Abstract
In this paper we consider the duality gap function g that measures the difference between the optimal values of the primal problem and of the dual problem in linear programming and in linear semi-infinite programming. We analyze its behavior when the data defining these problems may be perturbed, considering seven different scenarios. In particular we find some stability results by proving that, under mild conditions, either the duality gap of the perturbed problems is zero or + ∞ around the given data, or g has an infinite jump at it. We also give conditions guaranteeing that those data providing a finite duality gap are limits of sequences of data providing zero duality gap for sufficiently small perturbations, which is a generic result.
Highlights
Linear optimization consists in the minimization of a linear objective function subject to linear constraints
It is defined as the difference of the optimal value of the primal problem and of the optimal value of its dual problem, and it depends on the data defining these optimization problems
We will analyze the behavior of this duality gap when these data are perturbed; in doing this we will consider seven different scenarios related to the parameters that present admissible perturbations and the ones that remain fixed
Summary
Linear optimization consists in the minimization of a linear objective function subject to linear constraints. In the framework of continuous infinite linear programming, with decision space X being a Banach space, [22] introduces the gap function g (b, c) defined only for those pairs (b, c) providing primal and dual feasible problems (assumption guaranteeing that g (b, c) ∈ R, by the weak duality theorem) and keeping the gradient vectors at fixed. These results are applied to get conditions for the stability of the duality gap function and to give conditions guaranteeing that those data providing a finite duality gap are limits of sequences of data providing zero duality gap for sufficiently small perturbations.
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