Global minimization by reducing the duality gap

We derive a general principle demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated. The principle can be implemented in a Branch and Bound algorithm which computes an approximate global solution and a corresponding lower bound on the global optimal value. The algorithm involves decomposition and a nonsmooth local search. Numerical results for applying the algorithm to the pooling problem in oil refineries are given.Research supported by Shell Laboratorium, Amsterdam, and GIF—The German—Israel Foundation for Scientific Research and Development.     

Diewert-Crouzeix conjugation for general quasiconvex duality and applications

A complicated factor in quasiconvex duality is the appearance of extra parameters. In order to avoid these extra parameters, one often has to restrict the class of quasiconvex functions. In this paper, by using the Diewert-Crouzeix conjugation, we present a duality without an extra parameter for general quasiconvex minimization problem. As an application, we prove a decentralization by prices for the Von Neumann equilibrium problem.     

A generalization of Fenchel duality theory

This paper deals with an extention of Fenchel duality theory to fractional extremum problems, i.e., problems having a fractional objective function. The main result is obtained by regarding the classic Fenchel theorem as a decomposition property for the extremum of a sum of functions into a sum of extrema of functions, and then by extending it to the case where the addition is replaced by the quotient. This leads to a generalization of the classic concept of conjugate function. Several remarks are made about the conceivable further generalizations to other kinds of decomposition.     

Injective and projective objects in the category of locally compact modules over the ring of integral values of a global field

In the paper injective and projective objects in the category of locally compact modules over the ring of integral values of a global field are described together with the objects of this category possessing injective and projective resolvents. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 118–123, July, 1997. Translated by A. I. Shtern     

关于带一般混合边界条件的半线性反应扩散系统解的整体存在性问题

本文利用Ｇｒｏｎｗａｌｌ不等式及对偶技巧，研究了带一般混合边界条件的半线性反应扩散系统解的存在性问题，给出了一些控制条件，得到解的整体存在性，改进并推广了以前的结果。     

On the Existence and Convergence of the Central Path for Convex Programming and Some Duality Results

This paper gives several equivalent conditions which guarantee the existence of the weighted central paths for a given convex programming problem satisfying some mild conditions. When the objective and constraint functions of the problem are analytic, we also characterize the limiting behavior of these paths as they approach the set of optimal solutions. A duality relationship between a certain pair of logarithmic barrier problems is also discussed.     

Wave-particle duality: The mystery keeps unfolding

The current status of wave particle duality using single photon sources is briefly reviewed On sabbatical leave from S N Bose National Centre for Basic Sciences, Calcutta     

Duality in infinite dimensional linear programming

We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints. Examples include production planning and equipment replacement over an infinite horizon. We form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero. That is, we show, under this transversality condition, that optimal solutions are attained in both primal and dual problems and their optimal values are equal. The transversality condition, and hence strong duality, is established for an infinite horizon production planning problem.This material is based on work supported by the National Science Foundation under Grant No. ECS-8700836.     

On exact and approximate boundary controllabilities for the heat equation: A numerical approach

The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate boundary controllability problems for the heat equation. Using convex duality, we reduce the solution of the boundary control problems to the solution of identification problems for the initial data of an adjoint heat equation. To solve these identification problems, we use a combination of finite difference methods for the time discretization, finite element methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods discussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.