The Quasi-Newton Method for the Composite Multiobjective Optimization Problems
111This research was partially supported by the National Natural Science
Foundation of China (12271071, 11991024), the Team Project of Innovation
Leading Talent in Chongqing (CQYC20210309536), Contract System Project of
Chongqing Talent Plan (cstc2022ycjh-bgzxm0147), the Chongqing University
Innovation Research Group Project (CXQT20014) and the Basic and Advanced
Research Project of Chongqing ( cstc2021jcyj-msxmX0300).

Jian-Wen Peng222 Corresponding author. School of Mathematical Sciences,
Chongqing Normal University, Chongqing 401331, China. e-mail:
<EMAIL_ADDRESS>and Jen-Chih Yao333 Department of Applied Mathematics,
National Sun Yat-sen University, Kaohsiung, Taiwan 804. e-mail:
<EMAIL_ADDRESS>
Abstract In this paper, we introduce several new quasi-Newton methods for the
composite multiobjective optimization problems (in short, CMOP) with Armijo
line search. These multiobjective versions of quasi-Newton methods include
BFGS quasi-Newnon method, self-scaling BFGS quasi-Newnon method, and Huang
BFGS quasi-Newnon method. Under some suitable conditions, we show that each
accumulation point of the sequence generated by these algorithms, if exists,
is both a Pareto stationary point and a Pareto optimal point of (CMOP).

Keywords Composite Multiobjective optimization problem$~{}\cdot~{}$ quasi-
Newton method$~{}\cdot~{}$Pareto stationarity$~{}\cdot~{}$ Convergence
analysis

Mathematical Subject Classification 90C29, 90C53, 49M15

##  1 Introduction

In this work, let us consider the following composite multiobjective
optimization problems (in short, CMOP):

$\displaystyle\min\quad F(x)$ (1)
$\displaystyle\begin{array}[]{r@{\quad}r}s.t.&x\in{\mathbb{R}^{n}},\\\
\end{array}$

where $F:{\mathbb{R}^{n}}\to\mathbb{R}$ is a vector-valued function with
$F:={({f_{1}},..,{f_{\rm{m}}})^{T}}$ and $T$ denotes transpose. We assume that
each ${f_{i}}:{\mathbb{R}^{n}}\to\mathbb{R}$ is defined by

$\displaystyle{f_{i}}(x):={g_{i}}(x)+{h_{i}}(x),~{}~{}~{}i=1,...,m,$ (2)

where ${g_{i}}{\rm{:}}{\mathbb{R}^{n}}\to\mathbb{R}$ is a twice continuously
differentiable strongly convex function,
${h_{i}}:{\mathbb{R}^{n}}\to\mathbb{R}$ is convex but not necessarily
differentiable. It is worthy noting that if $h_{i}(x)\equiv 0$ for all
$x\in{\mathbb{R}^{n}}$ and $i=1,2,...,m$, then the above (CMOP) (i.e., (1)
with (2) ) reduces to the multiobjective optimization problems studied in [13,
12, 4, 6, 8, 11, 19].

Scalarization approach is one of the most popular strategies to solve the
multiobjective optimization problem, which transforms the multiobjective
optimization problem into one or several parameterized single-objective
problem (see [9, 10, 17] and the references therein). In general, the
converted problem and the primal multiobjective optimization problem enjoy
same optimal solutions under certain conditions. Nevertheless, Fliege el al.
[11] pointed out that the parameters in scalarization method are not known in
advance and the selection of parameters may result in unbounded scalar
problems even if the original multiobjective optimization problem has
solutions. In order to cope with these limitations, the descent methods for
multiobjective optimization problems has attracted wide attention in the
optimization field (see [13]). There are many kinds of descent methods for
multiobjective optimization problems, for examples, the steepest descent
method [12], the projection gradient method [4], the proximal point method
[6], the Newton’s method [11], quasi-Newton method [19, 20, 15], and the
subgradient method [8]. Recent years, some authors introduced and researched
the descent methods for the composite multiobjective optimization problem
(1.1), for examples, the proximal gradient method [22] , the Newton’s method
[1], proximal Newton methods [23], proximal quasi-Newton methods [18].
Inspired by above ideas, the main purpose of this paper is to introduce a new
quasi-Newton methods for (CMOP) in the case of Armijo line search.

The main contents of this paper are as follows: In Section 2, we give some
notations and some concepts about Pareto optimality and Pareto stationarity.
In Section 3, we propose BFGS quasi-Newnon method, self-scaling BFGS quasi-
Newnon method and Huang BFGS quasi-Newnon method for the (CMOP) with the
Armijo line search. We prove the global convergence of the proposed algorithms
in Section 4. In Section 5, some numerical experiments are also carried out to
verify the effectiveness of the proposed quasi-Newton methods and to show that
multiobjective version of the Huang BFGS proximal quasi-Newnon method is the
most effective among the gradient method, Newton method and the proposed
quasi-Newton methods for (CMOP) with an Armijo line search.

##  2  Preliminaries

Throughout this paper, $\mathbb{N}$ denotes the set of positive integers,
$\mathbb{R}^{n}$ denotes the $n$-dimentional Euclidean space. The Euclidean
norm in ${\mathbb{R}^{n}}$ will be denoted by $\left\|\cdot\right\|$. For two
vectors $u$ and $v$ in $\mathbb{R}^{m}$, by $u\leq v$, we mean that
${u_{i}}\leq{v_{i}}$ for all $i=1,...,m.$ by $u<v$, we mean that
${u_{i}}<{v_{i}}$ for all $i=1,...,m.$

We recall that the function $g:{\mathbb{R}^{n}}\to\mathbb{R}$ is strongly
convex or uniformly convex if there exists $\eta>0$ such that

$g(\lambda x+(1-\lambda)y)\leq\lambda
g(x)+(1-\lambda)g(y)-\frac{1}{2}\lambda(1-\lambda)\eta\left\|{x-y}\right\|^{2},$

for any $x,y\in{\mathbb{R}^{n}}$ and $\lambda\in[0,1]$ (see [16]). It is well
known that if $g:{\mathbb{R}^{n}}\to\mathbb{R}$ is twice continuously
differentiable, then $g$ is strongly convex if and only if there exists
$\eta>0$ such that ${\nabla^{2}}g(x)\succeq\eta I$, for any
$x\in{\mathbb{R}^{n}}$, where ${\nabla^{2}}g(x)$ denote the Hessian matrix of
$f$ at $x$. Hence if ${g}$ is strongly convex, then the Hessian matrix
${\nabla^{2}}{g}(x)$ is positive definite for any $x\in{\mathbb{R}^{n}}$. It
is clearly that the strong convexity of $g$ implies its strict convexity and
usual convexity.

