On Duality in Semi-Infinite Programming and Existence Theorems for Linear Inequalities

Abstract

Linear semi-infinite programming deals with the optimization of linear functionals on finite-dimensional spaces under infinitely many linear constraints. For such kind of programs, a positive duality gap can occur between them and their corresponding dual problems, which are linear programs posed on infinite-dimensional spaces. This paper exploits some recent existence theorems for systems of linear inequalities in order to obtain a complete classification of linear semi-infinite programming problems from the point of view of the duality gap and the viability of the discretization numerical approach. The elimination of the duality gap is also discussed. ᮊ 1999 Academic Press Ž .Ž .Ý Ý t t t t tgT t gT

Marco A. López, a Pioneer of Continuous Optimization in Spain

“…-In [32], the behavior of consistent semi-infinite linear inequality systems was analyzed with respect to many features of the solution set such as boundedness, dimension, description of its boundary, redundant inequalities, minimality, finite reduction, etc. -Duality in linear SIP, and its relationship with discretizability and reducibility, was deeply studied in [34], where it was proved that the Farkas-Minkowski property is crucial for the so-called uniform duality. -In [29], the powerful tool of the Farkas-Minkowski constraint qualification is exploited in the framework of convex SIP, exploring its relationship with stronger qualification conditions.…”