#  Bregman circumcenters: applications

Hui Ouyang  Mathematics, University of British Columbia, Kelowna, B.C. V1V
1V7, Canada. E-mail<EMAIL_ADDRESS>
(May 5, 2021)

###### Abstract

Recently, we systematically studied the basic theory of Bregman circumcenters
in another paper. In this work, we aim to investigate the application of
Bregman circumcenters.

Here, we propose the forward Bregman monotonicity which is a generalization of
the powerful Fejér monotonicity, and show a weak convergence result of the
forward Bregman monotone sequence. We also naturally introduce the Bregman
circumcenter mappings associated with a finite set of operators. Then we
provide sufficient conditions for the sequence of iterations of the forward
Bregman circumcenter mapping to be forward Bregman monotone. Furthermore, we
prove that the sequence of iterations of the forward Bregman circumcenter
mapping weakly converges to a point in the intersection of the fixed point
sets of relevant operators, which reduces to the known weak convergence result
of the circumcentered method under the Euclidean distance. In addition,
particular examples are provided to illustrate the Bregman isometry and
Browder’s demiclosedness principle, and our convergence result.

2020 Mathematics Subject Classification: Primary 90C48, 49M37, 47H05;
Secondary 90C25, 65K05, 52A41.

Keywords: Bregman distance, Legendre function, backward Bregman
(pseudo)-circumcenter, forward Bregman (pseudo)-circumcenter, fixed point set,
forward Bregman monotonicity, convergence, Bregman circumcenter method.

##  1 Introduction

Throughout the work, we assume that $\mathbb{N}=\\{0,1,2,\ldots\\}$ and
$\\{m,n\\}\subseteq\mathbb{N}\smallsetminus\\{0\\}$, and that

$\mathcal{H}\text{ is a real Hilbert space with inner product
}\langle\cdot,\cdot\rangle\text{ and induced norm }\lVert\cdot\rVert.$

Projection methods, including the method of cyclic projections based on
Bregman distances, are employed in many applications (see, e.g., [2], [3],
[4], and [18] and the references therein). The convergence rate of certain
circumcentered methods is no worse than that of the method of cyclic
projections under the Euclidean distance, and, in some cases, some
circumcentered methods converge much faster than the method of cyclic
projections, for solving the best approximation problem or the feasibility
problem (see, [9], [10], [11], [13], [14], [15], [16], [17], and [21] for
details).

In our recent work [12], we presented multiple beautiful theoretical results
on the Bregman circumcenter. Various examples under general Bregman distance
were also provided to illustrate our main results.

In this work, we aim to introduce the circumcenter mappings and methods, and
to investigate the convergence of circumcenter methods under general Bregman
distances.

We present the main results in this work below.

  * R1:

Theorem 3.3 characterizes the weak convergence of the forward Bregman monotone
sequence.

  * R2:

Theorem 4.12 shows that the sequence of iterations of the forward Bregman
circumcenter mapping convergences weakly to a point in the intersection of the
fixed point sets of operators inducing the mapping.

The remainder of the work is organized as follows.

In Section 2, we collect fundamental definitions and facts. In Section 3, we
introduce the forward Bregman monotonicity and investigate the weak
convergence of the forward Bregman monotone sequence. In Section 4, we
introduce the forward Bregman circumcenter mapping induced by a finite set of
operators, and specify sufficient conditions for the weak convergence of the
forward Bregman circumcenter method to a point in the intersection of fixed
point sets of the related operators. Moreover, we provide particular examples
to illuminate our hypotheses and main result under general Bregman distance.

We now turn to the notation used in this paper. $\Gamma_{0}(\mathcal{H})$ is
the set of proper closed convex functions from $\mathcal{H}$ to
$\left]-\infty,+\infty\right]$. Let
$f:\mathcal{H}\to\left]-\infty,+\infty\right]$ be proper. The _domain_
(_conjugate function, gradient, subgradient_ , respectively) of $f$ is denoted
by $\operatorname{dom}f$ ($f^{*}$, $\nabla f$, $\partial f$, respectively). We
say $f$ is _coercive_ if $\lim_{\lVert x\rVert\to+\infty}f(x)=+\infty$. Let
$C$ be a nonempty subset of $\mathcal{H}$. Its _interior_ and _boundary_ are
abbreviated by $\operatorname{int}C$ and $\operatorname{bd}C$, respectively.
$C$ is an _affine subspace_ of $\mathcal{H}$ if $C\neq\varnothing$ and
$(\forall\rho\in\mathbb{R})$ $\rho C+(1-\rho)C=C$. The smallest affine
subspace of $\mathcal{H}$ containing $C$ is denoted by $\operatorname{aff}\,C$
and called the _affine hull_ of $C$. The _best approximation operator_ (or
_projector_) onto $C$ under the Euclidean distance is denoted by
$\operatorname{P}_{C}$, that is, $(\forall x\in\mathcal{H})$
$\operatorname{P}_{C}x:=\operatorname{argmin}_{y\in C}\lVert x-y\rVert$.
$\iota_{C}$ is the _indicator function of $C$_, that is, $(\forall x\in C)$
$\iota_{C}(x)=0$ and $(\forall x\in\mathcal{H}\smallsetminus C)$
$\iota_{C}(x)=+\infty$. $\operatorname{Id}$ stands for the _identity mapping_.
For every $x\in\mathcal{H}$ and $\delta\in\mathbb{R}_{++}$, $B[x;\delta]$ is
the _closed ball with center at $x$ and with radius $\delta$_. Let
$A:\mathcal{H}\to 2^{\mathcal{H}}$ and let $x\in\mathcal{H}$. Then $A$ is
_locally bounded at $x$_ if there exists $\delta\in\mathbb{R}_{++}$ such that
$A(B[x;\delta])$ is bounded. Denote by
$\operatorname{Fix}A:=\\{x\in\mathcal{H}~{}:~{}x\in A(x)\\}$. For other
notation not explicitly defined here, we refer the reader to [5].

##  2 Bregman distances and projections

In this section, we collect some essential definitions and facts to be used
subsequently.

It is clear that if $f=\frac{1}{2}\lVert\cdot\rVert^{2}$ in the following
Definition 2.1, we recover the Euclidean distance
$\operatorname{D}:\mathcal{H}\times\mathcal{H}\to\left[0,+\infty\right]:(x,y)\mapsto\frac{1}{2}\lVert
x-y\rVert^{2}$.

######  Definition 2.1.

[2, Definitions 7.1 and 7.7] Suppose that $f\in\Gamma_{0}(\mathcal{H})$ with
$\operatorname{int}\operatorname{dom}f\neq\varnothing$ and that $f$ is Gâteaux
differentiable on $\operatorname{int}\operatorname{dom}f$. The _Bregman
distance $\operatorname{D}_{f}$ associated with $f$_ is defined by

$\displaystyle\operatorname{D}_{f}:\mathcal{H}\times\mathcal{H}\to\left[0,+\infty\right]:(x,y)\mapsto\begin{cases}f(x)-f(y)-\langle\nabla
f(y),x-y\rangle,\quad&\text{if }y\in\operatorname{int}\operatorname{dom}f;\\\
+\infty,\quad&\text{otherwise}.\end{cases}$

Moreover, let $C$ be a nonempty subset of $\mathcal{H}$. For every
$(x,y)\in\operatorname{dom}f\times\operatorname{int}\operatorname{dom}f$,
define the _backward Bregman projection_ (or simply _Bregman projection_) of
$y$ onto $C$ and _forward Bregman projection_ of $x$ onto $C$, respectively,
as

$\displaystyle\overleftarrow{\operatorname{P}}^{f}_{C}(y):=\left\\{u\in
C\cap\operatorname{dom}f~{}:~{}(\forall
c\in\operatorname{C})\operatorname{D}_{f}\left(u,y\right)\leq\operatorname{D}_{f}(c,y)\right\\},\text{and}$
$\displaystyle\overrightarrow{\operatorname{P}}^{f}_{C}(x):=\left\\{v\in
C\cap\operatorname{int}\operatorname{dom}f~{}:~{}(\forall
c\in\operatorname{C})\operatorname{D}_{f}\left(x,v\right)\leq\operatorname{D}_{f}(x,c)\right\\}.$

Abusing notation slightly, we shall write
$\overleftarrow{\operatorname{P}}^{f}_{C}(y)=u$ and
$\overrightarrow{\operatorname{P}}^{f}_{C}(x)=v$, if
$\overleftarrow{\operatorname{P}}^{f}_{C}(y)$ and
$\overrightarrow{\operatorname{P}}^{f}_{C}(x)$ happen to be the singletons
$\overleftarrow{\operatorname{P}}^{f}_{C}(y)=\\{u\\}$ and
$\overrightarrow{\operatorname{P}}^{f}_{C}(x)=\\{v\\}$, respectively.

The following definitions are compatible with their classical counterparts as
[1, Definitions 2.1, 2.3 and 2.8] in the finite-dimensional Euclidean space
(see, [2, Theorem 5.11] for more details).

######  Definition 2.2.

[2, Definition 5.2 and Theorem 5.6] Suppose that
$f\in\Gamma_{0}(\mathcal{H})$. We say $f$ is:

  1. (i)

_essentially smooth_ , $\operatorname{dom}\partial
f=\operatorname{int}\operatorname{dom}f\neq\varnothing$, $f$ is Gâteaux
differentiable on $\operatorname{int}\operatorname{dom}f$, and $\lVert\nabla
f(x_{k})\rVert\to+\infty$, for every sequence $(x_{k})_{k\in\mathbb{N}}$ in
$\operatorname{int}\operatorname{dom}f$ converging to some point in
$\operatorname{bd}\operatorname{dom}f$.

  2. (ii)

_essentially strictly convex_ , if $(\partial f)^{-1}$ is locally bounded on
its domain and $f$ is strictly convex on every convex subset of
$\operatorname{dom}\partial f$.

  3. (iii)

_Legendre_ (or a _Legendre function_ or a _convex function of Legendre type_),
if $f$ is both essentially smooth and essentially strictly convex.

######  Fact 2.3.

[2, Lemma 7.3] Suppose that $f\in\Gamma_{0}(\mathcal{H})$ with
$\operatorname{int}\operatorname{dom}f\neq\varnothing$ and that $f$ is Gâteaux
differentiable on $\operatorname{int}\operatorname{dom}f$. Let
$x\in\mathcal{H}$ and $y\in\operatorname{int}\operatorname{dom}f$. Then the
following statements hold.

  1. (i)

If $f$ is essentially strictly convex, then $D_{f}(\cdot,y)$ is coercive.

  2. (ii)

If $f$ is essentially strictly convex, then $D_{f}(x,y)=0\Leftrightarrow x=y$.

The following Facts 2.4 and 2.6 on the existence and uniqueness of backward
and forward Bregman projections are fundamental to some results in Section 4
below.

######  Fact 2.4.

[2, Corollary 7.9] Suppose that $f\in\Gamma_{0}(\mathcal{H})$ is Legendre,
that $C$ is a closed convex subset of $\mathcal{H}$ with
$C\cap\operatorname{int}\operatorname{dom}f\neq\varnothing$, and that
$y\in\operatorname{int}\operatorname{dom}f$. Then
$\overleftarrow{\operatorname{P}}^{f}_{C}(y)$ is a singleton contained in
$C\cap\operatorname{int}\operatorname{dom}f$.

Not surprisingly, not every Legendre function allows forward Bregman
projections and the backward and forward Bregman projections are different
notions (see, e.g., [1], [4] and [6] for details). In particular, we can find
well-known functions satisfying the hypotheses of the following Fact 2.6 in
[4, Examples 2.1 and 2.7]. We refer the interested readers to [1], [4] and [6]
for details on the backward and forward Bregman projections.

######  Definition 2.5.

[4, Definition 2.4] Suppose that $f\in\Gamma_{0}(\mathcal{H})$ is Legendre
such that $\operatorname{dom}f^{*}$ is open. We say _the function $f$ allows
forward Bregman projections_ if it satisfies the following properties.

  1. (i)

$\nabla^{2}f$ exists and is continuous on
$\operatorname{int}\operatorname{dom}f$.

  2. (ii)

$\operatorname{D}_{f}$ is convex on
$(\operatorname{int}\operatorname{dom}f)^{2}$.

  3. (iii)

For every $x\in\operatorname{int}\operatorname{dom}f$, $D_{f}(x,\cdot)$ is
strictly convex on $\operatorname{int}\operatorname{dom}f$.

