Computational Optimization and Applications

, Volume 51, Issue 3, pp 1001–1036 | Cite as

A constraint-reduced variant of Mehrotra’s predictor-corrector algorithm

  • Luke B. Winternitz
  • Stacey O. Nicholls
  • André L. Tits
  • Dianne P. O’Leary
Article

Abstract

Consider linear programs in dual standard form with n constraints and m variables. When typical interior-point algorithms are used for the solution of such problems, updating the iterates, using direct methods for solving the linear systems and assuming a dense constraint matrix A, requires \(\mathcal{O}(nm^{2})\) operations per iteration. When nm it is often the case that at each iteration most of the constraints are not very relevant for the construction of a good update and could be ignored to achieve computational savings. This idea was considered in the 1990s by Dantzig and Ye, Tone, Kaliski and Ye, den Hertog et al. and others. More recently, Tits et al. proposed a simple “constraint-reduction” scheme and proved global and local quadratic convergence for a dual-feasible primal-dual affine-scaling method modified according to that scheme. In the present work, similar convergence results are proved for a dual-feasible constraint-reduced variant of Mehrotra’s predictor-corrector algorithm, under less restrictive nondegeneracy assumptions. These stronger results extend to primal-dual affine scaling as a limiting case. Promising numerical results are reported.

As a special case, our analysis applies to standard (unreduced) primal-dual affine scaling. While we do not prove polynomial complexity, our algorithm allows for much larger steps than in previous convergence analyses of such algorithms.

Keywords

Linear programming Linear optimization Constraint reduction Primal-dual interior-point methods Mehrotra’s predictor corrector 

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Supplementary material

10589_2010_9389_MOESM1_ESM.pdf (156 kb)
Electronic Supplementary Material (appendix) to “A Constraint-Reduced Variant of Mehrotra’s Predictor-Corrector Algorithm” by L.B. Winternitz, S.O. Nicholls, A.L. Tits, and D.P. O’Leary. (156 KB)

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Luke B. Winternitz
    • 1
  • Stacey O. Nicholls
    • 2
  • André L. Tits
    • 3
  • Dianne P. O’Leary
    • 4
  1. 1.NASA – Goddard Space Flight CenterGreenbeltUSA
  2. 2.Applied Mathematics and Scientific Computing ProgramUniversity of MarylandCollege ParkUSA
  3. 3.Department of Electrical and Computer Engineering and the Institute for Systems ResearchUniversity of MarylandCollege ParkUSA
  4. 4.Department of Computer Science and Institute for Advanced Computer StudiesUniversity of MarylandCollege ParkUSA

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