An algorithm for solving large-scale nonlinear ' programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.

Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP -hard combinatorial optimization problems of simple structure such as the maxcut and graph bisection problems. In this work, we try to solve more complicated combinatorial problems such as the quadratic assignment, general graph partitioning and set partitioning problems. A tight SDP relaxation can be obtained by exploiting the geometrical structure of the convex hull of the feasible points of the original combinatorial problem. The analysis of the structure enables us to find the so-called "minimal face" and "gangster operator" of the SDP. This plays a significant role in simplifying the problem and enables us to derive a unified SDP relaxation for the three different problems. We develop an efficient "partial infeasible" primal-dual inter...

We present a relaxation for the set partitioning problem that combines the standard linear programming relaxation with a semidefinite programming relaxation. We include numerical results that illustrate the strength and efficiency of this relaxation. Contents 1 INTRODUCTION 2 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 SDP RELAXATION 3 3 NUMERICAL TESTS 8 4 SDP RELAXATION FOR LARGE SPARSE PROBLEMS 8 4.1 An SDP Relaxation with Block Structure . . . . . . . . . . . . . . . . . . . . . . 8 4.2 An Infeasible Primal-Dual Interior-Point Method . . . . . . . . . . . . . . . . . . 13 4.3 Preliminary Numerical Tests and Future Work . . . . . . . . . . . . . . . . . . . 15 A APPENDIX-Notation 16 This report is available by anonymous ftp at orion.uwaterloo.ca in the directory pub/henry/reports; or over WWW with URL ftp://orion.uwaterloo.ca/pub/henry/reports/ABSTRACTS.html y University of Waterloo, Department of Combinatorics and Optim...