Mond-Weir Type Duals Research Articles

Second-order symmetric duality in vector optimization involving (K,η)-pseudobonvexity

This paper presents new classes generalizations of second-order (K,η)-pseudobonvexity functions, strongly second-order (K,η)-pseudobonvexity functions, and establishes a pair Mond-Weir type duality multiobjective nonlinear programs over these new classes generalizations. The weak duality, strong duality, converse duality, and self-duality theorems are presents under these generalizations functions. Finally, we introduced two nontrivial numerical examples to illustrating results of the weak duality theorem and strong duality theorem.

On sufficiency and duality for multiobjective programming problems using convexificators

In this paper, we consider a multiobjective programming problem with inequality and set constraints. We derive sufficient conditions for the optimality of a feasible point under generalized invexity assumptions in terms of convexificators. We give an example to illustrate that the concept of invexity in terms of convexificators is weaker than invexity in terms of other subdifferentials. We formulateWolfe and Mond-Weir type duals for the nonsmooth multiobjective programming problem with inequality and set constraints in terms of convexificators. We establish weak, strong, converse, restricted converse and strict converse duality results under the assumptions of invexity and strict invexity using convexificators between the primal and the Wolfe dual. We derive the respective results between the primal and the Mond-Weir dual under the assumptions of generalized pseudoinvexity, strict pseudoinvexity and quasiinvexity in terms of convexificators. We also derive the relationship between a weak vector saddle-point and a weakly efficient solution of the multiobjective programming problem.

On semidifferentiable interval-valued programming problems

In this paper, we consider the semidifferentiable case of an interval-valued minimization problem and establish sufficient optimality conditions and Wolfe type as well as Mond–Weir type duality theorems under semilocal E-preinvex functions. Furthermore, we present saddle-point optimality criteria to relate an optimal solution of the semidifferentiable interval-valued programming problem and a saddle point of the Lagrangian function.

Karush-Kuhn-Tucker Optimality Conditions and Duality for Multiobjective Semi-Infinite Programming Via Tangential Subdifferentials

The aim of this article is to study Karush-Kuhn-Tucker optimality conditions and duality for nonsmooth multiobjective semi-infinite programing (MSIP). By using tangential subdifferentials and suitable generalized constraint qualifications, we establish necessary and sufficient optimality conditions for some types of efficient solutions of the MSIP. We also propose Wolfe and Mond-Weir type duality for the MSIP and explore weak and strong duality relations under generalized convexity.

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.

Vector optimization with generalized cone locally connected functions

In this paper properties of generalized cone locally (strictly) connected functions are studied. The concepts of cone locally (strictly) Pseudo connected and cone locally (naturally) quasi connected functions are introduced. Various interrelations between these function are discussed with the help of an example. Necessary optimality conditions are obtained for an approximate weak quasi efficient solution of a vector optimization problem over cones. Sufficient optimality conditions are also established using the above mentioned functions. Approximate Wolfe type and Mond–Weir type duals are formulated and duality theorems are proved.

On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (<i>F</i>, <i>ρ</i>)-convexity

<p style='text-indent:20px;'>Because interval-valued programming problem is used to tackle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena, this paper considers a non-differentiable interval-valued optimization problem in which objective and all constraint functions are interval-valued functions, and the involved endpoint functions in interval-valued functions are locally Lipschitz and Clarke sub-differentiable. A necessary optimality condition is first established. Some sufficient optimality conditions of the considered problem are derived for a feasible solution to be an efficient solution under the $G-(F, ρ)$ convexity assumption. Weak, strong, and converse duality theorems for Wolfe and Mond-Weir type duals are also obtained in order to relate the efficient solution of primal and dual inter-valued programs.

On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization

We introduce a new concept of generalized convexity at a given point for a family of real-valued functions and deduce nonsmooth sufficient optimality conditions for robust (weakly) efficient solutions. In addition, we present a robust duality theory and Mond–Weir type duality for an uncertain multiobjective optimization problem. Furthermore, some nonsmooth saddle-point theorems are obtained under our generalized convexity assumption. Finally we show the viability of our new concept of generalized convexity for robust optimization and portfolio optimization.

