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

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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 n≫m 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.

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Author information

Authors and Affiliations

  1. NASA – Goddard Space Flight Center, Mail Stop 596, Greenbelt, MD, 20771, USA

    Luke B. Winternitz

  2. Applied Mathematics and Scientific Computing Program, University of Maryland, College Park, MD, 20742, USA

    Stacey O. Nicholls

  3. Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland, College Park, MD, 20742, USA

    André L. Tits

  4. Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 20742, USA

    Dianne P. O’Leary

Authors
  1. Luke B. Winternitz
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  2. Stacey O. Nicholls
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  3. André L. Tits
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  4. Dianne P. O’Leary
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Corresponding author

Correspondence to André L. Tits.

Additional information

This work was supported by NSF grant DMI0422931 and DoE grants DEFG0204ER25655 and DESC0002218. The work of the first author was supported by NASA under the Goddard Space Flight Center Study Fellowship Program. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation, those of the US Department of Energy, or those of NASA.

Electronic Supplementary Material

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10589_2010_9389_MOESM1_ESM.pdf

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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Winternitz, L.B., Nicholls, S.O., Tits, A.L. et al. A constraint-reduced variant of Mehrotra’s predictor-corrector algorithm. Comput Optim Appl 51, 1001–1036 (2012). https://doi.org/10.1007/s10589-010-9389-4

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