Classifying convex extremum problems over linear topologies having separation properties

Abstract

It is shown that any convex or concave extremum problem possesses a subsidiary extremum problem which has certain homogeneous properties. Analogous to the given problem, the “homogenized” extremum problem seeks the minimum of a convex function or the maximum of a concave function over a convex domain. By using homogenized extremum problems, new relationships are developed between any given convex extremum problem (P) and a concave extremum problem (P ∗) (also having a convex domain), called the “dual” problem of (P). This is achieved by combining all possibilities in tabular form of (1) the values of the extremum functions and (2) the nature of the convex domains including perturbations of all problems (P), (P ∗), and each of their respective homogenized extremum problems. This detailed and refined classification is contrasted to the relationships obtainable by combining only the possible values of the extremum functions of the problems (P) and (P ∗) and the possible limiting values of these functions stemming from perturbations of the convex constraint domains of (P) and (P ∗), respectively. The extremum problems in this paper and classification results are set forth in real topologically paired vector spaces having the Hahn-Banach separation property.