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Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach

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  • Guoyong Gu
  • Bingsheng He
  • Xiaoming Yuan

Abstract

This paper focuses on some customized applications of the proximal point algorithm (PPA) to two classes of problems: the convex minimization problem with linear constraints and a generic or separable objective function, and a saddle-point problem. We treat these two classes of problems uniformly by a mixed variational inequality, and show how the application of PPA with customized metric proximal parameters can yield favorable algorithms which are able to make use of the models’ structures effectively. Our customized PPA revisit turns out to unify some algorithms including some existing ones in the literature and some new ones to be proposed. From the PPA perspective, we establish the global convergence and a worst-case O(1/t) convergence rate for this series of algorithms in a unified way. Copyright Springer Science+Business Media New York 2014

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  • Guoyong Gu & Bingsheng He & Xiaoming Yuan, 2014. "Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach," Computational Optimization and Applications, Springer, vol. 59(1), pages 135-161, October.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:1:p:135-161
    DOI: 10.1007/s10589-013-9616-x
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    References listed on IDEAS

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    1. B. S. He & X. L. Fu & Z. K. Jiang, 2009. "Proximal-Point Algorithm Using a Linear Proximal Term," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 299-319, May.
    2. Sun, Jie & Zhang, Su, 2010. "A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1210-1220, December.
    3. 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.
    4. Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, January.
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    Cited by:

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    2. Ying Gao & Wenxing Zhang, 2023. "An alternative extrapolation scheme of PDHGM for saddle point problem with nonlinear function," Computational Optimization and Applications, Springer, vol. 85(1), pages 263-291, May.
    3. Guoyong Gu & Junfeng Yang, 2024. "Tight Ergodic Sublinear Convergence Rate of the Relaxed Proximal Point Algorithm for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 373-387, July.
    4. Yanqin Bai & Xiao Han & Tong Chen & Hua Yu, 2015. "Quadratic kernel-free least squares support vector machine for target diseases classification," Journal of Combinatorial Optimization, Springer, vol. 30(4), pages 850-870, November.
    5. Feng Ma, 2019. "On relaxation of some customized proximal point algorithms for convex minimization: from variational inequality perspective," Computational Optimization and Applications, Springer, vol. 73(3), pages 871-901, July.
    6. Liusheng Hou & Hongjin He & Junfeng Yang, 2016. "A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA," Computational Optimization and Applications, Springer, vol. 63(1), pages 273-303, January.
    7. Eike Börgens & Christian Kanzow, 2019. "Regularized Jacobi-type ADMM-methods for a class of separable convex optimization problems in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 73(3), pages 755-790, July.

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