Duality theorems for marginal problems

Summary Given topological spaces X ₁, ..., X _n with product space X, probability measures _i on X _i together with a real function h on X define a marginal problem as well as a dual problem. Using an extended version of Choquet's theorem on capacities, an analogue of the classical duality theorem of linear programming is established, imposing only weak conditions on the topology of the spaces X _i and the measurability resp. boundedness of the function h. Applications concern, among others, measures with given support, stochastic order and general marginal problems.     

Duality for nonlinear multiple-criteria optimization problems

In this paper, we develop a duality theory for nonlinear multiple-criteria optimization problems. The theory associates to efficient points a matrix, rather than a vector, of dual variables. We introduce a saddle-point dual problem, study stability concepts and Kuhn-Tucker conditions, and provide an economic interpretation of the dual matrix. The results are compared to the classical approach of deriving duality, by applying nonlinear programming duality theory to a problem obtained by conveniently weighting the criteria. Possible directions for future research are discussed.This work was performed under Grant No. MCS-77-24654, National Science Foundation.The author is grateful to Professors S. C. Graves and T. L. Magnanti, and two anonymous referees for helpful comments on an earlier version of this paper.     

Duality for nonsmooth semi-infinite programming problems

This paper deals with nonsmooth semi-infinite programming problem which in recent years has become an important field of active research in mathematical programming. A semi-infinite programming problem is characterized by an infinite number of inequality constraints. We formulate Wolfe as well as Mond-Weir type duals for the nonsmooth semi-infinite programming problem and establish weak, strong and strict converse duality theorems relating the problem and the dual problems. To the best of our knowledge such results have not been done till now.     

Duality results for extended multifacility location problems

Abstract

Duality statements are presented for multifacility location problems as suggested by Drezner Hiu 1991, where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. We develop corresponding dual problems for the cases with and without set-up costs and present associated optimality conditions. In the concluding part of this note we use these optimality conditions for a geometrical characterization of the set of optimal solutions and consider for an illustration corresponding examples.     

Duality for constrained robust sum optimization problems

Mathematical Programming - Given an infinite family of extended real-valued functions $$f_{i}$$ , $$i\in I,$$ and a family $${\mathcal {H}}$$ of nonempty finite subsets of I,  the...     

Duality for optimization problems in Banach algebras

In this paper we consider Mond?CWeir type and Wolfe type duals for a general nonsmooth optimization problem in Banach algebras, and establish some duality results in the presence of a new class of functions, which is a generalization of the class of smooth KT-(p, r)-invex functions.     

Duality for variational problems with b-vex functions

Concept of B-vexity has been extended for continuous functions and applied to establish sufficiency and duality for variational problems. Both fixed end points and free boundary value problems are considered     

Duality for extended infinite monotropic optimization problems

Mathematical Programming - We establish necessary and sufficient conditions for strong duality of extended monotropic optimization problems with possibly infinite sum of separable functions. The...     

Duality theorems for convex problems with time delay

In this paper, we examine a class of convex problems of Bolza type, involving a time delay in the state. It encompasses a variety of time-delay problems arising in the calculus of variations and optimal control. A duality analysis is carried out which, among other things, leads to a characterization of minimizers in terms of the Euler-Lagrange inclusion. The results obtained improve in significant respects on what is achievable by techniques previously employed, based on elimination of the time delay by introduction of an infinite-dimensional state space or on the method of steps.     

Duality for location problems with unbounded unit balls

Given an optimization problem with a composite of a convex and componentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sufficient optimality conditions are derived using strong duality. Furthermore, as special case of this problem, we consider a location problem, where the “distances” are measured by gauges of closed convex sets. We prove that the geometric characterization of the set of optimal solutions for this location problem given by Hinojosa and Puerto in a recently published paper can be obtained via the presented dual problem. Finally, the Weber and the minmax location problems with gauges are given as applications.     

Duality in land rental problems

Land rental problems describe situations where one tenant demands land from several lessors. The way lessors rent their land can be seen as equivalent to a bankruptcy problem. We extend the idea of self-duality in bankruptcy problems to land rental problems. We provide a complete characterization of the family of rules that satisfy self-duality. Moreover, self-duality is enough to assure the proportional land share among lessors. Adding other reasonable properties, we pick up a single rule.     

Duality for implicit variational problems and numerical applications

The present paper deals with a dual characterization of the solutions of implicit variational problems. Some general results relating the solutions of the dual problem with those of the primal one are applied to variational and quasi-variational inequalities, Nash equilibria, saddle points and fixed points.

Finally a dual method for the numerical solutions of some quasi-variational inequalities is developed.     

Duality for multiobjective control problems with generalized invexity

Wolfe and Mond-Weir type duals for multiobjective control problems are formulated. Under pseudo-invexity/quasi-invexity assumptions on the functions involved, weak and strong duality theorems are proved to relate efficient solutions of the primal and dual problems.     

Duality theorems for a class of nonconvex programming problems

The object of this paper is to prove duality theorems for quasiconvex programming problems. The principal tool used is the transformation introduced by Manas for reducing a nonconvex programming problem to a convex programming problem. Duality in the case of linear, quadratic, and linear-fractional programming is a particular case of this general case.The authors are grateful to the referees for their kind suggestions.     

Duality theorems for convex problems of Lagrange with time-delay

Convex problems of Lagrange with time-delay in the state areexamined. The state x(t) is constrained in a convex set X(t)for each t. Optimal trajectories satisfy certain subdifferentialconditions which involve ‘multipliers’ that solvesome dual, convex problem.