Optimality and duality in non-differentiable interval valued multiobjective programming

In this paper, Fritz-John and Kuhn-Tucker type necessary and sufficient conditions for a non-differentiable interval valued multiobjective optimisation model are established. LU-convexity of interval valued functions is utilised to obtain interval efficient solution for the given problem. Moreover, weak, strong and strict converse duality theorems for Wolfe and Mond-Weir type duals are obtained in order to relate the interval efficient solution of primal and dual problems.

Variational problems and its duality in banach spaces

The concept of duality for the variational problems is introduced in general Banach spaces. Different forms of duality such as Mangasarian and Mond-Weir type are studied. Mond-Weir type duality is considered to weaken the convexity requirements. Many duality (weak, strong and converse) results are established under generalized convexity assumptions. Again, examples and counterexamples are discussed in support of the investigation.

Second-order optimality and duality in vector optimization over cones

In this paper, we introduce the notion of a second-order cone- convex function involving second-order directional derivative. Also, second-order cone-pseudoconvex, second-order cone-quasiconvex and other related functions are defined. Second-order optimality and Mond-Weir type duality results are derived for a vector optimization problem over conesnusing the introduced classes of functions.

Higher-order $$(\Phi , \rho )\hbox {-}V$$ ( Φ , ρ ) - V -invexity and duality for vector optimization problems

In this paper, we introduce the concept of higher-order \((\Phi , \rho )\hbox {-}V\)-invex function and prove duality theorems for higher-order Wolfe and Mond–Weir type duals of the vector optimization problem using aforesaid assumptions.

Generalized (Phi, Rho)-convexity in nonsmooth vector optimization over cones

In this paper, new classes of cone-generalized (Phi,Rho)-convex functions are introduced for a nonsmooth vector optimization problem over cones, which subsume several known studied classes. Using these generalized functions, various sufficient Karush-Kuhn-Tucker (KKT) type nonsmooth optimality conditions are established wherein Clarke's generalized gradient is used. Further, we prove duality results for both Wolfe and Mond-Weir type duals under various types of cone-generalized (Phi,Rho)-convexity assumptions.Phi,Rho

Nonsmooth Interval-Valued Optimization and Saddle-Point Optimality Criteria

In this article, we focus our attention on a nonsmooth interval-valued optimization problem and establish sufficient optimality conditions for a feasible solution to be an LU optimal solution under the invexity assumption. Appropriate duality theorems for Wolfe and Mond–Weir-type duals are presented in order to relate the LU optimal solution of primal and dual programs. Moreover, saddle-point-type optimality conditions are established in order to find relation between LU optimal solution of primal and saddle point of Lagrangian function.

Optimality and duality in vector optimization problems involving arcwise connected d-type-I functions over cones

In this paper arcwise cone connected d-type-I, quasi cone connected d-type-I, pseudo cone connected d-type-I and other related functions are defined for a vector optimization problem over cones. Sufficient optimality conditions are studied for this problem. Wolfe type, Mond Weir type and mixed type duals are formulated and duality results are established by using the above defined functions.

Sufficiency and duality for optimization problems involving interval-valued invex functions in parametric form

In this paper, we introduce the concepts of invexity, quasi-invexity and pseudo-invexity for interval-valued functions in parametric form. Sufficient optimality conditions for a class of interval-valued optimization problems are derived for feasible solution to be an efficient solution under proposed invexity assumptions. Furthermore, we formulate Wolfe and Mond–Weir type duals and establish appropriate duality theorems in order to relate the efficient solutions of primal and dual programs. Some examples are also constructed to illustrate the proposed invexity and weak duality theorems.

On minimax programming problems involving right upper-Dini-derivative functions

Abstract In this paper, we derive necessary and sufficient optimality conditions for a general minimax programming problem involving some classes of generalized convexities with the tool-right upper-Dini-derivative. Moreover, using the concept of optimality conditions, Mond-Weir type duality theory has been developed for such a minimax programming problem. MSC:26A51, 49J35, 90C32.

GeodesicB-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds

We introduce a class of functions called geodesicB-preinvex and geodesicB-invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi/pseudoB-preinvex and geodesic quasi/pseudoB-invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesicB-preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesicB-invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming.

Duality for a Control Problem Involving Support Functions

Mond-Weir type duality for control problem with support functions is investigated under generalized convexity conditions. Special cases are derived. A relationship between our results and those of nonlinear programming problem containing support functions is outlined.

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.