Necessary global optimality conditions for nonlinear programming problems with polynomial constraints |
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Authors: | V Jeyakumar G Y Li |
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Institution: | (1) Department of Applied Mathematics, University of New South Wales, Sydney, 2052, Australia;(2) Present address: School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, 3353, VIC, Australia;(3) Department of Mathematics, Chongqing Normal University, Chongqing, 400047, People’s Republic of China |
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Abstract: | In this paper, we develop necessary conditions for global optimality that apply to non-linear programming problems with polynomial
constraints which cover a broad range of optimization problems that arise in applications of continuous as well as discrete
optimization. In particular, we show that our optimality conditions readily apply to problems where the objective function
is the difference of polynomial and convex functions over polynomial constraints, and to classes of fractional programming
problems. Our necessary conditions become also sufficient for global optimality for polynomial programming problems. Our approach
makes use of polynomial over-estimators and, a polynomial version of a theorem of the alternative which is a variant of the
Positivstellensatz in semi-algebraic geometry. We discuss numerical examples to illustrate the significance of our optimality
conditions. |
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Keywords: | |
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