocean Significant Wave Height Estimation with Spatio-temporally Aware
Large Language Models

Zhe Li<EMAIL_ADDRESS>School of Data Science and Engineering, East
China Normal University ,  Ronghui Xu<EMAIL_ADDRESS>School of Data
Science and Engineering, East China Normal University ,  Jilin Hu
<EMAIL_ADDRESS>School of Data Science and Engineering, East China Normal
University ,  Zhong Peng<EMAIL_ADDRESS>The State Key Laboratory of
Estuarine and Coastal Research, ECNU ,  Xi Lu<EMAIL_ADDRESS>The State
Key Laboratory of Estuarine and Coastal Research, ECNU ,  Chenjuan Guo
<EMAIL_ADDRESS>School of Data Science and Engineering, East China
Normal University and  Bin Yang<EMAIL_ADDRESS>School of Data Science
and Engineering, East China Normal University

###### Abstract.

Significant wave height (SWH) is a vital metric in marine science, and
accurate SWH estimation is crucial for various applications, e.g., marine
energy development, fishery, early warning systems for potential risks, etc.
Traditional SWH estimation methods that are based on numerical models and
physical theories are hindered by computational inefficiencies. Recently,
machine learning has emerged as an appealing alternative to improve accuracy
and reduce computational time. However, due to limited observational
technology and high costs, the scarcity of real-world data restricts the
potential of machine learning models. To overcome these limitations, we
propose an ocean SWH estimation framework, namely Orca. Specifically, Orca
enhances the limited spatio-temporal reasoning abilities of classic LLMs with
a novel spatiotemporal aware encoding module. By segmenting the limited buoy
observational data temporally, encoding the buoys’ locations spatially, and
designing prompt templates, Orca capitalizes on the robust generalization
ability of LLMs to estimate significant wave height effectively with limited
data. Experimental results on the Gulf of Mexico demonstrate that Orca
achieves state-of-the-art performance in SWH estimation.

Significant Wave Height, Large Language Model, Prompt Fine-tuning

##  1\. Introduction

Significant wave height (SWH) is an essential metric in marine science,
reflecting the state of ocean activities. Since abnormal waves can cause
extensive disruptions, leading to production losses, human casualties, and
ecological damage, it is critical to accurately estimate the SWH and detect
anomalies to ensure safety in a variety of applications, such as maritime
navigation and marine energy development (Yang_Jin_Jia_Ye_2021, ;
Woo_Park_2021, ; DBLP:journals/pvldb/PedersenYJ20, ; Bu_Park_Yu_Camps_2022, ;
DBLP:conf/icde/KieuYGCZSJ22, ; DBLP:conf/icde/KieuYGJZHZ22, ; davidpvldb, ).

Figure 1. The buoy distribution in Gulf of Mexico.

Two categories of SWH estimation methods exist: traditional and machine
learning-based methods. Specifically, traditional methods involve numerical
models that simulate wave activities based on physical principles
(Pierson_Neumann_James, ; ris1995spectral, ; hwang2021simulation, ). Although
these models have a robust theoretical foundation, they require extensive
computational resources, which is too slow to estimate extremely high waves in
time. Moreover, they are difficult to adapt to additional impact factors,
which limits opportunities for performance improvement. Recently, machine
learning-based time series methods (DBLP:journals/pvldb/QiuHZWDZGZJSY24, ;
DBLP:journals/pacmmod/0002Z0KGJ23, ; DBLP:journals/pacmmod/Wu0ZG0J23, ;
kaivldb24, ; yunyaovldb24, ; haoicde24, ; MileTS, ) have emerged as a
promising alternative, offering improved accuracy and reduced computational
time in SWH estimation (yang2024swhformer, ; Qi_Majda_2020, ; AnandJS23, ;
wang2022significant, ). However, they face two primary challenges. Firstly,
the observed SWH data is too sparse. Due to the limitations in observational
technology (Qi_Majda_2020, ) and environmental factors, the deployed buoys in
the ocean are very sparse. Figure 1 shows the distribution of buoys in the
Gulf of Mexico, where only the red grids are with buoys. Such scarcity hinders
the development of machine learning-based methods. Secondly, since a wave is a
propagating dynamic disturbance of one or more quantities, the wave variations
have strong spatio-temporal correlations.  Thus, it is crucial to capture such
correlations in SWH estimation. However, current machine learning-based
methods fail to capture such intricate relationships and discern the wave
dynamic patterns implied in the data (Qi_Majda_2020, ; windspeed, ;
QureshiKZU17, ).

