Pair Of Dual Problems Research Articles

Optimality conditions for differentiable linearly constrained pseudoconvex programs

AbstractThe aim of this paper is to study optimality conditions for differentiable linearly constrained pseudoconvex programs. The stated results are based on new transversality conditions which can be used instead of complementarity ones. Necessary and sufficient optimality conditions are stated under suitable generalized convexity properties. Moreover, two different pairs of dual problems are proposed and weak and strong duality results proved. Finally, it is shown how transversality conditions can be applied to characterize optimality of convex quadratic problems and to efficiently solve a particular class of Max-Min problems

The Set of Target Vectors in a Semi-Infinite Linear Program with a Duality Gap

We propose a geometric method for the analysis of duality relations in a pair of semi-infinite linear programs (SILPs). The method is based on the use of the conic hull of the coefficients in the constraint system. A relation between the presence of a duality gap and the nonclosedness of the boundary of the conic hull of points in a multidimensional space is established. The geometric approach is used to construct an opposite pair of dual problems and to explore the duality relations for this pair. We construct a nontrivial example of a SILP in which the duality gap occurs for noncollinear target vectors.

Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations

Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations and the conjugacy of primal and dual value functions are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points.

Symmetric dual nonlinear programming and matrix game equivalence

We present some equivalences between some symmetric dualities in nonlinear programming and some matrix games. We study three pairs of dual problems: Wolfe, Mond–Weir and fractional types. For each pair, the paper presents two different zero-sum games whose Nash equilibria correspond to the solutions of the pair of nonlinear symmetric dual problems. And certain conclusions about symmetric dual fractional programming and its special cases are also treated.

A Parametrical Generalized Solution in Linear Vector Optimization

A parametrical representation of a linear problem of Pareto optimization is considered. The notion of a generalized solution is introduced for an appropriate pair of dual problems. The structure of a parametrical generalized solution is investigated. A relation between generalized and efficient solutions of problems is established.

Theorems of the Alternative and Duality

This paper investigates the relations between theorems of the alternative and the minimum norm duality theorem. A typical theorem of the alternative is associated with two systems of linear inequalities and/or equalities, a primal system and a dual one, asserting that either the primal system has a solution, or the dual system has a solution, but never both. On the other hand, the minimum norm duality theorem says that the minimum distance from a given point z to a convex set \(\mathbb{K}\) is equal to the maximum of the distances from z to the hyperplanes separating z and \(\mathbb{K}\). We consider the theorems of Farkas, Gale, Gordan, and Motzkin, as well as new theorems that characterize the optimality conditions of discrete l1-approximation problems and multifacility location problems. It is shown that, with proper choices of \(\mathbb{K}\), each of these theorems can be recast as a pair of dual problems: a primal steepest descent problem that resembles the original primal system, and a dual least–norm problem that resembles the original dual system. The norm that defines the least-norm problem is the dual norm with respect to that which defines the steepest descent problem. Moreover, let y solve the least norm problem and let r denote the corresponding residual vector. If r=0, which means that z ∈ \(\mathbb{K}\), then y solves the dual system. Otherwise, when r≠0 and z ∉ \(\mathbb{K}\), any dual vector of r solves both the steepest descent problem and the primal system. In other words, let x solve the steepest descent problem; then, r and x are aligned. These results hold for any norm on \(\mathbb{R}^n \). If the norm is smooth and strictly convex, then there are explicit rules for retrieving x from r and vice versa.

A duality theorem on a pair of simultaneous functional equations

Given P and Q convex compact sets in R k and R s , respectively, and u a continuous real valued function on P × Q, we consider the following pair of dual problems: Problem I—Minimize ƒ so that ƒ: P × Q → R and ƒ ⩾ Cav p Vex q × max(u, ƒ) . Problem II—Maximize g so that g: P × Q → R and g ⩽ Vex q × Cav p min( u, g). Here Cav p is the operation of concavification of a function with respect to the variable p ϵ P (for each fixed q ϵ Q). Similarly, Vex q is the operation of convexification with respect to q ϵ Q. Maximum and minimum are taken here in the partial ordering of pointwise comparison: ƒ ⩽ g means ƒ(p, q) ⩽ g(p, q) ∀(p, q) ϵ P × Q . It is proved here that both problems have the same solution which is also the unique simultaneous solution of the following pair of functional equations: (i) ƒ = Vex q max(u, ƒ) . (ii) ƒ = Cav p min(u, ƒ) . The problem arises in game theory, but the proof here is purely analytical and makes no use of game-theoretical concepts.

Limit Design in the Absense of a Given Layout: A Finite Element, Zero-One Programming Approach∗

ABSTRACT Limit design in three dimensions is discussed and formulated as a constrained minimax problem in kinematic and geometric variables. A finite element discretization is proposed which, combined with piecewise linearization of the yield surfaces, reduces the minimum weight design to a pair of dual problems in linear mixed zero one programming. The relevant duality theory is shown to be useful for the theoretical frame of the mechanical problem. Various ways of reducing the number of variables and constraints are pointed out, in order to make available algorithms economically applicable to practical situations.

Ein geometrischer beweis des dualitätsprlnzips in der linearen optimierung

Basing on a suitable geometric interpretation for a pair of dual problems the author performs a proof the duality- theoreirl in :ingar programming.Besides of simple notions and auxiliary means being necessary for the proof the separation theorem for convex sets is the only profounded result used in demonstrating the main theorem.