ε-Conjugate maps andε-conjugate duality in vector optimization with set-valued maps

Abstract

The aim of this article is to develop an approximate ()-conjugate duality theory for a general vector optimization problem with set-valued maps on the basis of -weak efficiency. We first introduce the concepts of -conjugate maps and -weak subgradients for set-valued maps and establish several properties of them. Then we introduce an -conjugate dual problem for the vector set-valued optimization problem. Finally, we establish a weak duality theorem and a strong duality theorem for the relationship between the primal and the dual problems. IntroductionIn the last two decades, much attention has been paid to approximate solutions of optimization problems. There are two reasons for this. First, optimization models are simplified representations instead of complete copies of real problems. Second, these models are solved usually by numerical methods (iterative procedures or heuristic algorithms) which produce approximations to the theoretical solutions.On multiobjective programming or vector optimization, a significant portion of the literature deals with approximate solution concepts, approximate optimality conditions, approximate multiplier rules, approximate saddle-point theorems, approximate duality theorems, properties of approximate solutions, etc. For example, Loridan [27] originally introduced the concept of approximate (-)efficiency in general vector optimization problems. White [49] introduced six alternative concepts of approximate solutions. Examinations or relations of these concepts as well as new definitions of approximate solutions were given by Va´lyi [46], Ne´meth [32], Tammer [38-40], Helbig and Pateva [13], Tanaka [41,42], Yokoyama [51,52], Li and Wang [2], Liu [25] and Gutie´rrez, Jime´nez and Novo [12]. Optimality conditions of approximate solutions were investigated by Yokoyama [50], Liu [24], Deng [6] and Dutta and Vetrivel [7]. Approximate multiplier

Duality related to approximate proper solutions of vector optimization problems

“…Later, Rong and Wu [25] applied the ideas and methods of Vályi in the setting of vector optimization problems with conesubconvexlike set-valued mappings, and they derived weak and strong duality results for approximate weak efficient solutions in the sense of Kutateladze (see [18]) of the primal problem. In the same framework, Jia and Li [17] established an approximate conjugate duality theory by means of approximate weak efficient solutions in the sense of Kutateladze. Recently, Sach et al [26] obtained approximate duality results for vector quasi-equilibrium problems with set-valued mappings by considering approximate Benson proper efficient solutions with respect to a vector (see Definition 2.2 with C(ε) = εq + D, q ∈ D\{0}).…”

“…For this reason it is important to study approximate solutions of vector optimization problems and, in particular, to develop approximate duality theories. Some works in this field are [11,14,17,19,23,[25][26][27]29].…”