To address the abovementioned two challenges, we propose an ocean SWH
estimation framework, namely Orca. In recent years, Large Language Models
(LLMs) have shown remarkable performance in few-shot learning scenarios,
matching the performance of task-specific models, e.g., arithmetic reasoning
(Wei0SBIXCLZ22, ; KojimaGRMI22, ), human mobility prediction (PromptCast, ),
and time series forecasting (pengiclr24, ; DBLP:journals/corr/abs-2402-02713,
). To tackle the issue of data sparsity in SWH estimation, we propose to use
LLM as the backbone of estimation.  Specifically, we invent a specific prompt
templates and embedding module to leverage the pre-trained LLM for SWH
estimation. With these designs, we aim to achieve accurate SWH estimations
using limited observed data by leveraging the robust generalization
capabilities of LLM. To enhance spatio-temporal reasoning, we segment buoy-
based data into overlapping temporal patches, and propose a novel spatial
encoding module. In addition, we integrate additional information from the
traditional numerical models by formulating a regularization term, such that
we can anchor the model and establish scientific knowledge. Finally, our
proposed model can not only overcome the limitations of traditional methods,
but also improve computational efficiency and the precision of SWH predictions
significantly.

The study makes four main contributions. Firstly, we propose a framework for
ocean SWH estimation based on LLMs with spatio-temporal awareness, Orca.
Secondly, to tackle the data sparsity in SWH estimation, we invent a specific
prompt templates and prompt embedding module to leverage LLM for estimation.
Next, to enhance the spatio-temporal reasoning capabilities of LLMs, we
propose a novel spatio-temporal aware encoding module to enable the detection
of implicit wave dynamic patterns within the data. Finally, we conduct
extensive experiments to demonstrate the computational efficiency and the
accuracy of the proposed model.

##  2\. Preliminaries

Significant Wave Height. Given a particular region and wave train, significant
wave height (SWH) is defined as the average height of the top one-third (1/3)
waves.

Buoy-based Data. Buoy-based data $\mathbf{X}\in\mathbb{R}^{F\times M\times T}$
encompasses the $F$ features collected by $M$ stationary oceanic buoys over
$T$ continuous time intervals.

Grid-based Significant Wave Height. Grid-based significant wave height (GSWH)
$\mathbf{Y}\in\mathbb{R}^{K\times J\times T}$ records the average SWH within
each grid area over $\mathrm{T}$ continuous time intervals, where $K$ and $J$
indicate the number of rows and columns, respectively.

Problem Definition. Given the buoy-based data $\mathbf{X}$ over $\mathrm{T}$
continuous time intervals, our goal is to estimate the GSWH values
$\mathbf{Y}$ of $\mathrm{T}$ continuous time intervals.

Figure 2. Overall framework of Orca.

##  3\. Methodology

###  3.1. Prompt Designing and Encoding

To guide LLMs towards an accurate comprehension of the data, we segment the
prompt into five distinct components, as depicted in Figure 2. ACTOR specifies
the role of the LLMs; INFORMATION conveys the dimensions of the input data;
TARGET clearly states the specific goals of the task; FEATURES specifies the
semantics of the input data, such as wind direction (WDIR); and DATA declares
the type of the input data, ensuring the LLMs accurately interprets numerical
values rather than mistaking them for strings.

The word embedding layer of LLMs takes input as the structured prompt
$P=\\{p_{1},p_{2},\ldots,p_{E}\\}$, resulting in
$\mathbf{P}=\\{\mathbf{p}_{1},\mathbf{p}_{2},\ldots,\mathbf{p}_{E}\\}$, where
$E$ is the length of the prompt, $\mathbf{p}_{i}\in\mathbb{R}^{D}$ is the
embedded vector of the corresponding token $p_{i}$, where $D$ is the dimension
of the semantic space of LLMs.

