Geometric Ideas in Nonlinear and Multicriteria Optimization

Kotkin, G.G. (1990). Geometric Ideas in Nonlinear and Multicriteria Optimization. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-90-046

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Abstract

Some geometric properties of the solution set for nonlinear and multicriteria programming problems and the related numeric algorithms are considered. The author deals with necessary and sufficient conditions for nonlinear programming problem stability (in the nonconvex case), with Pareto set stability, Pareto set connectedness conditions, with weak efficiency, efficiency and proper efficiency criteria. A study of numerical algorithms based on geometric properties of the so-called convolutions function is also considered. Necessary and sufficient convergence conditions for large classes of algorithms are presented and easy to check sufficient conditions are given. Further results deal with problems of using local unconstrained minimization algorithms to solve quasi-convex problems and the problem of using some convolution functions for constructing decision making procedures. New classes of inverse nonlinear programming problems are discussed and software implementations of DISO/PC-MCNLP are presented.

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