Mond-Weir Type Dual Problem Research Articles

Approximate proper efficiencies in nonsmooth semi-infinite multiobjective optimization problems

This article is devoted to studying a nonsmooth semiinfinite multiobjective optimization problem (SIMP) in terms of the Mordukhovich/limiting subdifferentials. We first establish necessary and sufficient conditions for an ε−quasi positively properly efficient solution of a problem (SIMP). We also investigate Mond-Weir type dual problems under assumptions of ε−quasi pseudo-generalized convexity. Next, we provide an application to a nonsmooth fractional semi-infinite multiobjective optimization problem. Finally, some examples are given to illustrate the obtained results. The obtained results improve or include some recent known ones.

Solving nonsmooth interval optimization problems based on interval-valued symmetric invexity

This paper focuses on a nonsmooth nonconvex interval-valued optimization problem. For this, we propose interval-valued symmetric invexity, interval-valued symmetric pseudo-invexity and interval-valued symmetric quasi-invexity in terms of the symmetric gH-differentiable interval-valued functions. Some important properties of these generalized convexities are also discussed. By utilizing these new concepts, we establish sufficient Karush–Kuhn–Tucker conditions for the considered problem. Further, the Wolfe and Mond–Weir type dual problems are associated and weak, strong and strict converse duality results have been derived. Finally, we apply the developed theory to a binary classification problem of interval data by support vector machine.

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.

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

This paper concentrates on studying multiobjective semi-infinite programming with vanishing constraints. Firstly, the necessary and sufficient Karush–Kuhn–Tucker optimality conditions for multiobjective semi-infinite programming with vanishing constraints are considered. Then, we formulate Wolfe and Mond–Weir type dual problems and establish duality relations under convexity assumptions. Some examples are proposed to verify our results.

Sufficiency and Duality for Interval-Valued Optimization Problems with Vanishing Constraints Using Weak Constraint Qualifications

In this paper, we are concerned with one of the difficult class of optimization problems called the interval-valued optimization problem with vanishing constraints. Sufficient optimality conditions for a LU optimal solution are derived under generalized convexity assumptions. Moreover, appropriate duality results are discussed for a Mond-Weir type dual problem. In addition, numerical examples are given to support the sufficient optimality conditions and weak duality theorem.

Higher-order Optimality Conditions and Duality for Approximate Solutions in Non-convex Set-valued Optimization

This paper deals with higher-order optimality conditions and duality theory for approximate solutions in vector optimization involving non-convex set-valued maps. Firstly, under the assumption of near cone-subconvexlikeness for set-valued maps, the higher necessary and sufficient optimality conditions in terms of Studniarski derivatives are derived for local weak approximate minimizers of a set-valued optimization problem. Then, applications to Mond-Weir type dual problem are presented.

A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems

In this paper, we introduce the notion of higher-order weak adjacent epiderivative for a set-valued map without lower-order approximating directions and obtain existence theorem and some properties of the epiderivative. Then by virtue of the epiderivative and Benson proper efficiency, we establish the higher-order Mond-Weir type dual problem for a set-valued optimization problem and obtain the corresponding weak duality, strong duality and converse duality theorems, respectively.

Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints

In this paper, we investigate a robust nonsmooth multiobjective optimization problem related to a multiobjective optimization with data uncertainty. We firstly introduce two kinds of generalized convex functions, which are not necessary to be convex. Robust necessary optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are established by a generalized alternative theorem and the robust constraint qualification. Further, robust sufficient optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are also derived. The Mond–Weir-type dual problem and Wolfe-type dual problem are formulated. Finally, we obtain the weak, strong and converse robust duality results between the primal one and its dual problems under the generalized convexity assumptions.

Optimality conditions and duality for minimax fractional programming problems with data uncertainty

In this paper, we consider minimax nondifferentiable fractional programming problems with data uncertainty in both the objective and constraints. Via robust optimization, we establish the necessary and sufficient optimality conditions for an uncertain minimax convex-concave fractional programming problem under the robust subdifferentiable constraint qualification. Making use of these optimality conditions, we further obtain strong duality results between the robust counterpart of this programming problem and the optimistic counterpart of its conventional Wolf type and Mond-Weir type dual problems. We also show that the optimistic counterpart of the Wolf type dual of an uncertain minimax linear fractional programming problem with scenario uncertainty (or interval uncertainty) in objective function and constraints is a simple linear programming, and show that the robust strong duality results in sense of Wolf type always hold for this linear minimax fractional programming problem.

Optimality Conditions for Nonregular Optimal Control Problems and Duality

ABSTRACTWe define a new class of optimal control problems and show that this class is the largest one of control problems where every admissible process that satisfies the Extended Pontryaguin Maximum Principle is an optimal solution of nonregular optimal control problems. In this class of problems the local and global minimum coincide. A dual problem is also proposed, which may be seen as a generalization of the Mond–Weir-type dual problem, and it is shown that the 2-invexity notion is a necessary and sufficient condition to establish weak, strong, and converse duality results between a nonregular optimal control problem and its dual problem. We also present an example to illustrate our results.

Optimality and duality in multiobjective programming involving support functions

In this paper a vector optimization problem (VOP) is considered where each component of objective and constraint function involves a term containing support function of a compact convex set. Weak and strong Kuhn−Tucker necessary optimality conditions for the problem are obtained under suitable constraint qualifications. Necessary and sufficient conditions are proved for a critical point to be a weak efficient or an efficient solution of the problem (VOP) assuming that the functions belong to different classes of pseudoinvex functions. Two Mond Weir type dual problems are considered for (VOP) and duality results are established.

Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators

In this paper, we consider semi-infinite mathematical programming problems with equilibrium constraints (SIMPEC). We establish necessary and sufficient optimality conditions for the SIMPEC, using convexificators. We study the Wolfe type dual problem for the SIMPEC under the \(\partial ^{*}\)-convexity assumption. A Mond–Weir type dual problem is also formulated and studied for the SIMPEC under the \(\partial ^{*}\)-convexity, \(\partial ^{*}\)-pseudoconvexity and \(\partial ^{*}\)-quasiconvexity assumptions. Weak duality theorems are established to relate the SIMPEC and two dual programs in the framework of convexificators. Further, strong duality theorems are obtained under generalized standard Abadie constraint qualification.

Optimality and duality in multiobjective programming involving higher order semilocally strong convexity

In this paper, we propose a generalisation of convexity, namely higher order semilocally strong convexity for a nonlinear multiobjective programming problem, where the function involved are semidifferentiable. The generalised Karush-Kuhn-Tucker necessary and sufficient optimality conditions are derived. Moreover, a general Mond-Weir type dual problem is presented for nonlinear multiobjective programming problem involving higher order semilocally strong convexity and usual duality theorems are discussed.

Optimality and duality in multiobjective programming involving higher order semilocally strong convexity

In this paper, we propose a generalisation of convexity, namely higher order semilocally strong convexity for a nonlinear multiobjective programming problem, where the function involved are semidifferentiable. The generalised Karush-Kuhn-Tucker necessary and sufficient optimality conditions are derived. Moreover, a general Mond-Weir type dual problem is presented for nonlinear multiobjective programming problem involving higher order semilocally strong convexity and usual duality theorems are discussed.

Necessary and Sufficient Conditions for Weak Quasi Efficiency in Nonsmooth Multiobjective Problems

ABSTRACTIn this article, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given and the relations between them are analyzed. We establish Kuhn-Tucker and strong Kuhn-Tucker necessary optimality conditions for (weak) quasi efficiency in terms of the Clarke subdifferential. By using two new classes of generalized convex functions, sufficient conditions for local (weak) quasi efficient are also provided. Furthermore, we study the Mond-Weir type dual problem and establish weak, strong and converse duality results.

Symmetric duality in non-differentiable programming using G-invexity

Mond-Weir and Wolfe type non-differentiable primal and dual symmetric problems in mathematical programming are formulated. In this paper, we use a generalisation of convexity, namely G-invexity, to prove weak, strong and converse duality relations for the two non-differentiable symmetric programming problems. Some previous known results for symmetric programming problems turn out to be special cases of the results established in the paper.

OPTIMALITY CONDITIONS AND DUALITY IN FRACTIONAL ROBUST OPTIMIZATION PROBLEMS

Abstract. In this paper, we consider a fractional robust optimizationproblem (FP) and give necessary optimality theorems for (FP). Estab-lishing a nonfractional optimization problem (NFP) equivalent to (FP),we formulate a Mond-Weir type dual problem for (FP) and prove dualitytheorems for (FP). 1. IntroductionConsider a fractional robust optimization problem:(FP) inf x2R n ˆf(x)g(x): h j (x;v j ) 0; 8v j 2V j ; j= 1; ;m˙;where v j are uncertain parameters and v j 2V j ; j= 1; ;mfor some convexcompact sets V j ˆR q ; j = 1; ;mand f : R n !R; g : R n !R andh j : R n R q !R; j = 1; ;mare continuously di erentiable functions.Assume that f(x) 0 and g(x) >0.Let F := fx2R n : h j (x;v j ) 0; 8v j 2V j ; j= 1; ;mgbe the robustfeasible set of (FP). Then we say that x is a robust solution of (FP) if x 2Fand f(x)g(x) = f(x )g(x ) for any x2F.Consider the following nonfractional optimization problem:(NFP) Minimize psubject to f(x) pg(x) 0;h j (x;v j ) 0; 8v j 2V j ; j= 1; ;m:Following the approaches in [8], we can establish an equivalent relationshipbetween (FP) and (NFP) as follows:

Optimality in continuous-time multiobjective optimization and vector variational-like inequalities

In this paper, we study the relationships between the Stampacchia and Minty vector variational inequalities and vector continuous-time programming problems under generalized invexity and monotonicity hypotheses. We extend the results given by other authors for the scalar case to vectorial one, and we show the equivalence of efficient and weak efficient solutions for the multiobjective continuous-time programming problem, and solutions of Stampacchia and Minty variational-like type inequality problems. Furthermore, we introduce a new concept of generalized invexity for continuous-time programming problems, namely, the Karush–Kuhn–Tucker (KKT)-pseudoinvexity-II. We prove that this new concept is a necessary and sufficient condition for a vector KKT solution to be an efficient solution for a multiobjective continuous-time programming problem. In this work, duality results for Mond–Weir type dual problems are obtained, using generalized KKT-pseudoinvexity-II. This paper gives an unified point of view for optimality results in mathematical programming or control or variational inequalities problems.

Optimality and duality for nonsmooth multiobjective optimization problems

In this paper, we consider a nonsmooth multiobjective programming problems including support functions with inequality and equality constraints. Necessary and sufficient optimality conditions are obtained by using higher-order strong convexity for Lipschitz functions. Mond-Weir type dual problem and duality theorems for a strict minimizer of order m are given.

Second Order (F, α, ρ, d, E)-convex function and the Duality Problem

A class of second order ( F , α , ρ , d , E )-convex functions and their generalization on functions involved, weak, strong, and converse duality theorems are established for a second order Mond-Weir type dual problem.