On Optimality Conditions for a Class of Nondifferentiable Multiobjective Programming Problems with Vanishing Constraints

In this paper, we are concerned with optimality conditions and duality results for nondifferentiable multiobjective fractional programming problems. Parametric necessary optimality conditions are established for such vector optimization problems in which each component of the involved functions is locally Lipschitz. Further, under the introduced concept of nondifferentiable $(b,\Psi ,\Phi ,\rho )$-univexity, the parametric sufficient optimality conditions are established for a new class of nonconvex multiobjective fractional programming problems. Furthermore, for the considered multiobjective fractional programming problem, its parametric vector dual problem in the sense of Schaible is defined. Then several duality theorems are also established under $(b,\Psi ,\Phi ,\rho )$% -univexity hypotheses.

In this paper, a new class of nonconvex nonsmooth multiobjective programming problems with both inequality and equality constraints is considered. Namely, the sufficient optimality conditions and Mond-Weir duality results are established for such nondifferentiable multicriteria optimization problems in which the involved functions are nondifferentiable (b,?,?,?)w-univex functions (not necessarly with respect to the same functions b and ? and ?). Then the aforesaid results developed here under (b,?,?, ?)w-uinvexity are applicable for a larger class of nonsmooth vector optimization problems than under other generalized convexity notions existing in the literature.

The purpose of this paper is to consider a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function contains a term involving the support function of a compact convex set. Based on the (C,α,ρ,d)-convexity, sufficient optimality conditions and duality results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problem are established. The results extend and improve the corresponding results in the literature.

Optimality conditions and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function

A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the definition of the generalized (F,?,?,d)-convex class about the Clarke's generalized gradient. Under the above generalized convexity assumption, necessary and sufficient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.

In this paper, new results, which exhibit some new applications for Mordukhovich's subdifferential in nonsmooth optimization and variational problems, are established. Nonsmooth (fractional) multiobjective optimization problems in special Banach spaces are studied, and some necessary and sufficient conditions for weak Pareto-optimality for these problems are introduced. Through this work, we introduce into nonsmooth optimization theory in Banach algebras a new class of mathematical programming problems, which generalizes the notion of smooth KT-$(p,r)$-invexity. Some optimality conditions regarding the generalized KT-$(p,r)$-invexity notion and Kuhn-Tucker points are provided. Also, we seek a connection between linear (semi-) infinite programming and nonlinear programming. Some sufficient conditions for (proper) optimality under invexity are provided. A nonsmooth variational problem corresponding to a considered multiobjective problem is defined and the relations between the provided variational problem and the considered optimization problem are studied. The final part of the paper is devoted to illustrating a penalization mechanism, using the distance function as a tool, to provide some conditions to the solutions of the nonsmooth variational inequality problems. All results of the paper have been established in the absence of gradient vectors, using the properties of Mordukhovich's subdifferential in Asplund spaces.

In this paper, we consider a class of nondifferentiable multiobjective fractional programs in which each component of the objective function contains a term involving the support function of a compact convex set. We establish necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems.

Optimality conditions and duality for nondifferentiable multiobjective programming problems involving d- r-type I functions

In this paper, a new class of nonconvex nonsmooth multiobjective programming problems with directionally differentiable functions is considered. The so-called G-V-type I objective and constraint functions and their generalizations are introduced for such nonsmooth vector optimization problems. Based upon these generalized invex functions, necessary and sufficient optimality conditions are established for directionally differentiable multiobjective programming problems. Thus, new Fritz John type and Karush-Kuhn-Tucker type necessary optimality conditions are proved for the considered directionally differentiable multiobjective programming problem. Further, weak, strong and converse duality theorems are also derived for Mond-Weir type vector dual programs.

ABSTRACTIn this article, the vector exact l1 penalty function method used for solving nonconvex nondifferentiable multiobjective programming problems is analyzed. In this method, the vector penalized optimization problem with the vector exact l1 penalty function is defined. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto optimal solutions of the considered nondifferentiable multiobjective programming problem and of the associated vector penalized optimization problem with the vector exact l1 penalty function. This equivalence is established for nondifferentiable invex vector optimization problems. Some examples of vector optimization problems are presented to illustrate the results established in the article.

Nondifferentiable multiobjective programming under generalized -invexity

In this paper, a new class of generalized invex functions named generalized uniform pseudoinvex (C, q) – type I, generalized uniform pseudoquasi-invex (C, q) – type I and generalized uniform quasipseudo-invex (C, q) – type I are defined by utilizing the Clarke subdifferential, where C is sublinear in the third argument. Then some sufficient optimality conditions are derived and proved for a class of nondifferentiable multiobjective programming problems involving the new generalized uniform invexity.

The vector exact l1 penalty method for nondifferentiable convex multiobjective programming problems

This paper introduces a class of new generalized functions of the concept of invex of σ(B, ϕ) − V − type I for nondifferentiable locally Lipschitz functions by using the tools of Clarke subdifferential. These functions are used to derive the sufficient optimality conditions for a class of nondifferentiable multiobjective nonlinear programming problems with inequality constraints where the objective and constraint functions are locally Lipschitz.

This paper obtains sufficient optimality conditions for a nonlinear nondifferentiable multiobjective semi-infinite programming problem involving generalized (C, α, ρ, d)-convex functions. The authors formulate Mond-Weir-type dual model for the nonlinear nondifferentiable multiobjective semi-infinite programming problem and establish weak, strong and strict converse duality theorems relating the primal and the dual problems.