######  Fact 2.6.

[4, Fact 2.6] Suppose that $\mathcal{H}=\mathbb{R}^{n}$ and
$f\in\Gamma_{0}(\mathcal{H})$ is Legendre such that $\operatorname{dom}f^{*}$
is open, that $f$ allows forward Bregman projections, and that $C$ is a closed
convex subset of $\mathcal{H}$ with
$C\cap\operatorname{int}\operatorname{dom}f\neq\varnothing$. Then $(\forall
x\in\operatorname{int}\operatorname{dom}f)$
$\overrightarrow{\operatorname{P}}^{f}_{C}(x)$ is a singleton contained in
$C\cap\operatorname{int}\operatorname{dom}f$.

##  3 Forward Bregman monotonicity

In this section, we show the weak convergence of the forward Bregman monotone
sequence, which plays a critical role in the proof of our main result in the
next section.

The following is a variant of the Bregman monotonicity defined in [3,
Definition 1.2]. Note that both [3, Definition 1.2] and the following
Definition 3.1 are natural generalizations of the classical Fejér
monotonicity. In view of the broad applications of Fejér monotonicity and
Bregman monotonicity in the proof of the convergence of iterative algorithms
(see, e.g., [5], [3], [20] and the references therein), we assume the forward
Bregman monotonicity defined in Definition 3.1 below is interesting on its
own, and probably can be used in many other iterative algorithms under general
Bregman distances.

######  Definition 3.1.

A sequence $(x_{k})_{k\in\mathbb{N}}$ in $\mathcal{H}$ is _forward Bregman
monotone with respect to a set $C\subseteq\mathcal{H}$_ if
$C\cap\operatorname{int}\operatorname{dom}f\neq\varnothing$,
$(x_{k})_{k\in\mathbb{N}}$ lies in $\operatorname{int}\operatorname{dom}f$,
and

$\displaystyle(\forall c\in
C\cap\operatorname{int}\operatorname{dom}f)~{}(\forall
k\in\mathbb{N})\quad\operatorname{D}_{f}(x_{k+1},c)\leq\operatorname{D}_{f}(x_{k},c).$

######  Fact 3.2.

Suppose that $f\in\Gamma_{0}(\mathcal{H})$ with
$\operatorname{int}\operatorname{dom}f\neq\varnothing$, that $f$ is Gâteaux
differentiable on $\operatorname{int}\operatorname{dom}f$, and that $f$ is
essentially strictly convex. Then $f$ is strictly convex on
$\operatorname{int}\operatorname{dom}f$, which is equivalent to that

$\displaystyle(\forall x\in\operatorname{int}\operatorname{dom}f)(\forall
y\in\operatorname{int}\operatorname{dom}f)\quad x\neq y\Rightarrow\langle
x-y,\nabla f(x)-\nabla f(y)\rangle>0.$

###### Proof.

Because $f$ is convex, by [5, Propositions 3.45(ii), 8.2 and 16.27],
$\operatorname{int}\operatorname{dom}f$ is a convex subset of
$\operatorname{dom}\partial f$. Then employ Definition 2.2(ii) to see that $f$
is strictly convex on $\operatorname{int}\operatorname{dom}f$.

We obtain the last equivalence by applying [5, Proposition 17.10] to the
function $f+\iota_{\operatorname{int}\operatorname{dom}f}$. ∎

######  Theorem 3.3.

Suppose that $f\in\Gamma_{0}(\mathcal{H})$ with
$\operatorname{int}\operatorname{dom}f\neq\varnothing$ and that $f$ is Gâteaux
differentiable on $\operatorname{int}\operatorname{dom}f$. Let $C$ be a closed
convex set in $\mathcal{H}$ with
$C\cap\operatorname{int}\operatorname{dom}f\neq\varnothing$, and let
$(x_{k})_{k\in\mathbb{N}}$ be in $\operatorname{int}\operatorname{dom}f$ and
be forward Bregman monotone with respect to $C$. Then the following statements
hold.

  1. (i)

$\left(\forall c\in C\cap\operatorname{int}\operatorname{dom}f\right)$
$\left(\operatorname{D}_{f}(x_{k},c)\right)_{k\in\mathbb{N}}$ is decreasing,
nonnegative and convergent.

  2. (ii)

Let $\\{y,z\\}\subseteq C\cap\operatorname{int}\operatorname{dom}f$. Then the
limit $\lim_{k\to\infty}\langle\nabla f(y)-\nabla f(z),x_{k}\rangle$ exists.

  3. (iii)

Suppose that $f$ is essentially strictly convex. Then
$(x_{k})_{k\in\mathbb{N}}$ is bounded.

  4. (iv)

Suppose that $f$ is essentially strictly convex. Then
$(x_{k})_{k\in\mathbb{N}}$ weakly converges to some point in
$C\cap\operatorname{int}\operatorname{dom}f$ if and only if all weak
sequential cluster points of $(x_{k})_{k\in\mathbb{N}}$ are in
$C\cap\operatorname{int}\operatorname{dom}f$.

###### Proof.

(i): This is clear from Definitions 3.1 and 2.1.

(ii): According to Definition 2.1, for every $k\in\mathbb{N}$,

$\displaystyle\operatorname{D}_{f}(x_{k},y)-\operatorname{D}_{f}(x_{k},z)=f(x_{k})-f(y)-\langle\nabla
f(y),x_{k}-y\rangle-\Big{(}f(x_{k})-f(z)-\langle\nabla
f(z),x_{k}-z\rangle\Big{)}$ $\displaystyle\Leftrightarrow$
$\displaystyle\langle\nabla f(z)-\nabla
f(y),x_{k}\rangle=\operatorname{D}_{f}(x_{k},y)-\operatorname{D}_{f}(x_{k},z)+f(y)-f(z)-\langle\nabla
f(y),y\rangle+\langle\nabla f(z),z\rangle.$

which, via (i), yields the desired result.

(iii): Let $c\in C\cap\operatorname{int}\operatorname{dom}f$. Then the
boundedness of $\left(\operatorname{D}_{f}(x_{k},c)\right)_{k\in\mathbb{N}}$
and Fact 2.3(i) imply that $(x_{k})_{k\in\mathbb{N}}$ is bounded.

(iv): “$\Rightarrow$”: This is trivial.

“$\Leftarrow$”: Suppose that all weak sequential cluster points of
$(x_{k})_{k\in\mathbb{N}}$ are in
$C\cap\operatorname{int}\operatorname{dom}f$. Because the boundedness of
$(x_{k})_{k\in\mathbb{N}}$ is proved in (iii) above, bearing [5, Lemma 2.46]
in mind, we know that it remains to prove that $(x_{k})_{k\in\mathbb{N}}$ has
at most one weak sequential cluster point in
$C\cap\operatorname{int}\operatorname{dom}f$. Assume that $\bar{x}$ and
$\hat{x}$ are two weak sequential cluster points of $(x_{k})_{k\in\mathbb{N}}$
in $C\cap\operatorname{int}\operatorname{dom}f$. Then there exist subsequences
$(x_{k_{j}})_{n\in\mathbb{N}}$ and $(x_{k_{t}})_{t\in\mathbb{N}}$ of
$(x_{k})_{k\in\mathbb{N}}$ such that
$x_{k_{j}}{\;\operatorname{\rightharpoonup}\;}\bar{x}$ and
$x_{k_{t}}{\;\operatorname{\rightharpoonup}\;}\hat{x}$. Then

$\displaystyle\langle\nabla f(\bar{x})-\nabla
f(\hat{x}),x_{k_{j}}\rangle\to\langle\nabla f(\bar{x})-\nabla
f(\hat{x}),\bar{x}\rangle;$ (3.1a) $\displaystyle\langle\nabla
f(\bar{x})-\nabla f(\hat{x}),x_{k_{t}}\rangle\to\langle\nabla
f(\bar{x})-\nabla f(\hat{x}),\hat{x}\rangle.$ (3.1b)

On the other hand, apply (ii) with $y=\bar{x}$ and $z=\hat{x}$ to deduce that
$\lim_{k\to\infty}\langle\nabla f(\bar{x})-\nabla f(\hat{x}),x_{k}\rangle$
exists. This combined with 3.1 implies that $\langle\nabla f(\bar{x})-\nabla
f(\hat{x}),\bar{x}\rangle=\langle\nabla f(\bar{x})-\nabla
f(\hat{x}),\hat{x}\rangle$, that is,

$\displaystyle\langle\nabla f(\bar{x})-\nabla
f(\hat{x}),\bar{x}-\hat{x}\rangle=0,$

which, combining with Fact 3.2, yields $\hat{x}=\bar{x}$. Altogether, the
proof is complete. ∎

##  4 Bregman circumcenter methods

As we mentioned in the introduction, projection methods based on Bregman
distances have broad applications and some circumcentered methods accelerate
the method of cyclic projections with Bregman projections for finding the best
approximation point onto or a feasibility point in the intersection of
finitely many closed convex sets. In this section, we introduce backward and
forward Bregman (pseudo-)circumcenter mappings and methods. We shall also
investigate the convergence of the forward Bregman circumcenter method.

Throughout this section, suppose that $f\in\Gamma_{0}(\mathcal{H})$ with
$\operatorname{int}\operatorname{dom}f\neq\varnothing$ and that $f$ is Gâteaux
differentiable on $\operatorname{int}\operatorname{dom}f$. Set
$\operatorname{I}:=\\{1,\ldots,m\\}$.

### Bregman circumcenter mappings

Henceforth, for every $\mathcal{D}\subseteq\mathcal{H}$,

$\displaystyle\text{denote by }\mathcal{P}(\mathcal{D})\text{ the set of all
nonempty subsets of }\mathcal{D}\text{ containing finitely many elements}.$

Suppose that $K\in\mathcal{P}(\operatorname{dom}f)$. Set

$\displaystyle\overrightarrow{E}_{f}(K):=\\{q\in\operatorname{int}\operatorname{dom}f~{}:~{}(\forall
x\in K)~{}\operatorname{D}_{f}(x,q)\text{ is a singleton}\\}.$

Suppose additionally $K\in\mathcal{P}(\operatorname{int}\operatorname{dom}f)$.
Denote by

$\displaystyle\overleftarrow{E}_{f}(K):=\\{p\in\operatorname{dom}f~{}:~{}(\forall
y\in K)~{}\operatorname{D}_{f}(p,y)\text{ is a singleton}\\}.$

######  Definition 4.1.

[12, Definition 3.5] Let
$K\in\mathcal{P}(\operatorname{int}\operatorname{dom}f)$.

  1. (i)

Define the _backward Bregman circumcenter operator $\overleftarrow{CC{}}$
w.r.t. $f$_ as

$\displaystyle\overleftarrow{CC{}}:\mathcal{P}(\operatorname{int}\operatorname{dom}f)\to
2^{\mathcal{H}}:K\mapsto\operatorname{aff}\,(K)\cap\overleftarrow{E}_{f}(K).$

  2. (ii)

Define the _backward Bregman pseudo-circumcenter operator
$\overleftarrow{CC{}}^{ps}$ w.r.t. $f$_ as

$\displaystyle\overleftarrow{CC{}}^{ps}:\mathcal{P}(\operatorname{int}\operatorname{dom}f)\to
2^{\mathcal{H}}:K\mapsto\operatorname{aff}\,(\nabla
f(K))\cap\overleftarrow{E}_{f}(K).$

In particular, for every
$K\in\mathcal{P}(\operatorname{int}\operatorname{dom}f)$, we call the element
in $\overleftarrow{CC{}}(K)$ and $\overleftarrow{CC{}}^{ps}(K)$ Backward
Bregman circumcenter and Backward Bregman pseudo-circumcenter of $K$,
respectively,

######  Definition 4.2.

[12, Definition 4.5] Let $K\in\mathcal{P}(\operatorname{dom}f)$.

  1. (i)

Define the _forward Bregman circumcenter operator w.r.t. $f$_ as

$\displaystyle\overrightarrow{CC{}}:\mathcal{P}(\operatorname{dom}f)\to
2^{\mathcal{H}}:K\mapsto\operatorname{aff}\,(K)\cap\overrightarrow{E}_{f}(K).$

  2. (ii)

Define the _forward Bregman pseudo-circumcenter operator w.r.t. $f$_ as

$\displaystyle\overrightarrow{CC{}}^{ps}:\mathcal{P}(\operatorname{dom}f)\to
2^{\mathcal{H}}:K\mapsto\left(\nabla
f^{*}\left(\operatorname{aff}\,K\right)\right)\cap\overrightarrow{E}_{f}(K).$

In particular, for every $K\in\mathcal{P}(\operatorname{dom}f)$, we call the
element in $\overrightarrow{CC{}}(K)$ and $\overrightarrow{CC{}}^{ps}(K)$
forward Bregman circumcenter and forward Bregman pseudo-circumcenter of $K$,
respectively.

