Nonsmooth Multiobjective Programming Problem Research Articles

Robust Optimality and Duality for Nonsmooth Multiobjective Programming Problems with Vanishing Constraints Under Data Uncertainty

This article investigates robust optimality conditions and duality results for a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC). Mathematical programming problems with vanishing constraints constitute a distinctive class of constrained optimization problems because of the presence of complementarity constraints. Moreover, uncertainties are inherent in various real-life problems. The aim of this article is to identify an optimal solution to an uncertain optimization problem with vanishing constraints that remains feasible in every possible future scenario. Stationary conditions are necessary conditions for optimality in mathematical programming problems with vanishing constraints. These conditions can be derived under various constraint qualifications. Employing the properties of convexificators, we introduce generalized standard Abadie constraint qualification (GS-ACQ) for the considered problem, UNMPVC. We introduce a generalized robust version of nonsmooth stationary conditions, namely a weakly stationary point, a Mordukhovich stationary point, and a strong stationary point (RS-stationary) for UNMPVC. By employing GS-ACQ, we establish the necessary conditions for a local weak Pareto solution of UNMPVC. Moreover, under generalized convexity assumptions, we derive sufficient optimality criteria for UNMPVC. Furthermore, we formulate the Wolfe-type and Mond–Weir-type robust dual models corresponding to the primal problem, UNMPVC.

Constraint qualifications and optimality conditions for nonsmooth multiobjective mathematical programming problems with vanishing constraints on Hadamard manifolds via convexificators

In this paper, we introduce the notion of convexificators in the framework of Hadamard manifolds. We derive some calculus rules for convexificators on Hadamard manifolds and employ them to investigate nonsmooth multiobjective programming problems with vanishing constraints on Hadamard manifolds (abbreviated as, (NMPVC)). The Abadie constraint qualification (abbreviated as, (ACQ)), as well as the generalized (ACQ) (abbreviated as, (GACQ)), are introduced for (NMPVC) in terms of convexificators. Further, (GACQ) is employed to establish the well-known Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary Pareto efficiency criteria for (NMPVC). Moreover, we introduce several other constraint qualifications for (NMPVC) in the Hadamard manifold setting using convexificators and establish interrelations among them. We have furnished non-trivial examples in the framework of some well-known Hadamard manifolds to illustrate the significance of our results. As far as the best of our knowledge is concerned, constraint qualifications and optimality conditions for (NMPVC) have not yet been studied in the framework of Hadamard manifolds via convexificators.

Constraint Qualifications for Nonsmooth Multiobjective Programming Problems with Switching Constraints on Hadamard Manifolds

Constraint Qualifications for Nonsmooth Multiobjective Programming Problems with Switching Constraints on Hadamard Manifolds

Constraint Qualifications and Optimality Criteria for Nonsmooth Multiobjective Programming Problems on Hadamard Manifolds

Constraint Qualifications and Optimality Criteria for Nonsmooth Multiobjective Programming Problems on Hadamard Manifolds

Optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints on Hadamard manifolds

This article is concerned with nonsmooth multiobjective semi-infinite programming problems with vanishing constraints in the setting of Hadamard manifolds (abbreviated as, (NMSIPVC)). We present the Abadie constraint qualification (abbreviated as, (ACQ)) and a modified version of (ACQ), namely, (NMSIPVC)-tailored Abadie constraint qualification (abbreviated as, (NMSIPVC-ACQ)), for (NMSIPVC). The Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary criteria of optimality for (NMSIPVC) are derived by employing (ACQ) and (NMSIPVC-ACQ). Moreover, sufficient criteria of optimality for (NMSIPVC) are derived under generalized geodesic convexity hypotheses and certain mild restrictions on the index sets. Further, we formulate Wolfe type and Mond-Weir type dual models related to (NMSIPVC) and derive several duality results that relate (NMSIPVC) and the corresponding dual models. Non-trivial numerical examples are incorporated to demonstrate the validity of the results established in this paper. To the best of our knowledge, this is for the first time that constraint qualifications, optimality conditions, and duality results have been investigated for (NMSIPVC) in the setting of Hadamard manifolds.

Continuous time non-smooth optimization through quasi efficiency

Abstract The importance of quasi efficiency lies in its versatile nature as it permits a definite tolerable error that depend on the decision variables. This has been a motivating factor for us to introduce the notion of quasi e cient solution for the non-smooth multiobjective continuous time programming problem. Necessary optimality conditions are derived for this problem. To derive sufficient optimality conditions, the concept of approximate convexity has been extended to continuous case in this paper. A mixed dual is proposed for which weak and strong duality results are proved.

On nonsmooth multiobjective semi-infinite programming with switching constraints using tangential subdifferentials

We investigate optimality conditions for a nonsmooth multiobjective semi-infinite programming problem subject to switching constraints. In particular, we employ a surrogate problem and a suitable constraint qualification to state necessary M-stationary conditions in terms of tangential subdifferentials. An example is given at the end to illustrate our main result.

