# Approximations for Pareto and Proper Pareto solutions and their KKT
conditions

P. Kesarwani111Department of Mathematics and Statistics, Indian Institute Of
Technology Kanpur,India, P.K.Shukla 222Institute AIFB, Karlsruhe Institute of
Technology, Karlsruhe, Germany, J. Dutta333Department of Economics Sciences,
Indian Institute Of Technology Kanpur, India and K. Deb444College of
Engineering, Michigan State University, Michigan, USA

###### Abstract

In this article, we view the approximate version of Pareto and weak Pareto
solutions of the multiobjective optimization problem through the lens of KKT
type conditions. We also focus on an improved version of Geoffrion proper
Pareto solutions and characterize them through saddle point and KKT type
conditions. We present an approximate version of the improved Geoffrion proper
solutions and propose our results in general settings.

##  1 Introduction

The importance of multiobjective optimization problems in various applications
in engineering, business and management can be hardly overstated. For a wide
range of applications in engineering design see, for example, [4].

From a theoretical point of view, the idea of multiobjective optimization
becomes challenging since we are speaking about minimizing/maximizing a
vector-valued function. In order to define the notion of a solution, we need
to depend on the partial order, which is often induced by a closed convex
pointed cone on the image space of the objective function. This leads to two
fundamental notions of solutions, namely the Pareto solutions and weak Pareto
solutions. The points corresponding to these solutions in the image space of
the objective function are often referred to as efficient solutions and weak
efficient solutions. The collection of all efficient solutions is often
referred to as the Pareto efficient frontier. We emphasize that the notion of
Pareto and weak Pareto solutions are global notions. Mathematically speaking,
it is not at all difficult to devise a local counterpart, and it is the global
aspect that is sought by the decision makers. Further, the idea of Pareto
solutions is often considered more relevant than the weak solutions from the
point of view of the applications. We refer the following monographs of
Ehrgott [10], Jahn [13], Luc [15], Chankong et al. [2] and the references
therein to see the development of multiobjective optimization over the past
several decades.
There are several approaches to solve multiobjective problems, for example,
scalarization methods, descent methods, metaheuristics and many more. But when
it comes to actual computation using the mentioned methods, the algorithms
always produce approximate solutions. Thus, it is essential to define notions
of approximate solutions and characterize their properties. There are various
notion of approximate solutions in the literature (see [14], [21], [9], [11],
[12]) which deals with characterizing introduced notions in greater details.
In this article, our main aim is to revisit the fundamental notion of
approximate Pareto solutions and a proper Pareto solution (which we shall
describe below). Further, we analyze these solutions through KKT type
conditions. This approach to studying KKT type conditions for approximate
solutions can lead to the development of stopping criteria for algorithms. It
can also be used to check the quality of the approximate solution produced by
any algorithm used to solve the multiobjective problem.
It is essential to a decision maker, who is taking some decisions based on
multiobjective optimization models need not necessarily be interested in all
the Pareto solutions of the problem at hand. In many cases, the decision maker
focuses on the part of the Pareto frontier in the image space, which
corresponds to a subset of the set of Pareto solutions. These subsets, when
chosen in a particular way, gives rise to various classes of proper Pareto
solutions (see[10]). Very recently, the authors discussed an improved version
of Geoffrion proper solutions in [19]. This solution notion is based on the
assumption that the decision maker, in practice, usually looks for those
proper solutions whose trade-off bound is bounded by a value preset by
her/him. The detailed analysis of such solutions and their approximate version
has been carried out and shown to be stable than the standard Geoffrion
solutions (for more details see [19]). In the present article, our major goal
is to analyse saddle point and KKT type conditions for these solutions.
The whole paper revolves around answering three questions in which first two
questions stem from an attempt to generalize two results, which are on
approximate solutions for scalar optimization problems which appeared in [8].
The first result concerns a scalar optimization problem with locally Lipschitz
data (see Theorem 3.2 in [8]) which says that if a sequence of points each
satisfying an approximate version of KKT conditions converges to point under a
suitable constraint qualification, then the limit of the sequence is a KKT
point. Thus, we have the following first question:

  * •

Q1: Can a similar kind of result be deduced for multiobjective optimization
problem?

Our second question stems from Theorem 3.7 in [8] in which the reversed result
of the Theorem 3.2 is asked. The result conclude an affirmative answer for the
reverse result which proves that for any local minimizer of an optimization
problem with suitable constraint qualification, there a sequence of points
converge to that local minima and there exists a subsequence of the main
sequence which satisfies some type of approximate KKT type conditions when
they are very near to the solution.

  * •

Q2: Can we generalize the reverse result in the multiobjective settings?
Further, do the locally Lipschitz data suffice, or we need more assumptions?
Can the convexity assumption give us better results?

Our third question is associated with the KKT-type conditions for the
approximate Geoffrion proper solutions with a preset bound.

  * •

Q3: Can we develope an approximate KKT type condition which can completely
characterize a Geoffrion proper solutions with a preset bound at least in the
convex case? Does the saddle point conditions completely characterize such
class of solutions?

The paper is organised as follows. In Section 2, we present the problem, basic
definitions and the technical tools from convex and non-smooth analysis
required in the article. In Section 3, we answer the first two questions
raised in this section and Section 4 assures the last question by trying to
develop the saddle point conditions and approximate KKT type conditions for
the improved Geoffrion proper solutions. We end our discussion by concluding
remarks in Section 5. We want to end this section by stating that most symbols
used in the article are fairly standard in the literature.

##  2 Preliminaries and basic tools

Let $A\subseteq\mathbb{R}^{n}$ be a given set, then closure and interior of
set $A$ is denoted by cl$A$ and int$A$ respectively. For vectors
$x,y\in\mathbb{R}^{n}$ the inner product given by $\langle x,y\rangle$. A set
$A\subset\mathbb{R}^{n}$ is a cone, if for each $a\in A$ and positive scalar
$\lambda$, $\lambda a\in A$. A cone $A$ is pointed, if $A\cap(-A)=\\{0\\}$. A
normal cone of a convex set $A$ at the point $x_{0}$, denoted by
$N_{A}(x_{0})$, is $N_{A}(x_{0})=\\{v\in\mathbb{R}^{n}:\langle
v,x-x_{0}\rangle\leq 0,\text{ for all }x\in U\\}$. We consider the following
form of multiobjective optimization problem (MOP) in this article:

$\displaystyle\min f(x):=(f_{1}(x),\ldots,f_{m}(x)),$ $\displaystyle{\rm
subject~{}to}~{}~{}g_{j}(x)\leq 0,~{}j=1,2,..,l.$

where each $f_{i}:\mathbb{R}^{n}\to\mathbb{R}$ and
$g_{j}:\mathbb{R}^{n}\to\mathbb{R}$. Let us denote the constraint set by
$X:=\\{x\in\mathbb{R}^{n}:g_{j}(x)\leq
0,~{}j=1,2,..,l\\}\subseteq\mathbb{R}^{n}$, $I:=\\{1,2,..,m\\}$,
$L:=\\{1,2,..,l\\}$. As we mentioned earlier that there are several notions
for approximate solutions but in this article, we consider the notion of
approximate solution introduced in Loridan [14]. We consider
$\epsilon\in\mathbb{R}^{m}_{+},$ i.e.,
$\epsilon=(\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{m})$, $\epsilon_{i}\geq
0$ for each $i\in I$ to formalize our notions. Our focus on this paper is on
$\epsilon$\- solutions of MOP. The partial order of image space
$f(X)\subseteq\mathbb{R}^{m}$ is induced by natural cone $\mathbb{R}^{m}_{+}$
in the following definition.

###### Definition 2.1

Given $\epsilon\in\mathbb{R}^{m}_{+}$, if there is no $x\in X$ such that
$f(x)+\epsilon-f(x^{*})\in-\mathbb{R}^{m}_{+}\setminus\\{0\\},$ then the point
$x^{*}\in X$ is said to be an $\epsilon$-Pareto optimal solution of MOP.
Further if there is no $x\in X$ such that $f(x)+\epsilon-f(x^{*})\in-{\rm
int}(\mathbb{R}^{m}_{+}),$ then the point $x^{*}$ is said to be a weak
$\epsilon$-Pareto optimal solution of MOP.

An $\epsilon$-Pareto (weak) optimal solution with $\epsilon=0$ is commonly
known as Pareto (weak) optimal solution. Though not always seen in the
literature the following notions of a local solutions are also relevant.

###### Definition 2.2

A point $x^{*}$ is said to be a loacl Pareto optimal solution of MOP if there
exists $\delta>0$ and no $x\in X\cap B_{\delta}(x^{*})$ such that,
$f(x)-f(x^{*})\in-\mathbb{R}^{m}_{+}\setminus\\{0\\},$ where
$B_{\delta}(x_{0})\subset\mathbb{R}^{n}$ is a ball of radius $\delta$.

The weak counter part of local solution can be defined in the similar fashion
as in Definition 2.2. We want to mention that in several situations we
consider the particular form of the vector $\epsilon\in\mathbb{R}^{m}_{+}$,
given by $\epsilon=\varepsilon e$, where $e=(1,1..,1)^{T}$ and
$\varepsilon\in\mathbb{R}_{+}$. In those cases, the solutions referred to as
the $\varepsilon e$-Pareto and $\varepsilon e$-weak Pareto solution
respectively. The set of all $\epsilon$-Pareto points is denoted by
${S}_{\epsilon}(f,X)$ and the set of all $\epsilon$-weak Pareto points as
$S_{w,\epsilon}(f,X)$.

###### Definition 2.3

Given $\epsilon\in\mathbb{R}^{n}_{+}$, a point $x_{0}\in X$ is called
$\epsilon$-Geoffrion proper solution of MOP if
$x_{0}\in\mathcal{S}_{\epsilon}(f,X)$ and if there exists a number $M>0$ such
that for all $i\in I$ and $x\in X$ satisfying
$f_{i}(x)<f_{i}(x_{0})-\epsilon_{i}$, there exists an index $j\in I$ such that
$f_{j}(x_{0})-\epsilon_{j}<f_{j}(x)$ and

$\displaystyle\frac{f_{i}(x_{0})-f_{i}(x)-\epsilon_{i}}{f_{j}(x)-f_{j}(x_{0})+\epsilon_{j}}\leq
M.$

The upper bound of the trade-off in the above definition is not known
beforehand and the definition only assures the existence of such a bound.
Further, it is clear form the definition that the trade-off varies as we
choose different proper points. The improved definition introduced in [19]
eliminates the dependence of the bound on the solution points. Let us state
the improved notion of Geoffrion proper solutions studied in [19].

