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.
Highlights
Convex programming deals with the minimization of a convex objective function over a convex set usually described by convex constraint functions
As a breakthrough to this, Lassere [1] showed that as far as KKT optimality conditions are concerned, the convexity of the constraint functions can be replaced by the convexity of the feasible set described by the constraints
This work of Lassere [1] has been carried forward to the non-smooth case by Dutta and Lalitha [5]. They considered the same problem (CP) with the only difference being that the function f is a non-differentiable convex function and the convex set F0 is described by local
Summary
Convex programming deals with the minimization of a convex objective function over a convex set usually described by convex constraint functions. This work of Lassere [1] has been carried forward to the non-smooth case by Dutta and Lalitha [5] They considered the same problem (CP) with the only difference being that the function f is a non-differentiable convex function and the convex set F0 is described by local. Dutta and Lalitha [5] introduced the following non-smooth version (ND2) of Lassere’s non-degeneracy condition: For all j 1, , m In this modified setting Dutta and Lalitha [5] concluded that if each g j is assumed to be regular in the sense of Clarke [6] and if the Slater constraint qualification and the non-degeneracy condition (ND2) hold, a feasible point x* is a global minimizer of f over F0 if and only if it is a KKT point. The overall aim of this paper is to extend Lassere’s [1] results to a vector optimization problem over cones
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