ON OPTIMALITY CONDITIONS FOR ABSTRACT CONVEX VECTOR OPTIMIZATION PROBLEMS

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1 J. Korean Math. Soc ), No. 4, pp ON OPTIMALITY CONDITIONS FOR ABSTRACT CONVEX VECTOR OPTIMIZATION PROBLEMS Ge Myng Lee and Kwang Baik Lee Reprinted from the Jornal of the Korean Mathematical Society Vol. 44, No. 4, Jly 2007 c 2007 The Korean Mathematical Society

2 J. Korean Math. Soc ), No. 4, pp ON OPTIMALITY CONDITIONS FOR ABSTRACT CONVEX VECTOR OPTIMIZATION PROBLEMS Ge Myng Lee and Kwang Baik Lee Abstract. A seqential optimality condition characterizing the efficient soltion withot any constraint qalification for an abstract convex vector optimization problem is given in seqential forms sing sbdifferentials and ɛ-sbdifferentials. Another seqential condition involving only the sbdifferentials, bt at nearby points to the efficient soltion for constraints, is also derived. Moreover, we present a proposition with a sfficient condition for an efficient soltion to be properly efficient, which are a generalization of the well-known Isermann reslt for a linear vector optimization problem. An example is given to illstrate the significance of or main reslts. Also, we give an example showing that the proper efficiency may not imply certain closeness assmption. 1. Introdction Vector optimization problem consists of vector valed objective fnction and the constrained set. There are three kinds of soltions for the problem, that is, properly, weakly) efficient soltion. To get an optimality condition for an efficient soltion of a vector optimization problem, we often formlate an corresponding scalar problem. However, it is so difficlt that sch scalar program satisfies a constraint qalification which we need to derive an optimality condition. Hence it is very important to investigate an optimality condition for an efficient soltion of a vector optimization problem which holds withot any constraint qalification. Recently Jeyakmar and Zaffaroni 16) established necessary and sfficient dal conditions for weakly and properly efficient soltions of an abstract convex vector optimization problems withot any constraint qalifications. The optimality conditions are given in asymptotic forms sing epigraphs of conjgate fnctions and sbdifferentials. Glover, Jayakmar and Rbinov 6) obtained necessary and sfficient dal conditions for efficient soltions of a Received March 6, Mathematics Sbject Classification. 90C29, 90C46. Key words and phrases. convex vector optimization problem, optimality conditions, efficient soltion, properly efficient soltion, convex fnction. This research was spported by Grant No. R from the Basic Research Program of KOSEF. 971 c 2007 The Korean Mathematical Society

3 972 GUE MYUNG LEE AND KWANG BAIK LEE finite-dimensional convex vector optimization problem withot any constraint qalification. Very recently, Jeyakmar, Lee and Dinh 13) gave seqential optimality conditions characterizing the soltion withot any constraint qalification for an abstract scalar convex optimization problem. On the other hand, Isermann 10) showed that every efficient soltion of a linear vector optimization problem is properly efficient. Many athors 2, 3, 7, 8, 17) have tried to get a sfficient condition to extend the Isermann reslt to several kinds of vector optimization problems. The aim of this paper is to present a seqential optimality condition characterizing the efficient soltion withot any constraint qalification for an abstract convex vector optimization problem, which is given in seqential forms sing sbdifferentials and ɛ-sbdifferentials, and to get a sfficient condition to extend the Isermann reslt to the problem. This paper is organized as follows. In Section 2, definitions and preliminary reslts are given for next sections. In Section 3, we obtain a seqential optimality condition characterizing the efficient soltion withot any constraint qalification for an abstract convex vector optimization problem, and derive another seqential condition involving only the sbdifferentials, bt at nearby points to the efficient soltion for constraints. Moreover, seqential conditions characterizing a properly efficient soltion and a weakly efficient soltion withot any constraint qalification are given. In Section 4, we present a preposition with a sfficient condition for an efficient soltion to be properly efficient, which are a generalization of the Isermann reslt. In Section 5, an example is given to illstrate the significance of or main reslts. Also, we give an example showing that the proper efficiency may not imply certain closeness assmption. 2. Preliminaries Now we give notations and preliminary reslts that will be sed later in this chapter. Throghot the chapter, nless otherwise stated, X is a reflexive Banach space, Y and Z are Banach spaces, and the cones K Y, T Z are closed and convex. The continos dal space to X will be denoted by X and will be endowed with the weak topology. When a seqence {x n} in X converges to x X in the weak topology, we denote it as w - lim x n = x. The positive polar of the cone K Y is the cone K + = {θ Y θk) 0 k K}, and the strict positive polar of the cone K is K +i = {θ Y θk) > 0 k K \ {0}}. If Y = R n and K is closed and pointed, then K +i and K +i = int K + 11), where int K + is the interior of the cone K +. The core of a sbset D of Z is defined by cored = {d D : x Z) ɛ > 0) λ ɛ, ɛ)d + λx D}. Definition 2.1. Let g : X R {+ } be a proper lower-semicontinos convex fnction.