Following P-Turning (liu2023gpt, ), we randomly generate a fixed number of
soft-prompt $Q=\\{q_{1},q_{2},\ldots,q_{R}\\}$, where $R$ is the length of
soft-prompt. Then, ${Q}$ is fed into a pretrained embedding, obtaining
$\mathbf{Q}=\\{\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{R}\\}$. We
utilize learnable lightweight networks to model the dependency relationships
between different embeddings:

(1)
$\mathbf{H}_{q}=\bm{\omega}_{1}(\mathrm{ReLU}(\bm{\omega}_{2}\mathrm{LSTM}(\mathbf{Q})+\bm{b}_{2}))+\bm{b}_{1},$

where $\bm{\omega}_{1}$, $\bm{b}_{1}$, $\bm{\omega}_{2}$, $\bm{b}_{2}$ are the
trainable parameters of the prompt encoding module.

Finally, we concatenate $\mathbf{H}_{q}$ and $\mathbf{P}$ to improve the
adaptability of the prompt to samples, obtaining the final prompt
representations
$\mathbf{H}_{prompt}=[\mathbf{H}_{q};\mathbf{P}]\in\mathbb{R}^{(R+E)\times
D}$.

###  3.2. Spatio-temporal Encoding

Spatial Encoding. As the phenomenon of wave variation at any specific maritime
coordinate is inherently influenced by the wave movements in its vicinity, it
is imperative to precise capture the spatial interconnections for both buoy
and grid-based data. However, LLMs has limited spatial reasoning capabilities.
To overcome this challenge and effectively capture the spatial correlations
essential, we introduce a spatial encoding module into our model.

Specifically, we allocate buoys to their corresponding grid locations, and the
locations of buoys are denoted as
$\mathbf{G}=\\{(\mathrm{u}_{i},\mathrm{v}_{i})\\}_{i=1}^{\mathrm{M}}$, where
$\mathrm{u}_{i}$ and $\mathrm{v}_{i}$ indicate the row and column grid indices
for the $i$-th buoy. To effectively represent the spatial relationships within
a multidimensional space, we employ the Z-order curve
(DBLP:conf/vldb/LeeZLL07, ) to map the buoy coordinates to a one-dimensional
binary embedding $\mathbf{Z}=\mathrm{Zorder}(\mathbf{G})$,
$\mathbf{Z}\in\mathbb{R}^{M\times A}$. $\mathrm{Zorder}(\cdot)$ is the mapping
function of the Z-order curve, and $A$ is defined by the number and size of
grids. Subsequently, $\mathbf{Z}$ is input into a fully connected layer to
generate the spatial embeddings
$\mathbf{H}_{loc}=\mathrm{ReLU}(\bm{\omega}_{3}\mathbf{Z}+\bm{b}_{3})$, where
$\bm{\omega}_{3}$, $\bm{b}_{3}$ are the trainable parameters of the spatial
encoding module, and $\mathbf{H}_{loc}\in\mathbb{R}^{\mathrm{M\times D}}$
represents the spatial embeddings. This module ensures that our model
accurately reflects the physical location of buoys in the estimations.

Temporal Encoding. To align the input buoy-based data with semantic space of
LLMs and consider temporal information simultaneously, we redesign the input
encoding layer, following a patch approach, which is used in time series
analytics (nie2022time, ).

Specifically, the observed data $\mathbf{X}$ of buoys is segmented into
overlapping patches $\mathbf{C}=\\{\mathbf{X}_{(i)}\\}_{i=1}^{S}$, where
$\mathbf{X}_{(i)}\in\mathbb{R}^{F\times M\times L}$,
${S}=\lfloor\frac{T-L}{W}\rfloor+2$ is the number of patches, and $W$, $L$ are
the stride and the patch length, respectively. We utilize a fully connected
layer to align $\mathbf{C}$ with the semantic space of LLMs, formulated as:

(2) $\mathbf{H}_{temp}=\mathrm{ReLU}(\bm{\omega}_{4}\mathbf{C}+\bm{b}_{4}),$

where $\mathbf{H}_{temp}\in\mathbb{R}^{S\times F\times M\times D}$ denotes the
representation of the buoys’ observed data, and $\bm{\omega}_{4}$,
$\bm{b}_{4}$ are the trainable parameters.

We concatenate the representations aforementioned to derive the final input
representation
$\mathbf{H}_{input}=[\mathbf{H}_{prompt};\mathbf{H}_{loc};\mathbf{H}_{temp}]\in\mathbb{R}^{\mathrm{I\times
F\times M\times D}}$, and $I=R+E+1+S$.

