Optimality and Duality for Multiobjective Semi-infinite Variational Problem Using Higher-Order B-type I Functions
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Optimality and Duality for Multiobjective Semi-infinite Variational Problem Using Higher-Order B-type I Functions Promila Kumar1 · Jyoti Dagar2 Received: 1 November 2018 / Revised: 2 March 2019 / Accepted: 10 September 2019 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract The notion of higher-order B-type I functional is introduced in this paper. This notion is utilized to study optimality and duality for multiobjective semi-infinite variational problem in which the index set of inequality constraints is an infinite set. The concept of efficiency is used as a tool for optimization. Mond–Weir type of dual is proposed for which weak, strong, and strict converse duality theorems are proved to relate efficient solutions of primal and dual problems. Keywords Semi-infinite · Variational problem · Efficient solution · Higher-order B-type I functions · Optimality and duality Mathematics Subject Classification 90C46 · 90C29 · 90C34
1 Introduction Semi-infinite programming problem is characterized by the optimization of an objective function of finitely many variables over a feasible region defined by an infinite number of constraints. One of the reason behind the popularity of these problems is their applicability in various fields such as control of robots, mechanical stress of materials, resource allocation in decentralized system, and air pollution abatements
Jyoti was supported by University Grant Commission Non-NET research fellowship, India (No. Schs/Non-NET/139/Ext-142/2015-16/1931).
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Jyoti Dagar [email protected] Promila Kumar [email protected]
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Department of Mathematics, Gargi College, University of Delhi, New Delhi 110049, India
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Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, New Delhi 110007, India
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P. Kumar, J. Dagar
[3–6]. For the recent work of semi-infinite programming, one may refer to Canovas et al. [7,8]. Certain restrictions are imposed on the functions to study duality and optimality for programming problems. Type I functions were introduced by Hanson and Mond [9] and were generalized to pseudo-type I and quasi-type I functions by Rueda and Hanson [10]. Kaul et al. [11] obtained optimality and duality results for multiobjective nonlinear programming problems involving type I and generalized type I functions. Semiinfinite multiobjective programming problems which are static in nature were studied by several authors [12–15] where they had used the notion of generalized convexity. Variational problem is a special type of dynamic optimization problem in which we have to find a piecewise smooth state vector x = x(t) which is brought from the initial state α to a final state γ in such a way that a specified function of x and x˙ is minimized, subject to a specified constraints. In practical situation, variational problems with more than a single objective function may occur, so it is beneficial to study multiobjective variation
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