Different Types of Ill-Posedness in Semi-Infinite Linear Optimization
MARCO A. LOPEZ CERDA
Universidad de Alicante

Abstract: We consider the parameter space of all the linear optimization
problems, in the Euclidean space, whose inequality constraint systems are
indexed by an arbitrary (possibly infinite) set T. No particular property is
assumed for the dependence of the problem's data on the index set, and the
parameter space of these problems is endowed with the uniform convergence
topology by means of a real-extended distance. A problem is said to be
ill-posed (according, for instance to Renegar) when rbitrarily small
perturbations may yield different types of problems (e.g.,
feasible/infeasible). In this talk we characterize the ill-posedness w.r.t.
primal and/or dual consistency (feasibility), boundedness (finiteness of the
optimal value), and solvablity (existence of optimal solutions). Based on
these characterizations, we provide some fomulas for the corresponding
distances to ill-posedness. These expressions are given in terms of the
distances, in the Euclidean space, of the origin to the boundaries of
certain convex sets. Applications to Lipschitzian behavior of the feasible
set and the optimal value are also given.

This talk is based on some joint work with M.J. Canovas, J. Parra and F.J. Toledo.