A Different Approach to Cone-Convex Optimization

Surjeet Kaur Suneja, Sunila Sharma, Meetu B. Grover, Malti Kapoor

Department of Mathematics, Miranda House College, University of Delhi, Delhi, India.
Department of Mathematics, Moti Lal Nehru College (M), University of Delhi, Delhi, India.

DOI: 10.4236/ajor.2013.36052                

Abstract

In classical convex optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary and sufficient for optimality if the objective as well as the constraint functions involved is convex. Recently, Lassere [1] considered a scalar programming problem and showed that if the convexity of the constraint functions is replaced by the convexity of the feasible set, this crucial feature of convex programming can still be preserved. In this paper, we generalize his results by making them applicable to vector optimization problems (VOP) over cones. We consider the minimization of a cone-convex function over a convex feasible set described by cone constraints that are not necessarily cone-convex. We show that if a Slater-type cone constraint qualification holds, then every weak minimizer of (VOP) is a KKT point and conversely every KKT point is a weak minimizer. Further a Mond-Weir type dual is formulated in the modified situation and various duality results are established.

Keywords

Convex Optimization; Cone-Convex Functions; KKT Conditions; Duality

Conflicts of Interest

The authors declare no conflicts of interest.

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