A Sup-Function Approach to Linear Semi-Infinite Optimization.

In this paper, we consider linear semi-infinite programming problems and the possibility of having no active constraint at a boundary point of the corresponding feasible set. This phenomenon causes serious troubles when one intends to construct the cone of feasible directions at this point. The main feature of our approach consists of exploiting the (sub)differential properties of the sup-function that allows replacing infinitely many constraints with a unique convex constraint. In the case where the set of active constraints is empty, this sup-function can present a quite abnormal behavior. To face this situation, two families of relaxed active constraint sets are introduced in the paper, and the relationship between them is studied in detail. Under the so-called strong Slater condition, we obtain two different formulas for the polar of the cone of feasible directions. They are derived through an extension of Valadier's formula for the subdifferential of a sup-function. Finally, a stability result is given for the most relevant mapping in our context. [ABSTRACT FROM AUTHOR]