Vector-Supercomputer Experiments with the Primal Affine Linear Programming Scaling Algorithm

Abstract

The vector-supercomputer CRAY series has provided significant speed and significant digits accuracy for solving difficult, large-scale, and ill-conditioned linear programming problems with the linear programming (LP) primal scaling algorithm of Dikin [Soviet Math. Dokl., 8 (1967), pp. 674–675]. Ranging from fully dense Chebyshev approximation-type matrices to highly sparse, relatively small, band-widths matrices generated from continuum mechanics problems, numerical results are presented that indicate good stability and objective function value accuracy. In a significant number of these experiments, a substantial speed-up in CPU time for obtaining good objective function accuracy has been obtained over a very stable implementation of the simplex method, termed LINOP. The companion QR-based scaling implementation, SHP, was applied to these dense problems, where high accuracy levels are required. A conjugate gradient-based implementation, termed HYBY, was applied to a discretized plane strain plasticity problem for ill-conditioned problems having up to 7700 equations in as many as 9000 variables. The FORTRAN code includes an effective stability enhancement over the original affine scaling algorithm. The sparse LP matrix generator for these problems is available from the author (kort@icaen.uiowa.edu) or from Professor E. Christiansen in Odense, Denmark (edc@imada.ou.dk).