###### Definition 2.4

Given $\epsilon\in\mathbb{R}^{n}_{+}$ and a scalar $\hat{M}>0$, a point
$x_{0}\in X$ is called $(\hat{M},\epsilon)$-Geoffrion proper solution of MOP
if $x_{0}\in\mathcal{S}_{\epsilon}(f,X)$ and for all $i\in I$ and $x\in X$
satisfying $f_{i}(x)<f_{i}(x_{0})-\epsilon_{i}$, there exists an index $j\in
I$ such that $f_{j}(x_{0})-\epsilon_{j}<f_{j}(x)$ and

$\displaystyle\frac{f_{i}(x_{0})-f_{i}(x)-\epsilon_{i}}{f_{j}(x)-f_{j}(x_{0})+\epsilon_{j}}\leq\hat{M}.$

Given $\hat{M}>0$, we shall denote the set of all $(\hat{M},\epsilon)$\-
Geoffrion proper as $\mathcal{G}_{\hat{M},\epsilon}(f,X)$. For $\epsilon=0$,
the set of exact $\hat{M}$\- Geoffrion proper is denoted by
$\mathcal{G}_{\hat{M}}(f,X)$. Now we shall present the Ekeland variation
principle for vector-valued functions which was introduced in [20] when the
ordering cone is $\mathbb{R}_{+}^{m}$. We first define the notion of lower
semicontinuity and boundness of vector-valued functions which will be needed
in the principle.

###### Definition 2.5

Let $f:U\rightarrow\mathbb{R}^{m}$ where $U$ is a non-empty subset of
$\mathbb{R}^{n}$. The function $f$ is $\mathbb{R}^{m}_{+}$-bounded below if
there exists $y\in\mathbb{R}^{m}$ such that $f(x)-y\in\mathbb{R}_{+}^{m}$ for
all $x\in U$. Let $c\in int(\mathbb{R}_{+}^{m})$, the function $f$ is
$(c,\mathbb{R}_{+}^{m})$\- lower semi continuous if for all $t\in\mathbb{R}$,
$\\{x\in U:tc-f(x)\in\mathbb{R}_{+}^{m}\\}$ is closed.

###### Theorem 2.6

Let $f:U\rightarrow\mathbb{R}^{m}$ where $U\subseteq\mathbb{R}^{m}$ be a
$(c_{0},\mathbb{R}^{m}_{+})$\- lower semi continuous function for $c_{0}\in
int(\mathbb{R}_{+}^{m})$ which is also $\mathbb{R}^{m}_{+}$-bounded below .
Further, suppose we are given $\rho>0$ and a point $x_{0}\in U$ such that,

$f(x)+\rho c_{0}-f(x_{0})\not\in-\mathbb{R}^{m}_{+}\setminus\\{0\\},\text{ for
all }x\in U.$ (2.1)

Then, there exists $\bar{x_{0}}=\bar{x_{0}}(\rho)\in U$ such that
$\|\bar{x_{0}}-x_{0}\|\leq\sqrt{\rho}$ and for all $x\in
U\setminus\\{\bar{x_{0}}\\}$

  1. 1.

$f(x)+\rho c_{0}-f(\bar{x_{0}})\not\in-int(\mathbb{R}^{m}_{+})$,

  2. 2.

$f(x)+\sqrt{\rho}\|\bar{x_{0}}-x\|c_{0}-f(\bar{x_{0}})\not\in-
int(\mathbb{R}^{m}_{+})$.

In this article, we rely on two major tools from non-smooth analysis, namely
the subdifferential of a convex function and the Clarke subdifferential of a
locally Lipschitz function. Though these notions are very well known in the
optimization community, we shall provide the definitions for completeness. We
shall however restrict ourselves to the class of functions which are finite-
valued function on $\mathbb{R}^{n}$.
Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a convex function, then the
subdifferential of $f$ at the point $x$ is a set of vectors in
$\mathbb{R}^{n}$, given as

$\partial f(x)=\\{v\in\mathbb{R}^{n}:f(y)-f(x)\geq\langle
v,y-x\rangle,~{}\text{for all}~{}y\in\mathbb{R}^{n}\\}.$

The subdifferential set is a non-empty, convex and compact for every
$x\in\mathbb{R}^{n}$. The subdifferential is also deeply linked with the
notion of the directional derivative of a convex function. The directional
derivative of a convex function at a given $x$ in the direction $h$ is given
as

$f^{\prime}(x,h)=\lim_{\lambda\downarrow 0}\frac{f(x+\lambda
h)-f(x)}{\lambda}$

This directional derivative exists for each $x$ and in each direction $h$,
and, the subdifferential of $f$ can be written as $\partial
f(x)=\\{v\in\mathbb{R}^{n}:~{}f^{\prime}(x,h)\geq\langle
v,h\rangle,~{}\text{for all}~{}h\in\mathbb{R}^{n}\\}.$ Thus each of these can
be recovered from the other. The generalized notion of derivative has
properties like the usual derivative of calculus. We will begin with the most
fundamental one, the sum rule. Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ are convex functions. Then

$\partial(f+g)(x)=\partial f(x)+\partial g(x).$ (2.2)

For more details on subdifferentail of convex functions see [1]. It is
important to note that a point $x_{0}$ is a global minimum of $f$ on
$\mathbb{R}^{n}$ if and only if $0\in\partial f(x_{0})$. Since subdifferential
is a generalized version of derivative, it has some limitation. The
$\varepsilon$-subdifferential is a relaxed version of the subdifferential
which is very useful tool in convex analysis and optimization. We begin with
defining the $\varepsilon$-subdifferential of convex function.

###### Definition 2.7

Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be convex function and
$\varepsilon\geq 0$. The $\varepsilon$-subdifferential of $f$ at the point $x$
is given as

$\partial_{\varepsilon}f(x)=\\{v\in\mathbb{R}^{n}:f(y)-f(x)\geq\langle
v,y-x\rangle-\varepsilon,~{}\text{for all}~{}y\in\mathbb{R}^{n}\\}.$

The elements of $\partial_{\varepsilon}f(x)$ are called
$\varepsilon$-gradients of $f$ at $x$ and
$\partial_{\varepsilon}f(x)\not=\emptyset$ for all $x\in\mathbb{R}^{n}$. A
point $x_{0}$ is called an $\varepsilon$-minimizer of $f$ on $\mathbb{R}^{n}$
if $f(y)-f(x)\geq-\varepsilon$, for all $y\in\mathbb{R}^{n}$. Thus $x_{0}$ is
an $\varepsilon$-minimizer of $f$ on $\mathbb{R}^{n}$ if and only if
$0\in\partial_{\varepsilon}f(x_{0})$. For complete description of properties
of $\varepsilon$-subdifferential see [5].
The subdifferential defined above is only defined for convex functions, so the
obvious question is to ask what about subdifferential of non-convex functions?
We now discuss subdifferential of a non-convex function which is locally
Lipschitz in nature. The relation of subdifferential and directional
derivative as above becomes a key to develop the notion of a subdifferential
for a locally Lipschitz functions.
A function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is Lipschitz around
$x\in\mathbb{R}^{n}$, if there exists a neighborhood $U_{x}$ of $x$ and
$L_{x}\geq 0$ such that $\|f(y)-f(z)\|\leq L_{x}\|y-z\|,$ for all $y,z\in
U_{x}$. The constant $L_{x}$ is the Lipschtiz constant of the function $f$ at
the point $x$. A function $f$ is said to be locally Lipschitz if $f$ is
Lipschitz around $x$ for any $x\in\mathbb{R}^{n}$. We shall focus in this
article on MOP with locally Lipschitz objective and constraint functions. We
now define the Clarke directional derivative of locally Lipschitz function $f$
at $x$ and in the direction $h\in\mathbb{R}^{n}$ as

$f^{\circ}(x,h)=\limsup_{y\to x,t\downarrow 0}\frac{f(y+th)-f(y)}{t}.$

The Clarke subdifferential of $f$ at $x\in\mathbb{R}^{n}$ is given as,

$\partial^{\circ}f(x)=\\{\xi\in\mathbb{R}^{n}:~{}f^{\circ}(x,h)\geq\langle\xi,h\rangle,~{}\text{for
all}~{}h\in\mathbb{R}^{n}\\}.$

For each $x\in\mathbb{R}^{n},$ the set $\partial^{\circ}f(x)$ is non-empty,
convex and compact. It is important to note that when function $f$ is convex,
then $\partial^{\circ}f(x)=\partial f(x),$ for all $x\in\mathbb{R}^{n}$. Same
as subdifferential for convex function, Clarke subdifferential has lots of
nice properties. If $x_{0}\in\mathbb{R}^{n}$ is a local minimum of $f$ over
$\mathbb{R}^{n}$, then $0\in\partial^{\circ}f(x_{0})$ (for proof see [17]). It
also satisfy sum rule but it gives only one side containment, i.e., for given
two locally Lipschtiz function $f$ and $g$, we have
$\partial^{\circ}(f+g)(x)\subset\partial^{\circ}f(x)+\partial^{\circ}g(x).$

##  3 Approximate KKT conditions

In this section, we begin by defining a notion of modified $\epsilon$-KKT
points which suits very well for the purpose of convex vector optimization
problem. This notion is motivated by a similar notion defined in [8] for
scalar optimization problem and [6] for convex vector optimization.