4 ON OPTIMALITY CONDITIONS 973 1he conjgate fnction of g, g : X R {+ } is defined by g v) = sp{vx) gx) x dom g}, where the domain of g, dom g, is given by dom g = {x X gx) < + }. 2he epigraph of g, epi g, is defined by epi g = {x, r) X R x dom g, gx) r}. 3he sbdifferential of g at a dom g is defined as the non-empty weak compact convex set ga) = {v X gx) ga) vx a), x dom g} and for ɛ 0, the ɛ-sbdifferential of g at a dom g is defined as the non-empty weak closed convex set ɛ ga) = {v X gx) ga) vx a) ɛ x dom g}. See Hiriart-Urrty and Lamarechal 9 for a detailed discssion on the ɛ-sbdifferential. Note that ɛ ga) = ga). ɛ>0 If g is sblinear i.e., convex and positively homogeneos of degree one), then ɛ g0) = g0) for all ɛ 0. If gx) = gx) k, x X, k R, then epi g = epi g +0, k). It is worth nothing that if g is sblinear, then epi g = g0) R +. Moreover, if g is sblinear and if gx) = gx) k, x X, k R, then epi g = g0) k, ). The following proposition describes the relationship between the epigraph of a conjgate fnction and the ɛ-sbdifferential and plays a key role in proving the main reslts. Proposition ) If g : X R {+ } is a proper lower semicontinos convex fnction and if a dom g, then epi g = { v X R : v va) + ɛ ga) ɛ ga)}, ɛ 0 where sperscript T denotes the transpose. The mapping g : X Z is T -convex if for every, v X and every t 0, 1 it holds: gt + 1 t)v) tg) 1 t)gv) T. For a continos T -convex mapping g it is easy to show that the set epiλg) is a convex cone 15). λ T +

5 974 GUE MYUNG LEE AND KWANG BAIK LEE Throghot this paper, we will consider the following abstract convex vector optimization problem: ACVP) Minimize fx) sbject to gx) T, where f : X Y and g : X Z are continos K-convex and T -convex fnctions respectively. Let S = {x X gx) T }. Then a S is said to be an efficient soltion of ACVP) if fs) fa) K) = {0}. We denote the set of all the efficient soltions of ACVP) by EffACVP). The point a S is called a properly efficient soltion of ACVP) if there exists a convex cone K sch that K \ {0} int K and fs) fa) K ) = {0}. We denote the set of all the properly efficient soltions of ACVP) by P reffacvp). The point a S is said to be a weakly efficient soltion of ACVP) if int K and fs) fa) int K) =. We denote the set of all the weakly efficient soltions of ACVP) by W EffACVP). It is clear that P reffacvp) EffACVP) W EffACVP). By separation theorem 11), we can obtain the following proposition: Proposition ) If in ACVP) the mapping f : X Y and g : X Z are K-convex and T -convex, respectively, then i) a P reffacvp) if and only if there exists θ K +i sch that θf)a) = min{θfx) x S}. ii) a W EffACVP) if and only if there exists θ K + \ {0} sch that θf)a) = min{θfx) x S}. Extending the reslt of Corley 4), we can obtain the following proposition: Proposition 2.3. Let θ K +i. If in ACVP) the mapping f : X Y and g : X Z are K-convex and T -convex respectively, then a EffACVP) if and only if a is an optimal soltion of the following scalar optimization problem: P) θ Minimize θf)x) sbject to gx) T fx) fa) K.