Suppose that $\operatorname{D}_{f}$ is symmetric, that is,
$\operatorname{dom}f=\operatorname{int}\operatorname{dom}f$ and
$(\forall\\{x,y\\}\subseteq\operatorname{dom}f)$
$\operatorname{D}_{f}(x,y)=\operatorname{D}_{f}(y,x)$. Then the backward
Bregman circumcenter and forward Bregman circumcenter are consistent. In
particular, if $f:=\frac{1}{2}\lVert\cdot\rVert^{2}$, then $\nabla
f=\operatorname{Id}$; and hence, the notions backward Bregman circumcenter,
backward Bregman pseudo-circumcenter, forward Bregman circumcenter and forward
Bregman pseudo-circumcenter are all the same and reduce to the circumcenter
defined in [7, Definition 3.4] under the Euclidean distance.

From now on, for every set-valued operator $A:\mathcal{H}\to 2^{\mathcal{H}}$,
if $A(x)$ is a singleton for some $x\in\mathcal{H}$, we sometimes by slight
abuse of notation allow $A(x)$ to stand for its unique element. The intended
meaning should be clear from the context.

######  Fact 4.3.

[12, Corollary 5.1] Suppose that

$\displaystyle
K:=\\{q_{0},q_{1},\ldots,q_{m}\\}\subseteq\operatorname{dom}f\text{ is
nonempty}.$

Then the following assertions hold.

  1. (i)

Suppose that $\mathcal{H}=\mathbb{R}^{n}$, that $f$ is Legendre such that
$\operatorname{dom}f^{*}$ is open, that
$\overleftarrow{E}_{f}(K)\neq\varnothing$, that $f$ allows forward Bregman
projections, that
$\operatorname{aff}\,(K)\subseteq\operatorname{int}\operatorname{dom}f$, and
that $\nabla f(\operatorname{aff}\,(K))$ is a closed affine subspace. Then
$\left(\forall z\in\overleftarrow{E}_{f}(K)\right)$
$\overrightarrow{\operatorname{P}}^{f}_{\operatorname{aff}\,(K)}(z)\in\overleftarrow{CC{}}(K)$.

  2. (ii)

Suppose that $\overleftarrow{E}_{f}(K)\neq\varnothing$ and that
$K\subseteq\operatorname{int}\operatorname{dom}f$ and
$\operatorname{aff}\,(\nabla f(K))\subseteq\operatorname{dom}f$. Then
$\left(\forall z\in\overleftarrow{E}_{f}(K)\right)$
$\overleftarrow{CC{}}^{ps}(K)=\operatorname{P}_{\operatorname{aff}\,(\nabla
f(K))}(z)$.

  3. (iii)

Suppose that $K\subseteq\operatorname{int}\operatorname{dom}f$ and
$\operatorname{aff}\,(\nabla f(K))\subseteq\operatorname{dom}f$, and that
$\nabla f(q_{0}),\nabla f(q_{1}),\ldots,\nabla f(q_{m})$ are affinely
independent. Then $\overleftarrow{CC{}}^{ps}(K)$ uniquely exists and has an
explicit formula.

  4. (iv)

Suppose that $f$ is Legendre, that
$\operatorname{aff}\,(K)\cap\operatorname{int}\operatorname{dom}f\neq\varnothing$,
and that $\overrightarrow{E}_{f}(K)\neq\varnothing$. Then $\left(\forall
z\in\overrightarrow{E}_{f}(K)\right)$
$\overleftarrow{\operatorname{P}}^{f}_{\operatorname{aff}\,(K)}(z)\in\overrightarrow{CC{}}(K)$.

  5. (v)

Suppose that $f$ is Legendre, and that
$\operatorname{aff}\,(K)\subseteq\operatorname{int}\operatorname{dom}f^{*}$
and $\overrightarrow{E}_{f}(K)\neq\varnothing$. Then $\left(\forall
z\in\overrightarrow{E}_{f}(K)\right)$ $\overrightarrow{CC{}}^{ps}(K)=\nabla
f^{*}\left(\operatorname{P}_{\operatorname{aff}\,(K)}(\nabla f(z))\right)$.

  6. (vi)

Suppose that $f$ is Legendre, that
$\operatorname{aff}\,(K)\subseteq\operatorname{int}\operatorname{dom}f^{*}$,
and that $q_{0},q_{1},\ldots,q_{m}$ are affinely independent. Then
$\overrightarrow{CC{}}^{ps}(K)$ uniquely exists and has an explicit formula.

Note that we have particular examples in [12] with functions
$f\neq\frac{1}{2}\lVert\cdot\rVert^{2}$ and sets $K$ such that the hypotheses
of each item of Fact 4.3 above hold.

######  Definition 4.4.

Let $t\in\mathbb{N}\smallsetminus\\{0\\}$. Suppose that $G_{1},\ldots,G_{t}$
are operators from $\mathcal{H}$ to $\mathcal{H}$. Set

$\displaystyle\mathcal{S}:=\\{G_{1},\ldots,G_{t}\\}\quad\text{and}\quad(\forall
x\in\mathcal{H})~{}\mathcal{S}(x):=\\{G_{1}x,\ldots,G_{t}x\\}.$

The _forward Bregman circumcenter mapping induced by $\mathcal{S}$_ is

$\overrightarrow{CC{missing}}_{\mathcal{S}}\colon\mathcal{H}\to
2^{\mathcal{H}}\colon y\mapsto\overrightarrow{CC{missing}}(\mathcal{S}(y)),$

that is, for every $y\in\mathcal{H}$, if the forward Bregman circumcenter of
the set $\mathcal{S}(y)$ defined in Definition 4.2(i) does not exist, then
$\overrightarrow{CC{missing}}_{\mathcal{S}}y=\varnothing$. Otherwise,
$\overrightarrow{CC{missing}}_{\mathcal{S}}y$ is the set of points $v$
satisfying the two conditions below:

  1. (i)

$v\in\operatorname{aff}\,(\mathcal{S}(x))\cap\operatorname{int}\operatorname{dom}f$,
and

  2. (ii)

$\operatorname{D}_{f}(G_{1}x,v)=\cdots=\operatorname{D}_{f}(G_{t}x,v)$.

Let $\mathcal{C}$ be a subset of $\mathcal{H}$. If $(\forall y\in\mathcal{C})$
$\overrightarrow{CC{missing}}_{\mathcal{S}}y$ contains at most one element, we
say that $\overrightarrow{CC{missing}}_{\mathcal{S}}$ is _at most single-
valued on $\mathcal{C}$_. Naturally, if $\mathcal{C}=\mathcal{H}$, then we
omit the phrase “on $\mathcal{C}$”.

Analogously, we define the _forward Bregman pseudo-circumcenter mapping
$\overrightarrow{CC{missing}}_{\mathcal{S}}^{ps}$ induced by $\mathcal{S}$_,
_backward Bregman circumcenter mapping
$\overleftarrow{CC{missing}}_{\mathcal{S}}$ induced by $\mathcal{S}$_, and
_backward Bregman pseudo-circumcenter mapping
$\overleftarrow{CC{missing}}_{\mathcal{S}}^{ps}$ induced by $\mathcal{S}$_
with replacing the forward Bregman circumcenter operator
$\overrightarrow{CC{missing}}$ above by the corresponding operator,
$\overrightarrow{CC{missing}}^{ps}$, $\overleftarrow{CC{missing}}$, and
$\overleftarrow{CC{missing}}^{ps}$, respectively.

We can also directly deduce the following result by [9, Theorem 3.3(ii)].

######  Corollary 4.5.

Suppose that $f:=\frac{1}{2}\lVert\cdot\rVert^{2}$. Let $(\forall
i\in\operatorname{I})$ $T_{i}:\mathcal{H}\rightarrow\mathcal{H}$ be a linear
isometry111A mapping $T:\mathcal{H}\to\mathcal{H}$ is said to be isometric or
an isometry if $(\forall x\in\mathcal{H})$ $(\forall y\in\mathcal{H})$ $\lVert
Tx-Ty\rVert=\lVert x-y\rVert$.. Set $\mathcal{S}:=\\{T_{1},\ldots,T_{m}\\}$
and $(\forall x\in\mathcal{H})$ $\mathcal{S}(x)=\\{T_{1}x,\ldots,T_{m}x\\}$.
Then
$\overrightarrow{CC{missing}}_{\mathcal{S}}(x)=\overrightarrow{CC{missing}}_{\mathcal{S}}^{ps}(x)=\overleftarrow{CC{missing}}_{\mathcal{S}}(x)=\overleftarrow{CC{missing}}_{\mathcal{S}}^{ps}(x)=\operatorname{P}_{\operatorname{aff}\,(\mathcal{S}(x))}(0)$.

###### Proof.

Notice that $\nabla f=\operatorname{Id}$ and $\operatorname{dom}f=\mathcal{H}$
in this case. Because, due to [12, Corollary 5.6(ii)],
$0\in\cap^{m}_{j=1}\operatorname{Fix}T_{j}\subseteq\overrightarrow{E}_{f}(K)=\overleftarrow{E}_{f}(K)$,
the required result follows easily from Fact 4.3(ii) or Fact 4.3(v). ∎

Henceforth, suppose $T_{1},\ldots,T_{m}$ are operators from $\mathcal{H}$ to
$\mathcal{H}$ with
$\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\neq\varnothing$.
Set

$\mathcal{S}:=\\{\operatorname{Id},T_{1},\ldots,T_{m}\\}\quad\text{and}\quad(\forall
x\in\mathcal{H})~{}\mathcal{S}(x):=\\{x,T_{1}x,\ldots,T_{m}x\\}.$

######  Lemma 4.6.

Suppose that $f$ is essentially strictly convex. Then the following statements
hold.

  1. (i)

$\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}=\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)$
and
$\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}^{ps}=\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\nabla
f^{*}$.

  2. (ii)

$\operatorname{Fix}\overleftarrow{CC{missing}}_{\mathcal{S}}=\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)$
and
$\operatorname{Fix}\overleftarrow{CC{missing}}_{\mathcal{S}}^{ps}=\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\nabla
f$.

###### Proof.

(i): Let $z\in\mathcal{H}$.

$\displaystyle
z\in\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}\Leftrightarrow$
$\displaystyle z\in\overrightarrow{CC{missing}}_{\mathcal{S}}z$
$\displaystyle\Leftrightarrow$ $\displaystyle
z\in\operatorname{aff}\,(\mathcal{S}(z))\cap\operatorname{int}\operatorname{dom}f\text{
and
}\operatorname{D}_{f}(z,z)=\operatorname{D}_{f}(T_{1}z,z)=\cdots=\operatorname{D}_{f}(T_{m}z,z)$
$\displaystyle\Leftrightarrow$ $\displaystyle
z\in\operatorname{int}\operatorname{dom}f\text{ and }(\forall
i\in\operatorname{I})~{}z\in\operatorname{Fix}T_{i}\quad(\text{by
\lx@cref{creftype~refnum}{fact:PropertD}\lx@cref{creftype~refnum}{fact:PropertD:=0}})$
$\displaystyle\Leftrightarrow$ $\displaystyle
z\in\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right),$

which guarantees that
$\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}=\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)$.

Similarly,

$\displaystyle
z\in\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}^{ps}\Leftrightarrow$
$\displaystyle z=\overrightarrow{CC{missing}}_{\mathcal{S}}^{ps}z$
$\displaystyle\Leftrightarrow$ $\displaystyle z\in\nabla
f^{*}\left(\operatorname{aff}\,\left(\mathcal{S}(z)\right)\right)\cap\operatorname{int}\operatorname{dom}f\text{
and
}\operatorname{D}_{f}(z,z)=\operatorname{D}_{f}(T_{1}z,z)=\cdots=\operatorname{D}_{f}(T_{m}z,z)$
$\displaystyle\Leftrightarrow$ $\displaystyle
z\in\operatorname{Fix}\left(\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S})\right)\right)\cap\operatorname{int}\operatorname{dom}f\text{
and }(\forall
i\in\operatorname{I})~{}z\in\operatorname{Fix}T_{i}\quad(\text{by
\lx@cref{creftype~refnum}{fact:PropertD}\lx@cref{creftype~refnum}{fact:PropertD:=0}})$
$\displaystyle\Leftrightarrow$ $\displaystyle
z\in\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\left(\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S})\right)\right),$

which deduces that
$\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}^{ps}=\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\left(\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S})\right)\right)$.

