Interior-point Linear Programming Solvers

We present an overview of available software for solving linear programming problems using interior-point methods. Some of the codes discussed include primal and dual simplex solvers as well, but we focus the discussion on the implementation of the interior-point solver. For each solver, we present types of problems solved, available distribution modes, input formats and modeling languages, as well as algorithmic details, including problem formulation, use of higher corrections, presolve techniques, ordering heuristics for symbolic Cholesky factorization, and the specifics of numerical factorization. We present an overview of available software for solving linear programming problems using interior-point methods. We consider both open-source and proprietary commercial codes, including BPMPD ([37], [36]), CLP [16], FortMP [15], GIPALS32 [45], GLPK [34], HOPDM ([24], [28]), IBM ILOG CPLEX [30], LINDO [31], LIPSOL [49], LOQO [48], Microsoft Solver Foundation [38], MOSEK [2], PCx [13], SAS/OR [44], and Xpress Optimizer [43]. Some of the codes discussed include primal and dual simplex solvers as well, but we focus the discussion on the implementation of the interior-point solver. For each solver, we present types of problems solved, available distribution modes, input formats and modeling languages, as well as algorithmic details, including problem formulation, use of higher corrections, presolve techniques, ordering heuristics for symbolic Cholesky factorization, and the specifics of numerical factorization. Many of the solvers allow user-control over some algorithmic details such as the type of presolve techniques applied to the problem and the choice of ordering heuristic. As the following discussion will show, the solvers tend to all have similar characteristics, and performance differences are generally due to subtle changes in the presolve phase, the implementation of the linear algebra routines, and other special considerations, such as scaling, to promote numerical stability. We start with a short discussion of input formats and modeling languages available for LPs. Then, we will present details of a wide-range of interior-point codes. 1. Input formats and Modeling Languages for Linear Programming For a standard LP of the form (1) maximize c x subject to Ax = b x ≥ 0, it suffices to pass the matrix A and the vectors b and c to the solver for a fulldescription of the problem. Of course, the general form of an LP can incorporate inequality constraints as well as bounds on variables, but such deviations from (1) Date: October 3, 2009.