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.The author is grateful to Professor W. Stadler and the referees for many valuable remarks and suggestions, which have enabled him to improve considerably the paper.     

Sufficient optimality conditions and duality for a quasiconvex programming problem

Under differentiability assumptions, Fritz John Sufficient optimality conditions are proved for a nonlinear programming problem in which the objective function is assumed to be quasiconvex and the constraint functions are assumed to quasiconcave/strictly pseudoconcave. Duality theorems are proved for Mond-Weir type duality under the above generalized convexity assumptions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are thankful to Professor B. Mond for suggestions that improved the original draft of the paper.     

On necessary and sufficient conditions in vector optimization

It is shown that some general multiplier rules are necessary conditions for vector optimization in infinite-dimensional spaces. Under additional convexity assumptions, these conditions are sufficient. As an application, the Pontryagin maximum principle for cooperative differential games is examined.The authors are grateful to Professor W. Stadler and the referees of the previous edition of this paper for their valuable remarks and suggestions, which have been very helpful in the preparation of this paper.     

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.     

On necessary optimality conditions in vector optimization problems

Necessary conditions of the multiplier rule type for vector optimization problems in Banach spaces are proved by using separation theorems and Ljusternik's theorem. The Pontryagin maximum principle for multiobjective control problems with state constraints is derived from these general conditions. The paper extends to vector optimization results established in the scalar case by Ioffe and Tihomirov.     

Sufficient conditions for the existence of multipliers and Lagrangian duality in abstract optimization problems

We consider the following optimization problem: in an abstract setX, find and elementx that minimizes a real functionf subject to the constraintsg(x)0 andh(x)=0, whereg andh are functions fromX into normed vector spaces. Assumptions concerning an overall convex structure for the problem in the image space, the existence of interior points in certain sets, and the normality of the constraints are formulated. A theorem of the alternative is proved for systems of equalities and inequalities, and an intrinsic multiplier rule and a Lagrangian saddle-point theorem (strong duality theorem) are obtained as consequences.     

Sufficient conditions for optimality of threat strategies in a differential game

The problem of defining threat strategies in nonzero-sum games is considered, and a definition of optimal threat strategies is proposed in the static case. This definition is then extended to differential games, and sufficient conditions for optimality of threat strategies are derived. These are then applied to a simple example. The definition proposed here is then compared with the definition of threat strategies given by Nash.     

Optimality conditions for weak efficiency to vector optimization problems with composed convex functions

We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.      

Necessary conditions for optimality for control problems with time delays appearing in both state and control variables

An integral maximum principle is developed for a class of nonlinear systems containing time delays in state and control variables. Its proof is based on the theory of quasiconvex families of functions, originally developed by Gamkrelidze and extended by Banks. This result is used to obtain a pointwise principle of the Pontryagin type.The authors wish to acknowledge Professor J. M. Blatt for suggesting this problem. Further, they also wish to acknowledge the referee of the paper for bringing to their attention the problems discussed in Section 6.     

Necessary and sufficient conditions in constrained optimization

Additional conditions are attached to the Kuhn-Tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization, under appropriate constraint qualifications. Necessary and sufficient conditions are also given for optimality of the dual problem. Duality and converse duality are treated accordingly.     

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.     

Graphical analysis of duality and the Kuhn-Tucker conditions in linear programming

Careful inspection of the geometry of the primal linear programming problem reveals the Kuhn-Tucker conditions as well as the dual. Many of the well-known special cases in duality are also seen from the geometry, as well as the complementary slackness conditions and shadow prices. The latter at demonstrated to differ from the dual variables in situations involving primal degeneracy. Virtually all the special relationships between linear programming and duality theory can be seen from the geometry of the primal and an elementary application of vector analysis.     

Optimality conditions and duality in subdifferentiable multiobjective fractional programming

Fritz John and Kuhn-Tucker necessary and sufficient conditions for a Pareto optimum of a subdifferentiable multiobjective fractional programming problem are derived without recourse to an equivalent convex program or parametric transformation. A dual problem is introduced and, under convexity assumptions, duality theorems are proved. Furthermore, a Lagrange multiplier theorem is established, a vector-valued ratio-type Lagrangian is introduced, and vector-valued saddle-point results are presented.The authors are thankful to the referees and Professor P. L. Yu for their many useful comments and suggestions which have improved the presentation of the paper.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are also thankful to the Dean's Office, Faculty of Management, University of Manitoba, for the financial support provided for the third author's visit to the Faculty.     

Saddle points and gap functions for vector equilibrium problems via conjugate duality in vector optimization

In this paper, two conjugate dual problems based on weak efficiency to a constrained vector optimization problem are introduced. Some inclusion relations between the dual objective mappings and the properties of the Lagrangian maps and their saddle points for primal problem are discussed. Gap functions for a vector equilibrium problem are established by using the weak and strong duality.     

First order optimality conditions in vector optimization involving stable functions

We study a nonsmooth vector optimization problem with an arbitrary feasible set or a feasible set defined by a generalized inequality constraint and an equality constraint. We assume that the involved functions are nondifferentiable. First, we provide some calculus rules for the contingent derivative in which the stability (a local Lipschitz property at a point) of the functions plays a crucial role. Second, another calculus rules are established for steady functions. Third, necessary optimality conditions are stated using tangent cones to the feasible set and the contingent derivative of the objective function. Finally, some necessary and sufficient conditions are presented through Lagrange multiplier rules.     

Lagrange duality,stability and subdifferentials in vector optimization

Abstract

Lagrange duality theorems for vector and set optimization problems which are based on a consequent usage of infimum and supremum (in the sense of greatest lower and least upper bounds with respect to a partial ordering) have been recently proven. In this note, we provide an alternative proof of strong duality for such problems via suitable stability and subdifferential notions. In contrast to most of the related results in the literature, the space of dual variables is the same as in the scalar case, i.e. a dual variable is a vector rather than an operator. We point out that duality with operators is an easy consequence of duality with vectors as dual variables.     

Sufficient conditions for Stackelberg and Nash strategies with memory

Sufficiency conditions for Stackelberg strategies for a class of deterministic differential games are derived when the players have recall of the previous trajectory. Sufficient conditions for Nash strategies when the players have recall of the trajectory are also derived. The state equation is linear, and the cost functional is quadratic. The admissible strategies are restricted to be affine in the information available.This work was supported in part by the Joint Services Electronics Program under Contract No. N00014-79-C-0424, in part by the National Science Foundation under Grant No. ECS-79-19396, and in part by Department of Energy under Contract No. EX-76-C-01-2088.     

Gap functions for generalized vector equilibrium problems via conjugate duality and applications

In this article, gap functions for a generalized vector equilibrium problem (GVEP) with explicit constraints are investigated. Under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are investigated by considering the different perturbations to GVEP. Then, gap functions for GVEP are established by using the weak and strong duality results. As application, the proposed approach is applied to construct gap functions for a vector optimization problem and a generalized vector variational inequality problem.     

Optimality and duality for a nonsmooth multiobjective optimization involving generalized type I functions

A nonsmooth multiobjective optimization problem involving generalized (F, α, ρ, d)-type I function is considered. Karush–Kuhn–Tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be an efficient or properly efficient solution. Duality results are obtained for mixed type dual under the aforesaid assumptions.