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Approximation Algorithms for Quadratic Programming

Author

Listed:
  • Minyue Fu

    (The University of Newcastle)

  • Zhi-Quan Luo

    (The University of Newcastle)

  • Yinyu Ye

    (The University of Newcastle)

Abstract

We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m=1, we rigorously show that an ∈-minimizer, where error ∈ ∈ (0, 1), can be obtained in polynomial time, meaning that the number of arithmetic operations is a polynomial in n, m, and log(1/∈). For m ≥ 2, we present a polynomial-time (1- $$\frac{1}{{m^2 }}$$ )-approximation algorithm as well as a semidefinite programming relaxation for this problem. In addition, we present approximation algorithms for solving QP under the box constraints and the assignment polytope constraints.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:2:y:1998:i:1:d:10.1023_a:1009739827008
    DOI: 10.1023/A:1009739827008
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    Citations

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    Cited by:

    1. Jiao, Hongwei & Liu, Sanyang & Lu, Nan, 2015. "A parametric linear relaxation algorithm for globally solving nonconvex quadratic programming," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 973-985.
    2. Nadav Hallak & Marc Teboulle, 2020. "Finding Second-Order Stationary Points in Constrained Minimization: A Feasible Direction Approach," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 480-503, August.
    3. Sturm, J.F. & Zhang, S., 2001. "On Cones of Nonnegative Quadratic Functions," Other publications TiSEM 075a6b4d-5b51-4153-a9c3-0, Tilburg University, School of Economics and Management.
    4. Samuel Burer & Kyungchan Park, 2024. "A Strengthened SDP Relaxation for Quadratic Optimization Over the Stiefel Manifold," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 320-339, July.
    5. Jos F. Sturm & Shuzhong Zhang, 2003. "On Cones of Nonnegative Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 246-267, May.
    6. Zhuoyi Xu & Linbin Li & Yong Xia, 2023. "A partial ellipsoidal approximation scheme for nonconvex homogeneous quadratic optimization with quadratic constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 98(1), pages 93-109, August.
    7. 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.
    8. Alexandre Belloni & Robert M. Freund & Santosh Vempala, 2009. "An Efficient Rescaled Perceptron Algorithm for Conic Systems," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 621-641, August.
    9. Zhuoyi Xu & Yong Xia & Jiulin Wang, 2021. "Cheaper relaxation and better approximation for multi-ball constrained quadratic optimization and extension," Journal of Global Optimization, Springer, vol. 80(2), pages 341-356, June.
    10. Kurt M. Anstreicher, 2024. "Solving Two-Trust-Region Subproblems Using Semidefinite Optimization with Eigenvector Branching," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 303-319, July.
    11. D. Henrion & S. Tarbouriech & D. Arzelier, 2001. "LMI Approximations for the Radius of the Intersection of Ellipsoids: Survey," Journal of Optimization Theory and Applications, Springer, vol. 108(1), pages 1-28, January.
    12. Tongli Zhang & Yong Xia, 2022. "Comment on “Approximation algorithms for quadratic programming”," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 1099-1103, September.

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