###### Definition 3.1

A feasible point $x_{0}\in X$ is said to be a modified $\varepsilon$-KKT point
of MOP if for a given $\varepsilon\in\mathbb{R}_{+}$, there exists
$x_{\varepsilon}$ such that $\|x_{0}-x_{\varepsilon}\|\leq\sqrt{\varepsilon}$
and there exists $u_{i}\in\partial^{\circ}f_{i}(x_{\varepsilon})$ for all
$i\in I$, $v_{r}\in\partial^{\circ}g_{r}(x_{\varepsilon})$ for all $r\in L$,
vectors $\lambda\in\mathbb{R}_{+}^{m}$ with $\|\lambda\|=1$ and
$\mu\in\mathbb{R}_{+}^{l}$ such that

$\displaystyle\left\|\sum\limits_{i\in
1}^{m}\lambda_{i}u_{i}+\sum\limits_{r=1}^{l}\mu_{r}v_{r}\right\|\leq\sqrt{\varepsilon},\text{
and, }\sum\limits_{r=1}^{r}\mu_{r}g_{r}(x_{0})\geq-\varepsilon.$

Now we are in position to answer the first two questions asked in the
introduction. To begin with we state two constraint qualifications, Slater
constraint qualification (SCQ for short) and Basic constraint qualification
(BCQ for short) which are used in the main results of this article (see [18]).

###### Definition 3.2

The MOP with constraint functions $g_{r}$ for all $r\in L$ to be convex
satisfies Slater constraint qualification if there exists $\hat{x}\in X$ such
that $g_{r}(\hat{x})<0$, for all $r\in L$.

###### Definition 3.3

The MOP with locally Lipschitz constraint functions $g_{r}$ for all $r\in L$
satisfies Basic Constraint Qualification (BCQ) at a point $\bar{x}$ if there
exists no $p\in\mathbb{R}^{l}_{+}\setminus\\{0\\}$ such that
$0\in\sum\limits_{r\in L}p_{r}\partial^{\circ}g_{r}(\bar{x})$.

The next theorem answers the first question (Q1) raised in this article which
says that if a sequence $x_{k}$ of modified $\varepsilon_{k}$-KKT points of
MOP converges to a point $x_{0}$ where basic constraint qualification holds at
$x_{0}$, then $x_{0}$ is a KKT point of MOP. Observe that we do not need
convexity of the objective functions to prove the following result whereas we
only require Lipschitz continuity of the objectives. It is important to note
that the similar kind of result has been discussed under convexity assumption
of objective functions in [6].

###### Theorem 3.4

Consider the problem MOP with locally Lipschitz data and let
$\\{\varepsilon_{k}\\}$ to be a decreasing sequence of positive real numbers
such that $\varepsilon_{k}\rightarrow 0$ as $k\rightarrow\infty$. Consider
$\\{x^{k}\\}$ to be a sequence of feasible points of MOP with
$x^{k}\rightarrow x_{0}$ as $k\rightarrow\infty$. Assume that for each $k$,
$x^{k}$ is a modified $\varepsilon_{k}$-KKT point of MOP. Further, assume that
the BCQ holds at $x_{0}$. Then $x_{0}$ is a KKT point of MOP.

Proof: Note that $x^{k}$’s are feasible points, i.e., $g_{r}(x^{k})\leq 0$ for
all $r\in L$ and $x^{k}\rightarrow x_{0}$. Hence, using the convexity of
$g_{r}$’s, we conclude that $g_{r}(x_{0})\leq 0$ for all $r\in L$. Hence,
$x_{0}$ is a feasible point of MOP. Now, as $x^{k}$ is a modified
$\varepsilon_{k}$\- KKT point, for each $k$, Definition 3.1 gives the
existence of a point $\hat{x^{k}}$ such that
$\|x^{k}-\hat{x^{k}}\|\leq\sqrt{\varepsilon_{k}}$, the existence of
$u_{i}^{k}\in\partial^{\circ}f_{i}(\hat{x^{k}})$ and
$v_{r}^{k}\in\partial^{\circ}g_{r}(\hat{x^{k}})$ for all $i\in I$ and $r\in
L$, and the vectors $\lambda^{k}\in\mathbb{R}_{+}^{m}$ and
$\mu^{k}\in\mathbb{R}_{+}^{l}$ with $\|\lambda^{k}\|=1$ such that

$\displaystyle\left\|\sum\limits_{i\in
I}\lambda^{k}_{i}u^{k}_{i}+\sum\limits_{r\in
L}\mu^{k}_{r}v_{r}^{k}\right\|\leq\sqrt{\varepsilon_{k}},\text{ and }$ (3.1)
$\displaystyle\sum\limits_{r\in
L}\mu_{r}^{k}g_{r}(x^{k})\geq-\varepsilon_{k}.$ (3.2)

We first claim that $\\{\mu^{k}\\}$ is bounded. To prove our claim, on the
contrary assume that $\\{\mu^{k}\\}$ is unbounded. Thus,
$\|\mu^{k}\|\rightarrow\infty$ as $k\rightarrow\infty$. Further, Equation
(3.1), can be re-written as

$\left\|\sum\limits_{i\in
I}\frac{\lambda^{k}_{i}}{\|\mu^{k}\|}u^{k}_{i}+\sum\limits_{r\in
L}\frac{\mu^{k}_{r}}{\|\mu^{k}\|}v_{r}^{k}\right\|\leq\frac{1}{\|\mu^{k}\|}\sqrt{\varepsilon_{k}}.$
(3.3)

Then, in Equation (3.3), we observe the following:

  1. 1.

As $\varepsilon_{k}$ converges to $0$, the same holds for
$\frac{1}{\|\mu^{k}\|}\sqrt{\varepsilon_{k}}$.

  2. 2.

Let $p_{r}^{k}=\frac{\mu_{r}^{k}}{\|\mu^{k}\|}\in\mathbb{R}_{+}$, for all
$r\in L$. As $\|p^{k}\|=1$, $\\{p^{k}\\}$ is a bounded sequence. So, by the
Bolzano-Weierstrass theorem, there exists a subsequence of $\\{p^{k}\\}$ which
converges to $\hat{p}\in\mathbb{R}^{l}_{+}$ with $\|\hat{p}\|=1$. In fact,
without loss of generality, we can assume that $p^{k}_{r}$ converges to
$\hat{p}_{r}$. Hence, for all $r\in L$

$\frac{\mu^{k}_{r}}{\|\mu^{k}\|}=p^{k}_{r}\rightarrow\hat{p}_{r},~{}as~{}k\rightarrow\infty.$
(3.4)

  3. 3.

As $f_{i}$’s are locally Lipschitz functions, their Clarke subdifferential are
locally bounded, i.e., for $x_{0}\in X$, there exists $\delta>0$ such that for
all $z\in B_{\delta}(x_{0})$, $\partial^{\circ}f_{i}(z)\subset K_{i}$, where,
for all $i\in I$, $K_{i}$’s are bounded sets on $\mathbb{R}^{n}$. Since
$x^{k}\rightarrow x_{0}$, there exists $k_{0}\in\mathbb{N}$ such that, for all
$k\geq k_{0}$, $x^{k}\in B_{\delta}(x_{0})$. Therefore, by choosing
$K=\bigcup\limits_{i\in I}{\tilde{K_{i}}}$ where
$\tilde{K_{i}}=K_{i}\cup\partial^{\circ}f_{i}({x^{1}})\cup\ldots\partial^{\circ}f_{i}({x^{k}_{0}})$,
we get $\partial^{\circ}f_{i}({x^{k}})\subset K$, for all $i\in I$ and $k\geq
0$. Hence, the sequence $\\{u_{i}^{k}\\}$, where
$u_{i}^{k}\in\partial^{\circ}f_{i}({x^{k}})$, is bounded for all $i\in I$.
Hence, using the fact that $\|\lambda^{k}\|=1$ and
$\|\mu^{k}\|\rightarrow\infty$, we deduce that for all $i\in I$,

$\displaystyle\frac{\lambda_{i}^{k}}{\|\mu^{k}\|}u_{i}^{k}\rightarrow
0,~{}as~{}k\rightarrow\infty.$ (3.5)

  4. 4.

An argument similar to the previous part implies that the sequence
$\\{v_{r}^{k}\\}$ where $v_{r}^{k}\in\partial^{\circ}g_{r}(\hat{x^{k}})$, for
each fixed $r\in L$, is bounded. Hence, the sequence $\\{v_{r}^{k}\\}$ has a
limit point, for all $r\in L$, say $\hat{v}_{r}$. Without loss of generality,
we can assume that for all $r\in L$,

$v_{r}^{k}\rightarrow\hat{v}_{r},\text{ as }k\rightarrow\infty.$ (3.6)

Since $\partial^{\circ}g_{r}$’s is graph closed and $\hat{x}_{k}\rightarrow
x_{0}$, one has $\hat{v}_{r}\in\partial^{\circ}g_{r}(x_{0})$ for all $r\in L$.

Now, take the limit as $k\rightarrow\infty$ in Inequality (3.3) and in view of
the above observations (3.4),(3.5) and (3.6), we get,

$\left\|\sum_{r\in L}\hat{p}_{r}\hat{v}_{r}\right\|\leq 0.$

Hence, we have $\sum\limits_{r\in L}\hat{p}_{r}\hat{v}_{r}=0$, where
$\hat{p}\in\mathbb{R}^{l}_{+}$ with $\|\hat{p}\|=1$ and
$\hat{v}_{r}\in\partial^{\circ}g_{r}(x_{0})$ for all $r\in L$. This
contradicts the assumption that BCQ holds at $x_{0}$. Therefore, we have shown
the correctness of our claim, i.e., the sequence $\\{\mu^{k}\\}$ is a bounded.