Let $f{\rm{:}}{\mathbb{R}^{n}}\to\mathbb{R}\cup\\{+\infty\\}$, and let $x\in
dom(f):=\\{x\in\mathbb{R}^{n}:f(x)<+\infty\\}$. Then the directional
derivative of $f$ at $x$ in the direction $d\in{\mathbb{R}^{n}}$ is defined to
be the limit

$f^{\prime}(x;d)=\mathop{lim}\limits_{\alpha\to{0^{+}}}\frac{{f(x+\alpha
d)-f(x)}}{\alpha},$

if it exists (see [21]). Clearly, if $f$ is differentiable at $x$, then
$f^{\prime}(x;d)=\nabla f{(x)^{T}}d$.

Definition 2.1 [13, 12] Recall that ${x^{*}}\in{{\mathbb{R}^{n}}}$ is an
efficient point or a Pareto optimum of (CMOP), if there is no
$x\in{{\mathbb{R}^{n}}}$ such that $F(x)\leq F({x^{*}})$ and $F(x)\neq
F({x^{*}})$. The set of all Pareto optimal values is called Pareto frontier.
Likewise, ${x^{*}}\in{{\mathbb{R}^{n}}}$ is a weakly efficient point or a
weakly Pareto optimum of (CMOP), if there is no $x\in{\mathbb{R}^{n}}$ such
that $F(x)<F({x^{*}})$.

It’s well known that the efficient point of (CMOP) is also a weakly efficient
point of (CMOP), and the converse is not true in general.

Definition 2.2 [22] We say that $\bar{x}\in{\mathbb{R}^{n}}$ is Pareto
stationary (or critical) of (CMOP), if and only if

$\mathop{\max}\limits_{i=1,...,m}{f^{\prime}_{i}}(\bar{x};d)\geq 0~{}~{}{\rm
for~{}all~{}}~{}d\in{\mathbb{R}^{n}}.$

It is worthy to noting that Definition 2.2 generalizes the corresponding ones
in [12].

We recall the following important result about the relationship of the weakly
efficient point (efficient point) and the Pareto stationary point of (CMOP):

Lemma 2.1 [11] (1) If $x\in{R^{n}}$ is a weakly efficient point of (CMOP) then
$x$ is Pareto stationary.

(2) Let every component ${f_{i}}$ of $F$ be convex. If $x\in{R^{n}}$ is a
Pareto stationary point of (CMOP), then $x$ is a weakly efficient point of
(CMOP).

(3) Let every component ${f_{i}}$ of $F$ be strictly convex. If $x\in{R^{n}}$
is a Pareto stationary point of (CMOP), then $x$ is an efficient point of
(CMOP).

##  3  Quasi-Newton methods for (CMOP)

In this section, we consider a new quasi-Newton method for (CMOP) with an
Armijo line search. Prior to that, we need the following result, which was
shown in [1].

Lemma 3.1 Suppose that $0\in
Co{\mathop{\cup}\limits_{j\in\\{{i=1,...,m}\\}}\partial f_{j}(x^{*})}$ for
some $x^{*}\in{\mathbb{R}^{n}}$, then $x^{*}$ is a critical point of the
(CMOP).

For $i=1,2,...,m,$ let $\nabla{g_{i}}(x)$ denote the gradient of ${g_{i}}$ at
$x$, ${B_{i}}(x)$ be an an approximation of ${\nabla^{2}}{g_{i}}(x)$, which
satisfies the following assumption:

Condition 3.1 For any fixed $x\in\mathbb{R}^{n}$ and for each
$i\in\\{1,2,...,m\\}$, there exists a constant $\sigma_{i}$ such that

$z^{T}{B_{i}}(x)z\geq\sigma_{i},\forall z\in\mathbb{R}^{n}.$

Remark 3.1 (i) Condition 3.1 implies that for any $x\in\mathbb{R}^{n}$ and for
each $i\in\\{1,2,...,m\\}$, there exists a constant $\sigma$ such that

$z^{T}{B_{i}}(x)z\geq\sigma,\forall z\in\mathbb{R}^{n},$

where $\sigma=\mathop{\min}\limits_{i=1,...,m}\sigma_{i}$.

(ii) If ${B_{i}}(x)=\nabla^{2}g_{i}(x)$, then Condition 3.1 holds true
automatically because the strong convexity of $g_{i}$.

Now, We define the function
$\theta(.,.):{\mathbb{R}^{n}}\times{\mathbb{R}^{n}}\to\mathbb{R}$ by

$\displaystyle{\theta(x,d):=\mathop{\max}\limits_{i=1,...,m}\theta_{i}(x,d)},$
(3)

where
$\theta_{i}(x,d)=\nabla{g_{i}}{(x)^{T}}d+\frac{1}{2}{d^{T}}{B_{i}}(x)d+{h_{i}}(x+d)-{h_{i}}(x).$

It is easy to see that for any fixed $x$, $\theta(x,d)$ is condinuous on $d$
since $\theta_{i}(x,d)$ is condinuous on $d$ for each $i\in\\{1,2,...,m\\}$.
It follows from Condition 3.1 and the convexity of ${h_{i}}$, we know that for
any fixed $x$ and for each $i\in\\{1,2,...,m\\}$, $\theta_{i}(x,d)$ is
$\sigma_{i}$-strongly convex in $d$. Therefore, for any fixed $x$,
$\theta(x,d)$ is $\sigma$-strongly convex in $d$ and $\theta(x,\bm{0})=0$,
where $\sigma=\mathop{\min}\limits_{i=1,...,m}\sigma_{i}$. We would like to
define the quasi-Newton direction at an iteration $k$ as
${d_{QN}^{k}}={d}({x^{k}})$, where

$\displaystyle{d}(x):=\mathop{argmin}\limits_{d\in{\mathbb{R}^{n}}}{\theta(x,d)}.$
(4)

For any fixed $x$, we can rewrite problem (4) as the following subproblem
$\Phi(x)$, which is to find a suitable descent direction of the (CMOP):

$\displaystyle\Phi(x):\mathop{\min_{d\in{\mathbb{R}^{n}}}}\theta(x,d).$ (5)

Remark 3.2 (i) For fixed $x\in{\mathbb{R}^{n}}$, it follows from the strongly
convexity of ${\theta(x,d)}$ in $d$ that (5) (i.e., $\Phi(x)$) has a unique
solution ${d}(x)$. Moreover, the optimal value of $\Phi(x)$ is denoted by
$\alpha(x)$, i.e.,

$\displaystyle\alpha(x)=\mathop{\min_{d\in{\mathbb{R}^{n}}}}\theta(x,d)=\theta(x,{d}(x)).$
(6)

(ii) For fixed point $x\in{\mathbb{R}^{n}}$, because of $\theta(x,\bm{0})=0$,
we have ${\theta(x,d(x))}\leq 0.$

Remark 3.3 (i) ${\theta(x,d)}$ is denoted by ${\theta_{x}}(d)$ in [18] and
$\theta(x,d)$ is exactly the ${\varphi_{\omega,x}}(d)$ in [18] with
$\omega=0$.

