A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming

Su-Hong Jiang; Min Li

doi:10.3934/jimo.2017052

Article Contents

2018, Volume 14, Issue 1: 397-412. Doi: 10.3934/jimo.2017052

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A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming

Su-Hong Jiang and
Min Li^,

School of Management and Engineering, Nanjing University, Nanjing 210093, China

_* Corresponding author: MIN LI
_* Corresponding author: MIN LI

Received: May 31, 2016

Revised: November 30, 2016

Early access: May 2017

Published: January 2018

This research was supported by National Science Foundation of China (Grant No. 11401300,71602083,71671085 and 11001053) and Program for New Century Excellent Talents in University (Grant No. NCET-12-0111).

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Abstract

We propose a modified splitting method for a linearly constrained minimization model whose objective function is the sum of three convex functions without coupled variables. Our work is mainly inspired by the recently proposed strictly contractive Peaceman-Rachford splitting method (SC-PRSM) for a two-block separable convex minimization model. For the new method, we prove its convergence and estimate its convergence rates measured by iteration complexity in the nonergodic sense. We also test the SC-PRSM on the continuous resource allocation problem, and the numerical results show that our method has a competitive performance with the direct extension of ADMM which usually works well in practice but may fail to converge in theory.

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Mathematics Subject Classification: Primary: 90C25; Secondary: 90B50.

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Figure 1. Evolutions of objective function values w.r.t. CPU 20:20:30

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Table 1. The function $\phi(s)$ for generating the cost function

Name	$\phi(s_i):\Re_+\rightarrow [0, \, \infty)$	Parameters
Linear cost	$w_i s_i$	$w_i \in U(1, \, 5)\\ k_i \in U(1, \, 5)$
Power cost	$k_i s_i^{q_i}$
Piecewise quadratic cost	$\displaystyle \left\{ \begin{array}{ll} k_is_i^2, ~~~~~~~~~~~~~~~~~~~{\rm {if}}\, s_i \leq w_i/\sqrt{2k_i}\\ w_i\sqrt{2k}s- {w_i^2/2}, ~~~~{\rm{otherwise}}. \end{array} \right. $

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