On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity
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On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity Bhuwan Chandra Joshi1
· Shashi Kant Mishra2 · Pankaj Kumar3
Received: 23 September 2018 / Revised: 20 July 2019 / Accepted: 2 August 2019 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, we consider semi-infinite mathematical programming problems with equilibrium constraints (SIMPPEC). By using the notion of convexificators, we establish sufficient optimality conditions for the SIMPPEC. We formulate Wolfe and Mond–Weir-type dual models for the SIMPPEC under the invexity and generalized invexity assumptions. Weak and strong duality theorems are established to relate the SIMPPEC and two dual programs in the framework of convexificators. Keywords Duality · Convexificators · Generalized invexity · Constraint qualification Mathematics Subject Classification 90C46 · 49J52 · 90C30
1 Introduction A semi-infinite programming (SIP) is an optimization problem in finitely many variables on a feasible set described by infinitely many constraints. There are many
The research of Shashi Kant Mishra was supported by Department of Science and Technology-Science and Engineering Research Board (No. MTR/2018/000121), India.
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Bhuwan Chandra Joshi [email protected] Shashi Kant Mishra [email protected] Pankaj Kumar [email protected]
1
Centre for Interdisciplinary Mathematical Sciences, Department of Science & Technology, Institute of Science, Banaras Hindu University, Varanasi 221005, India
2
Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India
3
Mahila Maha Vidyalaya, Banaras Hindu University, Varanasi 221005, India
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B. C. Joshi et al.
applications of SIP in various fields such as robust optimization, Chebyshev approximation, optimal control, robotics, transportation problems, mathematical physics, fuzzy sets, cooperative games, engineering design (see [1,2]). For basic theory, survey articles on SIP we refer to [3] and for monograph [4]. The notion of convexificators can be seen as a generalization of subdifferentials. Jeyakumar et al. [5] have shown that the Clarke subdifferentials [6], Michel–Penot subdifferentials [7], Ioffe–Mordukhovich subdifferentials [8] and Treiman subdifferentials [9] of a locally Lipschitz real-valued function are convexificators and these known subdifferentials may contain the convex hull of a convexificator. Convexificators are not necessarily compact or convex, unlike some of the subdifferentials which are compact and convex objects. We refer to the recent results [10–13] and the references therein for more details related to the convexificators. Usually, generalized convex functions have been introduced in order to weaken the convexity requirements as much as possible to obtain results related to optimization theory. One of the significant generalization of convex function is invex fun
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