Implementing proximal point methods for linear programming

We describe the application of proximal point methods to the linear programming problem. Two basic methods are discussed. The first, which has been investigated by Mangasarian and others, is essentially the well-known method of multipliers. This approach gives rise at each iteration to a weakly convex quadratic program which may be solved inexactly using a point-SOR technique. The second approach is based on the proximal method of multipliers, originally proposed by Rockafellar, for which the quadratic program at each iteration is strongly convex. A number of techniques are used to solve this subproblem, the most promising of which appears to be a two-metric gradient-projection approach. Convergence results are given, and some numerical experience is reported.This research was supported by National Science Foundation, Grant Nos. ASC-87-14009 and DMS-86-19903, and by Air Force Office of Scientific Research, Grant No. AFOSR-ISSA-87-0092. Part of this work was performed at Argonne National Laboratory. Computation was partly performed at the Pittsburgh Supercomputer Center under NSF Grant No. DMS-88-0033P and at the Argonne Advanced Computing Research Facility, whose support is gratefully acknowledged. The author thanks Olvi Mangasarian and Renato DeLeone for helpful discussions, and Jorge Moré for his valuable advice on the algorithms discussed in Section 3. The contributions of a referee, who provided helpful comments on an earlier version of this paper, are also acknowledged.