Wolfe Duality Research Articles

Wolfe duality and Mond–Weir duality via perturbations

Considering a general optimization problem, we attach to it by means of perturbation theory two dual problems having in the constraints a subdifferential inclusion relation. When the primal problem and the perturbation function are particularized different new dual problems are obtained. In the special case of a constrained optimization problem, the classical Wolfe and Mond–Weir duals, respectively, follow as particularizations of the general duals by using the Lagrange perturbation. Examples to show the differences between the new duals are given and a gate towards other generalized convexities is opened.

G-saddle point criteria and G-Wolfe duality in differentiate mathematical programming

In the paper, new saddle point criteria and Wolfe duality results are established for differentiable optimizations problems with nonconvex functions. The equivalence between the so-called G-saddle points and optima are proved for differentiable optimizations problems with G-invex functions. Further, a new Wolfe dual problem is formulated and the so-called G-Wolfe duality results are established under the assumption that the so-called G-Lagrange function is invex. The results obtained in the paper are illustrated by suitable optimization problems with nonconvex functions belonging to the classes of functions considered in the paper.

Separation of sets and Wolfe duality

Separation of sets and Wolfe duality

Separation of sets and Wolfe duality

Lagrangian duality can be derived from separation in the Image Space, namely the space where the images of the objective and constraining functions of the given extremum problem run. By exploiting such a result, we analyse the relationships between Wolfe and Mond-Weir duality and prove their equivalence in the Image Space under suitable generalized convexity assumptions.

Wolfe Duality for Interval-Valued Optimization

Weak and strong duality theorems in interval-valued optimization problem based on the formulation of the Wolfe primal and dual problems are derived. The solution concepts of the primal and dual problems are based on the concept of nondominated solution employed in vector optimization problems. The concepts of no duality gap in the weak and strong sense are also introduced, and strong duality theorems in the weak and strong sense are then derived.

Duality theory in fuzzy optimization problems formulated by the Wolfe’s primal and dual pair

The weak and strong duality theorems in fuzzy optimization problem based on the formulation of Wolfe's primal and dual pair problems are derived in this paper. The solution concepts of primal and dual problems are inspired by the nondominated solution concept employed in multiobjective programming problems, since the ordering among the fuzzy numbers introduced in this paper is a partial ordering. In order to consider the differentiation of a fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the Wolfe's dual problem can be formulated by considering the gradients of differentiable fuzzy- valued functions. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems in weak and strong sense are then derived naturally.

On interval-valued nonlinear programming problems

The Wolfe's duality theorems in interval-valued optimization problems are derived in this paper. Four kinds of interval-valued optimization problems are formulated. The Karush–Kuhn–Tucker optimality conditions for interval-valued optimization problems are derived for the purpose of proving the strong duality theorems. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems in weak and strong sense are then derived naturally.

Kuhn-tucker condition and wolfe duality of preinvex set-valued optimization

The optimality Kuhn-Tucker condition and the wolfe duality for the preinvex set-valued optimization are investigated. Firstly, the concepts of alpha-order G-invex set and the alpha-order S-preinvex set-valued function were introduced, from which the properties of the corresponding contingent cone and the alpha-order contingent derivative were studied. Finally, the optimality Kuhn-Tucker condition and the Wolfe duality theorem for the alpha-order S-preinvex set-valued optimization were presented with the help of the alpha-order contingent derivative.

Continuous Modular Design Problem: Analysis and Solution Algorithms

This article studies the continuous modular design (MD) problem. First, the article presents duality results for problem (MD) based upon the Wolfe duality theory for nonlinear programming. From these results, an optimality test for problem (MD) is derived that consists of solving a single, balanced transportation problem. Second, the article shows that two well-known optimization approaches, the generalized Benders decomposition and the separable programming approach of linear programming, each have the potential to solve efficiently large instances of problem (MD).

γ-preinvexity and γ-invexity in mathematical programming

In the paper, we introduce and characterize a new class of nonconvex functions in mathematical programming, named γ-preinvex functions, which are called γ-invex functions in the case of differentiability. The class of γ-reinvex functions is wider than that of preinvex functions and thus, in the case of differentiability, than the class of invex functions. Some optimality results are obtained under γ-preinvexity assumption for a constrained optimization problem with not necessarily differentiable functions. Further, a number of sufficiency conditions and Wolfe duality theorems are established under γ-invexity assumption. An alternative approach with a modified γ-invexity notion to optimality conditions and duality results is also considered.

Generalized (p, r)-Invexity in Mathematical Programming

New two classes of real differentiable functions, called (p, r)-Type I and (p, r)-Type II with respect to η, which represent a generalization of the notion of invexity, are introduced. Some examples of these functions are derived. The sufficient optimality conditions are obtained for a nonlinear programming problem involving (p, r)-Type I functions with respect to η and for an associated Wolfe dual problem in which involved functions are (p, 0)-Type II. It is also shown that the optimization problems possesing the considered notion of invexity need not to be equivalent to the class of optimization problems for which some notion of invexity guarantees that Karush–Kuhn–Tucker necessary conditions for optimality are also sufficient and the sufficiency for optimality in associated Wolfe dual problems.