As $\\{\mu^{k}\\}$ is a bounded sequence, an argument similar to the one
above, implies that there exist $\hat{\mu}\in\mathbb{R}^{l}_{+}$ such that
$\mu_{k}\rightarrow\hat{\mu}~{}as~{}k\rightarrow\infty$. Similarly, the
sequences $\\{\lambda^{k}\\}$ and $\\{u^{k}\\}$ have limit points, say
$\hat{\lambda}$ and $\hat{u}$, respectively, with $\|\hat{\lambda}\|=1$ and
$\lambda_{k}\rightarrow\hat{\lambda}$, $u^{k}\rightarrow\hat{u}$. Now taking
$k\rightarrow\infty$ in Inequality (3.1), we get $\|\sum\limits_{i\in
I}\hat{\lambda_{i}}\hat{u}_{i}+\sum\limits_{r\in
L}\hat{\mu_{r}}\hat{v}_{r}\|\leq 0.$ Thus

$\displaystyle\sum\limits_{i\in
I}\hat{\lambda_{i}}\hat{u}_{i}+\sum\limits_{r\in
L}\hat{\mu_{r}}\hat{v}_{r}=0,$ $\displaystyle\text{ where
}\hat{\lambda}\in\mathbb{R}^{m}_{+}\text{ with }\|\hat{\lambda}\|=1,$ (3.7)
$\displaystyle\hat{\mu}\in\mathbb{R}^{l}_{+},\hat{u}_{i}\in\partial^{\circ}f_{i}(x_{0})\text{
and }\hat{v}_{r}\in\partial^{\circ}g_{r}(x_{0}).$

Since, $x_{0}$ is a feasible point of MOP and $\hat{\mu}_{r}\geq 0$ for all
$r\in L$, we have $\sum\limits_{r\in L}\hat{\mu}_{r}g_{r}(x_{0})\leq 0.$
Taking $k\rightarrow\infty$ in Inequality (3.2), we get $\sum\limits_{r\in
L}\hat{\mu}_{r}g_{r}(x_{0})\geq 0$ and thus, we conclude that

$\sum\limits_{r\in L}\hat{\mu}_{r}g_{r}(x_{0})=0.$ (3.8)

The Inequalities (3.7) and (3.8) together imply that $x_{0}$ is a KKT point of
MOP. $\Box$

The next theorem deals with the second question asked in the article.
Basically, Q2 for multiobjective problem can be framed as follows: for every
local Pareto points of MOP, does there exists sequence which converges to the
point and the sequence has a subsequence which satisfies some type of
approximate KKT conditions? We answer this question in Theorem 3.6 for MOP
with locally Lipschitz objective function and Slater constraint qualification.
This result shows that we always have a sequence converging to a local Pareto
point of MOP with approximate KKT type of conditions which implies that the
idea of constructing approximate KKT type conditions is essential in
multiobjective theory. Note that Q2 has not been addressed in [6]. Before we
state theorem, we present the following lemma which will be needed in the
proof of the result. This lemma is a special case of Theorem 2.44 in [16].

###### Lemma 3.5

Let $A$ and $B$ be two non-empty subsets of $\mathbb{R}^{m}$. Let $A\cap
B\neq\emptyset$ and $\bar{x}\in A\cap B$. Assume that the following
qualification condition holds:

$N_{A}(\bar{x})\cap(-N_{B}(\bar{x}))=\\{0\\}.$

Then, $N_{A\cap B}(\bar{x})=N_{A}(\bar{x})+N_{B}(\bar{x})$.

###### Theorem 3.6

Consider the problem MOP with locally Lipschtiz objectives $f_{i}$’s for all
$i\in I$ and $g_{r}$’s for all $r\in L$ to be a convex functions which
satisfies the Slater constraint qualification. Further, assume that $x_{0}$ is
a local weak Pareto minima and consider $\\{\varepsilon_{k}\\}$ to be a
decreasing sequence of positive real numbers converging to $0$. Then, there
exists a sequence $\\{x^{k}\\}$ of feasible points converging to $x_{0}$ which
has a subsequence $\\{y^{k}\\}$ of $\\{x^{k}\\}$ such that for each $y^{k}$,
there exists $\hat{y}^{k}$ satisfying

  1. 1.

$\|y^{k}-\hat{y}^{k}\|\leq\sqrt{\varepsilon_{k}}$,

  2. 2.

there exists $u_{i}^{k}\in\partial^{\circ}f_{i}(\hat{y}^{k})$ and
$v_{r}^{k}\in\partial^{\circ}g_{r}(\hat{y}^{k})$, for all $i\in I$ and $r\in
L$, such that

$\displaystyle\left\|\sum\limits_{i\in
I}\lambda^{k}_{i}u^{k}_{i}+\sum\limits_{r\in
L}\mu^{k}_{r}v_{r}^{k}\right\|\leq\sqrt{\varepsilon_{k}},$ (3.9)
$\displaystyle\sum\limits_{r\in L}\mu_{r}^{k}g_{r}(\hat{y}^{k})=0,$ (3.10)

where $\lambda^{k}\in\mathbb{R}^{m}_{+}$ with $\|\lambda^{k}\|=1$ and
$\mu^{k}\in\mathbb{R}^{l}_{+}$.

Proof: By assumption, $x_{0}$ is a locally Pareto minimizer of MOP, i.e.,
there exists $\delta>0$ such that

$f(x)-f(x_{0})\not\in-\mathbb{R}^{m}_{+}\setminus\\{0\\},\text{ for all }x\in
V,$

equivalently,

$f(x)-f(x_{0})\in\tilde{W},\text{ for all }x\in V,$ (3.11)

where
$\tilde{W}:=\mathbb{R}^{m}\setminus(-\mathbb{R}^{m}_{+}\setminus\\{0\\})$ and
$V=X\cap\overline{B_{\delta}(x_{0})}$. The convexity of the constraint
functions $g_{r}$’s together with closed convex feasible set $X$ implies that
$V$ is a closed, convex and bounded set. As $x_{0}\in V$, there exists a
sequence $x^{k}$ in $X$ with $x^{k}$ converging to $x_{0}\in V$ and $x^{k}\in
V$, for all $k$ sufficiently large. We have broken the rest of the proof in
two steps. For the first step, we prove that there exists a sub-sequence
$\\{y^{k}\\}$ of $\\{x^{k}\\}$ such that $y^{k}\in V$ and is an
$\varepsilon_{k}e$-Pareto minima of MOP with feasible set as $V$ where
$e=(1,\dots,1)^{T}$ and $\varepsilon_{k}>0$.

As $f_{i}$’s, for $i\in I$, are locally Lipschitz, $f_{i}(x^{k})\rightarrow
f_{i}(x_{0})$ as $k\rightarrow\infty$, for all $i\in I$. So, for a given
$\varepsilon_{1}>0$, for each $i\in I$ there exist natural numbers
$N^{i}_{1}$, such that

$|f_{i}(x^{k})-f_{i}(x_{0})|<\varepsilon_{1},\text{ for all }k\geq N^{i}_{1}.$

Now choose $N_{1}=\max\\{N^{1}_{1},N^{2}_{1},\ldots,N^{m}_{1}\\}$. Thus, for
all $i\in I$

$|f_{i}(x^{k})-f_{i}(x_{0})|<\varepsilon_{1},\text{ for all }k\geq N_{1}.$
(3.12)

Choose $y^{1}=x^{N_{1}}$, then $|f_{i}(y^{1})-f_{i}(x_{0})|<\varepsilon_{1},$
or equivalently,

$f(x_{0})+e\varepsilon_{1}-f(y^{1})\in{int}(\mathbb{R}^{m}_{+}).$ (3.13)

Note that $\tilde{W}+int(\mathbb{R}^{m}_{+})\subseteq\tilde{W}$, hence, (3.11)
and (3.13) together gives

$f(x)+e\varepsilon_{1}-f(y^{1})\not\in-\mathbb{R}^{m}_{+}\setminus\\{0\\},\text{
for all }x\in V.$ (3.14)

Take $\varepsilon_{2}<\varepsilon_{1}$ and a similar argument applied to the
sequence $\\{x_{N_{1}},x_{N_{1}+1},x_{N_{1}+2},\ldots\\}$ gives an element
$y^{2}=x^{N_{2}}$, with $N_{2}>N_{1}$, such that
$f(x)+\varepsilon_{2}e-f(y^{2})\not\in-\mathbb{R}^{m}_{+}\setminus\\{0\\}\text{
for all }x\in V.$ Proceeding as above, gives a sub-sequence $\\{y^{k}\\}$ of
$\\{x^{k}\\}$ such that $y^{k}\in V$ and

$f(x)+\varepsilon_{k}e-f(y^{k})\not\in-\mathbb{R}^{m}_{+}\setminus\\{0\\},\text{
for all }x\in V.$ (3.15)

Hence, $y^{k}\in V$ is an $\varepsilon_{k}e$-Pareto minima of MOP with
feasible set as $V$. This completes the proof of the first step. We now come
to the second step to complete the proof.

Since each $f_{i}$ is locally Lipschitz, $f$ is ($e,\mathbb{R}^{m}_{+}$)-lower
semi continuous and $\mathbb{R}_{+}^{m}$-bounded below. Thus, the vector
Ekeland Variational Principle (Theorem 2.6) gives the existence of
$\hat{y}^{k}\in V$, for each $y^{k}\in V$, such that
$\|\hat{y}^{k}-y^{k}\|\leq\sqrt{\varepsilon_{k}},$ and for all $x\in
V\setminus\\{\hat{y}^{k}\\},$

  1. 1.

$f(x)+\varepsilon_{k}e-f(\hat{y}^{k})\not\in-int(\mathbb{R}_{+}^{m})$, and

  2. 2.

$f(x)+\sqrt{\varepsilon_{k}}\|\hat{y}^{k}-y^{k}\|e-f(\hat{y}^{k})\not\in-
int(\mathbb{R}^{m}_{+})$.