In addition, clearly, $\operatorname{Id}\in\mathcal{S}$ entails that
$\operatorname{Fix}\nabla f^{*}\subseteq\operatorname{Fix}\left(\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S})\right)\right)$. Hence,
$\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\nabla
f^{*}\subseteq\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\left(\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S})\right)\right)$. On the other
hand, $\left(\forall
y\in\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\left(\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S})\right)\right)\right)$ $y\in\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S}(y))\right)=\nabla f^{*}(y)$, which
necessitates that
$\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\left(\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S})\right)\right)\subseteq\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\nabla
f^{*}$. Therefore,
$\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\left(\nabla
f^{*}\left(\operatorname{aff}\,(\mathcal{S})\right)\right)=\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\nabla
f^{*}$.

Altogether,
$\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}^{ps}=\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\cap\operatorname{Fix}\nabla
f^{*}$.

(ii): The proof is similar to the proof of (i) above. ∎

### Bregman isometry and Browder’s Demiclosedness Principle

In this subsection, we generalize the traditional isometry and Browder’s
Demiclosedness Principle from the Euclidean distance to general Bregman
distances, and investigate the Bregman isometry and Browder’s Demiclosedness
Principle which play critical roles in the main result in this section. The
Bregman isometry and Browder’s Demiclosedness Principle are interesting in its
own right.

######  Definition 4.7.

Let $\mathcal{C}$ be a nonempty subset of $\mathcal{H}$ and let
$T:\mathcal{C}\to\mathcal{H}$. We say $T$ is _Bregman isometric_ (or a
_Bregman isometry_) w.r.t. $f$, if
$\left(\forall(x,y)\in\operatorname{dom}f\times\operatorname{int}\operatorname{dom}f\right)$
$\left(Tx,Ty\right)\in\operatorname{dom}f\times\operatorname{int}\operatorname{dom}f$,
and

$\displaystyle\left(\forall\\{x,y\\}\subseteq\operatorname{dom}f\times\operatorname{int}\operatorname{dom}f\right)~{}\operatorname{D}_{f}\left(Tx,Ty\right)=\operatorname{D}_{f}(x,y).$
(4.1)

In view of the definition, it is trivial that given an arbitrary function $g$,
the identity operator $\operatorname{Id}$ is Bregman isometric w.r.t. $g$.

Notice that if $f=\frac{1}{2}\lVert\cdot\rVert^{2}$, then the Bregman isometry
deduces the traditional isometric $($see, e.g., [9, Lemma 2.23] for examples
of isometries under the Euclidean distance$)$.

The following definition is a generalization of the well-known Browder’s
Demiclosedness Principle [19, Theorem 3(a)]. In view of [5, Corollary 4.25],
when $f=\frac{1}{2}\lVert\cdot\rVert^{2}$, the Bregman Browder’s
Demiclosedness Principle holds for all nonexpansive operators.

######  Definition 4.8.

Let $\mathcal{C}$ be a nonempty subset of $\mathcal{H}$, let
$T:\mathcal{C}\to\mathcal{H}$ and let
$x\in\operatorname{int}\operatorname{dom}f$. We say the _Bregman Browder’s
demiclosedness principle associated with $f$ holds at $x\in
C\cap\operatorname{int}\operatorname{dom}f$ for $T$_ if for every sequence
$\left(x_{k}\right)_{k\in\mathbb{N}}$ in
$\operatorname{int}\operatorname{dom}f$,
$\left(Tx_{k}\right)_{k\in\mathbb{N}}$ in
$\operatorname{int}\operatorname{dom}f$ and

$\displaystyle\begin{rcases}x_{k}{\;\operatorname{\rightharpoonup}\;}x\\\
\operatorname{D}_{f}(x_{k},Tx_{k})\to 0\end{rcases}\Rightarrow
x\in\operatorname{Fix}T.$ (4.2)

In addition, we say _$T$ is $f$-demiclosed_, if the Bregman Browder’s
demiclosedness principle associated with $f$ holds for every $x\in
C\cap\operatorname{int}\operatorname{dom}f$.

Because $\operatorname{Fix}\operatorname{Id}=\mathcal{H}$, given an arbitrary
function $g$, $\operatorname{Id}$ is $g$-demiclosed.

The following examples are used to illustrate the two new concepts above.

######  Example 4.9.

Suppose that $\mathcal{H}=\mathbb{R}^{n}$. Denote by $(\forall
x\in\mathbb{R}^{n})$ $x=(x_{i})^{n}_{i=1}$. Define
$f:x\mapsto-\sum^{n}_{i=1}\ln(x_{i})$, with
$\operatorname{dom}f=\left]0,+\infty\right[^{n}$, which is the Burg entropy.

Then the following statements hold.

  1. (i)

$f$ is Legendre such that $\operatorname{dom}f$ and $\operatorname{dom}f^{*}$
are open.

  2. (ii)

Define
$T:\mathbb{R}^{n}\to\mathbb{R}^{n}:(x_{i})_{i\in\operatorname{J}}\mapsto(c_{i}x_{i})_{i\in\operatorname{J}}$.
Set $\operatorname{J}:=\\{1,\ldots,n\\}$. Let
$(c_{i})_{i\in\operatorname{J}}\in\mathbb{R}^{n}$ with $(\forall
i\in\operatorname{J})$ $c_{i}\in\mathbb{R}_{++}$. Then:

    1. (a)

$T$ is Bregman isometric w.r.t. $f$.

    2. (b)

Suppose that there exists $j\in\operatorname{J}$ such that $c_{j}\neq 1$. Then
$T$ is $f$-demiclosed.

  3. (iii)

Suppose that $\mathcal{H}=\mathbb{R}$. Let $c\in\mathbb{R}_{++}$ and
$\mu\in\mathbb{R}_{++}\smallsetminus\\{1\\}$. Define
$T:\mathbb{R}\to\mathbb{R}:t\mapsto ct^{\mu}$. Then:

    1. (a)

$\operatorname{int}\operatorname{dom}f\cap\operatorname{Fix}T=\\{c^{-\frac{1}{\mu-1}}\\}$,
which is nonempty closed and convex.

    2. (b)

$T$ is $f$-demiclosed.

###### Proof.

(i): This is immediately form [4, Examples 2.1].

(ii)(a): This is clear from Definition 2.1 and the definitions of $f$ and $T$.

(ii)(b): Let $\left(x^{(k)}\right)_{k\in\mathbb{N}}$ be in
$\operatorname{dom}f$. Because $(\forall x\in\operatorname{dom}f)$
$Tx\in\operatorname{dom}f$, we have $\left(Tx^{k}\right)_{k\in\mathbb{N}}$ is
a sequence in $\operatorname{dom}f$.

Suppose that $x^{k}\to\bar{x}$ with
$\bar{x}:=\left(\bar{x}_{i}\right)_{i\in\operatorname{J}}\in\left]0,+\infty\right[^{n}$.
Denote by $(\forall k\in\mathbb{N})$
$x^{k}:=\left(x^{k}_{i}\right)_{i\in\operatorname{J}}$. Now,

$\displaystyle\operatorname{D}_{f}(x^{k},Tx^{k})$
$\displaystyle=f(x^{k})-f(Tx^{k})-\langle\nabla f(Tx^{k}),x^{k}-Tx^{x}\rangle$
$\displaystyle=-\sum^{n}_{i=1}\ln(x^{k}_{i})+\sum^{n}_{i=1}\ln(c_{i}x^{k}_{i})+\sum^{n}_{i=1}\frac{x^{k}_{i}-c_{i}x^{k}_{i}}{c_{i}x^{k}_{i}}$
$\displaystyle=\sum^{n}_{i=1}\ln\frac{c_{i}x^{k}_{i}}{x^{k}_{i}}+\sum^{n}_{i=1}\frac{x^{k}_{i}-c_{i}x^{k}_{i}}{c_{i}x^{k}_{i}},$

which implies that

$\displaystyle\operatorname{D}_{f}(x^{k},Tx^{k})\to\sum^{n}_{i=1}\left(\ln(c_{i})+\frac{1}{c_{i}}-1\right).$

On the other hand, consider the function
$g:\mathbb{R}_{++}\to\mathbb{R}:t\mapsto\ln(t)+\frac{1}{t}-1$. By some easy
calculus, $(\forall t\in\mathbb{R}_{++}\smallsetminus\\{1\\})$ $g(t)<g(1)=0$.
Because there exists $j\in\operatorname{J}$ such that $c_{j}\neq 1$, we know
that $\sum^{n}_{i=1}\left(\ln(c_{i})+\frac{1}{c_{i}}-1\right)<0$.

Hence, the hypothesis of 4.2 is always false, which deduces that 4.2 is true.
Therefore, due to Definition 4.8, $T$ is $f$-demiclosed.

(iii)(a): This is trivial.

(iii)(b): Let $a\in\left]0,+\infty\right[\,$. Let $(a_{k})_{k\in\mathbb{N}}$
be in $\operatorname{int}\operatorname{dom}f$ such that $a_{k}\to a$. Then
clearly $(Ta_{k})_{k\in\mathbb{N}}$ is a sequence in
$\left]0,+\infty\right[\,$. Suppose that
$\operatorname{D}_{f}(a_{k},Ta_{k})\to 0$. Note that

$\displaystyle\operatorname{D}_{f}(a_{k},Ta_{k})=-\ln(a_{k})+\ln(ca_{k}^{\mu})+\frac{a_{k}-ca_{k}^{\mu}}{ca_{k}^{\mu}}\to\ln(ca^{\mu})-\ln(a)+\frac{a^{1-\mu}}{c}-1.$

Denote by
$h:\mathbb{R}_{++}\to\mathbb{R}:t\mapsto\ln(ct^{\mu})-\ln(t)+\frac{t^{1-\mu}}{c}-1$.
Then $(\forall t\in\mathbb{R}_{++})$
$h^{\prime}(t)=(\mu-1)(\frac{1}{t}-\frac{1}{ct^{\mu}})$. Then by considering
the two cases $\mu>1$ and $\mu<1$ separately, we easily get that
$\ln(ca^{\mu})-\ln(a)+\frac{a^{1-\mu}}{c}-1=0$ implies that $a=ca^{\mu}$, that
is, $a=c^{-\frac{1}{\mu-1}}\in\operatorname{Fix}T$. Altogether, by Definition
4.8, $T$ is $f$-demiclosed. ∎

Consider our examples $f$ and $T$ in Example 4.9(ii). Notice that if there
exists $j\in\operatorname{J}$ such that $c_{j}\neq 1$, then
$\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)=\varnothing$.

The following example of $f$ and $T$ satisfies all requirements in our main
result Theorem 4.12 below.

######  Example 4.10.

Suppose that $\mathcal{H}=\mathbb{R}^{n}$. Denote by $(\forall
x\in\mathbb{R}^{n})$ $x=(x_{i})^{n}_{i=1}$. Define
$f:x\mapsto\sum^{n}_{i=1}x_{i}\ln(x_{i})+(1-x_{i})\ln(1-x_{i})$, with
$\operatorname{dom}f=\left[0,1\right]^{n}$, which is the Fermi-Dirac entropy.
Set $\operatorname{J}:=\\{1,\ldots,n\\}$. Then the following statements hold.

  1. (i)

$f$ is Legendre with $\operatorname{dom}f^{*}$ open, and allows forward
Bregman projections.

  2. (ii)

$\operatorname{Id}$ is $f$-demiclosed.

  3. (iii)

Define
$T:\mathbb{R}^{n}\to\mathbb{R}^{n}:(x_{i})^{n}_{i=1}\mapsto(1-x_{i})^{n}_{i=1}$.
Then:

    1. (a)

$T$ is Bregman isometric w.r.t. $f$.

    2. (b)

$T$ is $f$-demiclosed.

    3. (c)

$\operatorname{Fix}T\cap\operatorname{int}\operatorname{dom}f=\\{(\frac{1}{2})_{i\in\operatorname{J}}\\}$
is nonempty, closed and convex.

  4. (iv)

Let $\Lambda$ be a subset of $J$. Define $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$
by $\left(\forall x\in\mathbb{R}^{n}\right)$ $Tx:=(y_{i})^{n}_{i=1}$ where
$(\forall i\in\operatorname{J}\smallsetminus\Lambda)$ $y_{i}=x_{i}$ and
$(\forall i\in\Lambda)$ $y_{i}=(1-x_{i})$.

    1. (a)

$\operatorname{Fix}T=\left\\{x\in\mathbb{R}^{n}~{}:~{}(\forall
i\in\Lambda)x_{i}=\frac{1}{2}\right\\}$.

    2. (b)

$T$ is Bregman isometric w.r.t. $f$.

    3. (c)

$T$ is $f$-demiclosed.

