Stability in the Discretization of a Parametric Semi-Infinite Convex Inequality System

• Authors: María J. Cánovas;Marco A. López;Juan Parra
• Venue: Mathematics of Operations Research
• Year: 2002

In this paper, we consider a parametric family of convex inequality systems in the Euclidean space, with an arbitrary infinite index set,T, and convex constraints depending continuously on a parameter ranging in a separable metric space. No structure is assumed forT, and so the dependence of the constraints on the index has no particular property. In this context, the possibility of approaching the nominal system by means of sequences of finite subsystems associated to proximal parameters is analyzed. This possibility, of combining both approximation and discretization techniques, is formalized in terms of the lower semicontinuity of the feasible set mapping depending on a double parameter: The original one and the finite subset of indices (grid) itself. The paper characterizes this property in terms of the lower semicontiuity of the feasible set mapping depending only on the original parameter (and considering, then, all the constraints). Since in any approximation process we consider, as a last resort, a countable amount of constraints, the first step in this work consists of justifying the possibility of considering, without loss of generality, N(set of all natural numbers) as the proper index set. Moreover, in order to be able to consider any subset of indices as a new parameter, a suitable metric is introduced in the set of all the nonempty subsets of N, entailing desirable properties in relation to approximation strategies.