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PAL-Hom method for QP and an application to LP

Author

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  • Guoqiang Wang

    (Dalian University of Technology)

  • Bo Yu

    (Dalian University of Technology)

Abstract

In this paper, a proximal augmented Lagrangian homotopy (PAL-Hom) method for solving convex quadratic programming problems is proposed. This method takes the proximal augmented Lagrangian method as the outer iteration. To solve the proximal augmented Lagrangian subproblems, a homotopy method is presented as the inner iteration. The homotopy method tracks the piecewise-linear solution path of a parametric quadratic programming problem whose start problem takes an approximate solution as its solution and the target problem is the subproblem to be solved. To improve the performance of the homotopy method, the accelerated proximal gradient method is used to obtain a fairly good approximate solution that implies a good prediction of the optimal active set. Moreover, a sorting technique for the Cholesky factor update as well as an $$\varepsilon $$ ε -relaxation technique for checking primal-dual feasibility and correcting the active sets are presented to improve the efficiency and robustness of the homotopy method. Simultaneously, a proximal-point-based AL-Hom method which is shown to converge in finite number of steps, is applied to linear programming. Numerical experiments on randomly generated problems and the problems from the CUTEr and Netlib test collections, support vector machines (SVMs) and contact problems of elasticity demonstrate that PAL-Hom is faster than the active-set methods and the parametric active set methods and is competitive to the interior-point methods and the specialized algorithms designed for specific models (e.g., sequential minimal optimization method for SVMs).

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:73:y:2019:i:1:d:10.1007_s10589-019-00069-x
    DOI: 10.1007/s10589-019-00069-x
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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).
    2. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    3. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. 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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    Cited by:

    1. 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.

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