Non-smooth multi-objective fractional programming problem involving higher order functions

Optimality and Duality for Nonsmooth Multiobjective Fractional Programming with Generalized Invexity

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, 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

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, 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.

In this paper, we shall establish necessary and sufficient optimality conditions for a feasible solution to be efficient for a nonsmooth multiobjective fractional programming problem involving ?-pseudolinear functions. Furthermore, we shall show equivalence between efficiency and proper efficiency under certain boundedness condition. We have also obtained weak and strong duality results for corresponding Mond-Weir subgradient type dual problem. These results extend some earlier results on efficiency and duality to multiobjective fractional programming problems involving ?-pseudolinear and pseudolinear functions.

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.

Nonsmooth multiobjective mathematical programming problems with equilibrium constraints (NMMPEC) are studied in this article in the Hadamard manifold setting. In the context of (NMMPEC), the generalized Guignard constraint qualification (GGCQ) is formulated within the framework of Hadamard manifolds. Moreover, Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions are derived for (NMMPEC). Thereafter, we explore constraint qualifications (CQ) tailored to (NMMPEC) in the Hadamard manifold setting. Interrelations between these constraint qualifications are subsequently derived. It is further demonstrated that the proposed constraint qualifications, when satisfied, ensure that GGCQ holds. It is noteworthy that constraint qualifications and optimality conditions for (NMMPEC) have not been investigated in the Hadamard manifold setting.

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.

Convergence of exponential penalty function method for multiobjective fractional programming problems

On computational methods for solutions of multiobjective linear production programming games

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.

In this paper, we have considered a class of constrained nonsmooth multi objective programming problem involving semi-directionally differentiable functions from a view point of generalized convexity. A new generalized class of (dI—ρ—σ)-V-type I univex functions is introduced under which various weak, strong, converse and strict converse duality theorems are established for Mond-Weir type dual program in order to relate the efficient and weak efficient solutions of primal and dual problem. Also, we have illustrated through various non-trivial examples that this class extends many earlier studied classes in literature.

The present work defines interval and interval valued function in terms of parameters. Two types of multi-objective programming problems are considered: one is a general multi-objective interval fractional programming problem and the other is a parametric form of the first problem, where the objective and constraint functions are interval valued. Relationships between the solutions of both problems are developed, and some of the results are illustrated through numerical example.

A neural network approach to multiobjective and multilevel programming problems