(ii) If for $i=1,2,...,m$, ${B_{i}}(x)$ are replaced by
${\nabla^{2}}{g_{i}}(x)$, then $\theta(x,d)$ reduces to $Q(x,d)$ in [1] and
${d_{QN}^{k}}={d}({x^{k}})$ defined by (5) reduces to exactly the Newton
direction at an iteration $k$ in [1] which is defined as
${d_{N}^{k}}=:\mathop{argmin}\limits_{d\in{\mathbb{R}^{n}}}Q(x^{k},d)$.

We recall an important property of ${\theta(x,.)}$ as follows:

Lemma 3.2 ( see [18]) For fixed point $x\in{\mathbb{R}^{n}}$ and for all
$d\in{\mathbb{R}^{n}}$, the following equality holds:

$\theta^{\prime}((x,\bm{0});d)=\mathop{\max}\limits_{i=1,...,m}{f^{\prime}}_{i}(x;d).$

Denote $I(x,d)=\\{j\in\\{1,2,...,m\\}:\theta(x,d)=\theta_{j}(x,d)\\}.$ It
follows that
${d}(x)=\mathop{argmin}\limits_{d\in{\mathbb{R}^{n}}}{\theta(x,d)}$ and
$\alpha(x)=\theta(x,{d}(x))\leq\theta(x,\bm{0})=0$ that for every
$x\in{\mathbb{R}^{n}}$, $d(x)$ is a solution of $\Phi(x)$). Therefore,

$\bm{0}\in\partial_{d}\theta(x,{d}(x)).$

It follows from Corollary 3.5 in [3] that there exist
$w\in{\mathbb{R}_{+}^{|I(x,d(x))|}}$,
$\xi_{j}\in\partial_{d}h_{j}(x+d(x)),j\in I(x,d(x))$ such that the following
conditions hold:

$\displaystyle\sum_{j\in I(x,d(x))}w_{j}=1.$ (7)

$\displaystyle\sum_{j\in I(x,d(x))}w_{j}(\nabla
g_{j}(x)+B_{j}(x)d(x)+\xi_{j})=0.$ (8)

Let $\Xi_{m}:=\\{1,2,...,m\\}$ and substituting $w_{j}=0$ and
$\xi_{j}\in\partial_{d}h_{j}(x+d(x))$ for all $j\notin I(x,d(x)$, we can
obtain

$\displaystyle\sum_{j=1}^{m}w_{j}=1.$ (9)

$\displaystyle\sum_{j=1}^{m}w_{j}(\nabla g_{j}(x)+B_{j}(x)d(x)+\xi_{j})=0.$
(10)

$\displaystyle w_{j}\geq 0,~{}w_{j}(\nabla
g_{j}(x)^{T}d(x)+\frac{1}{2}d(x)^{T}B_{j}(x)d(x)+h_{j}(x+d(x))-h_{j}(x)-\alpha(x))=0,j\in\Xi_{m}.$
(11)

$\displaystyle\nabla
g_{j}(x)^{T}d(x)+\frac{1}{2}d(x)^{T}B_{j}(x)d(x)+h_{j}(x+d(x))-h_{j}(x)\leq\alpha(x),j\in\Xi_{m}.$
(12)

Therefore, we have the following result:

Lemma 3.3 If $d(x)$ is a solution of $\Phi(x)$ and
$\alpha(x)={\theta(x,d(x))}$, then

(i) there exist $w\in{\mathbb{R}_{+}^{|I(x,d(x))|}}$,
$\xi_{j}\in\partial_{d}h_{j}(x+d(x)),j\in I(x,d(x))$ such that such that
$(d(x),\alpha(x),w)$ satisfies (7)-(8).

(ii) there exists $w\in{\mathbb{R}_{+}^{m}}$ such that $(d(x),\alpha(x),w)$
satisfies (9)-(12).

The following lemma characterizes the Pareto stationarity of (CMOP) in terms
of ${d}(\cdot)$.

Lemma 3.4 Suppose that Condition 3.1 holds true, ${d}(x)$ and ${\alpha}(x)$
are defined in (4) and (6), respectively. Then, the following statements hold
true:

(1) If $x$ is a Pareto stationary point of (CMOP), then ${d}(x)=0$ and
${\alpha}(x)=0.$ Conversely, if ${d}(x)=0$ and ${\alpha}(x)=0$, then $x$ is a
Pareto stationary point of (CMOP).

(2) If $x$ is not a Pareto stationary point of (CMOP), then ${d}(x)\neq 0$ and
$\alpha(x)<0$. Conversely, if ${d}(x)\neq 0$ and ${\beta}(x)<0$, then $x$ is
not a Pareto stationary point of (CMOP).

######  Proof.

(1) Let $x$ be Pareto stationary of (CMOP). We will prove that ${d}(x)=0$ and
$\alpha(x)=0.$ On the contrary, we assume, that ${d}(x)\neq 0$ or
${\alpha}(x)<0.$ It follows from Remark 3.2 that ${d}(x)\neq 0$ if and only if
${\alpha}(x)<0$, which means that ${d}(x)\neq 0$ and ${\alpha}(x)<0.$ It
follows from (12) in Lemma 3.3(ii) and the positiveness of $B_{j}$ that

$\displaystyle\nabla
g_{j}(x)^{T}d(x)+h_{j}(x+d(x))-h_{j}(x)\leq\alpha(x)-\frac{1}{2}d(x)^{T}B_{j}(x)d(x)<0,j\in\Xi_{m}.$
(13)

It follows from the convexity of $h_{j}$ that for any $\lambda\in(0,1)$ that

$\displaystyle h_{j}(x+\lambda d(x))-h_{j}(x)\leq$
$\displaystyle\lambda(h_{j}(x+d(x))+(1-\lambda)h_{j}(x)-h_{j}(x)$
$\displaystyle=$ $\displaystyle\lambda[h_{j}(x+d(x))-h_{j}(x)],j\in\Xi_{m}.$
(14)

From (13) and (14), we obtain that

$\lambda\nabla g_{j}(x)^{T}d(x)+h_{j}(x+\lambda d(x))-h_{j}(x)<0,j\in\Xi_{m},$

which implies that

$\frac{1}{\lambda}[\lambda\nabla g_{j}(x)^{T}d(x)+h_{j}(x+\lambda
d(x))-h_{j}(x)]<0,j\in\Xi_{m}.$

Taking limit $\lambda\rightarrow 0^{+}$ in the above inequality we have
${f^{\prime}_{j}}(x;d(x))<0$ for each $j=1,2,...,m$. Hence, $x$ is not a
critical point of (CMOP), which is a contradiction.

