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Polynomially Solvable Cases of Binary Quadratic Programs

In: Optimization and Optimal Control

Author

Listed:
  • Duan Li

    (The Chinese University of Hong Kong)

  • Xiaoling Sun

    (Fudan University)

  • Shenshen Gu

    (Shanghai University)

  • Jianjun Gao

    (The Chinese University of Hong Kong)

  • Chunli Liu

    (Shanghai University of Finance and Economics)

Abstract

Summary We summarize in this chapter polynomially solvable subclasses of binary quadratic programming problems studied in the literature and report some new polynomially solvable subclasses revealed in our recent research. It is well known that the binary quadratic programming program is NP-hard in general. Identifying polynomially solvable subclasses of binary quadratic programming problems not only offers theoretical insight into the complicated nature of the problem but also provides platforms to design relaxation schemes for exact solution methods. We discuss and analyze in this chapter six polynomially solvable subclasses of binary quadratic programs, including problems with special structures in the matrix Q of the quadratic objective function, problems defined by a special graph or a logic circuit, and problems characterized by zero duality gap of the SDP relaxation. Examples and geometric illustrations are presented to provide algorithmic and intuitive insights into the problems.

Suggested Citation

  • Duan Li & Xiaoling Sun & Shenshen Gu & Jianjun Gao & Chunli Liu, 2010. "Polynomially Solvable Cases of Binary Quadratic Programs," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & Ider Tseveendorj (ed.), Optimization and Optimal Control, pages 199-225, Springer.
    • Handle: RePEc:spr:spochp:978-0-387-89496-6_11
    DOI: 10.1007/978-0-387-89496-6_11
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    Citations

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    Cited by:

    1. Chunli Liu & Jianjun Gao, 2015. "A polynomial case of convex integer quadratic programming problems with box integer constraints," Journal of Global Optimization, Springer, vol. 62(4), pages 661-674, August.
    2. Esteban Aguilera & Jins de Jong & Frank Phillipson & Skander Taamallah & Mischa Vos, 2024. "Multi-Objective Portfolio Optimization Using a Quantum Annealer," Mathematics, MDPI, vol. 12(9), pages 1-18, April.
    3. Frank Phillipson & Harshil Singh Bhatia, 2020. "Portfolio Optimisation Using the D-Wave Quantum Annealer," Papers 2012.01121, arXiv.org.
    4. X. Sun & C. Liu & D. Li & J. Gao, 2012. "On duality gap in binary quadratic programming," Journal of Global Optimization, Springer, vol. 53(2), pages 255-269, June.
    5. X. J. Zheng & X. L. Sun & D. Li, 2010. "Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 463-489, August.

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