###### Proof.

(i): This follows immediately form [4, Examples 2.1 and 2.7].

(ii): This is trivial.

(iii)(a): This follows easily from Definition 2.1 and the definitions of $f$
and $T$.

(iii)(b): Let $\bar{x}\in\left]0,1\right[^{n}\,$. Denote by
$\bar{x}=(\bar{x}_{i})^{n}_{i=1}$. Let $(x^{(k)})_{k\in\mathbb{N}}$ be a
sequence in $\left]0,1\right[^{n}$ such that $x^{(k)}\to\bar{x}$. Denote by
$(\forall k\in\mathbb{N})$ $x^{(k)}=(x^{(k)}_{i})_{i\in\operatorname{J}}$. In
view of the definition of $T$, $(Tx_{k})_{k\in\mathbb{N}}$ is also a sequence
in $\left]0,1\right[^{n}$. Suppose that $\operatorname{D}_{f}(x_{k},Tx_{k})\to
0$. Notice that

$\displaystyle\operatorname{D}_{f}(x_{k},Tx_{k})$
$\displaystyle=\sum^{n}_{i=1}\left(x^{(k)}_{i}\ln(x^{(k)}_{i})+(1-x^{(k)}_{i})\ln(1-x^{(k)}_{i})\right)-\sum^{n}_{i=1}\left((1-x^{(k)}_{i})\ln(1-x^{(k)}_{i})+x^{(k)}_{i}\ln(x^{(k)}_{i})\right)$
$\displaystyle\quad~{}-\sum^{n}_{i=1}\left(x^{(k)}_{i}-(1-x^{(k)}_{i})\right)\ln\left(\frac{1-x^{(k)}_{i}}{x^{(k)}_{i}}\right)$
$\displaystyle=-\sum^{n}_{i=1}\left(x^{(k)}_{i}-(1-x^{(k)}_{i})\right)\ln\left(\frac{1-x^{(k)}_{i}}{x^{(k)}_{i}}\right)\to-\sum^{n}_{i=1}\left(\bar{x}_{i}-(1-\bar{x}_{i})\right)\ln\left(\frac{1-\bar{x}_{i}}{\bar{x}_{i}}\right)$

Consider the function
$g:\mathbb{R}\to\mathbb{R}:t\mapsto-\left(t-(1-t)\right)\ln\left(\frac{1-t}{t}\right)$.
Because $(\forall t\in\left]0,1\right[\,)$ $g^{\prime\prime}(t)>0$ and
$g^{\prime}(\frac{1}{2})=0$, we know that $\left(\forall
t\in\left]0,1\right[\smallsetminus\\{\frac{1}{2}\\}\right)$ $g(t)>0$. Hence,
clearly,
$-\sum^{n}_{i=1}\left(\bar{x}_{i}-(1-\bar{x}_{i})\right)\ln\left(\frac{1-\bar{x}_{i}}{\bar{x}_{i}}\right)=0$
implies that
$\bar{x}=(\frac{1}{2})_{i\in\operatorname{J}}\in\operatorname{Fix}T$. Hence,
the assertion is true.

(iii)(c): The required results are trivial.

(iv): According to the definitions of $T$ and $f$, it is easy to see that

$\displaystyle(\forall x\in\operatorname{dom}f)(\forall
y\in\operatorname{dom}f)\quad
f(x)-f(Tx)=0\quad\text{and}\quad\Big{\langle}\nabla f(Ty),Tx-
Ty\Big{\rangle}=\Big{\langle}\nabla f(y),x-y\Big{\rangle}.$

(iv)(a): This is clear.

(iv)(b): This follows immediately from Definition 4.7 and Definition 2.1.

(iv)(c): Applying a proof similar to that of (iii)(b) and invoking (iv)(a), we
obtain the required result. ∎

### Convergence of forward Bregman circumcenter methods

Motivated by the Lemma 4.6, to find a point in the intersection
$\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}$, we prefer the backward
and forward Bregman circumcenter mappings induced by $\mathcal{S}$ rather than
the backward and forward Bregman pseudo-circumcenter mappings induced by
$\mathcal{S}$ with smaller fixed point sets. Moreover, notice that comparing
with the hypothesis of Fact 4.3(i) on the existence of the backward Bregman
circumcenter, we don’t have strong requirement for $f$ in the Fact 4.3(iv) on
the existence of the forward Bregman circumcenter.

Therefore, we consider only on the convergence of sequences of iterations
generated by forward Bregman circumcenter mappings in this work.

The _forward Bregman circumcenter method_ generates the sequence of iterations
of the forward Bregman circumcenter mapping.

According to [8, Theorem 4.3(ii)] and [9, Theorem 3.3(ii)], the circumcenter
mappings induced by finite sets of isometries (see [9, Definition 2.27] for
the exact definition), are special operators $G$ satisfying conditions in
Theorem 4.11 below.

######  Theorem 4.11.

Suppose that $f$ is Legendre. Let $\mathcal{C}$ be a nonempty subset of
$\operatorname{int}\operatorname{dom}f$. Suppose that
$G:\mathcal{C}\to\mathcal{C}$ satisfies that
$\operatorname{Fix}G\cap\operatorname{int}\operatorname{dom}f\neq\varnothing$
and $\operatorname{Fix}G$ is a closed convex subset of $\mathcal{H}$, and that
$(\forall x\in\mathcal{C}\cap\operatorname{int}\operatorname{dom}f)$ $(\forall
z\in\operatorname{Fix}G\cap\operatorname{int}\operatorname{dom}f)$
$G(x)=\overleftarrow{\operatorname{P}}^{f}_{\operatorname{aff}\,(\mathcal{S}(x))}z$.
Let $x\in\mathcal{C}\cap\operatorname{int}\operatorname{dom}f$. Then the
following statements hold.

  1. (i)

$(G^{k}x)_{k\in\mathbb{N}}$ is a well-defined sequence in
$\mathcal{C}\cap\operatorname{int}\operatorname{dom}f$.

  2. (ii)

$(G^{k}x)_{k\in\mathbb{N}}$ is forward Bregman monotone with respect to
$\operatorname{Fix}G$. Consequently, $\left(\forall
z\in\operatorname{Fix}G\cap\operatorname{int}\operatorname{dom}f\right)$
$\left(\operatorname{D}_{f}\left(G^{k}x,z\right)\right)_{k\in\mathbb{N}}$
converges.

  3. (iii)

$\lim_{k\to\infty}$ $\operatorname{D}_{f}\left(G^{k}x,G^{k+1}x\right)=0$.

  4. (iv)

Suppose that the Bregman Browder’s Demiclosedness Principle associated with
$f$ holds for $G$, and that all weak sequential cluster points of
$(G^{k}x)_{k\in\mathbb{N}}$ lie in $\operatorname{int}\operatorname{dom}f$.
Then $(G^{k}x)_{k\in\mathbb{N}}$ weakly converges to a point in
$\operatorname{int}\operatorname{dom}f\cap\operatorname{Fix}G$.

###### Proof.

Invoking the definition of the operator $G:\mathcal{C}\to\mathcal{C}$ and Fact
2.4,

$\displaystyle(\forall
y\in\mathcal{C}\cap\operatorname{int}\operatorname{dom}f)~{}(\forall
z\in\operatorname{Fix}G\cap\operatorname{int}\operatorname{dom}f)\quad
G(y)=\overleftarrow{\operatorname{P}}^{f}_{\operatorname{aff}\,(\mathcal{S}(y))}z\in\mathcal{C}\cap\operatorname{int}\operatorname{dom}f.$
(4.3)

Hence, use $y\in\operatorname{aff}\,(\mathcal{S}(y))$ and apply [12, Theorem
2.9(i)] with $U=\operatorname{aff}\,\left(\mathcal{S}(x)\right)$ to yield that

$\displaystyle(\forall
y\in\mathcal{C}\cap\operatorname{int}\operatorname{dom}f)~{}(\forall
z\in\operatorname{Fix}G\cap\operatorname{int}\operatorname{dom}f)\quad\operatorname{D}_{f}(y,z)=\operatorname{D}_{f}(y,G(y))+\operatorname{D}_{f}(G(y),z).$
(4.4)

(i): This is clear from 4.3 by induction.

(ii): Taking 4.4 and Definition 2.1 into account, we deduce that

$\displaystyle(\forall
y\in\mathcal{C}\cap\operatorname{int}\operatorname{dom}f)~{}\left(\forall
z\in\operatorname{int}\operatorname{dom}f\cap\operatorname{Fix}G\right)\quad\operatorname{D}_{f}(y,z)\geq\operatorname{D}_{f}\left(Gy,z\right).$
(4.5)

Employing (i) above and Definition 3.1, we know that
$(G^{k}x)_{k\in\mathbb{N}}$ is forward Bregman monotone with respect to
$\operatorname{Fix}G$, which, combining with Theorem 3.3(i), implies the last
assertion.

(iii): Let $z\in\operatorname{Fix}G\cap\operatorname{int}\operatorname{dom}f$.
For every $k\in\mathbb{N}$, substitute $y$ in 4.4 by $G^{k}x$ to deduce that
$\operatorname{D}_{f}\left(G^{k}x,z\right)=\operatorname{D}_{f}\left(G^{k}x,G^{k+1}x\right)+\operatorname{D}_{f}\left(G^{k+1}x,z\right)$
which implies that

$\displaystyle\sum_{k\in\mathbb{N}}\operatorname{D}_{f}\left(G^{k}x,G^{k+1}x\right)=\operatorname{D}_{f}\left(x,z\right)-\lim_{t\to\infty}\operatorname{D}_{f}\left(G^{t}x,z\right)<\infty,$

where the existence of the limit
$\lim_{t\to\infty}\operatorname{D}_{f}\left(G^{t}x,z\right)$ is from (ii).
Therefore, using the nonnegativity of the Bregman distance, we yield (iii).

(iv): Let $\bar{z}$ be a weak sequential cluster point of
$(G^{k}x)_{k\in\mathbb{N}}$. Then there exists a subsequence
$(G^{k_{j}}x)_{j\in\mathbb{N}}$ of $(G^{k}x)_{k\in\mathbb{N}}$ such that
$G^{k_{j}}x{\;\operatorname{\rightharpoonup}\;}\bar{z}$. Notice that, in view
of (iii) above, $\operatorname{D}_{f}\left(G^{k}x,G^{k+1}x\right)\to 0$. Due
to the assumption, $\bar{z}\in\operatorname{int}\operatorname{dom}f$. Hence,
utilizing the assumption that the Bregman Browder’s Demiclosedness Principle
holds for $G$, we observe that

$\displaystyle\begin{rcases}G^{k_{j}}x{\;\operatorname{\rightharpoonup}\;}\bar{z}\\\
\operatorname{D}_{f}\left(G^{k_{j}}x,G\left(G^{k_{j}}x\right)\right)\to
0\end{rcases}\Rightarrow\bar{z}\in\operatorname{Fix}G,$

which implies that all weak sequential cluster points of
$(G^{k}x)_{k\in\mathbb{N}}$ lie in
$\operatorname{Fix}G\cap\operatorname{int}\operatorname{dom}f$, since
$\bar{z}$ is an arbitrary cluster point of $(G^{k}x)_{k\in\mathbb{N}}$.
Therefore, via Theorem 3.3(iv), $(G^{k}x)_{k\in\mathbb{N}}$ weakly converges
to some point in
$\operatorname{int}\operatorname{dom}f\cap\operatorname{Fix}G$. ∎

The following Theorem 4.12(vi)(a) and Theorem 4.12(vi)(b) reduce to the
beautiful [8, Theorem 3.17] and [9, Theorem 4.7], respectively, when
$f=\frac{1}{2}\lVert\cdot\rVert^{2}$. Notice that the circumcenter mappings
studied in [9], [10], [11], and [21] under the Euclidean distance are all
single-valued operators, and that all of the examples of backward and forward
Bregman (pseudo)-circumcenters presented in [12] are singletons. So, our
assumption “$\overrightarrow{CC{missing}}_{\mathcal{S}}$ is at most single-
valued on $\operatorname{int}\operatorname{dom}f$” in the following Theorem
4.12 is not too weird. In addition, it is clear that if
$f=\frac{1}{2}\lVert\cdot\rVert^{2}$, then the following condition 4.6 is a
direct result from the triangle inequality.

######  Theorem 4.12.

Suppose that $f$ is Legendre, that
$\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)$
is nonempty closed and convex, that $(\forall i\in\operatorname{I})$ $T_{i}$
is Bregman isometric w.r.t. $f$, and that
$\overrightarrow{CC{missing}}_{\mathcal{S}}$ is at most single-valued on
$\operatorname{int}\operatorname{dom}f$. Let
$y\in\operatorname{int}\operatorname{dom}f$. Then the following statements
hold.