Thus from above, we conclude that $\hat{y}^{k}$ is a weak Pareto minimizer of
the problem

$\min\limits_{x\in V}g(x),\text{ where
}g(x)=f(x)+\sqrt{\varepsilon_{k}}\|x-\hat{y}^{k}\|e.$

Now, using the necessary optimality condition for the above multiobjective
problem, there exists $\lambda^{k}\in\mathbb{R}^{m}_{+}$ with
$\|\lambda^{k}\|=1$ such that

$0\in\sum\limits_{i\in
I}\lambda_{i}^{k}\partial^{\circ}g_{i}(\hat{y}^{k})+N_{V}(\hat{y}^{k}),$

where $N_{V}(\hat{y}^{k})$ is the normal cone to the set $V$ at $\hat{y}^{k}$.
For proof of above result see for example, page 137 of Chapter 5 in [7]. Now
applying sum rule for the Clarke subdifferential (see [3]) and using the fact
that subdifferential of the norm function at origin is the unit ball, we get

$0\in\sum\limits_{i\in
I}\lambda_{i}^{k}\partial^{\circ}f_{i}(\hat{y}^{k})+\sqrt{\varepsilon_{k}}B_{1}(0)+N_{V}(\hat{y}^{k}).$
(3.16)

Since $x^{k}\rightarrow x_{0}$ and $y^{k}$ is a sub-sequence of $\\{x^{k}\\}$,
$y^{k}\in X\cap B_{\delta}(x_{0})$, for sufficiently large $k$. As
$\hat{y}^{k}\in B_{\sqrt{\epsilon_{k}}}(y^{k})$ and $\epsilon_{k}\rightarrow
0$, for sufficiently large $k$, $B_{\sqrt{\epsilon_{k}}}(y^{k})\subset
B_{\delta}(x_{0})$. Hence, $\hat{y}^{k}\in B_{\delta}(x_{0})$, for $k$
sufficiently large.

Clearly, $X\cap\overline{B_{\delta}(x_{0})}\not=\emptyset$. We will now see
that the qualification condition for Lemma 3.5 holds in this case. Since
$\hat{y}^{k}\in B_{\delta}(x_{0})$, we see that
$\hat{y}^{k}\in{int}\overline{B_{\delta}(x_{0})}$, thus
$N_{\overline{B_{\delta}(x_{0})}}(\hat{y}^{k})=\\{0\\}$. Hence,
$N_{V}(\hat{y}^{k})\cap(-N_{\overline{B_{\delta}(x_{0})}}(\hat{y}^{k}))=\\{0\\}$.
Therefore, using Lemma 3.5, we conclude that

$N_{V}(\hat{y}^{k})=N_{X\cap\overline{B_{\delta}(x_{0})}}(\hat{y}^{k})=N_{X}(\hat{y}^{k})+N_{\overline{B_{\delta}(x_{0})}}(\hat{y}^{k}).$

Thus $N_{V}(\hat{y}^{k})=N_{X}(\hat{y}^{k}).$ Hence, we can rewrite (3.16) as

$0\in\sum\limits_{i\in
I}\lambda_{i}^{k}\partial^{\circ}f_{i}(\hat{y}^{k})+\sqrt{\varepsilon_{k}}B_{1}(0)+N_{X}(\hat{y}^{k}).$
(3.17)

Further as the Slater constraint qualification holds, using Corollary $23.7.1$
of [17],

$N_{X}(\hat{y}^{k})=\\{\sum\limits_{r\in
L}\mu_{r}^{k}v_{r}^{k}:v_{r}\in\partial g_{r}(\hat{y}^{k}),~{}\mu^{k}_{r}\geq
0,~{}\mu_{r}^{k}g_{r}^{k}(\hat{y}^{k})=0,~{}r\in L\\}.$

Now using the above form of $N_{X}(\hat{y}^{k})$ and (3.17), it is evident
that there exists $u_{i}^{k}\in\partial^{\circ}f_{i}(\hat{y}^{k})$ for all
$i\in I$, $v_{r}^{k}\in\partial g_{r}(\hat{y}^{k})$ for all $r\in L$ and
scalars $\lambda^{k}\in\mathbb{R}^{m}_{+}$ with $\|\lambda^{k}\|=1$,
$\mu^{k}\in\mathbb{R}^{l}_{+}$ such that (3.9) and (3.10) holds. This
completes the proof of the second part and hence the proof of the theorem is
complete. $\Box$

###### Remark 3.7

In the above theorem, the objective functions are taken to be locally
Lipschitz only. If the objective function $f_{i}$’s are convex as well, then
we have a more concrete result. To proof the next result we need the following
Lemma 3.8 and a result from [6] which will play a key role in proving the
Theorem 3.10.

###### Lemma 3.8

Consider the problem MOP with each objective functions $f_{i}$’s and
constraint function $g_{r}$’s to be convex. Then every local Pareto minima is
a global Pareto minima.

######  Theorem 3.9 (Theorem 3.6 of [6])

Let $x_{0}$ be a $\varepsilon e$-weak Pareto minima of the problem MOP with
each $f_{i}$’s and $g_{r}$’s to be convex functions and assume that Slater
constraint qualification holds. Then $x_{0}$ is a modified $\sigma$-KKT point
where $\sigma\in(0,\|e\|\varepsilon]$.

###### Theorem 3.10

Consider the problem MOP with each $f_{i}$ and $g_{r}$ being convex functions,
for all $i\in I$ and $r\in L$. Let $x_{0}$ be a Pareto minima and let the
Slater constraint qualification hold. Then, for decreasing sequence of
positive real numbers $\\{\varepsilon_{k}\\}$ converging to $0$, there exists
a feasible sequence $\\{x^{k}\\}$ converging to $x_{0}$ and a sub-sequence
$\\{y^{k}\\}$ of $\\{x^{k}\\}$ such that each $y^{k}$ is a modified
$\sigma_{k}$-KKT point with $\sigma_{k}\in(0,\;\|e\|\varepsilon_{k}]$.

Proof: Since the problem data is convex, local Pareto point is global. Now
proceed as in the proof of Theorem 3.6 to get a sub-sequence $\\{y^{k}\\}$ of
$\\{x^{k}\\}$ such that $y^{k}$ is a $\varepsilon_{k}e$-Pareto minima of MOP
with feasible set as $V$), where $V=X\cap\overline{B_{\delta}(x_{0})}$ with
$\delta>0$, i.e., $y^{k}$ is a local $\varepsilon_{k}e$-Pareto minima of MOP.
So, by using the assumption of convexity and Lemma 3.8, we conclude that
$y^{k}$ is a $\varepsilon_{k}e$-Pareto minima of MOP. Now using Theorem 3.9,
we conclude that $y^{k}$ is a modified $\sigma_{k}$-KKT point with
$\sigma_{k}\in(0,\;\|e\|\varepsilon_{k}]$. $\Box$

##  4 Approximate $\hat{M}$-Geoffrion solutions, Saddle points, and KKT
conditions

In this section, we analyze saddle point conditions and KKT type conditions
for the $(\hat{M},\epsilon)$-Geoffrion solutions which give a complete
characterization of the considered proper points. We also discuss a
scalarization rule for the $(\hat{M},\epsilon)$-Geoffrion solutions which is a
connecting bridge for deducing saddle point and KKT type conditions. Before
discussing the mentioned results, we shall observe that there is a
characterization of $(\hat{M},\epsilon)$-Geoffrion proper points by the system
of inequalities which appeared in [19]. For a given
$\epsilon\in\mathbb{R}^{m}_{+}$ and $\hat{M}>0$, consider $x_{0}\in X$, $i\in
I$ and define the following system of inequalities ($\mathcal{Q}_{i}(x_{0})$)
as

$\displaystyle\left\\{\begin{array}[]{ll}-f_{i}(x_{0})+f_{i}(x)+\epsilon_{i}<0,\\\
-f_{i}(x_{0})+f_{i}(x)+\epsilon_{i}<\hat{M}(f_{j}(x_{0})-f_{j}(x)-\epsilon_{j}),\text{
for all }j\in I\setminus\\{i\\}\\\ x\in{X}.\end{array}\right.$

###### Proposition 4.1

For given $\epsilon\in\mathbb{R}^{m}_{+}$ and $\hat{M}>0$, consider the
problem MOP. Then a point $x_{0}\in\mathcal{G}_{\hat{M},\epsilon}(f,X)$ if and
only if for each $i\in I$, the system $\mathcal{Q}_{i}(x_{0})$ is
inconsistent.

The above proposition follows from the definition of Proof of the
$(\hat{M},\epsilon)$-Geoffrion proper solutions, for complete proof, see [19].
Before discussing the saddle point conditions for the
$(\hat{M},\epsilon)$-Geoffrion proper solutions, let us discuss the
correspondence between $(M,\epsilon)$-Geoffrion proper solutions and solution
of the weighted sum scalar problem. As mentioned earlier, this correspondence
plays a pivotal role to prove main results of this section. To this end, let
for $s^{\ast}\in\mathbb{R}^{m}_{+}$, the weighted sum scalar problem
$P(s^{*})$ be defined as $\min\limits_{x\in X}~{}\langle s^{*},f(x)\rangle.$

###### Theorem 4.2

For a given $\epsilon\in\mathbb{R}^{m}_{+}$, $\hat{M}>0$, let $x_{0}$ is a
$\langle s^{\ast},\epsilon\rangle$-minimum of $P(s^{\ast})$, where
$s^{\ast}\in\text{int}(\mathbb{R}^{m}_{+})$. If
$\hat{M}\geq(m-1)\max\limits_{i,j}\\{\frac{s^{\ast}_{i}}{s^{\ast}_{j}}\\}$,
then $x_{0}$ is a $(\hat{M},\epsilon)$-Geoffrion proper solution of MOP, i.e.,
$x_{0}\in\mathcal{G}_{\hat{M},\epsilon}(f,X)$.