We now prove the converse. We assume that $d(x)=\bm{0}$ and ${\alpha}(x)=0$
hold true. It follows from Lemma 3.3(i) and
$I(x,\bm{0})=\\{1,2,...,m\\}=\Xi_{m}$ that there exist
$w\in\mathbb{R}_{+}^{m}$, $\xi_{j}\in\partial_{d}h_{j}(x+d(x)),j\in\Xi_{m}$
such that such that $(d(x),\alpha(x),w)$ satisfies

$\sum_{j=1}^{m}w_{j}=1,{\rm~{}~{}and~{}~{}}\sum_{j=1}^{m}w_{j}(\nabla
g_{j}(x)+\xi_{j})=0,$

which implies that

$\bm{0}\in Co{\mathop{\cup}\limits_{j\in\Xi_{m}}\partial f_{j}(x)}.$

Therefore, it follows from Lemma 3.1 that $x$ is a critical point of (CMOP).

(2) This statement is equivalent to statement (1). ∎

Lemma 3.5 Let $d(x)$ and ${\alpha}(x)$ be defined in (4) and (6),
respectively. Then, the mappings ${d}(\cdot)$ and ${\alpha}(\cdot)$ are
continuous.

######  Proof.

Clearly, the following function

${\theta(x,d):=\mathop{\max}\limits_{i=1,...,m}\theta_{i}(x,d)}=\mathop{\max}\limits_{i=1,...,m}\\{\nabla{g_{i}}{(x)^{T}}d+\frac{1}{2}{d^{T}}{B_{i}}(x)d+{h_{i}}(x+d)-{h_{i}}(x)\\}$

is continuous with respect to $x$ and $d$. Therefore, it follows from [2,
Proposition 23] that the optimal value function $\alpha(\cdot)$ is also
continuous. Furthermore, since the optimal set mapping ${d}(\cdot)$ is unique,
it follows from [14, Corollary 8.1] that ${d}(\cdot)$ is continuous. ∎

From Lemma 3.5, it is easy to see that the following result holds true.

Corollary 3.1 Let $x_{k}\in\mathbb{R}^{n}$ and $d(x_{k})$ be the solution of
$\Phi(x_{k})$.

(i) suppose that $\\{x_{k}\\}$ converges to $x^{*}$ and $d(x^{k})$ converges
to $d^{*}$, then $d^{*}=d(x^{*})$.

(ii)suppose that $\\{x_{k}\\}$ converges to $x^{*}$ and $\alpha(x^{k})$
converges to $\alpha^{*}$, then $\alpha^{*}=\alpha(x^{*})$.

Remark 3.4 (i) From Lemma 3.5, we know that $x$ is a Pareto stationary point
of (CMOP) if and only if ${d}(x)=0$, which extends and improves Lemma 3.2 in
[1] since the $\nabla^{2}g_{j}(x)$ has been replaced by a positive definite
matrix $B_{j}(x)$ for any $j\in\Xi_{m}$.

(ii) Corollary 3.1 extends and improves Theorem 3.1 in [1].

Theorem 3.1 Suppose that Condition 3.1 holds true.

(i) If $d(x)$ and $\alpha(x)$ are the solution and the optimal value of
$\Phi(x)$, respectively, then

$\displaystyle\alpha(x)\leq-\frac{\sigma}{2}\|d(x)\|^{2},$ (15)

where $\sigma=\mathop{\min}\limits_{i=1,...,m}\sigma_{i}$.

(ii) if $\rho\in(0,1)$ and $x$ is a non critical point of (CMOP), then the
following inequality holds true for every $\lambda>0$ sufficiently small,

$\displaystyle f_{j}(x+\lambda d(x))\leq f_{j}(x)+\rho\lambda\alpha(x)$ (16)

holds for any $j\in\Xi_{m}$.

Proof. (i) Suppose that $d(x)$ is a solution of $\Phi(x)$ and
$\alpha(x)=\theta(x,d(x))$. Then there exists $w\in{\mathbb{R}_{+}^{m}}$ such
that $(d(x),\alpha(x),w)$ satisfies (9)-(12). Since $h_{j}$ is convex and
$\xi_{j}\in\partial_{d}h_{j}(x+d(x))$,

$\displaystyle h_{j}(x+d(x))-h_{j}(x)\leq\xi_{j}^{T}d(x).$ (17)

Multiplying both sides of (10) by $d(x)$,

$\sum_{j=1}^{m}w_{j}[\nabla
g_{j}(x)^{T}d(x)+d(x)^{T}B_{j}(x)d(x)+\xi_{j}^{T}d(x)]=0.$

It follows from the above equality and (17) that

$\displaystyle\sum_{j=1}^{m}w_{j}[\nabla
g_{j}(x)^{T}d(x)+d(x)^{T}B_{j}(x)d(x)+h_{j}(x+d(x))-h_{j}(x)]\leq 0.$ (18)

Taking sum over $j\in\\{1,2,...,m\\}$ in (11) and using (9),

$\sum_{j=1}^{m}w_{j}[\nabla
g_{j}(x)^{T}d(x)+d(x)^{T}B_{j}(x)d(x)+h_{j}(x+d(x))-h_{j}(x)]=\sum_{j=1}^{m}w_{j}\frac{1}{2}d(x)^{T}B_{j}(x)d(x)+\alpha(x).$

It follows from (18) that

$\displaystyle\alpha(x)\leq-\sum_{j=1}^{m}w_{j}\frac{1}{2}d(x)^{T}B_{j}(x)d(x).$
(19)

From (19), (9), Condition 3.1 and the definition of $\sigma$, we obtain

$\alpha(x)\leq-\frac{\sigma}{2}\|d(x)\|^{2}.$

(ii) Suppose that $x$ is non critical. Then from Lemma 3.4, $d(x)\neq 0$. It
follows from (15) that $\alpha(x)<0$.