  1. (i)

$(\forall x\in\operatorname{dom}f)$
$\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\subseteq\overrightarrow{E}_{f}(\mathcal{S}(x))$.

  2. (ii)

$\left(\forall
z\in\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)\right)$
$\overrightarrow{CC{missing}}_{\mathcal{S}}y=\overleftarrow{\operatorname{P}}^{f}_{\operatorname{aff}\,(\mathcal{S}(y))}(z)\in\operatorname{int}\operatorname{dom}f$.

  3. (iii)

$(\overrightarrow{CC{missing}}_{\mathcal{S}}^{k}y)_{k\in\mathbb{N}}$ is a
well-defined sequence in $\operatorname{int}\operatorname{dom}f$.

  4. (iv)

$(\overrightarrow{CC{missing}}_{\mathcal{S}}^{k}y)_{k\in\mathbb{N}}$ is
forward Bregman monotone with respect to
$\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}$. Consequently, $(\forall
z\in\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right))$
$\left(\operatorname{D}_{f}\left(\overrightarrow{CC{missing}}_{\mathcal{S}}^{k}y,z\right)\right)_{k\in\mathbb{N}}$
converges.

  5. (v)

$\lim_{k\to\infty}$
$\operatorname{D}_{f}\left(\overrightarrow{CC{missing}}_{\mathcal{S}}^{k}y,\overrightarrow{CC{missing}}_{\mathcal{S}}^{k+1}y\right)=0$.

  6. (vi)

Suppose that $(\forall i\in\operatorname{I})$ $T_{i}$ is $f$-demiclosed, and
that for every sequence $(u_{k})_{k\in\mathbb{N}}$ in
$\operatorname{int}\operatorname{dom}f$,

$\displaystyle\begin{rcases}\operatorname{D}_{f}(u_{k},\overrightarrow{CC{missing}}_{\mathcal{S}}u_{k})\to
0\\\ (\forall
i\in\operatorname{I})~{}\operatorname{D}_{f}(T_{i}u_{k},\overrightarrow{CC{missing}}_{\mathcal{S}}u_{k})\to
0\end{rcases}\Rightarrow(\forall
i\in\operatorname{I})~{}\operatorname{D}_{f}(u_{k},T_{i}u_{k})\to 0.$ (4.6)

Then the following hold.

    1. (a)

$\overrightarrow{CC{missing}}_{\mathcal{S}}$ is $f$-demiclosed.

    2. (b)

Suppose that all weak sequential cluster points of
$(\overrightarrow{CC{missing}}_{\mathcal{S}}^{k}y)_{k\in\mathbb{N}}$ lie in
$\operatorname{int}\operatorname{dom}f$. Then
$(\overrightarrow{CC{missing}}_{\mathcal{S}}^{k}y)_{k\in\mathbb{N}}$ weakly
converges to some point in
$\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)$.

###### Proof.

Because $(\forall i\in\operatorname{I})$ $T_{i}$ is Bregman isometric, we know
that $(\forall i\in\operatorname{I})$
$\left(\forall(x,y)\in\operatorname{dom}f\times\operatorname{int}\operatorname{dom}f\right)$
$\left(T_{i}x,T_{i}y\right)\in\operatorname{dom}f\times\operatorname{int}\operatorname{dom}f$,
and that

$\displaystyle(\forall
i\in\operatorname{I})~{}\left(\forall\\{x,y\\}\subseteq\operatorname{dom}f\times\operatorname{int}\operatorname{dom}f\right)~{}\operatorname{D}_{f}\left(T_{i}x,T_{i}y\right)=\operatorname{D}_{f}(x,y).$
(4.7)

(i): Let $x\in\operatorname{dom}f$ and
$z\in\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)$.
Then

$\displaystyle(\forall
i\in\operatorname{I})~{}\operatorname{D}_{f}(x,z)=\operatorname{D}_{f}(T_{i}x,z)$
$\displaystyle\Leftrightarrow(\forall
i\in\operatorname{I})~{}\operatorname{D}_{f}(x,z)=\operatorname{D}_{f}(T_{i}x,T_{i}z)\quad(\text{by
}z\in\cap_{j\in\operatorname{I}}\operatorname{Fix}T_{j})$
$\displaystyle\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{eq:Tisometry}}}{{\Leftrightarrow}}(\forall
i\in\operatorname{I})~{}\operatorname{D}_{f}(x,z)=\operatorname{D}_{f}(x,z).$

Hence, $z\in\overrightarrow{E}_{f}(\mathcal{S}(x))$ by the definition of
$\overrightarrow{E}_{f}(\mathcal{S}(x))$ presented in Definition 4.2.

(ii): Let
$z\in\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)$.
Because $y\in\operatorname{int}\operatorname{dom}f$, we see that
$y\in\operatorname{aff}\,\left(\mathcal{S}(y)\right)\cap\operatorname{int}\operatorname{dom}f\neq\varnothing$,
and that, by (i) above, $z\in\overrightarrow{E}_{f}(\mathcal{S}(y))$. Applying
Fact 4.3(iv) with $K$ replaced by $\mathcal{S}(y)$, we deduce that
$\overleftarrow{\operatorname{P}}^{f}_{\operatorname{aff}\,(\mathcal{S}(y))}(z)\in\overrightarrow{CC{}}(\mathcal{S}(y))$.
Combine this with Definition 4.4, Fact 2.4, and the assumption that
$\overrightarrow{CC{missing}}_{\mathcal{S}}$ is at most single-valued, to
yield that
$\overrightarrow{CC{missing}}_{\mathcal{S}}y=\overrightarrow{CC{}}(\mathcal{S}(y))=\overleftarrow{\operatorname{P}}^{f}_{\operatorname{aff}\,(\mathcal{S}(y))}(z)\in\operatorname{int}\operatorname{dom}f$.

(iii)$\&$(iv)$\&$(v): By Lemma 4.6(i),
$\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}=\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)$.
Hence, (ii) implies that

$\displaystyle(\forall x\in\operatorname{int}\operatorname{dom}f)~{}(\forall
z\in\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}\cap\operatorname{int}\operatorname{dom}f)\quad\overrightarrow{CC{missing}}_{\mathcal{S}}x=\overleftarrow{\operatorname{P}}^{f}_{\operatorname{aff}\,(\mathcal{S}(x))}(z)\in\operatorname{int}\operatorname{dom}f.$

Therefore, the required results follow directly from Theorem
4.11(i)$\&$(ii)$\&$(iii), respectively, with
$\mathcal{C}=\operatorname{int}\operatorname{dom}f$ and
$G=\overrightarrow{CC{missing}}_{\mathcal{S}}$.

(vi)(a): Suppose that $x\in\operatorname{int}\operatorname{dom}f$ and that
$(x_{k})_{k\in\mathbb{N}}\subseteq\operatorname{int}\operatorname{dom}f$ such
that $x_{k}{\;\operatorname{\rightharpoonup}\;}x$. Notice that, due to (ii)
above,
$\\{\overrightarrow{CC{missing}}_{\mathcal{S}}x_{k}\\}_{k\in\mathbb{N}}\subseteq\operatorname{int}\operatorname{dom}f$.
Assume that
$\operatorname{D}_{f}(x_{k},\overrightarrow{CC{missing}}_{\mathcal{S}}x_{k})\to
0$. In view of Definition 4.4,

$\displaystyle(\forall i\in\operatorname{I})~{}(\forall
k\in\mathbb{N})\quad\operatorname{D}_{f}\left(x_{k},\overrightarrow{CC{missing}}_{\mathcal{S}}x_{k}\right)=\operatorname{D}_{f}\left(T_{i}x_{k},\overrightarrow{CC{missing}}_{\mathcal{S}}x_{k}\right),$

which, connecting with
$\operatorname{D}_{f}(x_{k},\overrightarrow{CC{missing}}_{\mathcal{S}}x_{k})\to
0$ and applying 4.6 with $(\forall k\in\mathbb{N})$ $u_{k}=x_{k}$, implies
that $(\forall i\in\operatorname{I})$
$\operatorname{D}_{f}(x_{k},T_{i}x_{k})\to 0$. Let $i\in\operatorname{I}$.
Note that $T_{i}$ is $f$-demiclosed. So the results
$x_{k}{\;\operatorname{\rightharpoonup}\;}x$ and
$\operatorname{D}_{f}(x_{k},T_{i}x_{k})\to 0$ imply that
$x\in\operatorname{Fix}T_{i}$. Because $i\in\operatorname{I}$ is chosen
arbitrarily, by Lemma 4.6(i),
$x\in\operatorname{int}\operatorname{dom}f\cap\left(\cap_{i\in\operatorname{I}}\operatorname{Fix}T_{i}\right)=\operatorname{Fix}\overrightarrow{CC{missing}}_{\mathcal{S}}$.

(vi)(b): Bearing (vi)(a) in mind, we observe that the desired convergence
follows from Theorem 4.11(iv) with
$\mathcal{C}=\operatorname{int}\operatorname{dom}f$ and
$G=\overrightarrow{CC{missing}}_{\mathcal{S}}$. ∎

To end this work, we revisit the operator $T$ and the function $f$ in Example
4.10, which satisfy all requirements in Theorem 4.12. In view of Example 4.10,
the following example illustrates Theorem 4.12, and demonstrates that the
forward Bregman circumcenter method finds the desired fixed point by one
iterate. In particular, the following Example 4.13(v)$\&$(vi) suggest the one
step convergence of the forward Bregman circumcenter method.

######  Example 4.13.

Suppose that $\mathcal{H}=\mathbb{R}^{n}$. Define
$f:x\mapsto\sum^{n}_{i=1}x_{i}\ln(x_{i})+(1-x_{i})\ln(1-x_{i})$, with
$\operatorname{dom}f=\left[0,1\right]^{n}$, which is the Fermi-Dirac entropy.
Let $\Lambda$ be a subset of $\operatorname{J}$. Define
$T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ by $\left(\forall
x\in\mathbb{R}^{n}\right)$ $Tx:=(y_{i})^{n}_{i=1}$ where $(\forall
i\in\operatorname{J}\smallsetminus\Lambda)$ $y_{i}=x_{i}$ and $(\forall
i\in\Lambda)$ $y_{i}=(1-x_{i})$. Let
$x:=(x_{i})_{i\in\operatorname{J}}\in\left[0,1\right]^{n}$. Denote by
$\Phi:=\\{i\in\operatorname{J}~{}:~{}x_{i}\neq\frac{1}{2}\\}$. Then the
following statements hold.

  1. (i)

$\overrightarrow{E}_{f}(\mathcal{S}(x))=\left\\{(p_{i})_{i\in\operatorname{J}}\in\left]0,1\right[^{n}~{}:~{}0=\sum_{i\in\Lambda}(2x_{i}-1)\ln\left(\frac{p_{i}}{1-p_{i}}\right)\right\\}$.