###  3.3. LLMs Fine-tuning

To leverage the knowledge learned by LLMs during pre-training, we only fine-
tune the positional encodings of the LLMs and freeze other parameters. We feed
$\mathbf{H}_{input}$ into LLMs and acquire the output of the last latent
layer:

(3) $\mathbf{H}_{LLM}=\mathrm{LLMs}(\mathbf{H}_{input}).$

Here, $\mathbf{H}_{LLM}\in\mathbb{R}^{I\times F\times M\times D}$ is the
output of LLMs. We perform average pooling on $\mathbf{H}_{LLM}$ to aggregate
the feature dimension:

(4) $\mathbf{H}_{pool}=\mathrm{AvgPool}(\mathbf{H}_{LLM}).$

$\mathbf{H}_{pool}$ is then flattened to a ($I\times M\times D$)-dimensional
embedding. We utilize a fully connected layer and a reshape function to obtain
the estimated GSWH values:

(5)
$\mathbf{\widehat{Y}}=\mathrm{Reshape}(\bm{\omega}_{5}\mathbf{H}_{pool}+\bm{b}_{5}),$

where $\mathrm{Reshape}(\cdot)$ is a function that reshapes a ($K\times
J\times T$)-dimensional embedding to estimated GSWH values
$\mathbf{\widehat{Y}}\in\mathbb{R}^{K\times J\times T}$. $\bm{\omega}_{5}$,
$\bm{b}_{5}$ are the weight and the bias of the layer, respectively.

###  3.4. Optimizing with Physical Regularization

We minimize the difference between the observed SWH values from buoys and the
estimated GSWH values. The loss function is:

(6)
$\mathcal{L}_{1}=\frac{1}{M}\sum_{i=1}^{M}(\mathrm{Y}_{u_{i},v_{i}}-\hat{\mathrm{Y}}_{u_{i},v_{i}})^{2},$

where $M$ is the number of buoys, $(u_{i},v_{i})$ is are the $2D$ coordinates
of the $i$-th buoy, $\mathrm{Y}_{u_{i},v_{i}}$ denotes the observed SWH values
of the $i$-th buoy, and $\hat{\mathrm{Y}}_{u_{i},v_{i}}$ indicates the
estimated SWH values of our proposed model.

To guide the model to follow physcial principles, we utilize data from the
numerical model to formulate a regularization term:

(7) $\mathcal{L}_{2}=\frac{1}{K\times
J}\sum_{i=1}^{K}\sum_{j=1}^{J}(\tilde{\mathrm{Y}}_{i,j}-\hat{\mathrm{Y}}_{i,j})^{2},$

where $\tilde{\mathrm{Y}}_{i,j}$ represents the GSWH value, generated by the
numerical model, for the grid at the $i$-th row and $j$-th column,
$\hat{\mathrm{Y}}_{i,j}$ is the corresponding estimated value produced by our
model.

Overall, the training objective can be formalized as:

(8) $\mathcal{L}=\mathcal{L}_{1}+\alpha\mathcal{L}_{2},$

where hyperparameter $\alpha$ is the weight of $\mathcal{L}_{2}$.

Table 1. Accuracy (best values in bold).

Model | MAE | MSE | RMSE
---|---|---|---
GWD (center2014user, ) | 0.3949 | 0.2000 | 0.4472
PatchTST (nie2022time, ) | 0.6652 | 0.9375 | 0.9682
GPT-2 (radford2019language, ) | 0.3962 | 0.2063 | 0.4542
GPT4TS (zhou2023one, ) | 0.3692 | 0.1796 | 0.4238
Orca | 0.2372 | 0.0838 | 0.2895

Figure 3. Ablation Studies.

Figure 4. Estimated SWH at buoy 42040.

##  4\. Experiments

Datasets and Baselines. In this paper, we focus on a defined rectangular
region within the Gulf of Mexico, spanning from 32°N to 18°N latitude and 98°W
to 78°W longitude. This area is uniformly segmented into $29\times 41$ grids,
with each grid cell representing a 0.5° (approximately 50 km) square. Then,
the Buoy-based data is from the National Data Buoy Center dataset
(NDBC)111https://www.ndbc.noaa.gov/, which can provide key measurements such
as surface wind speed, sea surface temperature, and significant wave height.
The corresponding numerical model data is from the Global Wave Database (GWD),
provided by the Estuary and Coastal
Laboratory222http://www.sklec.ecnu.edu.cn/, collects wave simulation data
generated by version 4.18 of the WaveWatch III (center2014user, ).

