Characterization of optimal solutions for nonlinear programming problems with conic constraints

Abstract

Convexity and generalized convexity play a central role in mathematical economics and optimization theory. So, the research on criteria for convexity or generalized convexity is one of the most important aspects in mathematical programming, in order to characterize the solutions set. Many efforts have been made in the few last years to weaken the convexity notions. In this article, taking in mind Craven's notion of K-invexity function (when K is a cone in ℝ ⁿ ) and Martin's notion of Karush–Kuhn–Tucker invexity (hereafter KKT-invexity), we define a new notion of generalized convexity that is both necessary and sufficient to ensure every KKT point is a global optimum for programming problems with conic constraints. This new definition is a generalization of KKT-invexity concept given by Martin and K-invexity function given by Craven. Moreover, it is the weakest to characterize the set of optimal solutions. The notions and results that exist in the literature up to now are particular instances of the ones presented here.

Keywords:

• conic constraints
• invexity
• Karush–Kuhn–Tucker conditions

AMS Subject Classifications:

• 90C30
• 49K27

Acknowledgements

The authors have been partially supported by M.E.C. (Spain), Project MTM2007-063432 and Fondecyt (Chile), project 1080628.