From Condition 3.1 and the convexity of $h_{i}$, we know that for
$i\in\Xi_{m}$ and for any $\lambda\in[0,1]$,

$\displaystyle{f_{i}}({x}+{\lambda}{d(x)})-{f_{i}}({x})$
$\displaystyle={g_{i}}({x}+{\lambda}{d(x)})-{g_{i}}({x})+{h_{i}}({x}+{\lambda}{d(x)})-{h_{i}}({x})$
$\displaystyle=\lambda(\nabla
g_{i}(x)^{T}d(x)+{h_{i}}({x}+{\lambda}{d(x)})-{h_{i}}({x}))+o(\lambda)$
$\displaystyle<\lambda(\nabla
g_{i}(x)^{T}d(x)+\frac{1}{2}d(x)^{T}B_{i}(x)d(x)+{h_{i}}({x}+{\lambda}{d(x)})-{h_{i}}({x}))+o(\lambda)$
$\displaystyle\leq\lambda\alpha(x)+o(\lambda).$

Then for $i\in\Xi_{m}$,

${f_{i}}({x}+{\lambda}{d(x)})-{f_{i}}({x})-\lambda\rho\alpha(x)\leq\lambda(1-\rho)\alpha(x)+o(\lambda).$

Since $\rho\in(0,1)$ and $\alpha(x)<0$, the right hand side term in the above
inequality becomes non positive for every $\lambda>0$ sufficiently small,
which implies that (16) holds true for every $\lambda>0$ sufficiently small.

Rewmark 3.5. If $B_{j}(x)\equiv\nabla^{2}g_{j}(x)$ for any $j\in\Xi_{m}$, then
by Theorem 3.1 we recover Theorem 3.2 in [1].

To solve for (CMOP), now we present our new BFGS quasi-Newton methods with
line searches. We compute the step length ${\lambda_{k}}>0$ by an Armijo rule.
Let $\rho\in(0,1)$ be a constant. If for each $i\in\Xi_{m}$, the following
inequality holds true

$\displaystyle{f_{i}}({x^{k}}+{\lambda_{k}}{d^{k}})\leq{f_{i}}({x^{k}})+{\lambda_{k}}\rho\alpha(x^{k}),$
(20)

then the ${\lambda_{k}}$ is accepted. Otherwise, we begin with
${\lambda_{k}}=1$ and if there exists $i\in\\{1,2,...,m\\}$ such that the
inequality (20) is not satisfied, we update

${\lambda_{k}}:=\zeta{\lambda_{k}},$

where $\zeta\in(0,1)$.

The following result illustrates that ${d^{k}}$ produced by the Armijo rule
procedure is a descent direction of (CMOP) at a nonstationary points
${x^{k}}$.

Lemma 3.6 Suppose that Condition 3.1 holds true. Let $\rho\in(0,1)$,
${d^{k}}:=d(x^{k})$ and $\alpha(x^{k})$ be the solution and the optimal value
of $\Psi(x^{k})$, respectively. If ${x^{k}}$ is not Pareto stationary, then
there exists some ${\bar{\lambda}_{k}}>0$ such that for each $i=1,...,m$ and
for any $\lambda\in(0,{\bar{\lambda}_{k}}]$, the following inequality holds
true

${f_{i}}({x^{k}}+\lambda{d^{k}})\leq{f_{i}}({x^{k}})+\lambda\rho\alpha(x^{k}).$

######  Proof.

Let $\lambda\in(0,1].$ It follows from the convexity of ${h_{i}}$ that for
each $i=1,...,m,$ we have

$\displaystyle{h_{i}}({x^{k}}+\lambda{d^{k}})-{h_{i}}({x^{k}})=$
$\displaystyle{h_{i}}((1-\lambda){x^{k}}+\lambda({x^{k}}+{d^{k}}))-{h_{i}}({x^{k}})$
$\displaystyle\leq$
$\displaystyle(1-\lambda){h_{i}}({x^{k}})+\lambda{h_{i}}({x^{k}}+{d^{k}})-{h_{i}}({x^{k}})$
$\displaystyle=$
$\displaystyle\lambda[{h_{i}}({x^{k}}+{d^{k}})-{h_{i}}({x^{k}})].$

From Condition 3.1 and the first-order Taylor expansion of ${g_{i}}$, we have
that for each $i\in\Xi_{m},$

$\displaystyle{g_{i}}({x^{k}}+\lambda{d^{k}})+{h_{i}}({x^{k}}+\lambda{d^{k}})$
$\displaystyle\leq$
$\displaystyle{g_{i}}({x^{k}})+\lambda\nabla{g_{i}}{({x^{k}})^{T}}{d^{k}}+\frac{1}{2}{(\lambda{d^{k}})^{T}}{B_{i}}({x^{k}})(\lambda{d^{k}})+{h_{i}}({x^{k}})+\lambda({h_{i}}({x^{k}}+{d^{k}})-{h_{i}}({x^{k}}))+o({\lambda})$
$\displaystyle=$
$\displaystyle{g_{i}}({x^{k}})+{h_{i}}({x^{k}})+\lambda[\nabla{g_{i}}{({x^{k}})^{T}}{d^{k}}+\frac{\lambda}{2}{({d^{k}})^{T}}{B_{i}}({x^{k}})({d^{k}})+{h_{i}}({x^{k}}+{d^{k}})-{h_{i}}({x^{k}})]+o({\lambda})$
$\displaystyle\leq$
$\displaystyle{g_{i}}({x^{k}})+{h_{i}}({x^{k}})+\lambda[\nabla{g_{i}}{({x^{k}})^{T}}{d^{k}}+\frac{1}{2}{({d^{k}})^{T}}{B_{i}}({x^{k}})({d^{k}})+{h_{i}}({x^{k}}+{d^{k}})-{h_{i}}({x^{k}})]+o({\lambda})$
$\displaystyle\leq$
$\displaystyle{g_{i}}({x^{k}})+{h_{i}}({x^{k}})+\lambda\alpha(x^{k})+o({\lambda})$
$\displaystyle=$
$\displaystyle{g_{i}}({x^{k}})+{h_{i}}({x^{k}})+\lambda\rho\alpha(x^{k})+\lambda\left[{(1-\rho)\alpha(x^{k})+\frac{{o({\lambda})}}{\lambda}}\right],$

where ${B_{i}}({x^{k}})$ is some approximation of
${\nabla^{2}}{g_{i}}({x^{k}}),i\in\Xi_{m},$ the second inequality follows from
the positive definiteness of ${B_{i}}({x^{k}})$ and $\lambda\in(0,1]$, and the
third one comes from the definition of $\alpha(x)$ in Remark 3.2. Since
${x^{k}}$ is not Pareto stationary, we have $\alpha(x^{k})<0$ from Lemma 3.5.
It follows from $\rho\in(0,1)$ that there exists some ${\bar{\lambda}_{k}}>0$
such that for each $i\in\Xi_{m},$

