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Strict benson proper-ε-efficiency in vector optimization with set-valued maps

In this paper the notion of Strict Benson proper-ε-efficient solution for a vector optimization problem with set-valued maps is introduced. The scalarization theorems and ε-Lagrangian multiplier theorems are established under the assumption of ic-cone-convexlikeness of set-valued maps. | Yugoslav Journal of Operations Research 25 (2015), Number 3, 387–395 DOI: 10.2298/YJOR131115007S STRICT BENSON PROPER-ε-EFFICIENCY IN VECTOR OPTIMIZATION WITH SET-VALUED MAPS Surjeet Kaur SUNEJA Department of Mathematics, University of Delhi, Delhi-110007, India surjeetsuneja@gmail.com Megha SHARMA∗ Department of Mathematics, University of Delhi, Delhi-110007, India mathmeghasharma@gmail.com Received: November 2013 / Accepted: March 2015 Abstract: In this paper the notion of Strict Benson proper-ε-efficient solution for a vector optimization problem with set-valued maps is introduced. The scalarization theorems and ε-Lagrangian multiplier theorems are established under the assumption of ic-cone-convexlikeness of set-valued maps. Keywords: Ic-cone-convexlikeness, Set-valued Maps, strict Benson proper-ε-efficiency, scalarization, ε-Lagrangian Multipliers. MSC: 90C26, 90C29, 90C30, 90C46. 1. INTRODUCTION In the study of vector optimization, the theory of efficiency plays an important role. Kuhn and Tucker [8] and later Geoffrion [6] observed that certain efficient points exhibit some abnormal properties and to eliminate such anomalous solutions in large set of efficient solutions, they introduced the concept of proper efficiency. Borwein [2] and Benson [1] proposed proper efficiency for vector ∗ Corresponding author 388 S. K. Suneja, M. Sharma / Strict Benson Proper-ε-Efficiency maximization problem over cones. Chen and Rong [4] and Li [9] gave characterization of Benson proper efficiency for vector optimization problems. Cheng and Fu [5] introduced the concept of strong efficiency in locally convex spaces. Various authors have studied approximate efficient solutions for vector optimization problems. Some of them are Liu [11], Chen and Huang [3] and Rong and Wu [12]. Li, Xu and Zhu [10] introduced ε-strictly efficient solutions for set-valued optimization problem. The purpose of this paper is to introduce the notion of Strict Benson proper-ε-efficient solution for vector optimization .