Optimality and duality for nonsmooth multiobjective optimization problems
© Bae and Kim; licensee Springer. 2013
Received: 24 July 2013
Accepted: 21 October 2013
Published: 22 November 2013
In this paper, we consider a nonsmooth multiobjective programming problems including support functions with inequality and equality constraints. Necessary and sufficient optimality conditions are obtained by using higher-order strong convexity for Lipschitz functions. Mond-Weir type dual problem and duality theorems for a strict minimizer of order m are given.
Nonlinear analysis problems are a new and vital area of optimization theory, mathematical physics, economics, engineering and functional analysis. Moreover, nonsmooth problems occur naturally and frequently in optimization.
In 1970, Rockafellar wrote in his book that practical applications are not necessarily differentiable in applied mathematics (see ). So, dealing with nondifferentiable mathematical programming problems was very important. Vial  studied strongly and weakly convex sets and ρ-convex functions.
Auslender  introduced the notion of lower second-order directional derivative and obtained necessary and sufficient conditions for a strict local minimizer. Based on Auslender’s results, Studniarski  proved necessary and sufficient conditions for the problem of the feasible set defined by an arbitrary set. Moreover, Ward  derived necessary and sufficient conditions for strict minimizer of order m in nondifferentiable scalar programs. Jimenez  introduced the notion of super-strict efficiency for vector problems and gave necessary conditions for strict minimality. Jimenez and Novo [7, 8] obtained first- and second-order optimality conditions for vector optimization problems. Bhatia  gave the higher-order strong convexity for Lipschitz functions and established optimality conditions for the new concept of strict minimizer of higher order for a multiobjective optimization problem.
Kim and Bae  formulated nondifferentiable multiobjective programs with the support functions. Also, Bae et al.  established duality theorems for nondifferentiable multiobjective programming problems under generalized convexity assumptions. Also, Kim and Lee  introduced the nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. They introduced Karush-Kuhn-Tucker type optimality conditions and established duality theorems for (weak) Pareto-optimal solutions. Recently, Bae and Kim  established optimality conditions and duality theorems for a nondifferentiable multiobjective programming problem with support functions.
In this paper, we consider nonsmooth multiobjective programming with inequality and equality constraints. In Section 2, we introduce the concept of a strict minimizer of order m and higher-order strong convexity for this problem. In Section 3, necessary and sufficient optimality theorems are established for a strict minimizer of order m under generalized strong convexity assumptions. In Section 4, we formulate a Mond-Weir type dual problem and obtain weak and strong duality theorems.
For , and have the usual meaning. Let be the n-dimensional Euclidean space, and let be its nonnegative orthant.
Definition 2.1 
where , and are locally Lipschitz functions, respectively, and X is the convex set of . For each , is a compact convex subset of .
Further, let be the feasible set of (MOP), be an open ball with center and radius ϵ and be the index set of active constraints at .
We introduce the following definitions due to Jimenez .
Definition 2.5 
Definition 2.6 
It is clear that .
We recall the notion of strong convexity of order m introduced by Lin and Fukushima in .
Proposition 2.1 
If , , are strongly convex of order m on a convex set X, then and are also strongly convex of order m on X, where , .
The following definition is due to the one in .
3 Optimality conditions
In this section, we establish Fritz John necessary optimality conditions, Karush-Kuhn-Tucker necessary optimality conditions and Karush-Kuhn-Tucker sufficient optimality condition for a strict minimizer of (MOP).
Theorem 3.1 (Fritz John necessary optimality conditions)
Theorem 3.2 (Karush-Kuhn-Tucker necessary optimality conditions)
for each . Since the basic regularity condition (BRC) holds at , we have , , , , , and , . This contradicts the fact that , , , , , and , , are not all simultaneously zero. Hence, . □
Theorem 3.3 (Karush-Kuhn-Tucker sufficient optimality conditions)
Assume further that , , are strongly convex of order m on X, , are strongly quasiconvex of order m on X and is strongly quasiconvex of order m on X. Then is a strict minimizer of order m for (MOP).
This is a contradiction to (3.6). □
Remark 3.1 Suppose that , are strongly convex of order m on X and that is strongly convex of order m on X. Then the conclusion of Theorem 3.3 also holds.
Proof It follows on the lines of Theorem 3.3. □
4 Duality theorems
Theorem 4.1 (Weak duality)
where , since . This is a contradiction to (4.8). □
Lemma 4.1 If , , are strongly convex of order m on X and is strongly convex of order m on X, then the same conclusion of Theorem 4.1 also holds.
Proof It follows on the lines of Theorem 4.1. □
Theorem 4.2 (Strong duality)
If is a strict minimizer of order m for (MOP) and the basic regularity condition (BRC) holds at , then there exist , , , , , and , , such that is a feasible solution of (MOD) and , . Moreover, if the assumptions of Theorem 4.1 are satisfied, then is a strict maximizer of order m for (MOD).
Thus, is a strict maximizer of order m for (MOD). □
Remark 4.1 Theorem 4.1 and Theorem 4.2 reduce to [, Theorem 4.1 and Theorem 4.2] in an inequality constraint case. More exactly, , , and , at the considered point in the framework of [, Theorem 4.1 and Theorem 4.2] are strongly convex of order m and strongly quasiconvex of order m, respectively.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2A10008908).
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