6 ON OPTIMALITY CONDITIONS 975 Proof. Sppose that a / EffACVP). Then there exists x 1 S sch that fx 1 ) fa) K \ {0}. Since θ K +i, θf)x 1 ) θf)a) < 0, which implies that a is not an optimal soltion of P) θ. Conversely, sppose that a EffACVP) and let x be any feasible soltion of P) θ. Then we have, fx) fa) K. Since a EffACVP), fx) = fa). Hence a is an optimal soltion of P) θ. The following lemmas are needed to prove the main reslts. Lemma , 14) Let T Z be a closed convex cone, let : X R be a continos linear mapping, and let g : X Z be a continos T -convex mapping. Sppose that the system gx) T is consistent. Let α R. Then the following statements are eqivalent: i) {x X ) : gx) T } {x X : x) α} ii) cl α λ T epiλg) ). + We can obtain the following Lemma 2.2. Lemma 2.2. Let f : X R be a continos convex fnction and g : X R {+ } be a proper lower semicontinos convex fnction. Then epif + g) = epi f + epi g. Lemma , 18) Let f : X R {+ } be a proper lower semicontinos convex fnction. Then for any real nmber ɛ > 0 and any x ɛ f x) there exist x ɛ X, x ɛ fx ɛ ) sch that x ɛ x ɛ, x ɛ x ɛ and fx ɛ ) x ɛ x ɛ x) f x) 2ɛ. 3. Optimality conditions Now we give seqential optimality conditions for properly, weakly) efficient soltions of the abstract convex vector optimization problem ACVP). Theorem 3.1. Let θ K +i and a S. Then the following are eqivalent: i) a EffACVP). ii) there exists θf)a) sch that a) cl epiλg) + λ T + epiµf) + 0, µf)a)).

7 976 GUE MYUNG LEE AND KWANG BAIK LEE iii) there exist θf)a), λ n T +, δ n 0, v n δn λ n g)a), µ n K +, ɛ n 0, w n ɛn µ n f)a) sch that + ω - lim v n + w n ) = 0, lim λ ng)a) = 0. Proof. a EffACVP) lim δ n = lim ɛ n = 0 and by Proposition 2.2) a is an optimal soltion of the problem P): P) Minimize θf)x) sbject to gx) T fx) fa) K. there exists θf)a) sch that x Z := {x X : gx) T, fx) fa) K}, x) a). gx) there exists θf)a) sch that x with T fx) fa) K), x) a). by Lemma 2.1) there exists θf)a) sch that a) cl λ,µ) T + K + epiλg + µf µfa)). by Lemma 2.2) there exists θf)a) sch that a) ) cl λ T + epiλg) + epiµf) + 0 µf)a) by Proposition 2.1) there exists θf)a) sch that a) { ) T v cl λ T δ 0 v + va) + δ λg)a) δ λg)a)} { + µ K ɛ 0 + ) +. 0 µf)a) w wa) + ɛ µf)a) : w ɛ µf)a)}. there exist θf)a), λ n T +, δ n 0, v n δn λ n g)a), µ n K +, ɛ n 0, w n ɛn µ n f)a) sch that = a)

8 lim + ON OPTIMALITY CONDITIONS 977 v n v n a) + δ n λ n g)a) + w n w n a) + ɛ n µ n f)a) 0 µ n f)a) ) iii). Theorem 3.2. Let θ K +i and a A. Then a EffACVP) if and only if there exist θf)a), λ n T +, µ n K +, x n X, s n λ n g + µ n f)x n ) sch that + w lim s n = 0, lim x n a = 0. lim λ ng + µ n f)x n ) µ n f)a) = 0 and Proof. Let a EffACVP). Then by Theorem 3.1, there exist θf)a), λ n T +, δ n 0, ɛ n 0, µ n K +, v n δn λ n g)a), w n ɛn µ n f)a) sch that + w - lim v n + w n ) = 0 lim δ n = lim ɛ n = 0 lim λ ng)a) = 0. Notice that it follows the definition of ɛ f that δn λ n g)a) + ɛn µ n f)a) αn λ n g + µ n f)a), where α n = δ n + ɛ n. Withot loss of generality, we may assme that α n > 0. Letting t n = v n + w n, t n αn λ n g + µ n f)a) and hence by Lemma 2.3, there exists x n X, s n λ n g + µ n f)x n ) sch that x n a α n s n t n α n and λ n g + µ n f)x n ) s n x n a) λ n g + µ n f)a) 2α n. Since α n 0 and s n x n a) 0, we have lim x n a = 0 + w - lim s n = 0 lim and λ n g + µ n f)x n ) µ n f)a) = 0.

