Approximations for Pareto and Proper Pareto solutions and their KKT conditions

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1 Approximations for Pareto and Proper Pareto solutions and their KKT conditions P. Kesarwani, P.K.Shukla, J. Dutta and K. Deb October 6, 2018 Abstract There has been numerous amount of studies on proper Pareto points in multiobjective optimization theory. Geoffrion proper points are one of the most prevalent form of proper optimality. Due to some convergence issues a restricted version of these proper points, Geoffrion proper points with preset bounds has been introduced recently. Since solution of any algorithm for multiobjective optimization problem gives rise to approximate solution set, in this article we study the approximate version of Geoffrion proper points with preset bounds. We investigate scalarization results and saddle point condition for these points. We also study the notion of approximate KKT condition for multiobjective optimization problem in general settings. Further, we discuss notion of approximate Benson proper point and develop KKT condition for the same. 1 Introduction There has been a growing interest among optimizers to analyze the nature of an approximate solution to an optimization problem. This stems from the fact that the most optimization problems cannot be solved in an exact manner. This fact about optimization has been very clearly stated in the beginning of the monograph on convex optimization by Nesterov [?]. In practice we know that our algorithms only return us some kind of approximate local/global solutions. Thus it is natural to develop a formal theory of approximate solutions and see to what extent their behavior parallels that of an exact solution. Optimizers have attempted to answer these questions over the past several decades, see for example [?], [?], [?] and [?]. In vector optimization the key to find a Pareto minimizer or a weak Pareto minimizer largely hinges on scalarization techniques(see for examples Ehrgott [?], Jahn [?], Luc [?], Chankong et al [?]). Using these techniques the solution that we generate is an approximate one. Hence it is natural to study the notion of approximate solutions in vector optimization. Several researchers have already analyzed various notions of approximate solution in vector optimization. See for example Dutte et al [?], Gutierrez et al [?,?], White [?], Rong et al [?]. It is also important to a decision maker, who is taking some decisions based on multiobjective optimization models need not necessarily be interested in all the pareto Department of Mathematics and Statistics, Indian Institute Of Technology Kanpur,India Institute AIFB, Karlsruhe Institute of Technology, Karlsruhe, Germany Department of Economics Sciences, Indian Institute Of Technology Kanpur, India College of Engineering, Michigan State University, Michigan, USA 1

2 solutions of the problem at hand. In many cases the decision maker just focuses on a 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 for example Ehrgott [?]). In real practice and more specifically in engineering there is a growing importance of the use of evolutionary algorithms for solving a multiobjective optimization problem(see for example Deb [?]). These evolutionary algorithms which are population based heuristics produces very quickly an approximation of the Pareto frontier in the image space. Thus the proper Pareto solutions that one gets in practice are in fact some kind of approximation of the proper Pareto solutions. Thus we are again faced with a situation where we need to build up an analysis for approximated proper Pareto solutions. When we consider the natural ordering cone or the non-negative cone then the first notion of a proper Pareto optimal solution was put forward by Geoffrion [?]. In [?], Geoffrion defined those Pareto optimal point as proper for which the objective function have a bounded trade off. The importance of this notion was clear when Geoffrion [?] tide this to the weighted-sum scalarization. If we take all the weights positive then solving the weighted-sum scalar problem will lead to a proper Pareto solution in the sense of Geoffrion [?]. The converse amazingly holds true if all objective functions are convex (see [?]). Such proper solutions are now referred to as the Geoffrion proper Pareto solution. It has been shown through an example in Eichfelder [?] that in many situations the ordering cone that we need to use is not the positive cone. How does one then talk of a proper Pareto solution? Thus the idea of Geoffrion had to be generalized into this new setting. This was achieved for example in Benson [?], Borwein [?] and Henig [?]. Around thirteen years ago in Shukla et al [?] it was shown that a more robust version of the Geoffrion proper Pareto solution can be in fact developed. The notion due to Geoffrion suffers from a drawback that the decision maker does not know before hand is how much is the trade-off bound. A decision maker in many situations might like to be concerned with only those proper solutions whose trade-off is bounded by a preset number provided by the decision maker herself/himself. In [?] such solutions are shown to be stable when approximate solution re considered. To be more precise a sequence of approximated Geoffrion proper Pareto solutions with a preset trade-off bound converges to a Geoffrion solution with the same trade-off bound. This property is not present if we consider the usual Geoffrion proper solutions. For more details on this new solution concept see [?]. 1.1 Our Aim The title of the paper reflects the fact that the celebrated Karush-Kuhn-Tucker condition(kkt for short) will play a fundamental role in this paper. There is no doubt that the KKT conditions play a pivotal role in scalar optimization. However its role in vector optimization is not clearly understood except for the fact that it provides a necessary condition. In fact if we look at the literature of multiobjective optimization we do not see any direct use of the KKT condition even in algorithms for solving linear multiobjective optimization problems. On the other hand as we have mentioned in most practical situations evolutionary algorithms are gaining prominence. Since these are heuristics it is important to check the quality of the points that we finally select as an approximation to Pareto points. To be more robust in our approach the KKT conditions can be profitably used as 2

3 a stopping criteria for such heuristics. In fact we can accept a point generated by a heuristic if it violates the KKT conditions with in a given margin of error. To the best of our knowledge such a use of the KKT condition for multiobjective optimization was first carried out in [?] (see also [?]). This motivated us to rethink on how to develope approximate versions of KKT condition for approximate Geoffrion proper solutions with a preset trade-off bound. In fact we show that such conditions are both necessary and sufficient when the problem data is convex. Thus at least in the convex case any point satisfying our approximate KKT conditions will be an approximate Geoffrion proper Pareto optimal point with a preset bound. This is a novel feature since usually the converse does not hold so easily even in the convex case when we just consider approximate weak Pareto solutions see for example Dutta et al [?]. Further we have developed a saddle point criteria for such type of solution points. We of course not only focus on Geoffrion solutions but also look at approximate version of Besnon proper Pareto solutions for which the necessary approximate KKT conditions are also sufficient when the problem data is cone-convex. thus both for the Geoffrion and Benson cases we have a complete characterization when the problem data is convex and when some qualification conditions are met. In order to make our presenation more complete we also study some approximate KKT conditions for approximate Pareto and weak Pareto points. The study of approximate KKT conditions for vector optimization problem done in a general setting was carried out for example in Durea et al [?]. We use their approach but provide results which are very different from that in [?]. A interesting result which we present here is a kind of converse. We show that if the problem data is locally Lipschitz or convex then if there is a sequence of points which are feasible and converges to a weak Pareto minimum, then there exists a subsequence of that sequence whose elements satisfy an approximate KKT type condition. We are yet to explore the complete implications of this result. 1.2 Organization of the Paper The paper is divided into five major sections which includes this introduction. In section 2 we introduce the standard notations used in this paper and also present the varios solution concept used in this paper. In this section we also introduce the notion of a Geoffrion proper Pareto solution with a fixed upper bound to the trade-offs. This new solution concept was introduced to the best of our knowledge by Shukla et al [?] and then was freshly looked into in Shukla et al [?]. In [?] new and important properties of the above mentioned solution concept was explored. In section 3 we focus on the usual notion of approximate weak Pareto solutions in the light of approximate KKT conditions. A key notion of approximate KKT condition introduced in this section is that of modified ɛ-kkt condition for multiobjective programming problem with locally Lipschitz data. This was motivated from its scalar counterpart introduced in Dutta et al [?]. The key result of this section is a result of the converse type. It says that if we consider a multiobjective programming problem with locally Lipschitz data and if there is a sequence converging to a weak local Pareto minimizer then there is a subsequence of the sequence which is feasible and satisfies a modified ɛ-kkt type condition under mild regularity assumption. This result is possible by the application of the Ekeland Variational for the vector case which is due to Tammer [?]. The key focus in section 4 is to develop an approximate version of the KKT condition for the approximate version of the Geoffrion proper Pareto solution with a fixed 3

