Journal
MATHEMATICAL PROGRAMMING
Volume 126, Issue 2, Pages 393-399Publisher
SPRINGER
DOI: 10.1007/s10107-009-0283-8
Keywords
Necessary global optimality conditions; Polynomial constraints; Polynomial over-estimators; Semi-algebraic geometry; Sums of squares polynomials
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Funding
- Australian Research Council
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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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