A New Primal Algorithm for Semi-Infinite Linear Programming

Abstract

We consider the following problem

$$minimise {c^T}x$$

$$subject to a{(s)^T}x > b(s), for all s in [0,1]$$

, where a and b are analytic functions. The usual method for solving this type of problem can be viewed as based on the simplex method applied to the dual problem. In this paper a new primal method is presented.

First a characterisation of the extreme points of the feasible set (the basic solutions) is given. Then the pivot operation is described. If we define

$$constr(x) = \{ s:a{(s)^T}x = b(s)\}$$

, then constr(x) can be assumed to be finite if x is a basic solution. The pivot operation takes one of two forms: either one of the points in constr(x) is dropped and a new point introduced, or one of the points in constr(x) is perturbed to the left or right.