9 978 GUE MYUNG LEE AND KWANG BAIK LEE Conversely, sppose that there exist θf)a), λ n T +, µ n K +, x n X, s n λ n g + µ n f)x n ) sch that + w - lim n = 0, λ n g + µ n f)x n ) µ n f)a) = 0 and lim lim x n a = 0. Since s n λ n g + µ n f)x n ) we have λ n g + µ n f) s n ) = s n x n ) λ n g + µ n f)x n ). So, Moreover, w - lim s n =, and Ths a) s n, s n x n ) λ n g + µ n f)x n ) + µ n f)a)) epiλ n g + µ n f) + 0, µ n f)a)). lim s n x n ) λ n g + µ n f)x n ) + µ n f)a) = lim s nx n ) = a). cl λ T +, Ths by Lemma 2.2, cl epiλg) + a) λ T + So, by Theorem 3.1, a EffACVP). epiλg + µf) + 0 µf)a) epiµf) 0 + µf)a) ).. By sing Propositions 2.1 and 2.2, and following methods of proofs in Theorems 3.1 and 3.2, we can obtain the following theorems. Theorem 3.3. The following are eqivalent: i) a P reffacvp). ii) there exist θ K +i, θf)a), ɛ n 0, λ n T +, v n ɛn λ n g)a) sch that + w - lim v n = 0, lim λ ng)a) = 0 and lim ɛ n = 0.

10 ON OPTIMALITY CONDITIONS 979 iii) there exist θ K +i, θf)a), ɛ n 0, λ n T +, x n X, v n λ n g)x n ) sch that + w - lim v n = 0, lim λ ngx n ) = 0 and lim x n = a. Theorem 3.4. The following are eqivalent: i) a W EffACVP). ii) there exist θ K + \ {0}, θf)a), ɛ n 0, λ n T +, v n ɛn λ n g)a) sch that + w - lim v n = 0, lim λ ng)a) = 0 and lim ɛ n = 0. iii) there exist θ K + \ {0}, θf)a), λ n T +, x n X, v n λ n g)x n ) sch that + w - lim v n = 0, lim λ ngx n ) = 0 and lim x n = a. Now we consider the closed cone constraint qalification which reqires that the convex cone epiλg) is weak closed. This constraint qalification λ S + holds nder the Robinson Reglarity Condition, that is, 0 coregx)+s) see 16) or the generalized Slater condition that int S is nonempty and x 0 X sch that gx 0 ) int S see 15). Following the proof in 13, we can obtain the following Khn-Tcker theorems the Lagrange mltiplier theorems) for ACVP) nder the closed cone constraint qalification. Theorem 3.5. Let a S and assme that the closed cone constraint qalification holds. Then the following are eqivalent: i) a P reffacvp). ii) there exist θ K +i, λ T + sch that 0 θf)a) + λg)a) and λg)a) = 0. Theorem 3.6. Let a S and assme that the closed cone constraint qalification holds. Then the following are eqivalent: i) a W EffACVP).

11 980 GUE MYUNG LEE AND KWANG BAIK LEE ii) there exist θ K +, λ T + sch that 0 θf)a) + λg)a) and λg)a) = Generalization of Isermann s reslt The following proposition, which is a generalization of the Isermann s reslt 10, 5, 19), gives a sfficient condition that an efficient soltion can be properly efficient. Proposition 4.1. Let a S and assme that epiλg) + epiµf) + 0, µf)a)) λ T + is closed in X R. Then if a EffACVP), a P reffacvp). Proof. Let a EffACVP) and θ K +i. Then by Lemma 2.2 and Theorem 3.1, there exists θf)a) sch that ) T ) epiλg + µf) 0 +. a) µf)a) λ T +, Ths there exist λ T + and µ K + sch that epiλg + µf) a) µfa) and hence λg+µf) ) a)µfa). Ths for any x X, x)λg+ µf)x) a)µfa). So, for any x S, 0 x)a)+µf)x)µf)a). Since θf)a), θf)x) θf)a) x a) and we have, for any x S, 0 θf)x) θf)a) + µf)x) µf)a) = θ + µ)fx) θ + µ)fa). Hence for any x S, θ + µ)fa) θ + µ)fx). Since θ K +i and µ K +, θ + µ K +i. So, by Proposition 2.2, a P reffacvp). The following proposition is the well-known Isermann theorem 10) for a linear vector optimization problem. Proposition 4.2. Let X = R n, Y = R p, Z = R m, K = R p +, T = R m + fx) = c 1 x,..., c p x) and gx) = a 1 x b 1,..., a m x b m ), where c i R n, i = 1,..., p, a j R m, j = 1,..., m and b j R, j = 1,..., m. Then P reffacvp) = EffACVP).