Proof: Let us assume on the contrary that
$x_{0}\notin\mathcal{G}_{\hat{M},\epsilon}(f,X)$. Therefore, from Proposition
4.1 we obtain an ${i}\in I$ such that $\mathcal{Q}_{i}(x_{0})$ is consistent.
Without loss of generality, we assume that $i=1$. Thus, the system
$\mathcal{Q}_{i}(x_{0})$, written as

$\displaystyle\left\\{\begin{array}[]{ll}-f_{1}(x_{0})+f_{1}(x)+\epsilon_{1}<0,\\\
-f_{1}(x_{0})+f_{1}(x)+\epsilon_{1}<\hat{M}(f_{j}(x_{0})-f_{j}(x)-\epsilon_{j}),\quad
j\in I\setminus\\{1\\}\\\ x\in{X}.\end{array}\right.$

has a solution. As $\hat{M}\geq(m-1)\\{\frac{s^{\ast}_{j}}{s^{\ast}_{i}}\\}$
for all $s^{\ast}\in\text{int}(\mathbb{R}^{m}_{+})$, the consistency of system
$\mathcal{Q}_{i}(x_{0})$ implies that

$s^{\ast}_{1}(-f_{1}(x_{0})+f_{1}(x)+\epsilon_{1})<s^{\ast}_{j}(m-1)(f_{j}(x_{0})-f_{j}(x)-\epsilon_{j}),\text{
for all }j\in I\setminus\\{1\\}.$

Summing the above equation for all $j\in I\setminus\\{1\\}$, we obtain that

$s^{\ast}_{1}(-f_{1}(x_{0})+f_{1}(x)+\epsilon_{1})<\sum_{j=2}^{m}s^{\ast}_{j}(f_{j}(x_{0})-f_{j}(x)-\epsilon_{j}),$

which further implies

$\langle s^{*},f(x_{0})\rangle-\langle s^{*},f(x)\rangle-\langle
s^{*},\epsilon\rangle>0.$ (4.3)

Since (4.3) is a contradiction to the $\langle
s^{\ast},\epsilon\rangle$-minimality of $P(s^{\ast})$. Therefore, the theorem
follows. $\Box$

All the solutions from $\mathcal{G}_{\hat{M},\epsilon}(f,X)$ satisfy an upper
trade-off bound of $\hat{M}$ (in the sense of Geoffrion-proper efficiency).
Smaller bounds are more relevant to the decision maker as they provide tighter
trade-offs among the criteria values. Therefore, it is of interest to find the
minimum ${M}$ such that $\mathcal{G}_{{M},\epsilon}(f,X)$ is non-empty. Under
the conditions of Theorem 4.2, we need minimum value of $\hat{M}$ equals
$m-1$, and this occurs when all components of $s^{\ast}$ are identical. The
next example shows that if conditions in Theorem 4.2 are not satisfied, then
even smaller values of $\hat{M}$ are possible. This is the case with non-
convex or discrete multicriteria optimization problems. In the following
example, we consider $\epsilon=0$ and find $\hat{M}$-Geoffrion proper points.

###### Example 4.3

Let
$X:=\\{(0,0,1)^{\top},\,(0,1,0)^{\top},\,(1,0,0)^{\top},(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})^{\top}\\}$,
$m=3$, and $f$ be the identity mapping. The sets $\mathcal{G}_{2}(f,X)$ and
$\mathcal{G}_{1}(f,X)$ can be easily computed as follows:

$\displaystyle\mathcal{G}_{2}(f,X)$ $\displaystyle=$
$\displaystyle\\{(0,0,1)^{\top},\,(0,1,0)^{\top},\,(1,0,0)^{\top},(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})^{\top}\\},$
$\displaystyle\mathcal{G}_{1}(f,X)$ $\displaystyle=$
$\displaystyle\\{(0,0,1)^{\top},\,(0,1,0)^{\top},\,(1,0,0)^{\top}\\}.$

Moreover, $\mathcal{G}_{M}(f,X)=\emptyset$ for $M<1$. Therefore, the minimum
value of ${M}$ is 1.

The converse of Theorem 4.2 also holds with convexity assumption on the
objective functions and the feasible set. Since, if for each $r\in L$, $g_{r}$
is convex, then the feasible set $X$ is a convex set. We have the following
result.

###### Theorem 4.4

Let us consider the problem MOP where for each $i\in I$ and $r\in L$, $f_{i}$
and $g_{r}$ are convex functions. If
$x_{0}\in\mathcal{G}_{\hat{M},\epsilon}(f,X)$, then there exists an
$s^{*}\in{\rm int}(\mathbb{R}^{m}_{+})$ such that $x_{0}$ is a $\langle
s^{*},\epsilon\rangle$-minimum of $P(s^{*})$.

Proof: Let $x_{0}\in\mathcal{G}_{\hat{M},\epsilon}(f,X)$. Then using
Proposition 4.1, we obtain that the system $\mathcal{Q}_{i}(x_{0})$ is
inconsistent, for each $i\in I$. Applying the Gordan’s Theorem of the
alternative (see [17]), we conclude, after some rearrangements, that for each
$i\in I$, there exists scalars $\lambda_{j}^{i}\geq 0$ with $\sum\limits_{j\in
I}\lambda_{j}^{i}=1$ such that, for all $x\in X$

$\displaystyle f_{i}(x)+\hat{M}\sum_{j\in
I,j\not=i}\lambda_{j}^{i}f_{j}(x)\geq f_{i}(x_{0})+\hat{M}\sum_{j\in
I,j\not=i}\lambda_{j}^{i}f_{j}(x_{0})-\left[\epsilon_{i}+\hat{M}\sum_{j\in
I,j\not=i}\lambda_{j}^{i}\epsilon_{j}\right].$

Therefore, by summing over all $i$, we get

$\displaystyle\sum_{i\in I}f_{i}(x)+\hat{M}\sum_{i\in I}\sum_{j\in
I,j\not=i}\lambda_{j}^{i}f_{j}(x)$ $\displaystyle\geq$
$\displaystyle\sum_{i\in I}f_{i}(x_{0})+\hat{M}\sum_{i\in I}\sum_{j\in
I,j\not=i}\lambda_{j}^{i}f_{j}(x_{0})$ $\displaystyle\hskip
25.29494pt-\sum_{i\in I}\left[\epsilon_{i}+\hat{M}\sum_{j\in
I,j\not=i}\lambda_{j}^{i}\epsilon_{j}\right].$

Hence, for all $x\in X$,

$\displaystyle\sum_{j\in I}\left[1+\hat{M}\sum_{i\in
I,i\not=j}\lambda_{j}^{i}\right]f_{j}(x)\geq\sum_{j\in
I}\left[1+\hat{M}\sum_{i\in
I,i\not=j}\lambda_{j}^{i}\right]f_{j}(x_{0})-\sum_{j\in
I}\left[1+\hat{M}\sum_{i\in I,i\not=j}\lambda_{j}^{i}\right]\epsilon_{j}.$

Setting $s_{j}=1+\hat{M}\sum\limits_{i\in I,i\not=j}\lambda_{j}^{i}$, gives
$s\in{int}(\mathbb{R}^{m}_{+})$ and $x_{0}$ is a $\langle
s,\epsilon\rangle$-minimum of $P(s)$. $\Box$

###### Remark 4.5

Theorem 4.4 can also be proved by noting the fact that each
$(\hat{M},\epsilon)$-Geoffrion proper point is $\epsilon$-Geoffrion proper
point with constant $\hat{M}>0$. Hence using Theorem 3.15 form [10], we can
deduce the above result. Now if we denote the set of $\langle
s^{*},\epsilon\rangle$-minimum of $P(s^{*})$ by $Sol_{\epsilon}(P(s^{*}))$,
then Theorem 4.2 and 4.4 implies that under convexity assumption on data and
for a given $\hat{M}$, there exists $s^{*}\in\text{int}(\mathbb{R}^{m}_{+})$
such that

$Sol_{\epsilon}(P(s^{*}))\subseteq\mathcal{G}_{\hat{M},\epsilon}(f,X)\subseteq\bigcup\limits_{s\in\text{int}(\mathbb{R}^{m}_{+})}Sol_{\epsilon}(P(s)).$

Now we come to the main attraction of this section, the saddle point
conditions for $(\hat{M},\epsilon)$-Geoffrion proper solutions. For this
study, we consider the problem MOP where each $f_{i}$, $i\in I$ and $g_{j}$,
$j\in L$ are a convex function. Whenever the data of problem is convex , we
shall denote the problem MOP as CMOP. Given $\hat{M}>0$, and any index $i\in
I$, we define the $(\hat{M},i)$-Lagrangian associated with CMOP as follows

$L^{\hat{M}}_{i}(x,\tau^{i},\mu^{i})=f_{i}(x)+\sum_{j\in
I,j\not=i}\tau^{i}_{j}\hat{M}f_{j}(x)+\sum_{r\in L}\mu_{r}^{i}g_{r}(x),$ (4.4)

where $\mu^{i}=(\mu^{i}_{1},\mu_{2}^{i},...,\mu^{i}_{l})\in\mathbb{R}^{l}_{+}$
and $\tau^{i}=(\tau^{i}_{1},\tau^{i}_{2},...,\tau^{i}_{m})\in S^{m}$ with
$S^{m}=\\{x\in\mathbb{R}^{m}:0\leq x_{i}\leq 1,i\in
I,\sum_{i=1}^{m}x_{i}=1\\},$ the unit simplex in $\mathbb{R}^{m}$. The
motivation behind considering the above Lagrangian comes from the $i$th-
objective Lagrangian problem defined in Chapter 4 of [2]. In [2], they used
the above Lagrangian form as a scalarization scheme of multiobjective
problems. In the same spirit as [2], we get a scalar structure of Lagrangian
functions which is comparatively easy than vector-valued Lagrangian to work
with. Our aim here is to show the key role played by the
$(\hat{M},i)$-Lagrangian in analyzing and characterizing the Geoffrion
($\hat{M},\epsilon$)-Proper solutions.

###### Theorem 4.6

For a given $\epsilon\in\mathbb{R}^{m}_{+}$ and $\hat{M}>0$, let us consider
the problem CMOP which satisfy the Slater constraint qualification. If
$x_{0}\in\mathcal{G}_{\hat{M},\epsilon}(f,X)$ then for each $i$, there exists
$\bar{\tau}^{i}\in S^{m}$, $\bar{\mu}^{i}\in\mathbb{R}^{l}_{+}$ such that for
all $x\in\mathbb{R}^{n}$ and $\mu\in\mathbb{R}_{+}^{m}$,

$(i)$ $L^{\hat{M}}_{i}(x_{0},\bar{\tau}^{i},\mu)-\bar{\epsilon}_{i}\leq
L^{\hat{M}}_{i}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i})\leq
L^{\hat{M}}_{i}(x,\bar{\tau}^{i},\bar{\mu}^{i})+\bar{\epsilon}_{i}$

($ii$) $\sum\limits_{r\in
L}\bar{\mu}_{r}^{i}g_{r}(x_{0})\geq-\bar{\epsilon}_{i}$,

where $\bar{\epsilon}_{i}=\epsilon_{i}+\sum\limits_{j=1,\\\
j\not=i}^{m}\tau^{i}_{j}\hat{M}\epsilon_{j}.$ Conversely if
$x_{0}\in\mathbb{R}^{n}$ be such that for each $i\in I$, there exists
$(\bar{\tau}^{i},\bar{\mu}^{i})\in S^{m}\times\mathbb{R}^{l}_{+}$ such that
$(i)$ and ($ii$) holds then $x_{0}\in\mathcal{G}_{\tilde{M},2\epsilon}(f,X)$,
where $\tilde{M}\geq(1+\hat{M})(m-1)$.