4 upper bound on the trade off which is necessary in general but is also sufficient if in particular when the problem data is convex. In this section we also present some approximate KKT conditions associated with the approximate version of some other classes of proper Pareto solution like the Benson proper solution when the ordering cone is not the non-negative cone itself. In section 5 we explore whether approximate version of the Geoffrion proper Pareto with a fixed upper bound to the trade-off can be characterized through an approximate version of saddle point condition when the problem data is convex. 2 Notations and Definitions Our notations are fairly standard. Let A R n be a given set then closure and interior of set A is denoted by cla and inta respectively. A vector x R n is usually written as a column vector x = (x 1, x 2,..., x n ) T. The inner product of two vectors is given by x, y or x T y. We will use these two notations of inner product is an interchangeable fashion. A set A is a cone if for each a A and positive scalar λ, λa A. A cone A is pointed if A ( A) = {0}. We shall write cone(a) to denote the cone generated by the set A which is given as cone(a) = {λa λ 0, a A}. A dual cone A of set A R n is a cone A = {y R n x, y 0, x A} and (A ) 0 is defined by (A ) 0 = {y R n x, y > 0, x A \ {0}}. Consider the following multiobjective optimization problem MOP: min f(x) := (f 1 (x),..., f m (x)), subject to g j (x) 0, j = 1, 2,.., l. where each f i : R n R and g j : R n R. Let us denote the constraint set by X := {x R n : g j (x) 0, j = 1, 2,.., l} R n, I := {1, 2,.., m}, L := {1, 2,.., l} and set of active indices at x as R(x) := {r L : g r (x) = 0}. Solving MOP requires a (binary) ordering relation on R m. Given an ordering relation induced by a cone, one can compare two m-dimensional vectors from the set f(x) := {f(x) x X} and define an optimality notion for MOP. Consider C R m to be a closed, convex and pointed cone and define a ordering relation C on R m. For x, y R m, we will say x C y if and only if x y C. By choosing different cone C, we can get different notions of optimality for MOP. These optimality notions is used in algorithms to find one or many optimal solutions of MOP. When there are objective functions which are conflicting in nature, there is no single feasible point which minimizes all the objective functions simultaneously. Thus, a concept of optimality, called (weak) Pareto-optimality is considered. Unless MOP has a special structure, e.g X being polyhedral and each f i being linear or quadratic, almost all the algorithms for solving MOP generates an infinite sequence of iterates (or an infinite sequence of a set of points) that converges to the set of optimal points of MOP. For computational reasons, it is usually necessary to terminate the algorithm after some finite number of iterations. This leads to a sub-optimal point or, depending upon the distance of the obtained point to the set of optimal points of MOP, an ɛ-optimal point. To formalize our notions, in what follows we will consider ɛ R m +, i.e. ɛ = (ɛ 1, ɛ 2,..., ɛ m ), ɛ i 0 for each i I. Our focus on this paper is on ɛ- solutions of MOP. In the following definition the order of image space f(x) is induced by natural cone C = R m +. 4

5 Definition 2.1 Given an ɛ R n +, if there is no x X such that f(x) + ɛ f(x ) R m + \ {0}, then point x X is said to be an ɛ-pareto optimal solution of MOP. Further if there is no x X such that f(x) + ɛ f(x ) int(r m +), then point x is said to be a weak ɛ-pareto optimal solution of MOP. Let us make it clear at this stage that all our notations in this paper, specifically those denoting solution stets are borrowed from our paper [?]. The current paper is in many ways a continuation of our study in [?]. 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 δ > 0 and there is no x X B δ (x ) such that, f(x) f(x ) R m + \{0}, where B δ (x 0 ) R n is a unit ball of radius δ. The weak counter part of local solution can be defined in the similar fashion as in Definition 2.1. We would like to mention that in several situations we would have to consider the particular form of the vector ɛ R m +, given by ɛ = εe, where e = (1, 1.., 1) T. In those cases the solutions would be reffered to as the ε-pareto and weak ε-pareto solution respectively. The set of all ɛ-pareto points will be denoted by S ɛ (f, X) and the set of all weak ɛ Pareto points as Sw(f, ɛ X). An (weak) ɛ-pareto optimal solution with ɛ = 0 is commonly known as (weak) Pareto optimal. For simplicity, the set of Pareto optimal solutions and the set of weak Pareto optimal solutions will be denoted by S(f, X) and S w (f, X), respectively. As argued at the beginning of this paper, different Pareto optimal solutions might have different properties that a decision maker may desire. The need to filter out bad Pareto optimal solutions lead to the notion of a properly efficient point. There are different notions of proper optimality (see a nice survey in [?]). For example, if a closed, convex and pointed cone C is used as the ordering cone, Benson proper optimality [?] is a widely-studied notion. In an earlier work [?], we introduced the following approximate version of Benson proper optimality. Definition 2.3 Given an ɛ R n +, a point x 0 X is called Benson ɛ-proper solutions (with respect to the ordering cone C) if cl(cone(f(x) + (C + ɛ) f(x 0 ))) ( C) = {0}. The set of all Benson ɛ-proper solutions will be denoted by SB ε (f, X, C). If ɛ = 0, this reduces to classical notion of Benson proper efficiency [?], and we will use the notation S B (f, X, C) instead of SB 0 (f, X, C). In the case of C = R m +, Benson ɛ-proper optimality is equivalent to the following notion of Geoffrion ɛ-proper optimality (see [?,?], for example). Definition 2.4 Given an ɛ R n +, a point x 0 X is called ɛ-geoffrion proper solution if x 0 S ɛ (f, X) and if there exists a number M > 0 such that for all (i, x) I X satisfying f i (x) < f i (x 0 ) ɛ i, there exists an index j I such that f j (x 0 ) ɛ j < f j (x) and f i (x 0 ) f i (x) ɛ i f j (x) f j (x 0 ) + ɛ j M. 5

