Reduction and Discrete Approximation in Linear Semi-Infinite Programming

Abstract

A problem (P):Inf {c'x∣x∈F(σ)}, where F{σ) is the solutions set of a linear semi-infinite system σ, is called reducible if there exists a finite subproblem with the same value. v(P). On the other hand we have that (P) is discreditable if for some sequence of finite subproblems of (P). Here we establish a classification theorem which shows out that (P) is always discreditable, unless cbelongs to the relative boundary of the moment cone associated with σ. In the last case we prove that, at least (P) can be approximated by means of a sequence of perturbed finite subproblems. The first part of the paper provides, then. a theory which extends and embeds earlier results. In the second part we prove that optimal solutions of (P) or, alternatively, descent directions, can be obtained by discretization, except in the uncertain case. The third part is devoted to the reducibility of (P), characterizing those systems for which (P) is reducible whichever the objective function is. At last we discuss the role of countable sub-problems.

Keywords:

• Semi-Infinite Programming approximation
• discretization
• finite reduction
• Haar's duality

AMS 1980 subject Classification:

• Primary:90 C 05
• Secondary:52 A 40
• Secondary:15 A 39

¹This research has been partially supported by the Foundation Garcia-Rogel(C.A.A.M).

¹This research has been partially supported by the Foundation Garcia-Rogel(C.A.A.M).

Notes

¹This research has been partially supported by the Foundation Garcia-Rogel(C.A.A.M).