Approximation of linear programs by Bregman's DF projections

Abstract

The motivation of this paper is twofold: to contribute to the theory of Bregman's D F projections and to point out its link to finding ϵ-optimal solutions to linear programming problems. A symmetric primal–dual pair of the D F projection problem is presented, we call it Young programming problem, because the usual inequality that relates the primal and dual objective functions reduces to the well-known Young inequality. Some basic properties of Young programs are derived and an associated linear program is defined in a natural way. It will be shown that an ϵ-optimal solution of the associated linear program is obtained by solving an ϵ-parametric Young program and the optimal solutions of ϵ-parametric Young programs converge to an optimal solution of the associated linear program when ϵ approaches 0. It may also be interesting that the explicit formulation of the dual program enables us to give a dual interpretation of row-action methods, a powerful tool for solving D F projection problems in general, and allows us to find new functions satisfying sufficient conditions for the convergence of row-action methods.