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Set optimization using improvement sets

In this paper we introduce a notion of minimal solutions for set-valued optimization problem in terms of improvement sets by unifying a solution notion, introduced by Kuroiwa for set-valued problems, and a notion of optimal solutions in terms of improvement sets, introduced by Chicco et al for vector optimization problems. We provide existence theorems for these solutions, and establish lower convergence of the minimal solution sets in the sense of Painlev´e-Kuratowski. | Yugoslav Journal of Operations Research 27 (2017), Number 2, 153–167 DOI: https://doi.org/10.2298/YJOR170115011D SET OPTIMIZATION USING IMPROVEMENT SETS M. DHINGRA Department of Mathematics, University of Delhi, Delhi 110007, India mansidhingra7@gmail.com C.S. LALITHA Department of Mathematics, University of Delhi South Campus, Benito Jaurez Road,New Delhi 110021, India cslalitha@maths.du.ac.in Received: January 2017 / Accepted: May 2017 Abstract: In this paper we introduce a notion of minimal solutions for set-valued optimization problem in terms of improvement sets by unifying a solution notion, introduced by Kuroiwa [15] for set-valued problems, and a notion of optimal solutions in terms of improvement sets, introduced by Chicco et al. [4] for vector optimization problems. We provide existence theorems for these solutions, and establish lower convergence of the minimal solution sets in the sense of Painlev´e-Kuratowski. Keywords: Set-valued Optimization, Improvement Set, Painlev´e-Kuratowski Convergence. MSC: 49J53, 90C30. 1. INTRODUCTION In the recent years much attention has been paid to various aspects of setvalued optimization problems due to their occurrence in the areas of decision making such as economics, differential inclusions, and optimal control [2, 11]. In the literature there are two types of well-known criteria of solutions for a set-valued optimization problem, the vector criterion and the set criterion. We consider a set-valued optimization problem (P) Min F (x) subject to x ∈ M, 154 M. Dhingra, C.S. Lalitha / Set Optimization Using Improvement Sets where F : X ⇒ Y is a set-valued map, X and Y are normed linear spaces, and M is a nonempty subset of X. Let K be a closed convex pointed cone in Y with nonempty interior. The vector criterion involves finding a point x ¯ ∈ M such that there exists y¯ ∈ F (¯ x) with (F (M ) − y¯) ∩ (−K\{0}) = ∅, S where F (M ) := x∈M F (x). Such an element x ¯ is said to be a minimal solution of (P), and the