Computational Experience and the Explanatory Value of Condition Measures for Linear Optimization

The modern theory of condition measures for convex optimization problems was initially developed for convex problems in the following conic format: (CPd) : z∗ := minx{cx | Ax− b ∈ CY , x ∈ CX}, and several aspects of the theory have now been extended to handle non-conic formats as well. In this theory, the (Renegar-) condition measure C(d) for (CPd) has been shown to be connected to bounds on a wide variety of behavioral and computational characteristics of (CPd), from sizes of optimal solutions to the complexity of algorithms for solving (CPd). Herein we test the practical relevance of the condition measure theory, as applied to linear optimization problems that one might typically encounter in practice. Using the NETLIB suite of linear optimization problems as a test bed, we found that 71% of the NETLIB suite problem instances have infinite condition measure. In order to examine condition measures of the problems that are the actual input to a modern IPM solver, we also computed condition measures for the NETLIB suite problems after pre-preprocessing by CPLEX 7.1. Here we found that 19% of the post-processed problem instances in the NETLIB suite have infinite condition measure, and that logC(d) of the postprocessed problems is fairly nicely distributed. Furthermore, among those problem instances with finite condition measure after pre-processing, there is a positive linear relationship between IPM iterations and logC(d) of the post-processed problem instances (significant at the 95% confidence level), and 42% of the variation in IPM iterations among these NETLIB suite problem instances is accounted for by logC(d) of the post-processed problem instances. ∗This research has been partially supported through the Singapore-MIT Alliance. †Industrial and Systems Engineering, University of Southern California, GER-247, Los Angeles, CA 90089-0193, USA, email: [email protected] ‡MIT Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142, USA, email: [email protected]