${g_{i}}({x^{k}}+\lambda{d^{k}})+{h_{i}}({x^{k}}+\lambda{d^{k}})\leq{g_{i}}({x^{k}})+{h_{i}}({x^{k}})+\lambda\rho\alpha(x^{k}),$

for any $\lambda\in(0,{\bar{\lambda}_{k}}]$. ∎

To simplify the notation we will use $B_{i}^{k}$ to denote ${B_{i}}({x^{k}})$
for all $i=1,...,m$ and $k=0,1,2,...$.

Now, we would like to state our new quasi-Newton methods with line searches
for (CMOP) as follows:

Algorithm 3.1

Step 1 Choose $\omega>0$, $\rho\in(0,1)$, $\zeta\in(0,1)$,
${x^{0}}\in{\mathbb{R}^{n}}$, symmetric positive definite matrix
$B_{i}^{0}\in{\mathbb{R}^{n\times n}},i=1,...,m$ and set $k:=0$;

Step 2 Compute $d(x^{k})$ and $\alpha(x^{k})$ by solving subproblem (4) with
$x={x^{k}}$, let ${d^{k}}:=d(x^{k})$;

Step 3 If ${d^{k}}=0$, then stop. Otherwise, proceed to the next step;

Step 4 Compute the step length ${\lambda_{k}}\in(0,1]$ as the maximum of the
following set

${\Lambda_{k}}:=\\{{\lambda}={\zeta^{j}}|j\in\mathbb{N},{f_{i}}({x^{k}}+{\lambda}{d^{k}})\leq{f_{i}}({x^{k}})+{\lambda}\rho\alpha(x^{k}),i=1,...,m\\};$

Step 5 Set ${x^{k+1}}={x^{k}}+{\lambda_{k}}{d^{k}}$, update $\\{B_{i}^{k}\\}$
by following BFGS update formula for each $i=1,2,...,m$

$\displaystyle
B_{i}^{k+1}=B_{i}^{k}-\frac{{B_{i}^{k}{s^{k}}{{({s^{k}})}^{T}}B_{i}^{k}}}{{{{({s^{k}})}^{T}}B_{i}^{k}{s^{k}}}}+\frac{{y_{i}^{k}{{(y_{i}^{k})}^{T}}}}{{{{({s^{k}})}^{T}}y_{i}^{k}}},$
(21)

where ${s^{k}}={x^{k+1}}-{x^{k}}={\lambda_{k}}{d^{k}}$,
$y_{i}^{k}=\nabla{g_{i}}({x^{k+1}})-\nabla{g_{i}}({x^{k}})$.

Step 5 Set $k:=k+1$, and go to Step 2.

Remark 3.6 (1) It follows from Lemma 3.4 that Algorithm 3.1 stops at Step 3
with a Pareto stationary point or produces an infinite sequence of
nonstationary points $\\{{x^{k}}\\}$. If Step 4 is reached in some iteration
$k$, it means that in Step 3, ${d^{k}}\neq 0$, or equivalently,
${\alpha}({x^{k}})<0.$ It follows from the Armijo condition and Lemma 3.6 that
objective value sequence $\\{{F}({x^{k}})\\}$ is
$\mathbb{R}_{++}^{m}$-decrease, i.e.,

${F}({x^{k+1}})<{F}({x^{k}})~{}{\rm for~{}all}~{}k.$

(2) It follows from [15] and the references therein that if $g_{i}$ is a
strongly convex function, then the matrix ${B_{i}^{k+1}}$ obtained from each
of the mentioned updating formula (21) for approximating the Hessian matrix
$\nabla^{2}g_{i}(x^{k+1})$ always preserves positive definiteness.

##  4  Convergence analysis

In this section, we prove that the sequences generated by Algorithm 3.1
converges to Pareto stationary points of (CMOP) and the Condition 3.1 will be
replaced by the following form which has been used in [24]:

Condition 4.1 For all $k$ and all $j=1,...,m$, we have
$z^{T}B_{j}(x^{k})z\geq\sigma\|z\|^{2}$.

Lemma 4.1 Suppose that Condition 4.1 holds true, $\\{{d^{k}}\\}$ is generated
by Algorithm 3.1 and that $\\{{f_{i}}({x^{k}})\\}$ is bounded from below for
all $i=1,...,m$. Then, it follows that

$\mathop{\lim}\limits_{k\to\infty}{\lambda_{k}}{\left\|{{d^{k}}}\right\|^{2}}=0.$

######  Proof.

It follows from Lemma 3.6 and step 4 of Algorithm 3.1 that there exists
$\lambda_{k}>0$ such that for each $i\in\Xi_{m}$,

${f_{i}}({x^{k}}+{\lambda_{k}}{d^{k}})\leq{f_{i}}({x^{k}})+\lambda_{k}\rho\alpha(x^{k}).$

Adding up the above inequality from $k=0$ to $k=\hat{k}$, where $\hat{k}$ is a
positive integer, we get

$\displaystyle~{}{f_{i}}({x^{\hat{k}+1}})\leq{f_{i}}({x^{0}})+\rho\sum\limits_{k=0}^{\hat{k}}{\lambda_{k}}\alpha(x^{k}).$
(22)

By (22) and Theorem 3.1, we get

$\displaystyle~{}{f_{i}}({x^{\hat{k}+1}})\leq{f_{i}}({x^{0}})-\frac{\sigma}{2}\rho\sum\limits_{k=0}^{\hat{k}}{\lambda_{k}}{\left\|{{d^{k}}}\right\|^{2}}.$
(23)

Since $\\{{f_{i}}({x^{k}})\\}$ is bounded from below for all $i=1,...,m$,
there exists ${\hat{f}_{i}}\in\mathbb{R}$ such that
${\hat{f}_{i}}\leq{f_{i}}({x^{k}})$ for all $i$ and $k$.

