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A New Global Optimization Scheme for Quadratic Programs with Low-Rank Nonconvexity

Author

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  • Xiaoli Cen

    (Key Laboratory of Ministry of Education for Information Mathematics and Behavior, School of Mathematical Sciences, Beihang University, Beijing 100191, China)

  • Yong Xia

    (Key Laboratory of Ministry of Education for Information Mathematics and Behavior, School of Mathematical Sciences, Beihang University, Beijing 100191, China)

Abstract

We consider the classical convex constrained nonconvex quadratic programming problem where the Hessian matrix of the objective to be minimized has r negative eigenvalues, denoted by (QP r ). Based on a biconvex programming reformulation in a slightly higher dimension, we propose a novel branch-and-bound algorithm to solve (QP 1 ) and show that it returns an ɛ -approximate solution of (QP 1 ) in at most O ( 1 / ɛ ) iterations. We further extend the new algorithm to solve the general (QP r ) with r > 1. Computational comparison shows the efficiency of our proposed global optimization method for small r . Finally, we extend the explicit relaxation approach for (QP 1 ) to (QP r ) with r > 1. Summary of Contribution: Nonconvex quadratic program (QP) is a classical optimization problem in operations research. This paper aims at globally solving the QP where the Hessian matrix of the objective to be minimized has r negative eigenvalues. It is known to be nondeterministic polynomial-time hard even when r = 1. This paper presents a novel algorithm to globally solve the QP for r = 1 and then extends to general r . Numerical results demonstrate the superiority of the proposed algorithm in comparison with state-of-the-art algorithms/software for small r .

Suggested Citation

  • Xiaoli Cen & Yong Xia, 2021. "A New Global Optimization Scheme for Quadratic Programs with Low-Rank Nonconvexity," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1368-1383, October.
    • Handle: RePEc:inm:orijoc:v:33:y:2021:i:4:p:1368-1383
    DOI: 10.1287/ijoc.2020.1017
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    References listed on IDEAS

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    1. Samuel Burer & Dieter Vandenbussche, 2009. "Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound," Computational Optimization and Applications, Springer, vol. 43(2), pages 181-195, June.
    2. Riccardo Cambini & Claudio Sodini, 2002. "A Finite Algorithm for a Particular D.C. Quadratic Programming Problem," Annals of Operations Research, Springer, vol. 117(1), pages 33-49, November.
    3. Cheng Lu & Zhibin Deng & Qingwei Jin, 2017. "An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints," Journal of Global Optimization, Springer, vol. 67(3), pages 475-493, March.
    4. Minyue Fu & Zhi-Quan Luo & Yinyu Ye, 1998. "Approximation Algorithms for Quadratic Programming," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 29-50, March.
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