On the Convergence of Inexact Predictor-Corrector Methods for Linear Programming

Abstract

Interior point methods (IPMs) are a common approach for solving linear programs (LPs) with strong theoretical guarantees and solid empirical performance. The time complexity of these methods is dominated by the cost of solving a linear system of equations at each iteration. In common applications of linear programming, particularly in machine learning and scientific computing, the size of this linear system can become prohibitively large, requiring the use of iterative solvers, which provide an approximate solution to the linear system. However, approximately solving the linear system at each iteration of an IPM invalidates the theoretical guarantees of common IPM analyses. To remedy this, we theoretically and empirically analyze (slightly modified) predictor-corrector IPMs when using approximate linear solvers: our approach guarantees that, when certain conditions are satisfied, the number of IPM iterations does not increase and that the final solution remains feasible. We also provide practical instantiations of approximate linear solvers that satisfy these conditions for special classes of constraint matrices using randomized linear algebra.

Proximal Stabilized Interior Point Methods and Low-Frequency-Update Preconditioning Techniques

“…The inexact PPM has been originally analysed in [28] but we follow here the developments of [20]. We consider an approximate version of the PPM scheme in (8) where (x k+1 , y k+1 ) satisfies the criterion (A 1 ) in [20], i.e.,…”

“…Theorem 2.1 summarizes the results we are going to use in this work (the statements are specialized for our case): Theorem 2.1 1. The sequence {(x k , y k )} k∈N generated by the recursion in (8) and using as inexactness criterion (a relaxed version of (10))…”

“…As previously mentioned, we factorize the Schur complement in (25) using Matlab ® 's ldl routine and, in our experiments, this factorization is recomputed if in the current PS-IPM step, GMRES has performed more than the 51% of the maximum allowed iterations in the solution of at least one of the two predictor-corrector systems. The stopping (absolute) tolerance for GMRES is set as min{10 −1 , 0.8μ} where μ is the current duality gap (the interested reader can see [8] for a recent analysis of inexact predictor-corrector schemes).…”