Optimality Conditions for Vector Optimization Problems of a Difference of Convex Mappings

Abstract

In this paper, we study optimality conditions for vector optimization problems of a difference of convex mappings

$$ (VP)\left\{\matrix{{\mathbb R}_{+}^{p} &-\hbox{Minimize} f(x)-g(x),\cr &\hbox{subject to the constraints} \cr & x\in C,l(x)\in -Q,Ax=b\cr &\hbox{and } h(x)-k(x)\in -{\mathbb R}_{+}^{m},}\right. $$

where $f:=(f_{1},\ldots,f_{p}),g:=(g_{1},\ldots,g_{p}) h:=(h_{1},\ldots,h_{m}), k:=(k_{1},\ldots,k_{m}), Q$ is a closed convex cone in a Banach space Z, l is a mapping Q-convex from a Banach space X into Z, A is a continuous linear operator from X into a Banach space $W, {\mathbb R}_{+}^{p}$ and ${\mathbb R}_{+}^{m}$ are respectively the nonnegative orthants of ${\mathbb R}^{p}$ and ${\mathbb R}^{m}$, C is a nonempty closed convex subset of X, b∈W, and the functions f _i ,g _i ,h _j and k _j are convex for i=1,...,p and j=1,ldots,m. Necessary optimality conditions for (VP) are established in terms of Lagrange-Fritz-John multipliers. When the set of constraints for (VP) is convex and under the generalized Slater constraint qualification introduced in Jeyakumar and Wolkowicz [11] , we derive necessary optimality conditions in terms of Lagrange-Karush-Kuhn-Tucker multipliers which are also sufficient whenever the functions g _i ,i=1,...,p are polyhedrals. Our approach consists in using a special scalarization function. A necessary optimality condition for convex vector maximization problem is derived. Also an application to vector fractional mathematical programming is given. Our contribution extends the results obtained in scalar optimization by Hiriart-Urruty [9] and improve substantially the few results known in vector case (see for instance: [11], [12] and [14]).