ABSTRACT
In all the preceding chapters we considered the convex programming problem (CP ) with the feasible set C of the form (3.1), that is,
C = {x ∈ Rn : gi(x) ≤ 0, i = 1, 2, . . . ,m},
where gi : Rn → R, i = 1, 2, . . . ,m, are convex functions. Observe that the problem involved only a finite number of constraints. Now in situations where the number of constraints involved is infinite, the problem extends to the class of semi-infinite programming problems. Such problems come into existence in many physical and social sciences models where it is necessary to consider the constraints on the state or the control of the system during a period of time. For examples from real-life scenarios where semi-infinite programming problem are involved, readers may refer to Hettich and Kortanek [57] and references therein. We consider the following convex semi-infinite programming problem,
inf f(x) subject to g(x, i) ≤ 0, i ∈ I (SIP ) where f, g(., i) : Rn → R, i ∈ I are convex functions with infinite index set I ⊂ Rm. The term “semi-infinite programming” is derived from the fact that the decision variable x is finite while the index set I is infinite. But before moving on with the derivation of KKT optimality conditions for (SIP ), we present some notations that will be used in subsequent sections.