6 Geoffrion ɛ-proper solution says that the trade-offs between objective functions at two points are bounded. It is interesting to ask whether the trade-offs are bounded above for all the points. This would mean that the same M can work for all the Geoffrion ɛ-proper solutions. This may not always be the case unless the set of all trade off bounds is bounded above. Now we state a practical version of above definition which has been introduced in Shukla et al [?]. Definition 2.5 Given an ɛ R n + and a scalar ˆM > 0, a point x 0 X is called ( ˆM, ɛ)-geoffrion proper solution if x 0 S ɛ (f, X) and for all (i, x) I X satisfying f i (x) < f i (x 0 ) ɛ i, there exists an index j I such that f j (x 0 ) ɛ j < f j (x) and f i (x 0 ) f i (x) ɛ i f j (x) f j (x 0 ) + ɛ j ˆM. Given ˆM > 0, we shall denote the set of all ( ˆM, ɛ)- Geoffrion proper and ɛ-geoffrion proper solution as G ˆM,ɛ (f, X) and G ɛ (f, X) respectively. For ɛ = 0, the set of exact ˆM- Geoffrion proper and Geoffrion proper solution is denoted by G ˆM(f, X) and G(f, X) respectively. Since in this article we are dealing with vector valued objective function, we need to understand the notion of continuity and boundedness of vector valued functions. Definition 2.6 Let C be a closed, convex and pointed cone with non empty interior and f : U R m where U is a non empty subset of R n. The function f is C-bounded below if there exists y R m such that f(x) C y for all x U. Let c int(c), the function f is (c, C)- lower semi continuous if for all t R, {x U : tc C f(x)} is closed. Now we state Ekeland variational principle for vector valued functions ( see [?]) which is going to play a important role in this article. Theorem 2.7 Let C be a closed convex and pointed cone with non empty interior say c 0 int(c) and f : U R m is (c 0, C)- lower semi continuous and C bounded below. Given ε R +, x 0 U satisfying f(x) + εc 0 f(x 0 ) C \ {0}, x U (2.1) Then there exists x 0 = x 0 (ε) U such that (a) f(x) + εc 0 f( x 0 ) int(c), x U \ { x 0 }, (b) x 0 x 0 ε, (c) f(x) + ε x 0 x c 0 f( x 0 ) int(c), x U \ { x 0 }. 2.1 Tools from non-smooth analysis 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 chosen here namely finite-valued function on R n. 6

7 Let f : R n R be a convex function, then the subdifferential of f at x is a set of vectors in R n, given as f(x) = {v R n : f(y) f(x) v, y x, y R n }. The subdifferential set is a non-empty, convex and compact for every x 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 (x, h) = lim λ 0 f(x + λh) f(x) λ This directional derivative exists for each x and in each direction h and we have f(x) = {v R n : f (x, h) v, h, h R n }. Thus each of these can be recovered from the other. Now the most common question to ask is whether 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 : R n R and g : R n R are convex functions. Then (f + g)(x) = f(x) + g(x). (2.2) For more details on subdifferentail of convex functions see for example [1]. It is important to note that a point x 0 is a global minimum of f on R n if and only if 0 f(x 0 ). Since subdifferential is a generalized version of derivative it has some limitation. The ε-subdifferential is a relaxed version of the subdifferential which is very useful tool in convex analysis and optimization. We begin with defining the ε-subdifferential of convex function. Definition 2.8 Let f : R n R be convex function and ε 0. The ε-subdifferential of f at x is given as ε f(x) = {v R n : f(y) f(x) v, y x ε, y R n }. The elements of ε f(x) are called ε-gradients of f at x and ε f(x) for all x R n. Property (2.2) holds true for ε-subdifferential as well. Thus for f, g convex real valued function and for ε, ε 1, ε 2 0, we have ε (f + g)(x) = ε=ε 1 +ε 2 ε f(x) + ε g(x). A point x 0 is called an ε-minimizer of f on R n if f(y) f(x) ε for all y R n. Thus x 0 is an ε-minimizer of f on R n if and only if 0 ε f(x 0 ). Above subdifferential 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 nonconvex function. It is the relation of subdifferential and directional derivative as above that becomes a key to develop the notion of a subdifferential for a locally Lipschitz function. A function f : R n R is Lipschitz around x R n, if there exists a neighborhood U x of x and L x 0 such that f(y) f(z) L x y z, y, z U x. 7

8 where L x is Lipschtiz constant of f at x. A function f is said to be locally Lipschitz if f is Lipschitz around x for any x R n. We now define the Clarke directional derivative of locally Lipschitz function f at x and in the direction h R n is given as f (x, h) = lim sup y x,t 0 f(y + th) f(y). t The Clarke subdifferential of f at x R n is given as, f(x) = {ξ R n : f (x, h) ξ, h, h R n }. For each x R n, the set f(x) is non-empty convex and compact. It is important to note that when function f is convex then f(x) = f(x), for all x R n. Same as subdifferential of convex function, Clarke subdifferential has sum rule but it gives only one side containment i.e. for given two locally Lipschtiz functionf, g, we have (f + g)(x) f(x) + g(x). Let us discuss some important properties of Clarke subdifferential which will be used in this article (for proofs see [?]). Let f : R n R be a locally Lipschitz function. Then (i) For each given x R n, the function f (x, h) is convex and positive homogeneous in h, (ii) f (x, h) is upper-semicontinuous function of (x, h). (iii) If x 0 R n is a local minimum of f over R n then 0 f(x 0 ). 3 Approximate KKT conditions we have already explained in the previous section the importance of approximate solutions of a multiobjective optimization problem from a practical point of view. Thus it is natural to ask whether these approximate solutions satisfy some kind of approximate KKT condition. In the literature there are several approaches to approximate KKT conditions (see for example [?],[?],[?]). In this section we begin by defining a notion of approximate KKT points which suits very well for the purpose of convex vector optimization problem. This notion is called modified ɛ-kkt points, which are motivated by a similar notion defined in Dutta et al [?] for scalar optimization problem and Durea et al [?] for vector optimization. The next question which is a natural one is as follows, Under what assumptions on the problem data, do an ɛ-weak Pareto point (ɛ-weak Pareto) satisfy the approximate KKT conditions and under what assumptions the converse can also be generated? We shall focus on these questions in this section. Through out our discussion in this section we shall also consider the m objective functions f 1, f 2,..., f m to be locally Lipschitz. We shall not repeat this assumption in the statements of our results. Definition 3.1 A feasible point x 0 X is said to be a modified ε-kkt point for a given ε R + if there exists x ε such that x 0 x ε ε and there exists u i f i (x ε ) 8

