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Operations Research Transactions >

2018 , Vol. 22 >Issue 1: 1 - 31

DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2018.01.001

My 20 years research on alternating directions method of multipliers

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  • 1. Department of Mathematics, Southern University Science and Technology, Shenzhen 518055, Guangdong, China; 2. Department of Mathematics, Nanjing University, Nanjing 210023, China

Received date: 2017-09-15

  Online published: 2018-03-15

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Abstract

My research on ADMM dates back to 1997 when I considered the problems from traffic network analysis. Over the last 10 years, the ADMM based on variational inequalities is widely used in optimization. This paper summarizes our research on ADMM over the last 20 years, particularly, the developments in splitting and contraction methods based on ADMM for convex optimization over the last 10 years. We list the main results as well as the motivations. Our analysis is based on the variational inequalities. All methods mentioned fall in a simple unified prediction-correction framework, in which the convergence analysis is quite simple. A through reading will acquaint you with the ADMM, while a more carefully reading may make you familiar with the tricks on constructing splitting methods according to the problem you met. We should notice that the ADMM originates from ALM and PPA, which are good at utilizing the splitting structure. However, it also inherits the intrinsic shortcomings of these first order methods.

Key words： convex optimization; monotone variational inequality; alternating directions method of multipliers; contractive properties; O(1/t) convergence rate; prediction-correction; unified framework

Cite this article

HE Bingsheng . My 20 years research on alternating directions method of multipliers[J]. Operations Research Transactions, 2018 , 22(1) : 1 -31 . DOI: 10.15960/j.cnki.issn.1007-6093.2018.01.001

References

[1] He B S. A new method for a class of linear variational inequalities [J]. Mathematical Programming, 1994, 66: 137-144.
[2] He B S. Solving a class of linear projection equations [J]. Numerische Mathematik, 1994, 68: 71-80.
[3] He B S. A class of projection and contraction methods for monotone variational inequalities
[J]. Applied Mathematics and Optimization, 1997, 35: 69-76.
[4] He B S, Liao L Z. Improvements of some projection methods for monotone nonlinear variational inequalities [J]. Journal of Optimization Theory and Applications, 2002, 112: 111-128.
[5] Chen D, Zhang Y N. A hybrid multi-objective scheme applied to redundant robot manipulators [J]. IEEE Transactions on Automation Science and Engineering, 2017, 14: 1337-1350.

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