The fundamental continuity theory of optimization on a compact space. I

Abstract

: Optimizers on a compact feasible region are abstractly specified and, as set-valued mappings, are studied for sufficient conditions yielding them (as well as certain associated maps and certain restrictions of all these) continuous, using function space methods.In particular, the study concerns the continuity of the set of optimal solutions as a function of the three arguments: (i) objective function used, (ii) an incentive (or "penalty/reward") function imposed, and (iii) an abstract "parameter".An interpretation of the mathematical apparatus is suggested and a brief gametheoretic illustration given. The set of all con-S tinuous mappings of S into T will be denoted by T . Given B C S and W c T , the set { f e T^I f (B) C W} will be denoted by (B , W) . (P (S) will denote the power set of S and t(S) will denote the set of nonempty closed subsets of S.Given B c: S ,^B) will denote the set {A e C(S) I AC B}. Throughout, R will be a non-empty set equipped with a total order^and a (Hausdorff) topology which is at least as fine as the order topology.Aim : Given non-empty topological spaces X = X^X" and Y, of which X is compact with X (compact and) Hausdorff, in this paper we seek sufficient conditions for the continuity of (a) the optimizers ex':