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A reduced proximal-point homotopy method for large-scale non-convex BQP

Author

Listed:
  • Xiubo Liang

    (Dalian University of Technology)

  • Guoqiang Wang

    (Business Research and Development department JD.com)

  • Bo Yu

    (Dalian University of Technology)

Abstract

In this paper, a reduced proximal-point homotopy (RPP-Hom) method is presented for large-scale non-convex box constrained quadratic programming (BQP) problems. As the outer iteration, at each step, the reduced proximal-point (RPP) algorithm applies the proximal point algorithm to a reduced BQP problem. The variables of the reduced subproblem include all free variables and variables at bound with respect to which the optimality conditions are violated. The RPP subproblem is solved by, as the inner iteration, an efficient piecewise linear homotopy path following method. A special termination criterion for the RPP algorithm is given and the global convergence as well as the locally linear convergence to a Karush-Kuhn-Tucker point is proved. Furthermore, a random perturbation procedure is given to modify RPP such that it converges to a local minimizer with probability 1. An accelerated version of RPP is also presented. Numerical experiments show that the RPP-Hom method outperforms the state-of-the-art algorithms for most of the benchmark problems, especially for training non-convex support vector machine.

Suggested Citation

  • Xiubo Liang & Guoqiang Wang & Bo Yu, 2022. "A reduced proximal-point homotopy method for large-scale non-convex BQP," Computational Optimization and Applications, Springer, vol. 81(2), pages 539-567, March.
    • Handle: RePEc:spr:coopap:v:81:y:2022:i:2:d:10.1007_s10589-021-00330-2
    DOI: 10.1007/s10589-021-00330-2
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    3. Guoqiang Wang & Bo Yu, 2019. "PAL-Hom method for QP and an application to LP," Computational Optimization and Applications, Springer, vol. 73(1), pages 311-352, May.
    4. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Klaus Ritter & Manfred Meyer, 1967. "A method for solving nonlinear maximum‐problems depending on parameters," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 14(2), pages 147-162.
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