Non-smooth Invariant Convex Multi-objective Optimal Conditions

In this paper, several problems of multiobjective programming are studied. For details, these results are given in the following.(1) This paper recalls the definitions of (weak) efficient points and all kinds of proper efficient points. we obtain the following results:i)Strict efficient points are Borwein proper efficient points.ii)Strict efficient points are Henig proper efficient points.iii)Strong efficient points are Borwein proper efficient points.iv)Strong efficient points are Henig proper efficient points.v)Strong efficient points are Hartley proper efficient points.(2) This paper considers nonsmooth multiobjective programming problems with inequality constrains involving locally Lipschitz functions. Several sufficient optimality conditions under various (generalized) invexity assumptions and certain regularity conditions are presented. In addition, This paper introduces a Wolfe-type dual and Mond-Weir type dual and establish duality relations under various (generalized) invexity and regularity conditions.(3) Optimality for mulitobjective programming problem having invex objective and constraint functions (with respect to the same functionη) is considered. An equivalent vector programming problem((VP|～)_η(x|ˉ)) is constructed by a modification of the objective function. Furthermore, anη-Lagrange function is introduced for a constructed multiobjective problem and modified saddle point results are presented. In addition, a new approach to a solution of a nonlinear constrained multiobjective programming problem is introduced. Anη-approximated problem associated with a primal multiobjective programming problem is presented that involvesη-approximated functions constituting the primal problem. The equivalence between efficient points for the original multiobjective programming problem(VP) and itsη-approximated optimization problem VP_η(x|ˉ) is established under invexity assumption.