9 for i I, v r g r (x ε ) for r = L, vectors λ R m + that with λ = 1 and µ R l + such λ i u i + and µ r v r ε, (3.1) r µ r g r (x 0 ) ε. (3.2) The following natural constraint qualification appears for example in Rockafellar and Wets [?]. This is called the Basic constraint qualification( BCQ for short). Definition 3.2 The problem MOP satisfies BCQ at x if there exists no p R l + \ {0} such that 0 l p r g r ( x). Theorem 3.3 Consider the problem MOP and let {ε k } to be a decreasing sequence of positive real numbers such that ε k 0 as k. Consider {x k } to be a sequence of feasible points of MOP with x k x 0 as k. Assume that for each k, x k is a modified ε k -KKT point of MOP. Further assume that the Basic constraint qualification holds at x 0. Then x 0 is a KKT point of MOP. Proof. By our assumption for each k, x k is a modified ε k - KKT point, so from Definition 3.1, there exists ˆx k such that x k ˆx k ε k and there exists u k i f i ( ˆx k ) for all i I, v k r g r ( ˆx k ) for all r L, vectors λ k R m + with λ k = 1 and µ k R l + such that λ k i u k i + µ k rvr k ε k, (3.3) µ k rg r (x k ) ε k. (3.4) First of all we will show that {µ k } is bounded. On the contrary assume that {µ k } is unbounded. Thus with out loss of generality we can write µ k as k. From equation (3.3), we have λ k i µ k uk i + µ k r µ k vk r 1 εk. (3.5) µ k Let p k = µk µ k Rl +. Since p k = 1, p k is a bounded sequence so by Bolzano- Weirstrass theorem there exists a subsequence of {p k } which converges to as ˆp R l + with ˆp = 1. In fact we need not relabel {p k } and assume that p k ˆp, without loss of generality. This shows that µ k µ k = pk ˆp, as k (3.6) By our assumption f i s and g r s are locally Lipschitz functions, it implies the Clarke subdifferential of f i s and g r s are locally bounded. Hence u k i f i ( ˆx k ) and v k r g r ( ˆx k ) are bounded for all i I and r L. Thus without loss of generality we can 9

10 assume that for each i I and r L there exist û i and ˆv r respectively such that for all i I and r L, u k i û i, and v k r ˆv r, as k. Since f i s and g r s are graph closed as well and x k x 0 that implies û i f i (x 0 ) for all i I and ˆv r g r (x 0 ) for all r L. Since the sequences {u k i } and {λ k } are bounded sequences for all i I, we have for all i I λ k i µ k uk i 0, as k. (3.7) Since ε k is decreasing sequence with ε k 0, it implies that ε k 0. So we have { ε k } to be a bounded sequence and hence εk 0, as k (3.8) µ k As k in (3.5) by using (3.6),(3.7) and (3.8), we get l ˆp rˆv r 0. Hence we have ˆp rˆv r = 0, where ˆp R l + with ˆp = 1 and ˆv r g r (x 0 ) for r L. This contradicts the assumption that BCQ holds at x 0. Therefore the sequence {µ k } is a bounded. Thus without loss of generality there exist ˆµ R l + such that µ k ˆµ as k. Now as λ k R m + with λ k = 1, the sequence {λ k } has a limit point say ˆλ with ˆλ = 1. With out loss of generality we can assume that λ k ˆλ. Now as k in (3.3), we get m ˆλ i û i + l ˆµ rˆv r 0. Thus ˆλ i û i + ˆµ rˆv r = 0, (3.9) where ˆλ R m + with ˆλ = 1, ˆµ R l +, û i f i (x 0 ) for i I and ˆv r g r (x 0 ) for r L. Now since x k are feasible points, g r (x k ) 0 for all r L and k. Using continuity property of g r s, we conclude that g r (x 0 ) 0 for all r L. Hence x 0 is a feasible point for MOP. Since ˆµ r 0 for all r L, we have in from (3.4), we get l ˆµ r g r (x 0 ) 0. Therefore we conclude that ˆµ r g r (x 0 ) 0. Now as k ˆµ r g r (x 0 ) = 0. (3.10) Thus (3.9) and (3.10) together implies that x 0 is a KKT point of MOP. An interesting question one might ask is the following. Does the previous theorem has some kind of converse? To be more precise we ask our self the following question. If we have sequence of iterates from the feasible set converge to a weak Pareto minimizer, then do these iterates satisfy some kind of approximate KKT conditions? The answer surprisingly turns out to be affirmative and thus it strengthens the whole premise of studying approximate version of KKT condition for multiobjective optimization problem. 10

11 Theorem 3.4 Consider the problem MOP where g 1,.., g l are convex functions. Let x 0 is a local weak Pareto minima and the Slater condition holds. Consider {ε k } to be a decreasing sequence of positive real numbers such that ε k 0 as k. Then there exists a feasible sequence {x k } with x k x 0 and a subsequence {y k } of {x k } such that for each y k there exists ŷ k with y k ŷ k ε k, u k i f i (ŷ k ) for all i I, v k r g r (ŷ k ) for all r L and λ k R m + with λ k = 1, µ k R l + such that λ k i u k i + µ k rvr k ε k, (3.11) µ k rg r (ŷ k ) = 0. (3.12) Proof. Since x 0 is a locally weak Pareto minimizer of MOP, by definition (2.2) there exists δ > 0 such that for all x V, f(x) f(x 0 ) int(r m +), (3.13) where V = X B δ (x 0 ). Since each g r s are convex functions, the feasible set X is closed and convex. Thus V is closed, convex and bounded. Now since x 0 V, there exists a sequence x k in X with x k x 0 and for k sufficiently large x k V. Since each f i s, i I are assumed to be locally Lipschitz it implies that f i (x k ) f i (x 0 ) as k for all i I. Now consider ε 1 > 0, since f i (x k ) f i (x 0 ) for all i I and it implies that for each i I there exists a natural number N i 1 such that for all k > N i 1, f i (x k ) f i (x 0 ) < ε 1. Now choose N 1 N, N 1 = max{n 1 1, N 2 1,..., N m 1 }. Then for all k > N 1 and i I, f i (x k ) f i (x 0 ) < ε l. Consider y 1 = x N1 +1, then f i (y 1 ) f i (x 0 ) < ε 1. This means that for all i I, Now (3.13) can be rephrased as, f i (y 1 ) < f i (x 0 ) + ε 1. (3.14) f(x) f(x 0 ) W, x V (3.15) where W = R m \ ( intr m +). It is important to note that W is a closed cone but a non-convex one. Further one has W + R m + W. Now using (3.14) we can write that Now adding (3.15) and (3.16) we have f(x 0 ) + eɛ 1 f(y 1 ) int(r m +). (3.16) f(x 0 ) + eɛ 1 f(y 1 ) int(r m +) W, x V. (3.17) Hence for x V, f(x 0 ) + eɛ 1 f(y 1 ) W. This shows that for x V, f(x 0 ) + eɛ 1 f(y 1 ) int(r m +). Hence y 1 is a weak ε 1 -Pareto minima of MOP over V. In similar way for each k N and ε k > 0, we can get a element y k {x k } with each y k to be a weak ε k -Pareto minima of MOP over V. Now consider y k which is a weak ε k -Pareto minima of MOP over V i.e. for all x V f(x) + ε k e f(y k ) int(r m +). (3.18) Since each f i is locally Lipschitz, we conclude that f is (e, R m +)-lower semi continuous and R m +-bounded below. Now using the vector Ekeland Variational Principle (2.7), for each y k there exists ŷ k V such that for all x V \ {ŷ k } 11

