Optimality and Duality for Invex Nonsmooth Multiobjective programming problems

Abstract

We consider nonsmooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions. Several sufficient optimality conditions under various (generalized) invexity assumptions and certain regularity conditions are presented. In addition, we introduce a Wolfetype dual and Mond-Weir-type dual and establish duality relations under various (generalized) invexity and regularity conditions.

A characterization of pseudoinvexity for the efficiency in non-differentiable multiobjective problems. Duality

“…With these results, we have generalized the characterizations about optimality provided by Martin [6] for scalar problems and have completed the work carried out in [15,16,14,18] for weakly efficient solutions or in [7,11,12] for efficient solutions. We have also illustrated these optimality results with examples.…”

“…Recently, there has been an increasing interest in developing optimality conditions and duality relations for non-differentiable multiobjective programming problems involving these kinds of functions. Giorgi and Guerraggio [9], Kaul et al [10], Kim and Schaible [11] or Nobakhtian [12,13] derive several sufficient optimality conditions under various generalized non-differentiable invexity assumptions. Sach, Kim and Lee [14] show that every generalized Kuhn-Tucker point in a vector optimization problem involving locally Lipschitz functions is a weakly efficient point if and only if this problem is KT-pseudoinvex, what is a generalization of the results given by Osuna et al [15,16].…”

Vector Exponential Penalty Function Method for Nondifferentiable Multiobjective Programming Problems

“…Note that in the case when r = 0, the definition of a (strictly) r -invex vector-valued function reduces to the definition of nondifferentiable (strictly) invex vector-valued function (see, for example, [33,34]). …”

“…Often, the feasible set of a multiobjective programming problem can be represented by functional inequalities and, therefore, we consider the nondifferentiable constrained vector optimization problem in the following form It is well known (see, for example, [33][34][35][36][37][38]) that the following conditions, known as the generalized form of the Karush-Kuhn-Tucker conditions, are necessary for a (weak) Pareto solution in the considered nondifferentiable vector optimization problem (VP). [30,36,39]) be satisfied at x.…”

Optimality and duality for proper and isolated efficiencies in multiobjective optimization

“…The interested reader is referred to [1][2][3][4] for further information about the theory of multiobjective optimization and to [5,6] for some of its applications in practice. Among many other things, optimality conditions and duality relations for multiobjective optimization problems involving locally Lipschitz functions have been investigated intensively by many researchers; see e.g., [7][8][9][10][11][12][13][14][15][16][17] and the references therein. One of the main tools here is to exploit the separation theorem of convex sets (see e.g., [18]) to establish necessary conditions for (weakly) efficient solutions of a multiobjective optimization problem, and to use various kinds of (generalized) convexity/or (generalized) invexity of functions to formulate sufficient conditions for such (weakly) efficient solutions.…”