It follows from (23) that

$\displaystyle\sum\limits_{k=0}^{\hat{k}}{{\lambda_{k}}{{\left\|{{d^{k}}}\right\|}^{2}}}\leq$
$\displaystyle\frac{2}{{\rho\sigma}}({f_{i}}({x^{0}})-{f_{i}}({x^{\hat{k}+1}}))$
$\displaystyle\leq$
$\displaystyle\frac{2}{{\rho\sigma}}({f_{i}}({x^{0}})-{\hat{f}_{i}}).$

Taking $\hat{k}\to\infty$, we have
$\sum\limits_{k=0}^{\infty}{{\lambda_{k}}{{\left\|{{d^{k}}}\right\|}^{2}}}<\infty$

and hence
$\mathop{\lim}\limits_{k\to\infty}{\lambda_{k}}{\left\|{{d^{k}}}\right\|^{2}}=0$.
∎

Theorem 4.1 (i) Assume that the Condition 4.1 holds true and that
$\\{{f_{i}}({x^{k}})\\}$ is bounded from below for all $i=1,...,m$, then every
accumulation point of the sequence $\\{{x^{k}}\\}$ generated by Algorithm 3.1,
if it exists, is both a Pareto stationary point and a Pareto optimum of
(CMOP).

(ii) Moreover, if the level set of ${F}$ in the sense that
$\\{x\in{\mathbb{R}^{n}}\mid F(x)\leq F({x^{0}})\\}$ is bounded, then
$\\{{x^{k}}\\}$ has accumulation points and they are all Pareto stationary
points and Pareto optimums of (CMOP).

######  Proof.

(i) Let $\bar{x}$ be an accumulation point of $\\{{x^{k}}\\}$ and let
$\\{{x^{{k_{j}}}}{\rm{\\}}}$ be a subsequence converging to $\bar{x}$. We now
prove that ${d}(\bar{x})=0$. On the contrary, we assume that ${d}(\bar{x})\neq
0$. Then, by Lemma 4.1, we get ${\lambda_{{k_{j}}}}\to 0$. It follows from the
definition of ${\lambda_{{k_{j}}}}$ in Step 4 of Algorithm 3.1 that for
sufficiently large $j$ there exists some ${i_{{k_{j}}}}\in\\{1,...,m\\}$ such
that

${f_{{i_{{k_{j}}}}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})>{f_{{i_{{k_{j}}}}}}({x^{{k_{j}}}})+{\zeta^{-1}}{\lambda_{{k_{j}}}}\rho\alpha(x^{k_{j}}).$

Since $i$ only takes finite number of values in $\\{1,...,m\\}$, we can assume
that ${i_{{k_{j}}}}=\bar{i}$ without loss of generality. We thus obtain

$\displaystyle\frac{{{f_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{f_{\bar{i}}}({x^{{k_{j}}}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}>\rho\alpha(x^{k_{j}}).$
(24)

It follows from $0<{\zeta^{-1}}{\lambda_{{k_{j}}}}<1$, the definition of
${\alpha(x)}$ and Theorem 23.1 in [21] that

$\displaystyle\alpha(x^{k_{j}})\geq$
$\displaystyle\nabla{g_{\bar{i}}}{({x^{{k_{j}}}})^{T}}{d^{{k_{j}}}}+\frac{1}{2}{({d^{{k_{j}}}})^{T}}{B_{\bar{i}}}({x^{{k_{j}}}})({d^{{k_{j}}}})+{h_{\bar{i}}}({x^{{k_{j}}}}+{d^{{k_{j}}}})-{h_{\bar{i}}}({x^{{k_{j}}}})$
$\displaystyle\geq$
$\displaystyle\frac{{{\zeta^{-1}}{\lambda_{{k_{j}}}}\nabla{g_{\bar{i}}}{{({x^{{k_{j}}}})}^{T}}{d^{{k_{j}}}}+\frac{1}{2}{\zeta^{-1}}{\lambda_{{k_{j}}}}{{({d^{{k_{j}}}})}^{T}}{B_{\bar{i}}}({x^{{k_{j}}}})({d^{{k_{j}}}})+{h_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{h_{\bar{i}}}({x^{{k_{j}}}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}$
$\displaystyle=$
$\displaystyle\frac{{{g_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})+{h_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{g_{\bar{i}}}({x^{{k_{j}}}})-{h_{\bar{i}}}({x^{{k_{j}}}})+o({{({\zeta^{-1}}{\lambda_{{k_{j}}}}\left\|{{d^{{k_{j}}}}}\right\|)}^{2}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}$
$\displaystyle=$
$\displaystyle\frac{{{f_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{f_{\bar{i}}}({x^{{k_{j}}}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}+\frac{{o({{({\zeta^{-1}}{\lambda_{{k_{j}}}}\left\|{{d^{{k_{j}}}}}\right\|)}^{2}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}},$

where ${B_{\bar{i}}}({x^{{k_{j}}}})$ is an approximation of
${\nabla^{2}}{g_{\bar{i}}}({x^{{k_{j}}}})$ which updated by (21). Thus, we
obtain

$\displaystyle\alpha(x^{k_{j}})\geq\frac{{{f_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{f_{\bar{i}}}({x^{{k_{j}}}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}+\frac{{o({{({\zeta^{-1}}{\lambda_{{k_{j}}}}\left\|{{d^{{k_{j}}}}}\right\|)}^{2}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}.$
(25)

It follows from (24) and (25) that

$\frac{{{f_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{f_{\bar{i}}}({x^{{k_{j}}}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}>\rho\frac{{{f_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{f_{\bar{i}}}({x^{{k_{j}}}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}+\rho\frac{{o({{({\zeta^{-1}}{\lambda_{{k_{j}}}}\left\|{{d^{{k_{j}}}}}\right\|)}^{2}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}.$

Therefore, we have

$\displaystyle\frac{{{f_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{f_{\bar{i}}}({x^{{k_{j}}}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}>(\frac{\rho}{{1-\rho}})\frac{{o({{({\zeta^{-1}}{\lambda_{{k_{j}}}}\left\|{{d^{{k_{j}}}}}\right\|)}^{2}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}.$
(26)

It follows from Theorem 3.1 that

$\alpha(x^{k_{j}})\leq-\frac{\sigma}{2}{\left\|{{d^{{k_{j}}}}}\right\|^{2}}.$

Since ${d^{{k_{j}}}}\to{d}(\bar{x})\neq 0$, by the above inequality, (25) and
(26), we know that there exists
$\gamma=\frac{\sigma}{2}{\left\|d(\bar{x})\right\|^{2}}>0$ such that