Proof: It is evident from Proposition 4.1 that if
$x_{0}\in\mathcal{G}_{\hat{M},\epsilon}(f,X)$, then for each $i\in I$, the
system $\mathcal{Q}_{i}(x_{0})$, re-written as

$\displaystyle-f_{i}(x_{0})+f_{i}(x)+\epsilon_{i}<0,$ $\displaystyle-
f_{i}(x_{0})+f_{i}(x)+\epsilon_{i}<M(f_{j}(x_{0})-f_{j}(x)-\epsilon_{j}),\text{
for all }j\in I\setminus\\{i\\}$ $\displaystyle g_{r}(x)\leq 0,~{}r\in L$

has no solution, for all $x\in\mathbb{R}^{n}$. It is easy to observe that the
system $\mathcal{Q}_{i}(x_{0})$ has no solution, if we replace $g_{r}\leq 0$
by $g_{r}<0$ for all $r\in L$. Now by applying the Gordan’s theorem of the
alternative (see [17]), there exists
$\tau^{i}=(\tau^{i}_{1},\ldots,\tau^{i}_{m})\in\mathbb{R}^{m}_{+}$ and
$\mu^{i}=(\mu^{i}_{1},\ldots,\mu^{i}_{l})\in\mathbb{R}^{l}_{+}$ with
$(\tau^{i},\mu^{i})\not=0$ such that for all $x\in\mathbb{R}^{n},$

$\tau^{i}_{i}(f_{i}(x)-f_{i}(x_{0})+\epsilon_{i})+\sum\limits_{j\in
I,j\not=i}\tau^{i}_{j}(f_{i}(x)+\hat{M}f_{j}(x)-f_{i}(x_{0})\\\
-\hat{M}f_{j}(x_{0})+\epsilon_{i}+\hat{M}\epsilon_{j})+\sum\limits_{r\in
L}\mu^{i}_{r}g_{r}(x)\geq 0.$

Hence, for all $x\in\mathbb{R}^{n},$

$\displaystyle\bigl{(}\sum\limits_{j\in
I}\tau^{i}_{j}\bigr{)}(f_{i}(x)-f_{i}(x_{0})+\epsilon_{i})+\sum\limits_{j\in
I,j\not=i}\left[\tau^{i}_{j}\hat{M}f_{j}(x)-\tau^{i}_{j}\hat{M}f_{j}(x_{0})+\tau^{i}_{j}\hat{M}\epsilon_{j}\right]$
$\displaystyle+\sum\limits_{r\in L}\mu^{i}_{r}g_{r}(x)\geq 0.$ (4.5)

Now, we first claim that $\tau^{i}=(\tau^{i}_{1},\ldots,\tau^{i}_{m})\not=0$.
For if, $\tau^{i}=0$ then $\mu^{i}\neq 0$ and Inequality (4) reduces to
$\sum\limits_{r\in L}\mu^{i}_{r}g_{r}(x)\geq 0$, for all $x\in\mathbb{R}^{n}$.
But, the Slater constraint qualification implies that there exists a point,
say $\hat{x}\in\mathbb{R}^{n}$, such that $g_{r}(\hat{x})<0$. As $\mu^{i}\neq
0$ and $\mu^{i}\in\mathbb{R}^{l}_{+}$, we obtain $\sum\limits_{r\in
L}\mu^{i}_{r}g_{r}(x)<0$, a contradiction to $\sum\limits_{r\in
L}\mu^{i}_{r}g_{r}(x)\geq 0.$ Hence, $\tau^{i}\not=0$ and thus
$\sum\limits_{j\in I}\tau^{i}_{j}>0$. Thus, dividing Inequality (4) by
$\sum\limits_{j\in I}\tau^{i}_{j}$, we get

$f_{i}(x)-f_{i}(x_{0})+\epsilon_{i}+\sum\limits_{j\in
I,j\not=i}[\bar{\tau}^{i}_{j}\hat{M}f_{j}(x)-\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0})+\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}]+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x)\geq 0,$ (4.6)

for all $x\in\mathbb{R}^{n}$, where
$\bar{\tau}^{i}_{j}=\frac{\tau^{i}_{j}}{\sum\limits_{j\in I}\tau^{i}_{j}}$ and
$\bar{\mu}^{i}_{r}=\frac{\mu^{i}_{r}}{\sum\limits_{j\in I}\tau^{i}_{j}}$. In
particular, for $x=x_{0}$, Inequality (4.6) gives
$\epsilon_{i}+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x_{0})\geq 0.$ By setting
$\bar{\epsilon_{i}}=\epsilon_{i}+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}$, we get Part ($ii$) as
$\sum\limits_{r\in L}\bar{\mu}^{i}_{r}g_{r}(x_{0})\geq-\bar{\epsilon_{i}}$.
Further, Inequality (4.6) reduces to, for all $x\in\mathbb{R}^{n}$,

$\displaystyle f_{i}(x)+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x)+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x)+\bar{\epsilon_{i}}\geq
f_{i}(x_{0})+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0}).$ (4.7)

As $x_{0}$ is feasible to CMOP, $\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x_{0})\leq 0$. Thus, Inequality (4.7) becomes

$\displaystyle f_{i}(x)+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x)+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x)+\bar{\epsilon_{i}}\geq
f_{i}(x_{0})+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0})+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x_{0}),$

which implies that for each $i\in I$ and for all $x\in\mathbb{R}^{n}$,

$\displaystyle
L_{i}^{\hat{M}}(x,\bar{\tau}^{i},\bar{\mu}^{i})+\bar{\epsilon_{i}}\geq
L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i}).$ (4.8)

Further, from Equation (4.4), we observe that for all $i\in I$ and any
$\mu\in\mathbb{R}^{l}_{+}$

$\displaystyle L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\mu)\leq
f(x_{0})+\sum\limits_{j\in I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0}),$

which can be written as $L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\mu)\leq
f(x_{0})+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0})+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x)+\bar{\epsilon}_{i}.$ Thus, for all
$x\in\mathbb{R}^{n}$ and $\mu\in\mathbb{R}^{l}_{+}$,

$\displaystyle L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\mu)\leq
L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i})+\bar{\epsilon_{i}}.$ (4.9)

The Inequalities (4.8) and (4.9) together prove Part ($i$). Now, for the
sufficient part, let us assume that for a given $x_{0}\in\mathbb{R}^{n}$ and
each $i\in I$ there exists $\bar{\tau}^{i}\in S^{m}$ and
$\bar{\mu}^{i}\in\mathbb{R}^{l}_{+}$ such that Conditions ($i$) and ($ii$)
hold. Our first step is to show that $x_{0}$ is feasible to CMOP. As we know
from $(i)$, for all $\mu\in\mathbb{R}^{l}_{+}$

$\displaystyle
L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\mu)-\bar{\epsilon_{i}}\leq
L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i}).$

Thus, $f_{i}(x_{0})+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0})+\sum\limits_{r\in
L}{\mu}_{r}g_{r}(x_{0})-\bar{\epsilon_{i}}\leq f_{i}(x_{0})+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0}).$ This shows that for all
$\mu\in\mathbb{R}^{l}_{+}$,

$\sum\limits_{r\in L}\mu_{r}g_{r}(x_{0})\leq\bar{\epsilon_{i}}.$ (4.10)

On the contrary, suppose $x_{0}$ is not feasible. Then, there exists $r_{0}\in
L$ such that $g_{r_{0}}(x_{0})>0$. Then, choose
$\mu=(0,\ldots,0,\mu_{r_{0}},0,\ldots,0)$, with $\mu_{r_{0}}>0$ and
sufficiently large such that $\mu_{r_{0}}g_{r_{0}}(x_{0})>\bar{\epsilon_{i}}.$
Note that this contradicts Inequality (4.10). Hence, we conclude that $x_{0}$
is a feasible solution of CMOP.

Now from right hand side of ($i$) we also have, for all $x\in\mathbb{R}^{n}$

$\displaystyle
L_{i}^{\hat{M}}(x,\bar{\tau}^{i},\bar{\mu}^{i})+\bar{\epsilon_{i}}\geq
L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i}).$ (4.11)

which implies

$f_{i}(x)+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x)+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x)+\epsilon_{i}+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}\geq f_{i}(x_{0})\\\
+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0})+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x_{0}).$

Now, for any feasible $x$, $\sum\limits_{r\in L}\bar{\mu}^{i}_{r}g_{r}(x)\leq
0$. Thus, from the above inequality we have,

$\displaystyle f_{i}(x)+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x)+\epsilon_{i}+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}\geq f_{i}(x_{0})$ (4.12)
$\displaystyle+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0})+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x_{0}).$

Using Condition ($ii$), we have

$f_{i}(x)+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x)+\epsilon_{i}+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}\geq f_{i}(x_{0})\\\
+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0})-(\epsilon_{i}+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}).$

Since, it holds for each $i$, by summing over all the $i$’s we get,

$\displaystyle\sum\limits_{i\in I}(1+\hat{M}\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j})f_{i}(x)+\sum\limits_{i\in
I}(1+\hat{M}\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j})(2\epsilon_{j})\geq\sum\limits_{i\in
I}(1+\hat{M}\sum\limits_{j\in I,j\not=i}\bar{\tau}^{i}_{j})f_{i}(x_{0}).$

Hence, $x_{0}$ is $\langle s,2\epsilon\rangle$-minimizer of $P(s)$, where
$s=(s_{1},\ldots,s_{m})$ with $s_{i}=1+\hat{M}\sum\limits_{k\in
I,k\not=i}\bar{\tau}^{i}_{k}$, for $i\in I$. Now since $\bar{\tau}^{i}\in
S^{m}$ for all $i,$ we have for all $i,j\in I$

