Nonsmooth Multiobjective Programming Problem Research Articles

Higher-order duality results for a new class of nonconvex nonsmooth multiobjective programming problems

In this paper, a nonconvex nonsmooth multiobjective programming problem is considered and two its higher-order duals are defined. Further, several duality results are established between the considered nonsmooth vector optimization problem and its dual models under assumptions that the involved functions are higher-order (??)-type I functions.

Strong Karush–Kuhn–Tucker optimality conditions for multiobjective semi-infinite programming via tangential subdifferential

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.

Isolated efficiency in nonsmooth semi-infinite multi-objective programming

ABSTRACTIn this paper, isolated efficient solutions of a given nonsmooth Multi-Objective Semi-Infinite Programming problem (MOSIP) are studied. Two new Data Qualifications (DQs) are introduced and it is shown that these DQs are, to a large extent, weaker than already known Constraint Qualifications (CQs). The relationships between isolated efficiency and some relevant notions existing in the literature, including robustness, are established. Various necessary and sufficient conditions for characterizing isolated efficient solutions of a general problem are derived. It is done invoking the tangent cones, the normal cones, the generalized directional derivatives, and some gap functions. Using these characterizations, the (strongly) perturbed Karush-Kuhn-Tucker (KKT) optimality conditions for MOSIP are analyzed. Furthermore, it is shown that each isolated efficient solution is a Geoffrion properly efficient solution under appropriate assumptions. Moreover, Kuhn-Tucker (KT) and Klinger properly efficient solutions for a nonsmooth MOSIP are defined and it is proved that each isolated efficient solution is a KT properly efficient solution in general, and a Klinger properly efficient solution under a DQ. Finally, in the last section, the largest isolated efficiency constant for a given isolated efficient solution is determined.

Sufficiency and Wolfe Type Duality for Nonsmooth Multiobjective Programming Problems

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.

On Constraint Qualifications for Multiobjective Programming Problems with Equilibrium Constraints

In this paper, we consider a nonsmooth multiobjective programming problem with equilibrium constraints. We present three constraint qualications (CQs) and investigate their relations. Furthermore, we derive two types of necessary optimality conditions under these CQs. In addition, some examples are given to clarify our result

On strong KKT type sufficient optimality conditions for multiobjective semi-infinite programming problems with vanishing constraints

In this paper, we consider a nonsmooth multiobjective semi-infinite programming problem with vanishing constraints (MOSIPVC). We introduce stationary conditions for the MOSIPVCs and establish the strong Karush-Kuhn-Tucker type sufficient optimality conditions for the MOSIPVC under generalized convexity assumptions.

Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators

The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function.

Vector critical points and generalized quasi-efficient solutions in nonsmooth multi-objective programming

In this work, several extended approximately invex vector-valued functions of higher order involving a generalized Jacobian are introduced, and some examples are presented to illustrate their existences. The notions of higher-order (weak) quasi-efficiency with respect to a function are proposed for a multi-objective programming. Under the introduced generalization of higher-order approximate invexities assumptions, we prove that the solutions of generalized vector variational-like inequalities in terms of the generalized Jacobian are the generalized quasi-efficient solutions of nonsmooth multi-objective programming problems. Moreover, the equivalent conditions are presented, namely, a vector critical point is a weakly quasi-efficient solution of higher order with respect to a function.

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

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a (weakly) LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval-objective function are convex.

Duality for a class of nonsmooth multiobjective programming problems using convexificators

As duality is an important and interesting feature of optimization problems, in this paper, we continue the effort of Long and Huang [X. J. Long, N. J. Huang, Optimality conditions for efficiency on nonsmooth multiobjective programming problems, Taiwanese J. Math., 18 (2014) 687-699] to discuss duality results of two types of dual models for a nonsmooth multiobjective programming problem using convexificators.

On semi-G-V-type I concepts for directionally differentiable multiobjective programming problems

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.

Multiobjective programming under nondifferentiable G-V-invexity

In the paper, new Fritz John type necessary optimality conditions and new Karush-Kuhn-Tucker type necessary opimality conditions are established for the considered nondifferentiable multiobjective programming problem involving locally Lipschitz functions. Proofs of them avoid the alternative theorem usually applied in such a case. The sufficiency of the introduced Karush-Kuhn-Tucker type necessary optimality conditions are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem are G-V-invex with respect to the same function ?. Further, the so-called nondifferentiable vector G-Mond-Weir dual problem is defined for the considered nonsmooth multiobjective programming problem. Under nondifferentiable G-V-invexity hypotheses, several duality results are established between the primal vector optimization problem and its G-dual problem in the sense of Mond-Weir.