12 ON OPTIMALITY CONDITIONS 981 Proof. Let ā EffACVP). Then we have, epiλg) + epiµf) + 0, µf)ā)) λ T + µ K { + ) { T ci aj = cone co : i = 1,..., p} : j = 1,..., m} c iā b j { ) } 0. 1 Ths the above set is closed, and hence it follows from Theorem 4.1, ā P reffacvp). Since P reffacvp) EffACVP), P reffacvp) = EffACVP). 5. Examples Now we give an example to illstrate Theorems 3.1 and 3.2, and Theorem 4.1. Example 5.1. Consider the following convex vector optimization problem: ACVP) Minimize fx) := x, x 2 ) sbject to gx) := x 0. Let T = R + and K = R + 2. Then 0 EffACVP), bt 0 / P reffacvp). 1) We will show that the condition ii) in Theorem 3.1 holds for ACVP) at a = 0. We can check that epiλg) + epiµf) + 0, µf)0)) λ T + = {x, y) R 2 : x < 0, y > 0} {x, y) R 2 : x 0, y 0}. Take θ = 1, 1). Then := θf)0) = 1 and Bt ) / λ T + epiλg) + ) cl epiλg) + λ T + Ths the condition ii) in Theorem 3.1 holds at a = 0. epiµf) + 0, µf)0)). epiµf) + 0, µf)0)). 2) We will show that the condition iii) in Theorem 3.1. holds for ACVP) at a = 0. Take θ = 1, 1), λ n = 1 n, δ n = 1 n, µ n = 0, 1 2 n), ɛ n = 1 2n. Then

13 982 GUE MYUNG LEE AND KWANG BAIK LEE δn λ n g)0) = { 1 n }, and ɛ n µ n f)0) = 1, 1. Let v n = 1 n and w n = 1. Then v n δn λ n g)0) and w n ɛn µ n f)0). Moreover, we have Ths the condition iii) holds at a = 0. θf)0) + lim v n + w n ) = 0, lim δ n = lim ɛ n = 0 and lim λ ng)0) = 0. 3) Now we will show that the conclsion of Theorem 3.2 holds for ACVP) at a = 0. Take θ = 1, 1), x n = 1 n, µ n = 0, 1 2 n), λ n = 0. Then lim x n = 0, θf)0) + lim nf + λ n g)x n ) = 1 + lim n 1 n ) = 0, and lim µ n f + λ n g)x n ) µ n f)0) 1 = lim 2 n 1 n )2 = 0. Hence the conclsion of Theorem 3.2 holds. 4) We will show that if a > 0, a P reffacvp). First we will prove that if a > 0, then there exists θf)a) sch that a) and epiλg) + λ T + Indeed, let a > 0, epiλg) + λ T + epiµf) + 0, µf)a)) = epiµf) + 0, µf)a)) epiµf) + 0, µf)a)) is closed. epiµf) + 0, µ 1 a + µ 2 a 2 ). ihe case that µ 1 0, µ 2 = 0 : epiµf) + 0, µ 1 a + µ 2 a 2 ) = {µ 1 } 0, ) + 0, µ 1 a) = {µ 1 } µ 1 a, ) iihe case that µ 1 0, µ 2 > 0, µ 1 a + µ 2 a 2 = 0 : epiµf) + 0, µ 1 a + µ 2 a 2 ) = epiµf) + 0, 0) and µf) v) = µ 1v) 2 4 µ1 a. Notice