$\frac{s_{i}}{s_{j}}=\frac{1+\hat{M}\sum\limits_{k\in
I,k\not=i}\bar{\tau}^{i}_{k}}{1+\hat{M}\sum\limits_{k\in
I,k\not=j}\bar{\tau}^{j}_{k}}=\frac{1+\hat{M}(1-\bar{\tau}^{i}_{i})}{1+\hat{M}(1-\bar{\tau}^{j}_{j})}\leq
1+\hat{M}.$

Since the above inequality is true for every $i$ and $j$, we have
$\max\limits_{i,j}\\{\frac{s_{i}}{s_{j}}\\}\leq 1+\hat{M}$. Now consider
$\tilde{M}\geq(1+\hat{M})(m-1)$ and using Theorem 4.2, we conclude that
$x_{0}\in\mathcal{G}_{\tilde{M},2\epsilon}(f,X).$ This completes the proof.
$\Box$

###### Remark 4.7

The saddle point type conditions are useful as a sufficient condition if the
number of objectives are only few in number. In fact, for sufficiency we can
have a much simpler condition which we now state. Let $x_{0}\in\mathbb{R}^{n}$
be a point that satisfies:
for each $i\in I$, there exists $\bar{\tau}^{i}\in S^{m}$ and
$\bar{\mu}^{i}\in\mathbb{R}^{l}_{+}$ such that for all
$\mu\in\mathbb{R}^{l}_{+}$ and $x\in\mathbb{R}^{n}$,

$(a)$ $L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\mu)-\epsilon_{i}\leq
L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i})\leq
L_{i}^{\hat{M}}(x,\bar{\tau}^{i},\bar{\mu}^{i})+\epsilon_{i},$

$(b)$ $\sum\limits_{r\in L}\bar{\mu}^{i}_{r}g_{r}(x_{0})\geq-{\epsilon_{i}}$.

Then, $x_{0}\in\mathcal{G}_{\tilde{M},2\epsilon}(f,X).$

In order to prove the above statement, note that
$\bar{\epsilon_{i}}=\epsilon_{i}+\sum\limits_{j=1,j\not=i}\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}$.
So, $\bar{\epsilon_{i}}\geq\epsilon_{i}$. Hence, Conditions ($a$) and ($b$)
above implies that Conditions ($i$) and ($ii$) of Theorem 4.6 are satisfied.
Therefore, we can simply apply the converse part of Theorem 4.6 to get
$x_{0}\in\mathcal{G}_{\tilde{M},2\bar{\epsilon}}(f,X),$ where
$\tilde{M}\geq(1+\hat{M})(m-1)$. Note that Condition ($a$) and ($b$) above are
much simpler as compared to checking Conditions ($i$) and ($ii$) as
$\bar{\epsilon_{i}}$ involves the multipliers $\bar{\tau}^{i}_{j}$. Hence, for
the sufficiency part of Theorem 4.6 which requires the verification of
Conditions $(i)$ and ($ii$), we will be using Conditions ($a$) and ($b$).

Of course from the necessary part of Theorem 4.6, we can also derive a
multiplier rule involving $\epsilon$-subdifferentials, however this rule will
be quite different. Observe that if
$x_{0}\in\mathcal{G}_{\hat{M},\epsilon}(f,X)$, then Condition ($i$) of Theorem
4.6 implies that for any $i\in I$ there exists $\bar{\tau}^{i}\in S^{m}$ and
$\bar{\mu}^{i}\in\mathbb{R}^{l}_{+}$ such that for all $x\in\mathbb{R}^{n}$,

$L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i})\leq
L_{i}^{\hat{M}}(x,\bar{\tau}^{i},\bar{\mu}^{i})+\bar{\epsilon_{i}},$

which implies that
$x_{0}\in\bar{\epsilon_{i}}-\underset{x\in\mathbb{R}^{n}}{\arg\min}L^{\hat{M}}_{i}(\;\cdot,\bar{\tau}^{i},\bar{\mu}^{i}),$
where $\bar{\epsilon}_{i}-\arg\min$ is the set of $\bar{\epsilon}_{i}$-minima
of the function $L_{i}^{\hat{M}}(x,\bar{\tau}^{i},\bar{\mu}^{i})$. Thus, for
each $i\in I$,
$0\in\partial_{\bar{\epsilon_{i}}}L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i}).$
In fact a more compact necessary condition of the KKT type is given as
follows,

$\displaystyle 0\in\sum_{i=\in
I}\partial_{\bar{\epsilon_{i}}}L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i})\quad\text{with}\quad\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x_{0})\geq-\bar{\epsilon_{i}}.$ (4.13)

###### Theorem 4.8

For a given $\epsilon\in\mathbb{R}^{m}_{+}$ and $\hat{M}>0$, let us consider
the problem CMOP. If $x_{0}\in\mathcal{G}_{\hat{M},{\epsilon}}(f,X)$, then
there exist vectors $\bar{\tau}^{i}\in S^{m}$ and
$\bar{\mu}^{i}\in\mathbb{R}^{l}_{+}$, $i\in I$ such that

$(A)$ $0\in\sum\limits_{i\in
I}\partial_{\bar{\epsilon_{i}}}L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i}),$

$(B)$
$\sum\limits_{r=1}^{l}\bar{\mu}^{i}_{r}g_{r}(x_{0})\geq-\bar{\epsilon_{i}}$,

where $\bar{\epsilon_{i}}=\epsilon_{i}+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}\epsilon_{j}$, $i\in I$. Conversely, if
$x_{0}\in X$ be a point for which there exist vectors
$(\bar{\tau}^{i},\bar{\mu}^{i})\in S^{m}\times\mathbb{R}^{l}_{+}$, $i\in I$
such that ($A$) and ($B$) hold then
$x_{0}\in\mathcal{G}_{\tilde{M},{2\epsilon}}(f,X)$, where
$\tilde{M}=(1+\hat{M})(m-1)$.

Proof: The necessary part has already been done in above remark. For
sufficient part, let conditions $(A)$ and $(B)$ hold for $x_{0}\in X$. This
means that there exists
$\bar{v}^{i}\in\partial_{\bar{\epsilon_{i}}}L_{i}^{\hat{M}}(x_{0},\bar{\tau}^{i},\bar{\mu}^{i})$
for all $i\in I$ such that

$0=\bar{v}^{1}+\bar{v}^{2}+\ldots+\bar{v}^{m}.$ (4.14)

Thus, from definition of $\epsilon$-subdifferential, for each $i\in I$,

$L^{\hat{M}}_{i}(x,\bar{\tau},\bar{\mu}^{i})-L^{\hat{M}}_{i}(x,\bar{\tau}^{i},\bar{\mu}^{i})\geq\langle\bar{v}^{i},x-x_{0}\rangle-\bar{\epsilon}^{i}.$

Hence,

$\sum\limits_{i\in
I}L^{\hat{M}}_{i}(x,\bar{\tau},\bar{\mu}^{i})-\sum\limits_{i\in
I}L^{\hat{M}}_{i}(x,\bar{\tau}^{i},\bar{\mu}^{i})\geq\langle\sum\limits_{i\in
I}\bar{v}^{i},x-x_{0}\rangle-\sum\limits_{i\in I}\bar{\epsilon}^{i}.$

Now using Equation (4.14), we get

$\sum\limits_{i\in I}(f_{i}(x)+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x)+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x))-\sum\limits_{i\in I}(f_{i}(x_{0})\\\
+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0})+\sum\limits_{r\in
L}\bar{\mu}^{i}_{r}g_{r}(x_{0}))\geq-\sum\limits_{i\in I}\bar{\epsilon}_{i}.$

So, if $x$ is a feasible point then using Condition ($B$), the above
inequality reduces to

$\displaystyle\sum\limits_{i\in I}(f_{i}(x)+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x))\geq\sum\limits_{i\in
I}(f_{i}(x_{0})+\sum\limits_{j\in
I,j\not=i}\bar{\tau}^{i}_{j}\hat{M}f_{j}(x_{0}))-\sum\limits_{i\in
I}2\bar{\epsilon}_{i},$

which can be rewritten as

$\displaystyle\sum\limits_{i\in I}(1+\sum\limits_{i\in
I,i\not=j}\bar{\tau}^{i}_{j}\hat{M})f_{i}(x)\geq\sum\limits_{i\in
I}(1+\sum\limits_{i\in
I,i\not=j}\bar{\tau}^{i}_{j}\hat{M})f_{i}(x_{0})-\sum\limits_{i\in
I}(1+\sum\limits_{i\in I,i\not=j}\bar{\tau}^{i}_{j}\hat{M})2\epsilon_{i}.$

Hence, $x_{0}$ is $\langle s,2\epsilon\rangle$-minimizer of $P(s)$ where
$s_{i}=1+\hat{M}\sum\limits_{k\in I,k\not=i}\bar{\tau}^{i}_{k}$. Now using the
same argument as in Theorem 4.2, we conclude that
$x_{0}\in\mathcal{G}_{\tilde{M},{2\epsilon}}(f,X)$, where
$\tilde{M}=(1+\hat{M})(m-1)$. This completes the proof. $\Box$

##  5 Concluding remarks

To analyze the behaviour of an optimization problem from the viewpoint of KKT
conditions is deep-rooted in psyche of researchers in optimization theory.
Though KKT conditions may not have been used very heavily in multiobjective
optimization, but they can, however, act very well as a tool to develop
stopping criteria. In this article, we characterize approximate versions of
Pareto and proper Pareto solution using KKT type conditions. In fact, in the
convex case, we achieve a complete characterization, for example, Theorem 3.6
demonstrates that a sequence of points which converge to weak Pareto minimizer
has a subsequence where each point satisfies an approximate version of the KKT
conditions. This result thus demonstrates the reason why approximate KKT type
conditions can be used as stopping criteria.
The analysis of the approximate versions of the $\hat{M}$-Geoffrion proper
solutions in terms of approximate KKT conditions is a starting point for
building stopping criteria to identify such points. Our future research would
involve more computational studies by using these optimality conditions as a
stopping criterion.

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