We compare Orca with various competitive baseline models including a
traditional method using physical principles GWD (center2014user, ), a SOTA
time series forecasting method PatchTST (nie2022time, ), a large language
model without spatio-temporal aware encoding GPT-2 (radford2019language, ),
and a LLM based time series forecasting method GPT4TS (zhou2023one, ).

Experiment Settings. All the experiments are conducted on a high-performance
NVIDIA RTX3090 24GB GPU. We utilize GPT-2 (radford2019language, ) as the
backbone network for our model. The training process is executed over a
maximum of 50 epochs. The learning rate and $\alpha$ are set at 0.001 and 0.3,
respectively. We split NDBC dataset into non-overlapping train, validation,
and test sets with a ratio of 8:1:1. The GWD dataset is only used for
training. The time interval is set to 3-hour for both datasets. We conduct the
evaluation using buoy-based data from the test set as the ground truth, and
employ MSE, RMSE, and MAE as metrics.

Performance comparison. Table 4 shows the comparative results of various
models on SWH estimation. Time series models, specifically PatchTST, exhibit
limitations in capturing spatial relationships and dynamic wave patterns,
particularly in data-scarce environments, leading to poor performance.
Conversely, applying GPT-2 directly or fine-tuning it with GPT4TS approach
allows LLMs to achieve results comparable to the numerical model GWD. This
validates the efficacy of LLMs in few-shot learning scenarios. Notably, Orca
achieves the best performance, affirming the superior impact of spatio-
temporal encoding module and designed prompts in refining LLMs for SWH
estimation.

Ablation Studies. To verify the impact of Orca’s key components, we conduct
ablation studies, exploring the following variants: (1) w/ light prompt: Orca
utilizing a light prompt: “ACTOR: You are a marine scientist. TARGET: I will
provide you with several sets of data describing the marine environment.
Please utilize your own expertise and this data to predict the significant
wave heights.” (2) w/o fea+location: It excludes both the FEATURES from the
prompt and the spatial encoding module within Orca. (3) w/o fea: It removes
FEATURES in prompt. (4) w/o location: It removes the spatial encoding module.

Figure 4 shows that different variants yield distinct performance. The
performance of w/ light prompt, while worse than Ocra, outperforms the
numerical model GWD, indicating that prompt fine-tuning can significantly
enhance the generalization capabilities of LLMs for this task. Moreover, the
removal of FEATURES or the spatial encoding module from Orca (i.e., w/o
fea+location, w/o fea, and w/o location) results in a noticeable decrease in
performance. This decline confirms the vital role of both the designed prompts
and the spatial encoding module in achieving accurate SWH estimations,
underscoring their indispensability for the model’s success.

Time Efficiency Analysis. The time efficiency of the traditional model for SWH
estimation is notably low, taking about 4 minutes to estimate one day’s SWH on
a high-performance 64-core computer. In contrast, Orca showcases remarkable
computational efficiency, completing short-term (1 day) estimation in just
0.0095 seconds. Even for medium-term (8 days) and long-term (28 days)
estimations, Orca remains this efficiency, taking only 5.5527 seconds and
27.3071 seconds, respectively. Thus, Orca is significantly more
computationally efficient than traditional models in SWH estimation.

Visualization. We evaluate the estimated SWH from various models over 8
consecutive time steps at buoy 42040, comparing them with the ground truth
(see Figure 4). The estimations of Orca closely match the ground truth,
effectively capturing notable fluctuations. In contrast, since the modeling of
GWD is limited to fixed factors, its trend is overly smooth, failing to
capture significant fluctuations. Similarly, other baseline models tend to
underestimate SWH values, and fail to accurately capture wave fluctuations.

We use heat maps to show GSWH distribution in the target sea area at a certain
moment (see Figure 5). Each grid cell presents a GSWH value. The data reveals
that wave activity intensifies with increasing distance from the coast. Orca
is not only able to capture similar results as the numerical model GWD, but
also more pronounced fluctuations, further confirming the differences between
the two models in Figure 4.

(a) GWD

(b) Orca

Figure 5. The heat map of GSWH at a certain moment.