12 (a) f(x) + ε k e f(ŷ k ) int(r m +), (b) ŷ k y k ε k, (c) f(x) + ε k ŷ k y k e f(ŷ k ) int(r m +). Thus from (c) above we conclude that ŷ k is weak Pareto minimizer of the problem min f(x) + ε k x ŷ k e x V Now using necessary optimality condition for the above multiobjective problem, there exists λ k R m + with λ k = 1 such that 0 λ k i (f i + ε k x ŷ k )(ŷ k ) + N V (ŷ k ), where f i (ŷ k ) denotes Clarke subdifferential of function f i at the point ŷ k and N V (ŷ k ) is normal cone of V at ŷ k. Now applying sum rule for the Clarke subdifferential (see Clarke [?])and using the fact that subdifferential of the norm function at origin is the unit ball, we get 0 λ k i f i (ŷ k ) + ε k B R n + N V (ŷ k ). (3.19) Since x k x 0 and y k {x k }, for k sufficiently large we can conclude that y k X B δ (x 0 ). Further from (ii) above we see that ŷ k B ɛ k (y k ). Now we choose k large enough so that B ɛ k (y k ) B δ (x 0 ), proving that ŷ k B δ (x 0 ). Now since X B δ (x 0 ) (as x 0 X and x 0 B δ (x 0 )), we conclude that ri(x B δ (x 0 )). Further using Theorem 6.5 in Rockafellar [?], we conclude that ri(x B δ (x 0 )) = ri(x) ri(b δ (x 0 )). Now using Theorem 23.8 in Rockafellar [?] we have N V (ŷ k ) = N X Bδ (x 0 ) (ŷ k) = N X (ŷ k ) + N Bδ (x 0 ) (ŷ k). As ŷ k B δ (x 0 ) = intb δ (x 0 ), we see that N Bδ (x 0 ) (ŷ k) = {0}. Thus N V (ŷ k ) = N X (ŷ k ). Hence we can rewrite (3.19) as 0 λ k i f i (ŷ k ) + ε k B R n + N X (ŷ k ). (3.20) Further as the Slater condition holds, using corollary from Rockafellar [?], we conclude that N X (ŷ k ) = { l µ k rvr k : v r g r (ŷ k ), µ k r 0, µ k rgr k (ŷ k ) = 0, r L}. Now using above form of N X (ŷ k ) and (3.20), we conclude that there exists u k i f i (ŷ k ) for all i I, vr k g r (ŷ k ) for all r L and scalars λ k R m + with λ k = 1, µ k R l + such that (3.11) and (3.12) holds. Thus the theorem follows. Remark 3.5 In the above theorem the objective functions are taken to be locally Lipschitz only. If objective function f i s are convex as well then we have more concrete result then above. To proof this result we need the following lemma. Lemma 3.6 Consider a MOP problem with each objective functions f 1, f 2,.., f m s to be convex. Then every local weak Pareto minima is a global weak Pareto minima. 12

13 Proof. Let x o is a local weak Pareto minima of MOP i.e. there exists δ > 0 such that f(x) f(x 0 ) int(r m +), x X B δ (x 0 ). On contrary assume that x 0 is not global weak minima i.e. there exists x X such that f(x ) f(x 0 ) int(r m +), (3.21) which implies f i (x ) < f i (x 0 ) for all i I. Since each f i s are convex functions, we have for t R + and using (3.21), f i ((1 t)x + tx 0 ) (1 t)f i (x ) + tf i (x 0 ) < f i (x 0 ), i I. (3.22) Now we can always choose t R + such that (1 t)x + tx 0 B δ (x 0 ). For this particular t, equation (3.22) is a contradiction to the fact that x 0 is local weak Pareto minima. Theorem 3.7 Consider the problem MOP with each f 1,.., f m and g 1,.., g l to be convex and locally Lipschitz functions. Let x 0 is a local weak Pareto minima and Slater condition holds. Consider {ε k } to be a decreasing sequence of positive real numbers such that ε k 0 as k. Then there exists a feasible sequence {x k } with x k x 0 and a subsequence {y k } of {x k } such that each y k is a modified ε k -KKT point. Proof. Follow the proof of above theorem till equation (3.18) which implies that y k is a weak ε k -Pareto minima of MOP over V = X B δ (x 0 ) with δ > 0 i.e. y k is a local weak ε k -Pareto minima of MOP. By using the assumption of convexity and Lemma 3.6, we conclude that y k is a weak ε k -Pareto minima of MOP i.e. f(x) + ε k e f(y k ) int(r m +), x X, Now using Theorem 3.6 of Durea et al [?], we conclude that y k is a modified ε k -KKT point. 4 Approximate ˆM-Geoffrion solutions, Saddle points and KKT conditions In this section we analyze saddle point conditions and KKT condition for approximate ˆM-Geoffrion solutions which gives a complete characterization of the considered proper points. We also discuss the correspondence between ˆM-Geoffrion solutions and solution of a weighted sum scalar problem. Before discussing the mentioned results we shall observe that there is a characterization of Geoffrion properly efficient points by a system of inequalities. For a given ɛ R m + and ˆM > 0, consider x 0 X, i I and define the following system of inequalities (Q i ) as f i (x 0 ) + f i (x) + ɛ i < 0, f i (x 0 ) + f i (x) + ɛ i < M(f j (x 0 ) f j (x) ɛ j ), x X. j I \ {i} Proposition 4.1 For given ɛ R m + and ˆM > 0 consider the problem MOP. Then a point x 0 G ˆM,ɛ (f, X) if and only if for each i I, the system Q i is inconsistent. 13