  2. (ii)

$\overrightarrow{CC{missing}}_{\mathcal{S}}(x)=\left\\{(p_{i})_{i\in\operatorname{J}}\in\Big{(}\left]0,1\right[^{n}\cap\operatorname{aff}\,\\{x,1-x\\}\Big{)}~{}:~{}0=\sum_{i\in\Lambda}(2x_{i}-1)\ln\left(\frac{p_{i}}{1-p_{i}}\right)\right\\}$.
Moreover,

$\displaystyle\left\\{(p_{i})_{i\in\operatorname{J}}\in\Big{(}\left]0,1\right[^{n}\cap\operatorname{aff}\,\\{x,1-x\\}\Big{)}~{}:~{}(\forall
i\in\Lambda\smallsetminus\Phi)p_{i}=\frac{1}{2}\right\\}\subseteq\overrightarrow{CC{missing}}_{\mathcal{S}}(x);$
$\displaystyle\varnothing\neq\left\\{(p_{i})_{i\in\operatorname{J}}\in\Big{(}\left]0,1\right[^{n}\cap\operatorname{aff}\,\\{x,1-x\\}\Big{)}~{}:~{}(\forall
i\in\Lambda)p_{i}=\frac{1}{2}\right\\}\subseteq\operatorname{Fix}T\cap\operatorname{int}\operatorname{dom}f\cap\overrightarrow{CC{missing}}_{\mathcal{S}}(x).$

  3. (iii)

Suppose that $(\forall i\in\Lambda\smallsetminus\Phi)$ $x_{i}>\frac{1}{2}$ or
$(\forall i\in\Lambda\smallsetminus\Phi)$ $x_{i}<\frac{1}{2}$. Then

$\displaystyle\overrightarrow{CC{missing}}_{\mathcal{S}}(x)=\left\\{(p_{i})_{i\in\operatorname{J}}\in\Big{(}\left]0,1\right[^{n}\cap\operatorname{aff}\,\\{x,1-x\\}\Big{)}~{}:~{}(\forall
i\in\Lambda\smallsetminus\Phi)p_{i}=\frac{1}{2}\right\\}.$

  4. (iv)

Suppose that $(\forall i\in\Lambda\smallsetminus\Phi)$ $x_{i}>\frac{1}{2}$ or
$(\forall i\in\Lambda\smallsetminus\Phi)$ $x_{i}<\frac{1}{2}$, and that
$\Lambda\cap\Phi=\varnothing$. Then

$\displaystyle\overrightarrow{CC{missing}}_{\mathcal{S}}(x)=\left\\{(p_{i})_{i\in\operatorname{J}}\in\Big{(}\left]0,1\right[^{n}\cap\operatorname{aff}\,\\{x,1-x\\}\Big{)}~{}:~{}(\forall
i\in\Lambda)p_{i}=\frac{1}{2}\right\\}\subseteq\operatorname{Fix}T\cap\operatorname{int}\operatorname{dom}f.$

  5. (v)

Suppose that $\Lambda$ is a singleton, say $\Lambda:=\\{i_{0}\\}$. Then if
$x_{i_{0}}=\frac{1}{2}$, then
$\overrightarrow{CC{missing}}_{\mathcal{S}}(x)=\left]0,1\right[^{n}\cap\operatorname{aff}\,\\{x,1-x\\}$;
if $x_{i_{0}}\neq\frac{1}{2}$, then

$\displaystyle\overrightarrow{CC{missing}}_{\mathcal{S}}(x)=\left\\{(p_{i})_{i\in\operatorname{J}}\in\left]0,1\right[^{n}\cap\operatorname{aff}\,\\{x,1-x\\}~{}:~{}p_{i_{0}}=\frac{1}{2}\right\\}\subseteq\operatorname{Fix}T\cap\operatorname{int}\operatorname{dom}f.$

  6. (vi)

Suppose that $\mathcal{H}=\mathbb{R}$. Then

$\displaystyle\left(\forall
x\in\left[0,1\right]\smallsetminus\left\\{\frac{1}{2}\right\\}\right)\quad\overrightarrow{CC{missing}}_{\mathcal{S}}(x)=\left\\{\frac{1}{2}\right\\}=\operatorname{int}\operatorname{dom}f\cap\operatorname{Fix}T.$

Consequently, the forward Bregman circumcenter method converges to the desired
point in $\operatorname{int}\operatorname{dom}f\cap\operatorname{Fix}T$ in one
iterate if the initial point is not the desired solution

###### Proof.

(i): This is easy form the related definitions. Alternatively, we can deduce
the required result by applying [12, Example 4.8(ii)(a)] with
$\mathcal{S}=\\{x,Tx\\}$.

(ii): Based on Example 4.10(iii)(b),

$\displaystyle\operatorname{Fix}T\cap\operatorname{int}\operatorname{dom}f=\left\\{x\in\left]0,1\right[^{n}~{}:~{}(\forall
i\in\Lambda)x_{i}=\frac{1}{2}\right\\}.$

Hence, this is clear from Definition 4.4.

(iii): This follows immediately form (ii).

(iv)$\&$(v)$\&$(vi): The required results follows easily from (iii). ∎

We shall explore more particular examples of $T$ satisfying the requirements
in Theorem 4.12 under general Bregman distances associated with
$f\neq\frac{1}{2}\lVert\cdot\rVert^{2}$ in future work.

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# Raman spectra of hydrocarbons under extreme conditions of pressure and
temperature: a first-principles study

Rui Hou  Department of Physics, Hong Kong University of Science and
Technology, Hong Kong, China  Ding Pan<EMAIL_ADDRESS>Department of Physics
and Department of Chemistry, Hong Kong University of Science and Technology,
Hong Kong, China  HKUST Fok Ying Tung Research Institute, Guangzhou, China

###### Abstract

Hydrocarbons are of great importance in carbon-bearing fluids in deep Earth
and in ice giant planets at extreme pressure (P)-temperature (T) conditions.
Raman spectroscopy is a powerful tool to study the chemical speciation of
hydrocarbons; however, it is challenging to interpret Raman data at extreme
conditions. Here, we performed ab initio molecular dynamics simulations
coupled with the modern theory of polarization to calculate Raman spectra of
methane, ethane, and propane up to 48 GPa and 2000 K. Our method includes
anharmonic temperature effects. We studied the pressure and temperature
effects on the Raman bands, and identified the characteristic Raman modes for
the C-C and C-C-C bonds. Our result may help to interpret in-situ Raman data
of hydrocarbons at extreme P-T conditions, with important implications for
understanding the deep carbon cycle inside Earth and the compositions of ice
giant planets.

##  I introduction

Methane (CH4) and other light hydrocarbons are important components of reduced
carbon-hydrogen-oxygen (C-H-O) fluids inside deep Earth Manning  _et al._
(2013); Sephton and Hazen (2013). Methane is also a major constituent of deep
interiors of two ice giant planets in the solar system: Uranus and Neptune
Hubbard (1981); Podolak  _et al._ (1995). The abiotic hydrocarbons may
participate in the formation of petroleum deposits; this controversial
hypothesis can be dated back to the time of Dimitri Mendeleev Mendeleev
(1877); Sephton and Hazen (2013). The properties of hydrocarbons under extreme
pressure (P) and temperature (T) conditions play crucial roles in the deep
carbon cycle in Earth Sephton and Hazen (2013), and in shaping the structure
and dynamics of ice giant planets. Previously, it was assumed that
hydrocarbons were simply mixed with other small volatile molecules, e.g., H2O,
CO2, H2, in C-H-O fluids Zhang and Duan (2009), or with H2O and NH3 inside
Uranus and Neptune Podolak  _et al._ (1995). However, many recent theoretical
and experimental studies show that methane and other hydrocarbons may further
react at extreme P-T conditions (e.g., Ancilotto  _et al._ (1997); Hirai  _et
al._ (2009); Kolesnikov  _et al._ (2009); Gao  _et al._ (2010); Spanu  _et
al._ (2011a); Lobanov  _et al._ (2013)). Using ab initio molecular dynamics
(AIMD) simulations, Ancilotto et al. found that at 100 GPa and 4000 K, methane
dissociates into a mixture of methane, ethane (C2H6), and propane (C3H8)
Ancilotto  _et al._ (1997). Spanu et al. predicted that higher hydrocarbons
become thermodynamically more favored than methane at above 4 GPa and
1000$\sim$2000 K Spanu  _et al._ (2011b). Experiments with laser-heated
diamond anvil cells reported that ethane, propane, and higher hydrocarbons may
form out of methane at pressures higher than 2 GPa and temperatures above 1000
K Kolesnikov  _et al._ (2009); Hirai  _et al._ (2009).

Raman spectroscopy is the most widely used experimental tool to detect
chemical speciation in C-H-O fluids at _both_ high-P _and_ high-T conditions
Goncharov (2012). Raman spectra of vibrations provide fingerprints to identify
molecules and ions in in-situ experiments Goncharov (2012); Vandenabeele
(2013). However, it is very challenging to measure Raman spectra under extreme
P-T conditions, because many high energy levels are excited at high
temperatures and the signals from diamond anvils may interfere; for example,
the peaks near 1332 cm-1 and 2400 cm-1 are attributed to the first- and the
second-order scattering from diamond anvils, respectively Goncharov  _et al._
(2005). Thus, theoretically studying the P-T dependence of Raman bands is of
great importance to the interpretation of Raman spectra under extreme
conditions Lobanov  _et al._ (2013); Petrov (2018). In first-principles
calculations, we usually calculate the change of electronic polarizability
along a normal vibrational mode under the harmonic approximation to obtain the
Raman spectra of hydrocarbons in the gas or solid phase (e.g., Lin  _et al._
(2007, 2011); Ramya  _et al._ (2012)), where the temperature is treated as 0
K; however, for C-H fluids at the supercritical state, the anharmonic and
temperature effects can not be ignored Putrino and Parrinello (2002); Wan  _et
al._ (2013); Pan and Galli (2020).

Here, we performed AIMD simulations coupled with density functional
perturbation theory to calculate the Raman spectra of methane, ethane, and
propane at 13.4$\sim$48 GPa and 1445$\sim$2000 K. The electronic
polarizability was calculated on the fly along with AIMD simulations. We
studied how pressure and temperature affect Raman spectra, and assigned Raman
bands at the molecular scale. We discussed possible characteristic Raman
signals for the formation of higher hydrocarbons. Our work has important
implications for studying chemical speciation of hydrocarbons at extreme P-T
conditions found in deep Earth and ice giant planets.

##  II Methods

We carried out AIMD simulations using the Qbox package Gygi (2008); qbo (2008)
with the Born–Oppenheimer approximation and a time step of 5 a.u. We used the
Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional Perdew  _et al._
(1996) and the HSCV norm-conserving pseudopotentials Hamann  _et al._ (1979);
Vanderbilt (1985); pps . The kinetic energy cutoff of plane waves is 65 Ry,
which was increased to 145 Ry to calculate pressure. We applied the Bussi-
Donadio-Parrinello thermostat to control temperature Bussi  _et al._ (2007).
For methane, ethane, and propane, the simulation boxes with periodic boundary
conditions contain 56, 32, and 23 molecules, respectively.

We performed on-the-fly calculations using density functional perturbation
theory (DFPT) Baroni  _et al._ (2001) implemented in the Qbox package along
with AIMD simulations to calculate electronic polarizability, which were
conducted every 25 MD steps after about 10 ps equilibration. Each MD
trajectory is longer than 130 ps. We computed Raman spectra of supercritical
fluids using the Fourier transfer of the autocorrelation functions of
polarizabilities. The isotropic and anisotropic Raman spectra are calculated
respectively by

$\displaystyle R_{\mathrm{iso}}(\omega)\propto\frac{\hbar\omega}{k_{B}T}\int
dte^{-i\omega
t}\frac{\left\langle\bar{\alpha}^{\mathrm{}}(0)\bar{\alpha}^{\mathrm{}}(t)\right\rangle}{\left\langle\bar{\alpha}^{\mathrm{}}(0)\bar{\alpha}^{\mathrm{}}(0)\right\rangle}$
(1) $\displaystyle
R_{\mathrm{aniso}}(\omega)\propto\frac{2}{15}\frac{\hbar\omega}{k_{B}T}\int
dte^{-i\omega
t}\frac{\left\langle\Tr(\beta^{\mathrm{}}(0)\beta^{\mathrm{}}(t))\right\rangle}{\left\langle\Tr(\beta^{\mathrm{}}(0)\beta^{\mathrm{}}(0))\right\rangle},$
(2)

where $\hbar$ is the Planck constant, $\omega$ is the Raman frequency, $k_{B}$
is the Boltzmann constant, Tr stands for the trace of matrix. $\bar{\alpha}$
and $\beta$ denote the isotropic and anisotropic components of the
polarizability tensor $\alpha$, $\bar{\alpha}=\frac{1}{3}Tr\alpha$ and
$\beta=\alpha-\bar{\alpha}I$, respectively. In high-P and high-T experiments,
we measured unpolarized Raman spectra, which are obtained from the linear
combination of Eqs. (1) and (2):

$R_{unpol}=R_{iso}+\frac{7}{4}R_{aniso}.$ (3)

We further smoothed Raman spectra using Gaussian broadening, whose full width
at half maximum is 25 cm-1.

For the gas and solid phases, DFPT calculations were carried out using the
Quantum Espresso packageGiannozzi  _et al._ (2009). For methane, ethane, and
propane in the gas phase, the mean absolute error of calculated Raman
frequencies is 23 cm-1 compared with experimental values, indicating that our
computational setups are reliable.

##  III Results and Discussion

Fig. 1 shows the mean squared displacements of methane, ethane, and propane at
13.4 GPa, 1445 K; 48 GPa, 1445 K; and 48 GPa, 2000 K. All the alkanes are in
the supercritical state, except methane at 48 GPa and 1445 K, which is an
amorphous solid. It is interesting to see that methane freezes at 1445 K,
whereas propane does not. At ambient pressure, the melting point of methane is
90.7 K, while propane melts at 85.5 K Thalladi and Boese (2000). In fact, the
melting point of propane is unusually lower than other n-alkanes, which may be
due to the low packing efficiency of propane molecules Thalladi and Boese
(2000).

