Duality for a class of nonsmooth multiobjective programming problems using convexificators

Optimality and Duality for Nonsmooth Multiobjective Fractional Programming with Generalized Invexity

In this paper, a new generalised class of higher order (F, α, ρ, d)-V-type I function is introduced for a non-smooth multi-objective fractional programming problem involving support functions. The newly defined class extends several known classes in the literature has been justified through a non-trivial example. In the framework of new concept, we determine conditions under which a fractional function becomes higher order (F, α, ρ, d)-V-type I function and do some computational work to substantiate the analysis. Further, we establish Karush-Kuhn-Tucker type sufficient optimality conditions and derive various duality results for higher order Mond-Weir type and Schaible type dual programs.

In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concept of invex of order type II for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. The new functions are used to derive the sufficient optimality condition for a class of nonsmooth multiobjective programming problems. Utilizing the sufficient optimality conditions, weak and strong duality theorems are established for Wolfe type duality model.

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.

We consider non-smooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions. Several optimality conditions under various (generalized) V-invexity assumptions and certain regularity conditions are presented. Further, we introduce a Mond–Weir type dual and establish duality relations under the aforesaid assumptions.

The optimality conditions of major constraints nonsmooth programming with multi-objectives are studied. The objective function of this class of multi-objective programming is the sum of a convex vector function and a differentiable vector function. The constraint condition is major cone constraint in Euclid space. By using of the structure representation of major cone constraint of the given problem, the Fritz John condition of Pareto weakly effective solution is obtained by Gordan theorem. Meanwhile, the Kuhn-Tucker condition for Pareto weakly effective solution under Slater constraint qualification is given. The results are very useful to design its numerical methods.

This paper considers the following nonsmooth multiobjective programming problem: (P) weak min f(x)=(f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (x), hellip, f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (x)), s.t. -g(x)isinS, h(x)=0, where xisinR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , f(x)isinR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> , g(x)isinR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , h(x)isinR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> , SsubR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> is a closed convex cone with interior, and f, g, h satisfy the local Lipschitz condition. The necessary Lagrangian condition is established for a weak minimum of (P), and the proof is different from the proofs given by B.D.Craven, Ying Meiqian and other authors. Necessary conditions of Kuhn-Tucker type are obtained and sufficient conditions are discussed.

In this paper, nonsmooth univex, nonsmooth quasiunivex, and nonsmooth pseudounivex functions are introduced. By utilizing these new concepts, sufficient optimality conditions for a weakly efficient solution of the nonsmooth multiobjective programming problem are established. Weak and strong duality theorems are also derived for Mond-Weir type multiobjective dual programs.

Optimality Conditions for Efficiency in Locally Lipschitz Vector Equilibrium Problem with Constraints in Terms of Michel–Penot’s Directional Derivatives

A new class of (not necessarily differentiable) multiobjective fractional programming problems with E-differentiable functions is considered. The so-called parametric E-Karush–Kuhn–Tucker necessary optimality conditions and, under E-convexity hypotheses, sufficient E-optimality conditions are established for such nonsmooth vector optimization problems. Further, various duality models are formulated for the considered E-differentiable multiobjective fractional programming problems and several E-duality results are derived also under appropriate E-convexity hypotheses.

The main aim of this paper is to study strong Karush–Kuhn–Tucker (KKT) optimality conditions for nonsmooth multiobjective semi-infinite programming (MSIP) problems. By using tangential subdifferential and suitable regularity conditions, we establish some strong necessary optimality conditions for some types of efficient solutions of nonsmooth MSIP problems. In addition to the theoretical results, some examples are provided to illustrate the advantages of our outcomes.

In this paper, we present a multi-objective fractional probabilistic programming problem where the parameters in the objective functions and right-hand side constraints are multi-choice in nature. Multi-choice multi-objective objective fractional probabilistic programming problem is formulated. The multi-choice parameters in the right-hand side of constraints are random variables following Cauchy distribution. Multi-choice multi-objective fractional probabilistic programming problem is transformed to deterministic equivalent multi-choice multi-objective fractional programming problem using chance-constrained programming problem. Lagrange interpolating polynomial approach is proposed to deal with multi-choice parameters. The multi-choice multi-objective fractional programming problem is transformed to non-linear mixed integer multi-objective programming problem. An \(\epsilon -\) constrained method is applied to find the compromise solutions of non-linear mixed integer multi-objective programming problem using different values of \(\epsilon \). LINGO software is applied to find the optimal solution of single objective non-linear mixed integer programming problem. Finally, a numerical example is given to illustrate the proposed programming problem.KeywordsMulti-choice programming problemMulti-objective programming problemProbabilistic programming problemFractional programming problem\(\epsilon \)-constraint method

A new method for evaluating generalized derivatives in nonsmooth problems is reviewed. Lexicographic directional (LD-)derivatives are a recently developed tool in nonsmooth analysis for evaluating generalized derivative elements in a tractable and robust way. Applicable to problems in both steady-state and dynamic settings, LD-derivatives exhibit a number of advantages over current theory and algorithms. As highlighted in this article, the LD-derivative approach now admits a suitable theory for inverse and implicit functions, nonsmooth dynamical systems and optimization problems, among others. Moreover, this technique includes an extension of the standard vector forward mode of automatic differentiation (AD) and acts as the natural extension of classical calculus results to the nonsmooth case in many ways. The theory of LD-derivatives is placed in the context of state-of-the-art methods in nonsmooth analysis, with an application in multistream heat exchanger modelling and design used to illustrate the usefulness of the approach.

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.

Necessary and sufficient saddle-point- and stationary-point-type proper efficiency conditions are established for a class of continuous-time multiobjective fractional programming problems with convex operator inequality and affine operator equality constraints. Subsequently, the form and contents of these proper efficiency results are utilized for constracting a Lagrangian-type, two parametric, and four semiparametric duality models, and for proving appropriate duality theorems. These proper efficiency and duality results contain, as special cases, similar results for continuous-time programming problems with nonfractional multiple, single fractional, and conventional objective functions, which are patricular cases of the principal problem studied in this paper.