Iterative Refinement for Linear Programming

Abstract

We describe an iterative refinement procedure for computing extended precision or exact solutions to linear programming problems (LPs). Arbitrarily precise solutions can be computed by solving a sequence of closely related LPs with limited precision arithmetic. The LPs solved share the same constraint matrix as the original problem instance and are transformed only by modification of the objective function, righthand side, and variable bounds. Exact computation is used to compute and store the exact representation of the transformed problems, while numeric computation is used for solving LPs. At all steps of the algorithm the LP bases encountered in the transformed problems correspond directly to LP bases in the original problem description. We show that this algorithm is effective in practice for computing extended precision solutions and that it leads to a direct improvement of the best known methods for solving LPs exactly over the rational numbers. Our implementation is publically available as an extension of the academic LP solver SoPlex.

Reliable and efficient solution of genome-scale models of Metabolism and macromolecular Expression

“…19 is devoted to multiscale metabolic networks, showing significant improvement relative to CPLEX5. Our work is complementary and confirms the value of enhancing the simplex solver in refs 17, 18, 19, 20 to employ quadruple-precision computation, as we have done here.…”

“…Exact solvers employ rational arithmetic, and have been applied to important problems1314151718192038. Quad precision and variable-precision floating-point have also been mentioned1338.…”

“…Here we name the approach DQQ and report experiments with an analogous but cheaper DRR procedure based on conventional iterative refinement of all linear equations arising in the simplex method (see Methods section and Supplementary Information). We also became aware of the work of Gleixner et al 17181920. and their thorough and successful implementation of iterative refinement in SoPlex80 bit.…”

“…However, we learned that DRR may lose ground during periodic refactorizations of the simplex basis matrix B , if the current B is nearly singular and “basis repair” becomes necessary. Our DQQ and DRR experience points to the need for an optional Quad version of the basic SoPlex solver to ensure maximum reliability of the refinement approach in refs 17, 18, 19, 20. Meanwhile, DQQ will be effective on a wide range of problems as long as step D finishes naturally or is limited to a reasonable number of iterations before steps Q1 and Q2 take over.…”

Linear programming using limited-precision oracles

“…Algorithm 1 states the basic iterative refinement procedure introduced in [16]. For clarity of presentation, in contrast to [16], Algorithm 1 uses equal primal and dual scaling factors and tracks the maximum violation of primal feasibility, dual feasibility, and complementary slackness in the single parameter δ k . The basic convergence result, restated here as Lemma 3, carries over from [16].…”