On Approximate Efficiency for Nonsmooth Robust Vector Optimization Problems
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ON APPROXIMATE EFFICIENCY FOR NONSMOOTH ROBUST VECTOR OPTIMIZATION PROBLEMS∗ Tadeusz ANTCZAK† Faculty of Mathematics and Computer Science, University of L´ od´z, Banacha 22, 90-238 L´ od´z, Poland E-mail : [email protected]
Yogendra PANDEY Department of Mathematics, Satish Chandra College, Ballia 277001, India
Vinay SINGH Department of Mathematics, National Institute of Technology, Aizawl-796012, Mizoram, India
Shashi Kant MISHRA Department of Mathematics, Banaras Hindu University, Varanasi-221005, India Abstract In this article, we use the robust optimization approach (also called the worst-case approach) for finding ǫ-efficient solutions of the robust multiobjective optimization problem defined as a robust (worst-case) counterpart for the considered nonsmooth multiobjective programming problem with the uncertainty in both the objective and constraint functions. Namely, we establish both necessary and sufficient optimality conditions for a feasible solution to be an ǫ-efficient solution (an approximate efficient solution) of the considered robust multiobjective optimization problem. We also use a scalarizing method in proving these optimality conditions. Key words
Robust optimization approach; robust multiobjective optimization; ǫ-efficient solution; ǫ-optimality conditions; scalarization
2010 MR Subject Classification
1
90C46; 90C29; 90C30; 49J52
Introduction
Robust optimization methodology (the worst-case approach) is a powerful approach for examining and solving optimization problems under data uncertainty. In robust optimization, the data is uncertain but bounded, that is, the data is varying in a given uncertainty set, and we choose the best solution among the robust feasible ones; for detail, we refer to [1–6, ∗ Received August 15, 2018; revised April 18, 2019. The research of Yogendra Pandey and Vinay Singh are supported by the Science and Engineering Research Board, a statutory body of the Department of Science and Technology (DST), Government of India, through file no. PDF/2016/001113 and SCIENCE & ENGINEERING RESEARCH BOARD (SERB-DST) through project reference no. EMR/2016/002756, respectively. † Corresponding author
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
9, 12, 20–26, 28, 29, 31, 32, 38]. Ben-Tal et al. [5] introduced the concept of the uncertain linear optimization problem and its robust counterpart, and discussed the computational issues. Also, Bertsimas et al. [6] characterized the robust counterpart of a linear mathematical programming problem with uncertainty set described by an arbitrary norm. Jeyakumar and Li [23, 24] presented basic theory and applications of an uncertain linear mathematical program problem. Jeyakumar and Li [25] derived a robust theorem of the alternative for parameterized convex inequality systems using conjugate analysis and introduced duality theory for convex mathematical programming problems in the face of data uncertainty via robust optimiza
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