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An LPCC approach to nonconvex quadratic programs

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Abstract

Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving two linear programs with linear complementarity constraints, i.e., LPCCs. Specifically, this task can be divided into two LPCCs: the first confirms whether the QP is bounded below on the feasible set and, if not, computes a feasible ray on which the QP is unbounded; the second LPCC computes a globally optimal solution if it exists, by identifying a stationary point that yields the best quadratic objective value. In turn, the global resolution of these LPCCs can be accomplished by a parameter-free, mixed integer-programming based, finitely terminating algorithm developed recently by the authors, which can be enhanced in this context by a new kind of valid cut derived from the second-order conditions of the QP and by exploiting the special structure of the LPCCs. Throughout, our treatment makes no boundedness assumption of the QP; this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption. The general theory is illustrated by 3 classes of indefinite problems: QPs with simple upper and lower bounds (existence of optimal solutions is guaranteed); same QPs with an additional inequality constraint (extending the case of simple bound constraints); and nonnegatively constrained copositive QPs (no guarantee of the existence of an optimal solution). We also present numerical results to support the special cuts obtained due to the QP connection.

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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

Authors
  1. Jing Hu
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  2. John E. Mitchell
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  3. Jong-Shi Pang
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Correspondence to John E. Mitchell.

Additional information

The work of 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 Pang was supported by the Office of Naval Research under grant no. N00014-06-1-0014 and by the Air Force Office of Sponsored Research under grant FA9550-08-1-0061.

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Hu, J., Mitchell, J.E. & Pang, JS. An LPCC approach to nonconvex quadratic programs. Math. Program. 133, 243–277 (2012). https://doi.org/10.1007/s10107-010-0426-y

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