14 Proof of the above proposition can be found in [?]. Before discussing the saddle point condition for approximate ˆM Geoffrion proper point let us discuss the correspondence between (M, ɛ)-geoffrion proper solutions and solution of weighted sum scalar problem. This correspondence plays a pivotal role to prove saddle point conditions for (M, ɛ)-geoffrion proper solutions. To this end, let for an s R m, the weighted sum scalar problem P (s ) be defined as follows: min x X s, f(x). (P (s )) Theorem 4.2 For a given ɛ R m + and ˆM > 0, let x 0 is a s, ɛ -minimum of P (s ), where s int(r m +). If ˆM (m 1) max { s i }, then x i,j s 0 G j ˆM,ɛ (f, X). Proof. Let us assume on the contrary that x 0 / G ˆM,ɛ (f, X). Therefore, from Proposition 4.1 we obtain an i I such that Q i is consistent. Without loss of generality, we assume that i = 1. Thus, the system Q i, written as f 1 (x 0 ) + f 1 (x) + ɛ 1 < 0, f 1 (x 0 ) + f 1 (x) + ɛ 1 < ˆM(f j (x 0 ) f j (x) ɛ j ), j I \ {1} x X. has a solution. As ˆM (m 1){ s j s i } for all s intr m +, the consistency of system Q i implies that s 1( f 1 (x 0 ) + f 1 (x) + ɛ 1 ) < s j(m 1)(f j (x 0 ) f j (x) ɛ j ), j I \ {1}. Summing the above equation for all j I \ {1}, we obtain that s 1( f 1 (x 0 ) + f 1 (x) + ɛ 1 ) < s j(f j (x 0 ) f j (x) ɛ j ), which further implies j=2 s, f(x 0 ) s, f(x) s, ɛ > 0. (4.1) Since (4.1) is a contradiction to the s, ɛ -minimality of P (s ). Therefore, the theorem follows. All the solutions from G ˆM,ɛ (f, X) satisfy an upper trade-off bound of ˆM (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. It would, therefore, be of interest to find the minimum M such that G M,ɛ (f, X) is non-empty. Under the conditions of Theorem 4.2, we need minimum value of ˆM equals m 1, and this occurs when all components of s are identical. The next examples show that if conditions in Theorem 4.2 are not satisfied, then even smaller values of ˆM are possible. This is the case with non-convex or discrete multicriteria optimization problems. In the following examples we consider ɛ = 0 and find ˆM-Geoffrion proper points Example 4.3 Let X := {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1/ 3, 1/ 3, 1/ 3) }, m = 3, and f be the identity mapping. The sets G 2 (f, X) and G 1 (f, X) can be easily computed as follows: G 2 (f, X) = {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1/ 3, 1/ 3, 1/ 3) }, G 1 (f, X) = {(0, 0, 1), (0, 1, 0), (1, 0, 0) }. Moreover, G M (f, X) = for M < 1. Therefore, the minimum value of M is 1. 14

15 Example 4.4 Let us the consider the DTLZ2 test problem from [?]. This test problem has 12 variables, 3 objectives, and a concave efficient front. The set of Paretooptimal solutions are given by S(f, X) := { x [0, 1] 12 f 1 (x) 2 + f 2 (x) 2 + f 3 (x) 2 = 1, f 1 (x), f 2 (x), f 3 (x) 0 }. Let x, y, z X := [0, 1] 12 be three Pareto-optimal solutions such that f(x) = (0, 0, 1), f(y) = (0, 1, 0), f(z) = (1, 0, 0) and we aim to check if f(x) belongs to G 1 (f, X) or not. From [?, Lemma 2], we know that G 1 (f, X) = G 1 (f, S(f, X)). Therefore, we consider the trade-offs of f(x) with respect to all efficient points, i.e. all points from the set f(g 1 (f, S(f, X))). Let, for δ [0, 1], S δ := {x S(f, X) f 3 (x) = 1 δ}. From the definition of S(f, X), it follows that f(s δ ) := {(u, v, w) R 3 w = 1 δ, u 2 + v 2 1 (1 δ) 2 }. The minimum value of the Geoffrion M bound, denoted by ˆM, can now be computed as follows: ˆM = max δ [0,1] δ max (u,v,w) S δ = max δ{u, v} δ [0,1] δ (1 δ) = 1. 2 The converse of Theorem 4.2 also holds with convexity assumption on objective functions and feasible set. Since if for each i, g i is convex then the feasible set X is convex set. We have the following result. Theorem 4.5 Let for each i I and r L, f i and g r are convex functions. If x 0 G ˆM,ɛ (f, X), then there exists an s int(r m +) such that x 0 is an s, ɛ -minimum of P (s ). Proof. Let x 0 G ˆM,ɛ (f, X). Then using Proposition 4.1 we obtain that the system Γ (x 0, ɛ, i, ˆM, X) is inconsistent for each i I. Applying the Gordan s Theorem of the alternative we conclude, after some rearrangements, that for each i I, there exists scalars λ i i 0, λ i j 0, m j=1 λi j = 1 such that f i (x) + ˆM j i λ i jf j (x) f i (x 0 ) + ˆM j i λ i jf j (x 0 ) ( ɛ i + ˆM j i λ i jɛ j ) for all x X. Therefore f i (x) + ˆM λ i jf j (x) j i f i (x 0 ) + ˆM λ i jf j (x 0 ) j i ( ɛ i + ˆM ) λ i jɛ j. j i 15