On strong KKT type sufficient optimality conditions for nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints

In this paper, we consider a nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints (MOSIMPECs). We introduce the concept of Mordukhovich stationary point for the nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints in terms of the Clarke subdifferentials. Further, we establish that the M-stationary conditions introduced in this paper are strong KKT type sufficient optimality conditions for the nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints under generalized invexity assumptions. We also illustrate our result with an example.

Sufficiency and duality for nonsmooth multiobjective fractional programming problems involving $$(\Phi ,\rho ,\alpha )$$ ( Φ , ρ , α ) -V-invexity

In this paper, we consider a class of nonsmooth multiobjective fractional programming problems in which the involved functions are locally Lipschitz. A new concept of invexity for locally Lipschitz vector-valued functions is introduced, called \((\Phi ,\rho ,\alpha )\)-V-invexity. Based upon the \((\Phi ,\rho ,\alpha )\)-V-invex functions, the optimality conditions for a feasible solulion to be an efficient solution are derived. Weak, strong, and strict converse duality theorems for Mond–Weir type dual are also proved under the aforesaid functions.

Sufficiency and Duality for Nonsmooth Multiobjective Programming Problems Involving Generalized (Φ, ρ)-V-Type I Functions

In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concepts of (Φ, ρ)-V-type I, (pseudo, quasi) (Φ, ρ)-V-type I and (quasi, pseudo) (Φ, ρ)-V-type I functions, in which the involved functions are locally Lipschitz. Based upon these generalized (Φ, ρ)-V-type I functions, the sufficient optimality conditions for weak efficiency, efficiency and proper efficiency are derived. Mond-Weir duality results are also established under the aforesaid functions.

Saddle point criteria and Wolfe duality in nonsmooth (Φ, ρ)-invex vector optimization problems with inequality and equality constraints

In this paper, saddle point criteria and Wolfe duality theorems are established for a new class of nondifferentiable vector optimization problems with inequality and equality constraints. The results are proved under nondifferentiable (Φ, ρ)-invexity and related scalar and vector-valued Lagrangians defined for the considered nonsmooth multiobjective programming problem. It turns out that the results are established for such vector optimization problems in which not all functions constituting a vector optimization problem possess the fundamental property of invexity and the most of generalized invexity notions previously defined in the literature.

Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of (φ,α,ρ,σ)-dI-V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems for Mond-Weir type multiobjective dual program.

OPTIMALITY CONDITIONS FOR EFFICIENCY ON NONSMOOTH MULTIOBJECTIVE PROGRAMMING PROBLEMS

In this paper, a nonsmooth multiobjective programming problem is introduced and studied. By using the generalized Guignard constraint qualification, some stronger Kuhn-Tucker type necessary optimality conditions for efficiency in terms of convexificators are established, in which we are not assuming that the objective functions are directionally differentiable. Moreover, some conditions which ensure that a feasible solution is an efficient solution to nonsmooth multiobjective programming problems are also given. The results presented in this paper improve the corresponding results in the literature.

Duality and Saddle Point for Multiobjective Semi-infinite Programming

This paper is devoted to study the mixed dual models for a class of non-smooth multiobjective semi-infinite programming problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. Weak duality conclusions are derived and proved for mixed type multiobjective dual programs, using the generalized uniform convexity on the functions involved. Some previous duality results for differentiable semi-infinite programming problems turn out to be special cases for the results described in the paper. Furthermore, the vector saddle point theory is discussed for the semi-infinite programming problems. The results extend and improve the corresponding results in the literature.

Non-smooth multiobjective programming problem involving (d<SUB align="right">I - ρ - σ)-V-type I functions

In this paper, we consider a multiobjective optimisation problem where the objective and constraint functions involved are non-differentiable. We introduce a new class of functions namely (dI −ρ − σ)-V-type I functions and illustrate through non-trivial examples that the class introduced is non-empty. We then obtain sufficient optimality conditions under the newly introduced class of functions and derive various weak, strong, converse and strict converse duality results for Wolfe type and Mond-Weir type dual programmes in order to relate the efficient and weak efficient solutions of primal and dual problems.