14 ON OPTIMALITY CONDITIONS 983 that µ 2 = µ 1 a and vµ1)2 4 µ1 a av v R. Hence µ 1 0,µ 2=0 µ 2 >0,µ 1 a+µ 2 a 2 =0 epiµf) + 0, µ 1 a + µ 2 a 2 ) epiµf) + 0, 0) = {v, α) R 2 : α av}. iiihe case that µ 1 0, µ 2 > 0, µ 1 a + µ 2 a 2 0 : µf )v) = µ 1v) 2 4µ 2 and vµ 1) 2 4µ 2 µ 1 a + µ 2 a 2 + av = v µ 1 + 2µ 2 a) 2 0 v R. Hence epiµf) + 0, µf)a)) = {v, α) R 2 : α av}. So, λ T + epiλg) + epiµf) + 0, µf)a)) = {x, y) R 2 : x 0, y 0} + {x, y) R 2 : y ax} = {x, y) R 2 : y ax} and hence this set is closed. Since θf)x) = x + x 2, θf)a) = {1 2a}. Let = 1 2a. Then µf)a)) epiλg) + epiµf) + 0,. a) λ T + Ths by Theorem 3.1, a Ef facvp). Moreover, by Proposition 4.1, a P reffacvp). We give an example to show that a P reffacvp) may not imply the closeness assmption of Proposition 4.1. Example 5.2. Let fx) = x, x) and gx) = the following vector convex optimization problem: { 0 if x 0 x 2 if x > 0. Consider ACVP) Minimize fx) sbject to gx) 0. Then 0 P reffacvp). epiµf) + 0, µf)0)) = {x, y) R 2 : x 0, y 0} µ R 2 +

15 984 GUE MYUNG LEE AND KWANG BAIK LEE and epiλg) + λ 0 µ R 2 + epiµf) + 0, µf)0)) = {x, y) : x 0, y 0} {x, y) : x > 0, y > 0}. Therefore epiλg) + epiµf) + 0, µf)0)) is not closed. λ 0 µ R 2 + References 1 A. Brondsted and R. T. Rockafellar, On the sbdifferentiability of convex fnctions, Proc. Amer. Math. Soc ), K. L. Chew and E. U. Choo, Psedolinearity and efficiency, Math. Program ), no. 2, E. U. Choo, Proper efficiency and the linear fractional vector maximm problem, Oper. Res ), H. W. Corley, A new scalar eqivalence for Pareto optimization, IEEE Trans. Atomat. Control ), no. 4, M. Ehrgott, Mlticriteria Optimization, Lectre Notes in Economics and Mathematical Systems 491, Springer-Verlag Berlin Heidelberg, B. M. Glover, V. Jeyakmar, and A. M. Rbinov, Dal conditions characterizing optimality for convex mlti-objective programs, Math. Program ), no. 1, Ser. A, T. R. Glati and M. A. Islam, Efficiency in linear fractional vector maximization problem with nonlinear constraints, Optimization ), no. 4, , Efficiency and proper efficiency in nonlinear vector maximm problems, Eropean J. Oper. Res ), no. 3, J. B. Hiriart-Urrty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Volmes I and II, Springer-Verlag, Berlin Heidelberg, H. Isermann, Proper efficiency and the linear vector maximm problem, Oper. Res ), no. 1, J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Verlag Peter D. Lang, Frankfrt am Main, Germany, V. Jeyakmar, Asymptotic dal conditions characterizing optimality for convex programs, J. Optim. Theory Appl ), no. 1, V. Jeyakmar, G. M. Lee, and N. Dinh, New seqential Lagrange mltiplier conditions characterizing optimality withot constraint qalification for convex programs, SIAM J. Optim ), no. 2, , Soltion sets of convex vector minimization problems, to appear in Eropean J. Oper. Res.. 15 V. Jeyakmar, A. M. Rbinov, B. M. Glover, and Y. Iskizka, Ineqality systems and global optimization, J. Math. Anal. Appl ), no. 3, V. Jeyakmar and A. Zaffaroni, Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization, Nmer. Fnct. Anal. Optimiz ), no. 3-4, G. M. Lee, On efficiency in nonlinear fractional vector maximization problem, Optimization ), no. 1, L. Thibalt, Seqential convex sbdifferential calcls and seqential Lagrange mltipliers, SIAM J. Control Optim ), no. 4, Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of Mltiobjective Optimization, Academic Press, Inc., 1985.

16 ON OPTIMALITY CONDITIONS 985 Ge Myng Lee Department of Applied Mathematics Pkyong National University Psan , Korea address: Kwang Baik Lee Department of Applied Mathematics Pkyong National University Psan , Korea address:

On Multiobjective Duality For Variational Problems

On Multiobjective Duality For Variational Problems The Open Operational Research Jornal, 202, 6, -8 On Mltiobjective Dality For Variational Problems. Hsain *,, Bilal Ahmad 2 and Z. Jabeen 3 Open Access Department of Mathematics, Jaypee University of Engineering