Characterizing Set Containments Involving Infinite Convex Constraints and Reverse-Convex Constraints

Dual characterizations of the containment of a closed convex set, defined by infinite convex constraints, in an arbitrary polyhedral set, in a reverse-convex set, defined by convex constraints, and in another convex set, defined by finite convex constraints, are given. A special case of these dual characterizations has played a key role in generating knowledge-based support vector machine classifiers which are powerful tools in data classification and mining. The conditions in these dual characterizations reduce to simple nonasymptotic conditions under Slater's constraint qualification.