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

The modern theory of condition numbers for convex optimization problems was initially developed for convex problems in the following conic format: (CPd) : z∗ := min x {ctx | Ax− b ∈ CY , x ∈ CX} . The condition number C(d) for (CPd) has been shown in theory to be connected to a wide variety of behavioral and computational characteristics of (CPd), from sizes of optimal solutions to the complexity of algorithms for solving (CPd). The goal of this paper is to develop some computational experience and test the practical relevance of condition numbers for linear optimization on problem instances that one might encounter in practice. We used the NETLIB suite of linear optimization problems as a test bed for condition number computation and analysis. Our computational results indicate that 72% of the NETLIB suite problem instances are ill-conditioned. However, after pre-processing heuristics are applied, only 19% of the post-processed problem instances are ill-conditioned, and logC(d) of the finitely-conditioned post-processed problems is fairly nicely distributed. We also show that the number of IPM iterations needed to solve the problems in the NETLIB suite varies roughly linearly (and monotonically) with logC(d) of the post-processed problem instances. Empirical evidence yields a positive linear relationship between IPM iterations and logC(d) for the post-processed problem instances, significant at the 95% confidence level. Furthermore, 42% of the variation in IPM iterations among the NETLIB suite problem instances is accounted for by logC(d) of the problem instances after pre-processing. ∗MIT Operations Research Center, 77 Massachusetts Avenue, Bldg. E40-149, Cambridge, MA 02139, USA, email: [email protected] †MIT Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142, USA, email: [email protected]