Reducing Concave Programs with Some Linear Constraints

Abstract

The problem of maximizing a concave function over a general convex set subject to linear inequality constraints is reduced to a finite sequence of sub-problems involving linear equality constraints. This reduction can be expected to be computationally useful when there are but a few constraints, or when at most a few constraints are binding at the optimal solution of the original problem, or when prior (though possibly fallible) information is available concerning which constraints are likely to be binding. For quadratic programs the procedure specializes to an improved version of the Theil-van de Panne method. Computational considerations and experience are discussed, and a graphical example is given. The theory and viewpoint developed herein provide the foundation for related reduction procedures that may prove computationally useful even for large problems in the absence of a priori information.