On relations between nonsmooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities

In this article, we study nonsmooth interval-valued multiobjective programming problem and generalized Stampacchia vector variational inequality with its weak form for interval-valued functions. Using the tools of Mordukhovich subdifferential, we define some new classes of generalized approximate LU-convex functions. These functions are then employed to establish the relations between the solutions of generalized vector variational inequalities and the approximate LU-efficient solutions of the nonsmooth interval-valued multiobjective programming problem. Moreover, we identify the Kuhn–Tucker vector critical points of the considered nonsmooth interval-valued multiobjective programming problem. Under suitable constraint qualification, we establish the equivalence among local approximate LU-efficient points, Kuhn–Tucker vector critical points and the solutions of generalized Stampacchia vector variational inequalities. The results of this paper extend and sharpen the corresponding results of [Giannessi F. On Minty variational principle. In: Giannessi F, Komlósi S, Rapcśak T. editors. New trends in mathematical programming. Dordrecht: Kluwer Academic Publishers; 1997. p. 93–99], [Golestani M, Sadeghi H, Tavan Y. Nonsmooth multiobjective problems and generalized vector variational inequalities using quasi-efficiency. J Optim Theory Appl. 2018;179(3):896–916], [Lee GM, Lee KB. Vector variational inequalities for nondifferentiable convex vector optimization problems. J Global Optim. 2005;32(4):597–612], [Mishra SK, Upadhyay BB. Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity. 2013;17(4):1071–1083], [Upadhyay BB, Mohapatra RN, Mishra SK. On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers. Adv Nonlinear Var Inequal. 2017;20(2):1–12], [Zhang J, Zheng Q,Ma X, Li L. Relationships between interval-valued vector optimization problems and vector variational inequalities. Fuzzy Optim Decis Mak. 2016;15(1):33–55] for nonsmooth interval-valued multiobjective programming problem by using the powerful tool of Mordukhovich subdifferential.

Robust optimality conditions and duality for nonsmooth multiobjective fractional semi-infinite programming problems with uncertain data

In this article, some Karush-Kuhn-Tucker type robust optimality conditions and duality for an uncertain nonsmooth multiobjective fractional semi-infinite programming problem ((UMFP), for short) are established. First, we provide, by combining robust optimization and the robust limiting constraint qualification, robust necessary optimality conditions in terms of Mordukhovich's subdifferentials. Under suitable assumptions on the generalized convexity/the strictly generalized convexity, robust necessary optimality condition becomes robust sufficient optimality condition. Second, we formulate types of Mond-Weir and Wolfe robust dual problem for (UMFP) via the Mordukhovich subdifferentials. Finally, as an application, we establish weak/strong/converse robust duality theorems for the problem (UMFP) and its Mond-Weir and Wolfe types dual problem. Some illustrative examples are also provided for our findings.

Karush-Kuhn-Tucker optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints

Karush-Kuhn-Tucker optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints

Optimality conditions for nonsmooth multiobjective bilevel optimization using tangential subdifferentials

In combining the value function approach and tangential subdifferentials, we establish necessary optimality conditions of a nonsmooth multiobjective bilevel programming problem under a suitable constraint qualification. The upper level objectives and constraint functions are neither assumed to be necessarily locally Lipschitz nor convex.

Optimality conditions and duality in terms of convexificators for multiobjective bilevel programming problem with equilibrium constraints

This paper is devoted to the investigation of a nonsmooth multiobjective bilevel programming problem with equilibrium constraints ((MBPP) for short) in terms of convexificators in finite-dimensional spaces. We present necessary optimality conditions for the local weak efficient solution to such problem. Under the Mangasarian–Fromovitz and generalized standard Abadie type constraint qualification in the sense of convexificators, we establish as an application the Wolfe and Mond-Weir type dual problem for the problem (MBPP). Besides, we provide strong and weak duality theorems for the original problem and its Wolfe and Mond–Weir type dual problem under suitable assumptions on the $$\partial ^*$$ -convexity and the upper semi-regularity of objective and constraint functions. Illustrative examples are also proposed to demonstrate the main results of the paper.

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

Mixed Type Duality for Nonsmooth Multiobjective Semi-Infinite Programming Problems

The mixed type duality for nonsmooth multiobjective semi-infinite programming problems was studied. Firstly, by the Lagrange function, the definitions of weakly efficient solutions and efficient solutions to the mixed type duality were introduced. Secondly, by means of the Dini-pseudoconvexity, the weak duality theorems, the strong duality theorems and the converse duality theorems were obtained. The results generalize the main results in previous literatures.

Nonsmooth semi-infinite E-convex multi-objective programming with support functions

In this paper, we study a nonsmooth semi-infinite multi-objective E-convex programming problem involving support functions. We derive sufficient optimality conditions for this problem. We formulate Mond-Weir type dual for the considered problem and establish weak and strong duality theorems under E-convexity assumptions.

On Approximate Efficiency for Nonsmooth Robust Vector Optimization Problems

In this article, we use the robust optimization approach (also called the worst-case approach) for finding e-efficient solutions of the robust multiobjective optimization problem defined as a robust (worst-case) counterpart for the considered nonsmooth multiobjective programming problem with the uncertainty in both the objective and constraint functions. Namely, we establish both necessary and sufficient optimality conditions for a feasible solution to be an e-efficient solution (an approximate efficient solution) of the considered robust multiobjective optimization problem. We also use a scalarizing method in proving these optimality conditions.

Weak Slater Qualification for Nonconvex Multiobjective Semi-infinite Programming

We consider a nonsmooth multiobjective semi-infinite programming problem with a feasible set defined by inequality constraints, MSIP for short. First, we introduce the weak Slater constraint qualification and derive the Karush–Kuhn–Tucker types necessary and sufficient conditions for (weakly, properly) efficient solution of the considered problem. Then, we introduce a dual of Mond–Weir type and present (weak and strong) duality results for MSIP. All results are given in terms of Clarke subdifferential.

Optimality and duality for nonsmooth semi-infinite E-convex multi-objective programming with support functions

In this paper, we study a nonsmooth semi-infinite multi-objective E-convex programming problem involving support functions. We derive sufficient optimality conditions for the primal problem. We formulate Mond-Weir type dual for the primal problem and establish weak and strong duality theorems under various generalized E-convexity assumptions.

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

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.

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

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