Optimality and Wolfe duality for multiobjective bectovex programming

In this paper we extend, the concept of a sublinear functional to a becto-sublinear functional and the concepts of F -convex and generalized F -convex functions to bectovex and generalized bectovex functions, respectively. Next, we introduce a primal multiobjective programming problem, and assuming the functions to be bectovex (generalized bectovex), establish sufficient conditions for efficient and properly efficient solutions of the primal. Furthermore, we introduce a Wolfe type multiobjective dual, establish the duality results relating the dual and the primal, and relate this to certain vector saddle point of a vector-valued Lagrangian. We also show that the duality for certain multiobjective fractional program follows as a special case of the main problem.

The Minimum Covering l_pb -Hypersphere Problem

The minimu vering hypersphere problem is defined as to find a hypersphere of minimum radius enclosing a finite set of given points in R^n. A hypersphere is a setS(c,r)={x ∈ R^n : d(x,c) ≤ r} , where c is the center of S, r is the radius of S and d(x,c) is the Euclidean distance between x and c, i.e.,d(x,c)=l_2(x-c) . We consider the extension of this problem when d(x, c) is given by anyl_pb -norm, where 1 0,j=1,...,n , then S(c,r) is called anl_pb -hypersphere, in particular for p=2 and b_j=1, j=1,...,n, we obtain thel_2 -norm. We study some properties and propose some primal and dual algorithms for the extended problem , which are based on the feasible directions method and on the Wolfe duality theory. By computational experiments, we compare the proposed algorithms and show that they can be used to approximate the smallestl_pb -hypersphere enclosing a large set of points inR^n .

(Φ1, Φ2) Optimality And Duality Under Differentiablity

(Φ1, Φ2)-convexity is applied to develop optimality conditions of Fritz John type and Kuhn-Tucker type under differentiability for a minimization problem with real valued objective and inequality constraints. A dual of the Mond-Weir type is considered and a number of weak and strong duality results are established. Weak and strong duality theorems are also given in the framework of Wolfe duality

Sufficient optimality conditions and duality in vector optimization with invex-convexlike functions

We prove the Kuhn-Tucker sufficient optimality condition, the Wolfe duality, and a modified Mond-Weir duality for vector optimization problems involving various types of invex-convexlike functions. The class of such functins contains many known generalized convex functions. As applications, we demonstrate that, under invex-convexlikeness assumptions, the Pontryagin maximum principle is a sufficient optimality condition for cooperative differential games. The Wolfe duality is established for these games.

Invex-convexlike functions and duality

We define a class of invex-convexlike functions, which contains all convex, pseudoconvex, invex, and convexlike functions, and prove that the Kuhn-Tucker sufficient optimality condition and the Wolfe duality hold for problems involving such functions. Applications in control theory are given.

A modified wolfe dual for weak vector minimization

A version of the Wolfe dual problem is constructed for constained weak minimization of a vector objective function, in finite or infinite dimensions (e.g. continuous programming) The usual convex requirements are weakened to invex. Weak duality is replaced by an inclusion, constructed using the cone defining the weak minimum. Relations with Pareto (or proper Pareto) minima are discussed.

Duality for a Non‐Convex Program in a Real Banach Space

AbstractThis paper works out a direct duality theorem for a mathematical program in Banach space having ratio of a convex and a linear functional as its objective function. For a convex program in real BANACH space, RITTER [1] extended the WOLFE's duality theorem for a convex program in EUCLIDEAN space. It however turns out that WOLFE's direct duality theorem does not hold for a pseudoconvex program in general [2, p. 158]. BECTOR [3] succeeded in proving WOLFE's direct duality theorem for a mathematical program in EUCLIDEAN space in which the pseudo‐convex objective function is the ratio of a convex and a linear function. The present paper aims at extending this result for programs in BANACH space. The approach to establish the direct duality theorem, however, is the same as that of RITTER, i.e. using the weak duality theorem and the Kuhn‐Tucker necessary conditions for optimality.

On Symmetric Duality in Nonlinear Programming

In this study we generalize the formulation of symmetric duality introduced by Dantzig, Eisenberg, and Cottle to include the case where the constraints of the inequality type are defined via closed convex cones and their polars. The new formulation retains the symmetric properties of the original programs. Under suitable convexity/concavity assumptions we generalize the known results about symmetric duality. The case where the function involved is strongly convex/strongly concave is also treated and Karamardian's result in this case is generalized. As a result, we show that every strongly convex function achieves a minimum value over any closed convex cone at a unique point. Some special cases of symmetric programs are then considered, leading to generalizations of Wolfe's duality as well as generalizations of quadratic and linear programming formulations.