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On relaxation of some customized proximal point algorithms for convex minimization: from variational inequality perspective

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  • Feng Ma

    (High-Tech Institute of Xi’an)

Abstract

The proximal point algorithm (PPA) is a fundamental method for convex programming. When applying the PPA to solve linearly constrained convex problems, we may prefer to choose an appropriate metric matrix to define the proximal regularization, so that the computational burden of the resulted PPA can be reduced, and sometimes even admit closed form or efficient solutions. This idea results in the so-called customized PPA (also known as preconditioned PPA), and it covers the linearized ALM, the primal-dual hybrid gradient algorithm, ADMM as special cases. Since each customized PPA owes special structures and has popular applications, it is interesting to ask wether we can design a simple relaxation strategy for these algorithms. In this paper we treat these customized PPA algorithms uniformly by a mixed variational inequality approach, and propose a new relaxation strategy for these customized PPA algorithms. Our idea is based on correcting the dual variables individually and does not rely on relaxing the primal variables. This is very different from previous works. From variational inequality perspective, we prove the global convergence and establish a worst-case convergence rate for these relaxed PPA algorithms. Finally, we demonstrate the performance improvements by some numerical results.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:73:y:2019:i:3:d:10.1007_s10589-019-00091-z
    DOI: 10.1007/s10589-019-00091-z
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    References listed on IDEAS

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    1. 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.
    2. Bingsheng He & Xiaoming Yuan, 2018. "A class of ADMM-based algorithms for three-block separable convex programming," Computational Optimization and Applications, Springer, vol. 70(3), pages 791-826, July.
    3. Yuan Shen & Hongyong Wang, 2016. "New Augmented Lagrangian-Based Proximal Point Algorithm for Convex Optimization with Equality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 251-261, October.
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