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On exact and inexact RLT and SDP-RLT relaxations of quadratic programs with box constraints

Author

Listed:
  • Yuzhou Qiu

    (The University of Edinburgh)

  • E. Alper Yıldırım

    (The University of Edinburgh)

Abstract

Quadratic programs with box constraints involve minimizing a possibly nonconvex quadratic function subject to lower and upper bounds on each variable. This is a well-known NP-hard problem that frequently arises in various applications. We focus on two convex relaxations, namely the reformulation–linearization technique (RLT) relaxation and the SDP-RLT relaxation obtained by combining the Shor relaxation with the RLT relaxation. Both relaxations yield lower bounds on the optimal value of a quadratic program with box constraints. We show that each component of each vertex of the RLT relaxation lies in the set $$\{0,\frac{1}{2},1\}$$ { 0 , 1 2 , 1 } . We present complete algebraic descriptions of the set of instances that admit exact RLT relaxations as well as those that admit exact SDP-RLT relaxations. We show that our descriptions can be converted into algorithms for efficiently constructing instances with (1) exact RLT relaxations, (2) inexact RLT relaxations, (3) exact SDP-RLT relaxations, and (4) exact SDP-RLT but inexact RLT relaxations. Our preliminary computational experiments illustrate that our algorithms are capable of generating computationally challenging instances for state-of-the-art solvers.

Suggested Citation

  • Yuzhou Qiu & E. Alper Yıldırım, 2024. "On exact and inexact RLT and SDP-RLT relaxations of quadratic programs with box constraints," Journal of Global Optimization, Springer, vol. 90(2), pages 293-322, October.
    • Handle: RePEc:spr:jglopt:v:90:y:2024:i:2:d:10.1007_s10898-024-01407-y
    DOI: 10.1007/s10898-024-01407-y
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    References listed on IDEAS

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    1. M. Raghavachari, 1969. "On Connections Between Zero-One Integer Programming and Concave Programming Under Linear Constraints," Operations Research, INFORMS, vol. 17(4), pages 680-684, August.
    2. Godai Azuma & Mituhiro Fukuda & Sunyoung Kim & Makoto Yamashita, 2022. "Exact SDP relaxations of quadratically constrained quadratic programs with forest structures," Journal of Global Optimization, Springer, vol. 82(2), pages 243-262, February.
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