In our simulations, a few C-H bonds broke temporarily, but we did not see any
new chemical bond form. Experimentally, Nellis et al. reported no chemical
reactions in shock wave experiments up to 26 GPa and 3200 K Nellis  _et al._
(2001), whereas Lobanov et al. found that heavier alkanes and unsaturated
hydrocarbons may form in laser-heated diamond anvil cells above 1500 K Lobanov
_et al._ (2013). Theoretically, Spanu et al. found that the mixtures of higher
hydrocarbons and hydrogen become thermodynamically more stable than methane at
above 4 GPa and 1000$\sim$2000 K using AIMD simulations and free energy
calculations, but without metal catalysts or unsaturated diamond surfaces they
did not see chemical reactions in that P-T range Spanu  _et al._ (2011b).
Thus, we conclude that activation barriers of alkanes may be too high, so we
could not see chemical reactivity within limited simulation time.

Fig. 2 shows the unpolarized Raman spectra of hydrocarbons, from which we know
how Raman bands change with pressure and temperature. The C-H bond stretching
region is between 2800 and 3200 cm-1. The stretching band downshifts with
increasing temperature or decreasing pressure. For methane at 48 GPa and
1445$\sim$2000 K, the stretching band splits into two peaks, which was found
as an indicator of orientational ordering of molecules in compressed methane
crystals at least up to 30 GPa Bini  _et al._ (1995); Bini and Pratesi (1997),
suggesting that CH4 molecules may have some orientational correlation under
such extreme P-T conditions.

Fig. 3 shows the angle distribution of two C-H bonds respectively from two
nearest methane molecules. For the crystalline methane (space group:
$P2_{1}/c$ Lin  _et al._ (2011)), the angle distribution has two main peaks at
70.5∘ and 109.5∘, and the former peak is taller than the latter one. In gas
phase, the angle is randomly distributed, so the probability density function
is $\frac{\pi}{360^{\circ}}\sin\theta$. At 48 GPa and 2000 K, the angle
distribution becomes asymmetric, different from the $\sin\theta$ function,
indicating a certain angular correlation between neighboring molecules.

The C-H stretching bands of hydrocarbons largely overlap, whereas the Raman
bands in the low frequency region between 600 and 2000 cm-1 show distinct
peaks for methane, ethane, and propane, so the Raman modes in this region are
often used to identify hydrocarbons in experiment Buldakov  _et al._ (2013).
Methane only shows one peak between 1400 and 1600 cm-1, which upshifts with
increasing temperature. For an isolated methane molecule, the calculated Raman
active mode in this range appears at $\sim$1505 cm-1, which corresponds to the
degenerate deformation $\nu_{2}$ Shimanouchi  _et al._ (1972). We also
performed the Fourier transform of autocorrelation functions of C-H bonds and
H-C-H angles, respectively:

$\displaystyle A_{CH}(w)=\int
dte^{-iwt}\frac{\left\langle\dot{d}_{CH}(0)\dot{d}_{CH}(t)\right\rangle}{\left\langle\dot{d}_{CH}(0)\dot{d}_{CH}(0)\right\rangle}$
(4) $\displaystyle A_{HCH}(w)=\int
dte^{-iwt}\frac{\left\langle\dot{\theta}_{HCH}(0)\dot{\theta}_{HCH}(t)\right\rangle}{\left\langle\dot{\theta}_{HCH}(0)\dot{\theta}_{HCH}(0)\right\rangle},$
(5)

where $d_{CH}$ is the C-H bond length and $\theta_{HCH}$ is the H-C-H bond
angle. As shown in Fig. 4(A), there is no C-H stretching mode between 1000 and
1800 cm-1. The H-C-H angle spectrum has two peaks: $\nu_{4}$ at 1269 cm-1 and
$\nu_{2}$ at 1511 cm-1. We can not identify the $\nu_{4}$ mode in the
calculated Raman spectrum in Fig. 4(A), because the Raman cross section of
$\nu_{2}$ is about 28 times as much as that of $\nu_{4}$ for an isolated CH4
molecule.

Fig. 4 (B) shows the Raman spectra of ethane between 800 and 1800 cm-1. For an
isolated ethane molecule, there are two notable Raman modes in our
calculation: $\nu_{3}$ at 987 cm-1 and $\nu_{8}$ at 1455 cm-1, corresponding
to the C-C stretching and CH3 degenerate deformation modes, respectively
Shimanouchi  _et al._ (1972). At 48 GPa and 1445 K, both of the peaks upshift.
In addition to the C-H bond and H-C-H angle spectra, we also calculated the
C-C bond spectrum:

$A_{CC}(w)=\int
dte^{-iwt}\frac{\left\langle\dot{d}_{CC}(0)\dot{d}_{CC}(t)\right\rangle}{\left\langle\dot{d}_{CC}(0)\dot{d}_{CC}(0)\right\rangle},$
(6)

where $d_{CC}$ is the C-C bond length. It has a main peak at 1145 cm-1, which
overlaps with the $\nu_{3}$ band, indicating that the $\nu_{3}$ mode is indeed
the C-C stretching mode. The $\nu_{8}$ mode is found in the vicinity of the
H-C-H angle spectrum, suggesting that the CH3 degenerate deforming is relevant
to the variation of H-C-H angles. Comparing Fig. 4 (A) with (B), we found that
the $\nu_{2}$ band of methane overlaps with the $\nu_{8}$ band of ethane, and
the $\nu_{3}$ band of ethane is missing in the Raman spectra of methane, so
this band can be used as a characteristic signal for the C-C bond formation
under high P-T conditions. The $\nu_{3}$ mode upshifts with increasing
pressure, and downshifts with increasing temperature. The pressure effect is
more obvious than the temperature effect in the P-T range studied here, as
shown in Fig. 2.

Fig. 4 (C) shows the Raman spectra of propane between 800 and 1800 cm-1. In
our calculation, an isolated propane molecule has notable Raman modes as
follows: C-C stretching mode $\nu_{8}$ at 860 cm-1, C-C stretching mode
$\nu_{20}$ at 1048 cm-1, CH3 rocking mode $\nu_{7}$ at 1137 cm-1, CH2 twisting
mode $\nu_{12}$ at 1274 cm-1, and CH3 degenerate deforming mode $\nu_{11}$ at
1435 cm-1 Shimanouchi  _et al._ (1972). At 48 GPa and 1445 K, we can identify
the $\nu_{12}$ and $\nu_{11}$ modes in the H-C-H angle spectrum. The C-C bond
spectrum has three peaks: $\nu_{8}$, $\nu_{20}$, and $\nu_{7}$. We also
calculated the C-C-C angle ($\theta_{CCC}$) spectrum:

$A_{CCC}(w)=\int
dte^{-iwt}\frac{\left\langle\dot{\theta}_{CCC}(0)\dot{\theta}_{CCC}(t)\right\rangle}{\left\langle\dot{\theta}_{CCC}(0)\dot{\theta}_{CCC}(0)\right\rangle},$
(7)

which has one peak overlapping with the $\nu_{20}$ mode, indicating that this
CC stretching mode also considerably changes the C-C-C bond angle. The
$\nu_{8}$ band does not show up in the spectra of methane and ethane, so it
can be used to verify the formation of the C-C-C bonds.

We further divided Raman spectra into inter- and intramolecular contributions
with the help of maximally localized Wannier functions Gygi  _et al._ (2003),
which provide local molecular orbitals to partition electron density into
molecules Marzari  _et al._ (2012). The effective molecular polarizability,
$\alpha_{mol}^{eff}$, was obtained by:

$\alpha_{mol}^{eff}\vec{E}=-e\int_{\Omega}\vec{r}\Delta\rho_{mol}d\vec{r},$
(8)

where $\vec{E}$ is the macroscopic electric field, $e$ is the elementary
charge, $\Delta\rho_{mol}$ is the electron polarization density of the
molecule induced by the electric field, and the integral is over the whole
simulation box ($\Omega$).

The autocorrelation functions in Eqs. (1) and (2) are respectively written as,

$\displaystyle\langle\bar{\alpha}(0)\bar{\alpha}(t)\rangle$
$\displaystyle=\left\langle\sum_{\genfrac{}{}{0.0pt}{2}{m=1}{m\neq
n}}^{N_{\mathrm{mol}}}\sum_{n=1}^{N_{\mathrm{mol}}}\bar{\alpha}_{\mathrm{mol},m}^{\mathrm{eff}}(0)\bar{\alpha}_{\mathrm{mol},n}^{\mathrm{eff}}(t)+\sum_{m=1}^{N_{\mathrm{mol}}}\bar{\alpha}_{\mathrm{mol},m}^{\mathrm{eff}}(0)\bar{\alpha}_{\mathrm{mol},m}^{\mathrm{eff}}(t)\right\rangle$
(9) $\displaystyle\langle\Tr(\beta(0)\beta(t))\rangle$
$\displaystyle=\left\langle\sum_{\genfrac{}{}{0.0pt}{2}{m=1}{m\neq
n}}^{N_{\mathrm{mol}}}\sum_{n=1}^{N_{\mathrm{mol}}}\Tr(\beta_{\mathrm{mol},m}^{\mathrm{eff}}(0)\beta_{\mathrm{mol},n}^{\mathrm{eff}}(t))+\sum_{m=1}^{N_{\mathrm{mol}}}\Tr(\beta_{\mathrm{mol},m}^{\mathrm{eff}}(0)\beta_{\mathrm{mol},m}^{\mathrm{eff}}(t))\right\rangle$
(10)

where $N_{\mathrm{mol}}$ is the total number of hydrocarbon molecules. We
applied the Fourier transform on the first and second terms of Eqs. (9)(10) to
get the inter- and intramolecular Raman spectra, respectively.

Fig. 5 shows the inter- and intramolecular Raman spectra in the high frequency
region. The intensities of intermolecular contributions are much larger than
the intramolecular ones, indicating strong molecular interactions under
extreme P-T conditions. The intramolecular spectra have only one main peak,
whereas there appears to be a few peaks in the intermolecular spectra, which
may come from multiple intermolecular couplings between hydrocarbon molecules.

##  IV Conclusion

In summary, we calculated the unpolarized Raman spectra of methane, ethane and
propane at 13.4$\sim$48 GPa and 1445$\sim$2000 K using ab initio molecular
dynamics simulations coupled with density functional perturbation theory. Our
method considers the Raman selection rule and anharmonic temperature effects.
We monitored how pressure and temperature change the Raman bands, and
understood the Raman bands at the molecular scale. Particularly, we identified
the characteristic Raman bands for the formation of C-C and C-C-C bonds. We
found that the C-H stretching bands of supercritical hydrocarbons in the high-
frequency region split due to the intermolecular interactions. Our results may
help to interpret in-situ Raman measurements of hydrocarbons at extreme P-T
conditions, which are of great importance in understanding the deep carbon
cycle inside Earth and the compositions of ice giant planets.

##  V acknowledgments

###### Acknowledgements.

D.P. acknowledges support from the Croucher Foundation through the Croucher
Innovation Award, Hong Kong Research Grants Council (Projects ECS-26305017 and
GRF-16307618), National Natural Science Foundation of China (Project 11774072
and Excellent Young Scientists Fund), and the Alfred P. Sloan Foundation
through the Deep Carbon Observatory. Part of this work was carried out using
computational resources from the National Supercomputer Center in Guangzhou,
China.

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Figure 1: Mean squared displacements of carbon atoms in (A) methane, (B)
ethane, and (C)propane at extreme P-T conditions. Figure 2: Unpolarized Raman
spectra of (A) methane, (B) ethane, and (C) propane at extreme P-T conditions.
The Raman intensity is multiplied by 10 below 2000 cm-1. Figure 3: Angle
distribution of two C-H bonds respectively from two nearest methane molecules.
Three methane phases are compared: gas, supercritical fluid (SCF) at 48 GPa
and 2000 K, and solid (phase group: $P2_{1}/c$). The probability densities of
the gas and liquid phases are multiplied by 10.  Figure 4:  Raman band
assignments. The Raman, C-H bond ($d_{CH}$), H-C-H angle ($\theta_{HCH}$), C-C
bond ($d_{CC}$), C-C-C angle ($\theta_{CCC}$) spectra are compared. The Raman
spectra of methane, ethane, and propane are multiplied by 0.333, 0.200, and
0.067, respectively. The pressure is 48 GPa and the temperature is 1445 K.
Figure 5: Intramolecular ( left panel ) and intermolecular ( right panel )
contributions to unpolarized Raman spectra at extreme P-T conditions: (A)
methane, (B) ethane, and (C) propane.