16 Hence, for all x X ( 1 + ˆM ) λ i j f j (x) j=1 i j ( 1 + ˆM i j j=1 λ i j ) f j (x 0 ) ( 1 + ˆM i j j=1 λ i j ) ɛ j. Setting s j = (1 + ˆM i j λi j), we conclude that s int(r m +) and that x 0 is an s, ɛ -minimum of P (s ). Remark 4.6 Theorem 4.5 can also be proved by noting the fact that each ( ˆM, ɛ)- Geoffrion proper point is ɛ-geoffrion proper point with constant ˆM > 0. Hence using Theorem 3.15 form [?] we can deduce the above result. Now if we denote the set of s, ɛ -minimum of P (s ) by Sol ɛ (P (s )) under convexity assumption on data, then Theorem 4.2 and 4.5 implies that for a given ˆM, there exists s intr m + such that Sol ɛ (P (s )) G ˆM,ɛ (f, X) Sol ɛ (P (s)). s intr m + Now we come to the main attraction of this section, the saddle point conditions for ( ˆM, ɛ)-geoffrion proper solutions. For this study we consider the problem MOP where each f i, i I and g j, j L are a convex function. Whenever the data of problem is convex, we shall denote the problem MOP as CMOP. Given ˆM > 0, and any index i I we define the ( ˆM, i)-lagrangian associated with CMOP as follows L ˆM i (x, τ i, µ i ) = f i (x) + τ i ˆMf j j (x) + j i µ i rg r (x), (4.2) where τ i = (τ1, i τ2, i..., τm) i S m and µ i = (µ i 1, µ i 2,..., µ i l ) Rl +. Here S m is the unit simplex in R m given as S m = {x R m : 0 x i 1, i I, x i = 1} (4.3) Our aim here is to show the key role played by the ( ˆM, i)-lagrangian in analyzing or rather characterizing the Geoffrion ( ˆM, ɛ)-proper solution. Theorem 4.7 For a given ɛ R m + and ˆM > 0, let us consider the problem CMOP which satisfy the Slater condition, i.e. there exists ˆx R n s,t, g r (ˆx) < 0 for all r L. If x 0 G ˆM,ɛ (f, X) then for each i, there exists τ i S m, µ i R l + such that for all x R n and µ R m +, (i) L ˆM i (x 0, τ i, µ) ɛ i L ˆM i (x 0, τ i, µ i ) L ˆM i (x, τ i, µ i ) + ɛ i (ii) µ i rg r (x 0 ) ɛ i, where ɛ i = ɛ i + m j=1,j i τ i j ˆMɛ j. Conversely if x 0 R n be such that for each i I, there exists ( τ i, µ i ) S m R l + such that (i) and (ii) holds then x 0 G M,2ɛ (f, X), where M (1 + ˆM)(m 1). Proof. Let x 0 G ˆM,ɛ (f, X). Then using Proposition 4.1, we conclude that for each i the system Q i, f i (x 0 ) + f i (x) + ɛ i < 0, f i (x 0 ) + f i (x) + ɛ i < M(f j (x 0 ) f j (x) ɛ j ), j I \ {i} g r (x) 0, r L 16

17 has no solution for all x R n. It is easy to observe that the system Q i has no solution if we replace g r 0 by g r < 0 for all r L. Now by applying the Gordan s theorem of alternative, there exists τ i = (τ1, i..., τm) i R m + and µ i = (µ i 1,..., µ i l ) Rl + with (τ i, µ i ) 0 such that for all x R n, τi i (f i (x) f i (x 0 ) + ɛ i ) + τj(f i i (x) + ˆMf j (x) f i (x 0 ) ˆMf j (x 0 ) + ɛ i + ˆMɛ j ) j i + Hence for all x R n, µ i rg r (x) 0. ( τj)[f i i (x) f i (x 0 ) + ɛ i ] + [τ i ˆMf j j (x) τ i ˆMf j j (x 0 ) + τ i ˆMɛ j j ] j=1 j i + µ i rg r (x) 0. (4.4) Now first we claim that τ i = (τ1, i..., τm) i 0. So on the contrary let τ i = 0 and then from (4.4), we have for all x R n, µ i rg r (x) 0. Since the Slater condition holds with ˆx R n and µ i 0 we have µ i rg r (x) < 0, which contradicts the fact that µ i rg r (x) 0. Hence τ i 0 and thus m τj i > 0. Thus dividing both sides of (4.4) with m τj i we have for all x R n, j=1 f i (x) f i (x 0 ) + ɛ i + [ τ i ˆMf j j (x) τ i ˆMf j j (x 0 ) + τ i ˆMɛ j j ] + j i j=1 µ i rg r (x) 0, (4.5) where τ i j = τ j i τj i j=1 and µ i r = µi r τj i j=1. Now set x = x 0 in (4.5) to get ɛ i + j i τ j i ˆMɛ j + µ i rg r (x 0 ) 0, By setting ɛ i = ɛ i + j i τ i j ˆMɛ j, we get µ i rg r (x 0 ) ɛ i which proves part (ii) of the theorem. Now from (4.5) we have for all x R n, f i (x) + j i τ j i ˆMf j (x) + µ i rg r (x) + ɛ i f i (x 0 ) + j i τ i j ˆMf j (x 0 ). (4.6) Further as x 0 is feasible to CMOP, we have l µ i rg r (x 0 ) 0. Thus from (4.6) we get f i (x) + j i τ j i ˆMf j (x) + µ i rg r (x) + ɛ i f i (x 0 ) + j i 17 τ j i ˆMf j (x 0 ) + µ i rg r (x 0 ),

18 which implies for each i and for all x R n, L ˆM i (x, τ i, µ i ) + ɛ i L ˆM i (x 0, τ i, µ i ). (4.7) Further from (4.2) we observe that for all i and any µ R l + L ˆM i (x 0, τ i, µ) f(x 0 ) + j i τ i j ˆMf j (x 0 ), which can be written as L ˆM i (x 0, τ i, µ) f(x 0 ) + τ i ˆMf j j (x 0 ) + l µ i rg r (x) + ɛ i. Thus j i for all x R n and µ R l +, L ˆM i (x 0, τ i, µ) L ˆM i (x 0, τ i, µ i ) + ɛ i. (4.8) The equation (4.7) and (4.8) together proves part (i) of the theorem. Now for the sufficient part, let us assume that for a given x 0 R n and each i I there exists τ i S m and µ i R l + such that (i) and (ii) holds. Our first step would be to show that x 0 is feasible to CMOP. As we know from (i), for all µ R l + Thus L ˆM i (x 0, τ i, µ) ɛ i L ˆM i (x 0, τ i, µ i ). f i (x 0 ) + j i τ i j ˆMf j (x 0 ) + µ r g r (x 0 ) ɛ i f i (x 0 ) + j i τ i j ˆMf j (x 0 ). This shows that for all µ R l +, µ r g r (x 0 ) ɛ i, (4.9) On the contrary assume that x 0 is not feasible, thus there exists r 0 L such that g r0 (x 0 ) > 0. Consider in particular µ = (0,..., 0, µ r0, 0,..., 0), with µ r0 > 0. Therefore from (4.9), we have µ r0 g r0 (x 0 ) ɛ i. (4.10) But if we continue to increase the value of µ r0, there exists a value of µ r0 beyond which µ r0 g r0 (x 0 ) > ɛ i. This contradicts (4.10) and 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 R n which implies L ˆM i (x, τ i, µ i ) + ɛ i L ˆM i (x 0, τ i, µ i ). (4.11) f i (x) + j i τ j i ˆMf j (x) + µ i rg r (x) + ɛ i + j i 18 τ j i ˆMɛ j f i (x 0 ) + τ i ˆMf j j (x 0 ) j i + µ i rg r (x 0 ).