##  5\. Conclusions

This paper presents a solution to the significant wave height (SWH) estimation
problem in scenarios of limited data. Firstly, we introduce LLMs as the
backbone to leverage their few-shot learning ability. We design a specific
prompt templates and embedding module to adapt LLMs to the SWH estimation
task. Secondly, we segment buoy-based data into overlapping temporal patches,
and employ a novel spatial encoding module to improve spatio-temporal
reasoning. Extensive experiments confirm the effectiveness of our proposed
model, Orca. In the future, we aim to boost Orca’s efficiency, facilitating
both one-step and multi-step SWH predictions.

###### Acknowledgements.

This work was partially supported by National Natural Science Foundation of
China (62372179).

## References

  * [1] Shuai Yang, Shuanggen Jin, Yan Jia, et al.  Significant wave height estimation from joint CYGNSS DDMA and LES observations.  Sensors, 21(18):6123, 2021.
  * [2] Hye Jin Woo and Kyung Ae Park.  Estimation of extreme significant wave height in the northwest pacific using satellite altimeter data focused on typhoons.  Remote Sensing, 13(6):1063, 2021.
  * [3] Simon Aagaard Pedersen, Bin Yang, and Christian S. Jensen.  Anytime stochastic routing with hybrid learning.  Proc. VLDB Endow., 13(9):1555–1567, 2020.
  * [4] Jinwei Bu, Hyuk Park, Kegen Yu, et al.  Estimation of significant wave height using the features of cygnss delay doppler map.  In Proceedings of the International Geoscience and Remote Sensing Symposium, 2022.
  * [5] Tung Kieu, Bin Yang, Chenjuan Guo, Razvan-Gabriel Cirstea, Yan Zhao, Yale Song, and Christian S. Jensen.  Anomaly detection in time series with robust variational quasi-recurrent autoencoders.  In ICDE, pages 1342–1354, 2022.
  * [6] Tung Kieu, Bin Yang, Chenjuan Guo, Christian S. Jensen, Yan Zhao, Feiteng Huang, and Kai Zheng.  Robust and explainable autoencoders for unsupervised time series outlier detection.  In ICDE, pages 3038–3050, 2022.
  * [7] David Campos, Tung Kieu, Chenjuan Guo, Feiteng Huang, Kai Zheng, Bin Yang, and Christian S. Jensen.  Unsupervised time series outlier detection with diversity-driven convolutional ensembles.  Proc. VLDB Endow., 15(3):611–623, 2022.
  * [8] Willard J. Pierson, Gerhard Neuman, and Richard W. James.  Practical methods for observing and forecasting ocean waves by means of wave spectra and statistics.  1955\.
  * [9] R. C. Ris, L. H. Holthuijsen, and N. Booij.  A spectral model for waves in the near shore zone.  Coastal Engineering, pages 68–78, 1995.
  * [10] Paul A. Hwang, Jeffrey D. Ouellette, Jakov V. Toporkov, et al.  A simulation study of significant wave height retrieval from bistatic scattering of signals of ppportunity.  Transactions on Geoscience and Remote Sensing, 19:1–5, 2021.
  * [11] Xiangfei Qiu, Jilin Hu, Lekui Zhou, Xingjian Wu, Junyang Du, Buang Zhang, Chenjuan Guo, Aoying Zhou, Christian S. Jensen, Zhenli Sheng, and Bin Yang.  TFB: towards comprehensive and fair benchmarking of time series forecasting methods.  Proc. VLDB Endow., 17(9):2363–2377, 2024.
  * [12] David Campos, Miao Zhang, Bin Yang, Tung Kieu, Chenjuan Guo, and Christian S. Jensen.  LightTS: Lightweight time series classification with adaptive ensemble distillation.  Proc. ACM Manag. Data, 1(2):171:1–171:27, 2023.
  * [13] Xinle Wu, Dalin Zhang, Miao Zhang, Chenjuan Guo, Bin Yang, and Christian S. Jensen.  AutoCTS+: Joint neural architecture and hyperparameter search for correlated time series forecasting.  Proc. ACM Manag. Data, 1(1):97:1–97:26, 2023.
  * [14] Kai Zhao, Chenjuan Guo, Peng Han, Miao Zhang, Yunyao Cheng, and Bin Yang.  Multiple time series forecasting with dynamic graph modeling.  Proc. VLDB Endow., 2024.
