Characterization of total ill-posedness in linear semi-infinite optimization

Abstract

This paper deals with the stability of linear semi-infinite programming (LSIP, for short) problems. We characterize those LSIP problems from which we can obtain, under small perturbations in the data, different types of problems, namely, inconsistent, consistent unsolvable, and solvable problems. The problems of this class are highly unstable and, for this reason, we say that they are totally ill-posed. The characterization that we provide here is of geometrical nature, and it depends exclusively on the original data (i.e., on the coefficients of the nominal LSIP problem). Our results cover the case of linear programming problems, and they are mainly obtained via a new formula for the subdifferential mapping of the support function.

Subdifferential Calculus Rules in Convex Analysis: A Unifying Approach Via Pointwise Supremum Functions

“…It has no equivalence in the classical theory of differentiable analysis and constitutes a largely used tool in convex optimization, in theory as well as in practice (see, for instance, [1], [10], and the references therein). In [5] and [8] certain specific techniques relying on the supremum function were applied in the framework of semi-infinite linear optimization.…”

Valadier-like formulas for the supremum function II: The compactly indexed case

“…Thus, taking into account (15), we are in position to apply (8) to the lsc proper functions g t , and we get…”

“…We also refer to [14], and references therein, for the nonconvex case. The supremum function plays a crucial role in many fields, including semi-infinite optimization ( [8], [20]).…”