19 Now for any feasible x, µ i rg r (x) 0. Thus from the above inequality we have, f i (x) + j i τ i j ˆMf j (x) + ɛ i + j i τ i j ˆMɛ j f i (x 0 ) + j i τ i j ˆMf j (x 0 ) + µ i rg r (x 0 ). (4.12) Using (ii) we have f i (x) + j i τ i j ˆMf j (x) + ɛ i + j i τ i j ˆMɛ j f i (x 0 ) + j i τ i j ˆMf j (x 0 ) (ɛ i + j i τ i j ˆMɛ j ). Since it holds for each i, by summing over all the i s we get τ i j)f i (x) + τ i j)(2ɛ j ) τ i j)f i (x 0 ). (1 + ˆM j i (1 + ˆM j i (1 + ˆM j i Hence x 0 is s, 2ɛ -minimizer of P (s ) where the vector s has the form s i = 1 + ˆM m τ k i, i = 1, 2,.., m. Now since τ i S m for all i, we have for all i, j I k=1,k i s i s j = 1 + ˆM m τ k i k=1,k i τ j k k=1,k j 1 + ˆM m = 1 + ˆM(1 τ i i ) 1 + ˆM(1 τ j j ) 1 + ˆM. Since the above inequality is true for every i and j, we have max i,j { s i } 1 + ˆM. s j Now consider M (1 + ˆM)(m 1) and using Theorem 4.2 we conclude that x 0 G M,2ɛ (f, X). This completes the proof. Remark 4.8 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. If x 0 R n be such that for each i I, there exists τ i S m and µ i R l + such that for all µ R l + and x R n, (a) L ˆM i (x 0, τ i, µ) ɛ i L ˆM i (x 0, τ i, µ i ) L ˆM i (x, τ i, µ i ) + ɛ i, (b) µ i rg r (x 0 ) ɛ i, holds. Then x 0 G M,2ɛ (f, X). In order to prove the statement we mentioned above,note that ɛ i = ɛ i + j i τ i j ˆMɛ j, thus ɛ i ɛ i. Hence from (a) and (b) above, we get (i) and (ii) of Theorem 4.7. Now we can simply apply the converse part of Theorem 4.7 to show that x 0 G M,2 ɛ (f, X), M (1+ ˆM)(m 1) Note that the condition (a) and (b) above are much more simpler to check compared to (i) and (ii) since ɛ i involves the multipliers τ i j. Hence for the sufficient part of the above theorem we will be using item (a) and (b) in place of (i) and (ii). 19

20 Of course from the necessary part of Theorem 4.7 we can also derive a multiplier rule involving ɛ-subdifferentials, however this rule will be quite different. Observe that if x 0 G ˆM,ɛ (f, X), then form Part (i) of Theorem 4.7 we have for any i I there exists τ i S m and µ i R l + such that for all x R n L ˆM i (x 0, τ i, µ i ) L ˆM i (x, τ i, µ i ) + ɛ i, which implies that x 0 ɛ i arg min x R n L ˆM i (., τ i, µ i ). Thus for each i I, 0 ɛil ˆM i (x 0, τ i, µ i ). In fact a more compact necessary condition of the KKT type is given as follows, 0 ɛil ˆM i (x 0, τ i, µ i ) with µ i rg r (x 0 ) ɛ i. (4.13) Theorem 4.9 For a given ɛ R m + and ˆM > 0, let us consider the problem CMOP. If x 0 G ˆM,ɛ (f, X), then there exists m vectors τ i S m and µ i R l +, i I such that (i) 0 ɛil ˆM i (x 0, τ i, µ i ), (ii) µ i rg r (x 0 ) ɛ i. where ɛ i = ɛ i + j i τ i j ˆMɛ j, i I. Conversely if x 0 X be such that there exists vectors ( τ i, µ i ) S m R l +, i I such that (i) and (ii) holds then x 0 G M,ɛ (f, X), where M = (1 + ˆM)(m 1). Proof. The necessary part has already been done in above remark. For sufficient part, let (i) and (ii) holds for x 0 X. This means that there exists v i ɛil ˆM i (x 0, τ i, µ i ) for all i I such that 0 = v 1 + v v m. (4.14) Thus from definition of ɛ-subdifferential we have for each i I, L ˆM i (x, τ, µ i ) L ˆM i (x, τ i, µ i ) v i, x x 0 ɛ i. Thus L ˆM i (x, τ, µ i ) Now using (4.14), we get L ˆM i (x, τ i, µ i ) v i, x x 0 ɛ i. (f i (x) + j i τ j i ˆMf j (x) + µ i rg r (x)) τ j i ˆMf j (x 0 ) (f i (x 0 ) + j i + µ i rg r (x 0 )) ɛ i. For feasible point x and using part (ii), we get (f i (x) + j i τ i j ˆMf j (x)) (f i (x 0 ) + j i τ i j ˆMf j (x 0 )) ɛ i, 20

21 which can be rewritten as τ i j ˆM)f i (x) τ i j ˆM)f i (x 0 ) τ i j ˆM)ɛ i. (1 + i j (1 + i j (1 + i j Hence x 0 is s, ɛ -minimizer of P (s ) where s i = 1 + ˆM minimizer of P (s ) where s i = 1 + ˆM m j=1,j i m k=1,k i τ i k. Hence x 0 is s, ɛ - τ k i. Now using the same argument as in Theorem 4.2, we conclude that x 0 G M,ɛ (f, X), where M = (1 + ˆM)(m 1). This completes the proof. 5 Approximate KKT conditions for Approximate Benson solutions In this section we would like to focus on Benson ɛ-proper solutions. Our aim in this section is to develop scalarization rules for these approximate proper solutions. Scalarization rules are one of the keys to optimality condition in vector optimization. It is important to note that when C = R n +, Benson ɛ-proper Pareto solutions reduce to Geoffrion ɛ-proper solutions. We would like to note that our result are different from the optimality condition for approximate solutions studied in Dutta and Vetrivel [?]. To develop scalarization techniques for Benson ɛ-proper, we will make use of the following cone separation result. Proposition 5.1 (Borwein, [?]) Let K, N be closed, convex cones in R m and let N K = {0}. If the dual cone K has nonempty interior, then (K ) 0 ( N ). (5.1) The next two theorems present and analyze a scalarization technique for Benson ɛ-proper solutions. Theorem 5.2 Let C R m be a cone such that (C ) 0 and let s (C ) 0. If x 0 is an s, ɛ -minimum of P (s ), then x 0 SB ɛ (f, X, C). Proof. Let h 0 and h cl(cone(f(x) + (C + ε) f(x 0 ))). Our aim is to establish that h C. From the definition of h, there exists a sequence {h n } such that h n cone(f(x) + (C + ɛ) f(x 0 ))) and h n = t n (f(x n ) + s n + ɛ f(x 0 )) h, with t n 0, x n X, and s n C. Now since x 0 is an s, ɛ -minimum of P (s ), for each n it holds that This shows that s, f(x 0 ) s, f(x n ) + s, ɛ. s, f(x n ) + ɛ f(x 0 ) 0. (5.2) 21