reducedLP swMATH ID: 4821 Software Authors: André L. Tits, P.-A. Absil, William P. Woessner Description: Constraint reduction for linear programs with many inequality constraints Consider solving a linear program in standard form where the constraint matrix A is m×n, with n≫m≫1. Such problems arise, for example, as the result of finely discretizing a semi-infinite program. The cost per iteration of typical primal-dual interior-point methods on such problems is O(m 2 n). We propose to reduce that cost by replacing the normal equation matrix, AD 2 A T , where D is a diagonal matrix, with a ”reduced” version (of same dimension), A Q D Q 2 A Q T , where Q is an index set including the indices of M most nearly active (or most violated) dual constraints at the current iterate, with M≥m a prescribed integer. This can result in a speedup of close to n/|Q| at each iteration. Promising numerical results are reported for constraint-reduced versions of a dual-feasible affine-scaling algorithm and of Mehrotra’s predictor-corrector method [S. Mehrotra, SIAM J. Optim. 2, No. 4, 575–601 (1992; Zbl 0773.90047)]. In particular, while it could be expected that neglecting a large portion of the constraints, especially at early iterations, may result in a significant deterioration of the search direction, it appears that the total number of iterations typically remains essentially constant as the size of the reduced constraint set is decreased down to some threshold. In some cases this threshold is a small fraction of the total set. In the case of the affine-scaling algorithm, global convergence and local quadratic convergence are proved. Homepage: http://www.inma.ucl.ac.be/~absil/Publi/reducedLP.htm Related Software: rMPC; SDPT3; NewtonKKTqp; LIPSOL; CVX; SeDuMi; Algorithm 566; minpack; STRSCNE; SDPA; SUTIL; PRMLT; LINPACK; Cg; GPGPU; MINOS Cited in: 12 Documents Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Constraint reduction for linear programs with many inequality constraints. Zbl 1112.90049Tits, André L.; Absil, P.-A.; Woessner, William P. 2006 all top 5 Cited by 17 Authors 5 O’Leary, Dianne P. 5 Tits, André Leon 3 Jung, Jin Hyuk 2 Park, Sungwoo 1 Absil, Pierre-Antoine 1 Cao, Yankai 1 Gu, Ran 1 Laird, Carl D. 1 Laiu, M. Paul 1 Mehrotra, Sanjay 1 Nicholls, Stacey O. 1 Peng, Jigen 1 Song, Xueli 1 Winternitz, Luke B. 1 Woessner, William P. 1 Yuan, Ya-xiang 1 Zavala, Victor M. all top 5 Cited in 6 Serials 4 Computational Optimization and Applications 2 Journal of Optimization Theory and Applications 2 SIAM Journal on Optimization 2 ETNA. Electronic Transactions on Numerical Analysis 1 Acta Mathematica Sinica. English Series 1 Nonlinear Analysis. Real World Applications Cited in 5 Fields 11 Operations research, mathematical programming (90-XX) 4 Numerical analysis (65-XX) 2 Computer science (68-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Ordinary differential equations (34-XX) Citations by Year