Projections of Convex Programs with Unattained Infima

Abstract

Convex programs with closed objective function and closed feasible region are classified as degenerate if the objective function and the feasible region have a common direction of recession. For each degenerate program, a reduced form is defined by projecting the feasible region and the objective function epigraph on the orthogonal complement of the recession directions. A finite sequence of such reductions yields a nondegenerate problem for which the infimum is attained on a bounded set. Under a very mild condition the infimum of the reduced problem is equal to that of the original problem. It is shown that the objective and constraint functions of the “projected” problem may be obtained by calculating limits of the objective and constraint functions in the directions of recession. These results generalize the concept of degeneracy and reduction to canonical form which was originally developed for posynomial geometric programming.