$-\gamma\geq\lim_{j\rightarrow\infty}\alpha(x^{k_{j}})\\\
\geq\lim_{j\rightarrow\infty}[\frac{{{f_{\bar{i}}}({x^{{k_{j}}}}+{\zeta^{-1}}{\lambda_{{k_{j}}}}{d^{{k_{j}}}})-{f_{\bar{i}}}({x^{{k_{j}}}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}+\frac{{o({{({\zeta^{-1}}{\lambda_{{k_{j}}}}\left\|{{d^{{k_{j}}}}}\right\|)}^{2}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}]$
$\geq\lim_{j\rightarrow\infty}[(\frac{\rho}{{1-\rho}})\frac{{o({{({\zeta^{-1}}{\lambda_{{k_{j}}}}\left\|{{d^{{k_{j}}}}}\right\|)}^{2}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}+\frac{{o({{({\zeta^{-1}}{\lambda_{{k_{j}}}}\left\|{{d^{{k_{j}}}}}\right\|)}^{2}})}}{{{\zeta^{-1}}{\lambda_{{k_{j}}}}}}]=0,$

which contradicts the fact that $\gamma>0$. Therefore, we conclude that
${d}(\bar{x})=0$.

It follows from Lemma 3.4 that $\bar{x}$ is Pareto stationary point of (CMOP).
By Lemma 2.1 and the strong convexity of $f_{i}$, we know that $\bar{x}$ is
also a Pareto optimum of (CMOP).

(ii)It follows from Remark 4.1(1) that the objective value sequence
$\\{{F}({x^{k}})\\}$ is $\mathbb{R}_{++}^{m}$-decrease. Moreover, since the
set $\\{x\in{\mathbb{R}^{n}}\mid F(x)\leq F({x^{0}})\\}$ is bounded, Thus, the
sequence $\\{{x^{k}}\\}$ generated by Algorithm 3.1 is contained in the above
set and so it is also bounded and has at least one accumulation point, which
is a Pareto stationary point and a Pareto optimum of (CMOP) according to the
first statement. ∎

##  5  Conclusion

First, for the composite multiobjective optimization problems (in short,
CMOP), where each objective function is the sum of a twice continuously
differentiable strongly convex function and a proper convex and lower
semicontinuous but not necessarily differentiable function, the BFGS quasi-
Newnon method with Armijo line search are introduced. Secondly, under
appropriate conditions, we prove that each cluster point of the sequence
generated by the BFGS quasi-Newton algorithms is both a Pareto stationary
point and a Pareto optimum of (CMOP). Thirdly, numerical experiments are
performed to verify the effectiveness of the proposed algorithms. In the
future, there still exists an interesting topics about the convergence rate of
the proposed algorithms.

##  References

  * [1] M. A. T. Ansary, A Newton-type proximal gradient method for nonlinear multi-objective optimization problems, OPTIMIZATION METHODS $\&$ SOFTWARE https://doi.org/10.1080/10556788.2022.2157000
  * [2] J. P. Aubin, and I. Ekeland, Applied Nonlinear Analisis, John Wiley $\&$Sons (1984).
  * [3] A. Bagirov, N. Karmitsa M. M. Mäkelä, Introduction to Nonsmooth Optimization, Theory, Practice and Software, Springer International Publishing, Switzerland, 2014
  * [4] Bello Cruz, J.Y., Lucambio P rez, L.R., Melo, J.G.: Convergence of the projected gradient method for quasiconvex multiobjective optimization. Nonlinear Anal. 74(16), 5268–5273 (2011)
  * [5] Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)
  * [6] Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)
  * [7] Chen, G., Teboulle, M.: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM J. Optim. 3(3), 538–543 (1993)
  * [8] Da Cruz Neto, J.X., Da Silva, G.J.P., Ferreira, O.P., Lopes, J.O.: A subgradient method for multiobjective optimization. Comput. Optim. Appl. 54(3), 461–472 (2013)
  * [9] Eichfelder G.: Adaptive Scalarization Methods in Multiobjective Optimization[M], Springer -Verlag Berlin Heidelberg, 2008
  * [10] Ehrgott, M.: Multicriteria Optimization[M], Springer, Berlin Heidelberg, 2005
  * [11] Fliege, J., Grana Drummond, L.M., Svaiter, B.F.: Newton s method for multiobjective optimization. SIAM J. Optim. 20(2), 602–626 (2009)
  * [12] Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)
  * [13] Fukuda, E.H., Grana Drummond, L.M.: A survey on multiobjective descent methods. Pesquisa Operacional 34(3), 585–620 (2014)
  * [14] Hogan, W.W.: Point-to-set maps in mathematical programming. SIAM Rev. 15(3), 591–603 (1973)
  * [15] Morovati V., Basirzadeh H., Pourkarimi L.: Quasi-Newton methods for multiobjective optimization problems, 4OR, 16(3), 261 C294(2018)
  * [16] Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd ed. Springer Science and Business Media, LLC, New York (2006)
  * [17] Nakayama H., Yun Y., Yoon M.: Sequential Approximate Multiobjective Optimization Using Computational Intelligence[M]. Springer -Verlag Berlin Heidelberg, 2009
  * [18] Peng J. W., Ren J. J. C. Yao, Proximal Quasi-Newton Methods for the Composite Multiobjective Optimization Problems, J. Nonlinear Convex Anal., To appear
  * [19] Povalej, Z.: Quasi-Newton’s method for multiobjective optimization. J. Comput. Appl. Math. 255, 765–777 (2014)
  * [20] S. Qu, M. Goh, and F.T.S. Chan, Quasi-Newton methods for solving multiobjective optimization, Oper. Res. Lett. bf 39, 397- C399 (2011)
  * [21] Rockafellar, R.T.: Convex Analysis. University Press, Princeton (1970)
  * [22] Tanabe, H., Fukuda, E.H., Yamashita, N.: Proximal gradient methods for multiobjective optimization and their applications. Comput. Optim. Appl. 72, 339–361 (2019)
  * [23] Ren, J., Peng, J.W.: Proximal Newton Methods for Multiobjective Optimization Problems. Acta Mathematicae Applicatae Sinica. 45(2), 222–237 (2022)
  * [24] Mahdavi-Amiri, N., Salehi Sadaghiani, F.: A superlinearly convergent nonmonotone quasi-Newton method for unconstrained multiobjective optimization, Optimization Methods $\&$ Software, 35(6) 1223-1247 (2020)