  * [15] Yunyao Cheng, Peng Chen, Chenjuan Guo, Kai Zhao, Qingsong Wen, Bin Yang, and Christian S. Jensen.  Weakly guided adaptation for robust time series forecasting.  Proc. VLDB Endow., 2024.
  * [16] Hao Miao, Yan Zhao, Chenjuan Guo, Bin Yang, Zheng Kai, Feiteng Huang, Jiandong Xie, and Christian S. Jensen.  A unified replay-based continuous learning framework for spatio-temporal prediction on streaming data.  ICDE, 2024.
  * [17] Razvan-Gabriel Cirstea, Bin Yang, and Chenjuan Guo.  Graph attention recurrent neural networks for correlated time series forecasting.  In MileTS19@KDD, 2019.
  * [18] Zhiding Yang and Weimin Huang.  Swhformer: A vision transformer for significant wave height estimation from nautical radar images.  Transactions on Geoscience and Remote Sensing, 62:1–13, 2024.
  * [19] Di Qi and Andrew J. Majda.  Using machine learning to predict extreme events in complex systems.  Proceedings of the National Academy of Sciences, page 52–59, 2020.
  * [20] Pritam Anand, Shantanu Jain, and Harsh Savaliya.  New improved wave hybrid models for short-term significant wave height forecasting.  IEEE Access, 11:109841–109855, 2023.
  * [21] Changyang Wang, Kegen Yu, Kefei Zhang, et al.  Significant wave height retrieval based on multivariable regression models developed with CYGNSS data.  Transactions on Geoscience and Remote Sensing, 61:1–15, 2022.
  * [22] T.G. Barbounis and J.B. Theocharis.  Locally recurrent neural networks optimal filtering algorithms: application to wind speed prediction using spatial correlation.  In Proceedings of the International Joint Conference on Neural Networks, volume 5, pages 2711–2716, 2005.
  * [23] Aqsa Saeed Qureshi, Asifullah Khan, Aneela Zameer, et al.  Wind power prediction using deep neural network based meta regression and transfer learning.  Applied Soft Computing, 58:742–755, 2017.
  * [24] Jason Wei, Xuezhi Wang, Dale Schuurmans, et al.  Chain-of-thought prompting elicits reasoning in large language models.  In Proceesings of the Advances in Neural Information Processing Systems, 2022.
  * [25] Takeshi Kojima, Shixiang Shane Gu, Machel Reid, et al.  Large language models are zero-shot reasoners.  In Proceedings of the Advances in Neural Information Processing Systems, 2022.
  * [26] Hao Xue and Flora D. Salim.  Promptcast: A new prompt-based learning paradigm for time series forecasting.  IEEE Transactions on Knowledge and Data Engineering, pages 1–14, 2023.
  * [27] Peng Chen, Yingying Zhang, Yunyao Cheng, Yang Shu, Yihang Wang, Qingsong Wen, Bin Yang, and Chenjuan Guo.  Pathformer: Multi-scale transformers with adaptive pathways for time series forecasting.  ICLR, 2024.
  * [28] Ming Jin, Yifan Zhang, Wei Chen, Kexin Zhang, Yuxuan Liang, Bin Yang, Jindong Wang, Shirui Pan, and Qingsong Wen.  Position paper: What can large language models tell us about time series analysis.  CoRR, abs/2402.02713, 2024.
  * [29] Xiao Liu, Yanan Zheng, Du Zhengxiao, et al.  GPT understands, too.  AI Open, 2023.
  * [30] Ken C. K. Lee, Baihua Zheng, Huajing Li, et al.  Approaching the skyline in Z order.  In Proceedings of the International Conference on Very Large Data Bases, pages 279–290, 2007.
  * [31] Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, et al.  A time series is worth 64 words: Long-term forecasting with transformers.  In Proceedings of the International Conference on Learning Representations, 2023.
  * [32] Environmental Modeling Center.  User manual and system documentation of WAVEWATCH III R version 4.18.  2014\.
  * [33] Alec Radford, Jeffrey Wu, Rewon Child, et al.  Language models are unsupervised multitask learners.  OpenAI Blog, 1(8):9, 2019.
  * [34] Tian Zhou, Peisong Niu, Xue Wang, Liang Sun, et al.  One fits all: Power general time series analysis by pretrained LM.  In Proceesings of the Advances in Neural Information Processing Systems, 2023.