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On linear programs with linear complementarity constraints

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Abstract

The paper is a manifestation of the fundamental importance of the linear program with linear complementarity constraints (LPCC) in disjunctive and hierarchical programming as well as in some novel paradigms of mathematical programming. In addition to providing a unified framework for bilevel and inverse linear optimization, nonconvex piecewise linear programming, indefinite quadratic programs, quantile minimization, and ℓ 0 minimization, the LPCC provides a gateway to a mathematical program with equilibrium constraints, which itself is an important class of constrained optimization problems that has broad applications. We describe several approaches for the global resolution of the LPCC, including a logical Benders approach that can be applied to problems that may be infeasible or unbounded.

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Authors and Affiliations

  1. Market Analytics, Inc., 500 Davis Street, Suite 1001, Evanston, IL, 60201, USA

    Jing Hu

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180-1590, USA

    John E. Mitchell

  3. Department of Industrial and Enterprise Systems Engineering, University of Illinois, Urbana, IL, 61801, USA

    Jong-Shi Pang

  4. Department of Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY, 12180-1590, USA

    Bin Yu

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  1. Jing Hu
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  2. John E. Mitchell
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  3. Jong-Shi Pang
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  4. Bin Yu
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Correspondence to Jong-Shi Pang.

Additional information

It is our great pleasure to dedicate this work to Professor Richard W. Cottle on the occasion of his 75th birthday in 2009. Professor Cottle is the father of the linear complementarity problem (LCP) [16]. The linear program with linear complementarity constraints (LPCC) treated in this paper is a natural extension of the LCP; our hope is that the LPCC will one day become as fundamental as the LCP, thereby continuing Professor Cottle’s legacy, bringing it to new heights, and extending its breadth. The work of the author John E. Mitchell was supported by the National Science Foundation under grant DMS-0715446 and by the Air Force Office of Sponsored Research under grant FA9550-08-1-0081. The work of the author Jong-Shi Pang was supported by the Office of Naval Research under grant no. N00014-06-1-0014, by the Air Force Office of Sponsored Research under grant FA9550-08-1-0061, and by the National Science Foundation under grant CMMI-0969600.

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Hu, J., Mitchell, J.E., Pang, JS. et al. On linear programs with linear complementarity constraints. J Glob Optim 53, 29–51 (2012). https://doi.org/10.1007/s10898-010-9644-3

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