A proximal alternating direction method of multipliers for a minimization problem with nonconvex constraints

In this paper, a proximal alternating direction method of multipliers is proposed for solving a minimization problem with Lipschitz nonconvex constraints. Such problems are raised in many engineering fields, such as the analytical global placement of very large scale integrated circuit design. The proposed method is essentially a new application of the classical proximal alternating direction method of multipliers. We prove that, under some suitable conditions, any subsequence of the sequence generated by the proposed method globally converges to a Karush–Kuhn–Tucker point of the problem. We also present a practical implementation of the method using a certain self-adaptive rule of the proximal parameters. The proposed method is used as a global placement method in a placer of very large scale integrated circuit design. Preliminary numerical results indicate that, compared with some state-of-the-art global placement methods, the proposed method is applicable and efficient.

[1]  Paul Tseng,et al.  Partial Proximal Minimization Algorithms for Convex Pprogramming , 1994, SIAM J. Optim..

[2]  Masao Fukushima,et al.  Some Reformulations and Applications of the Alternating Direction Method of Multipliers , 1994 .

[3]  Anatoly S. Antipin.  Gradient Approach of Computing Fixed Points of Equilibrium Problems , 2002, J. Glob. Optim..

[4]  Chris Chu.  CHAPTER 11 – Placement , 2009 .

[5]  Alexander Kaplan,et al.  Proximal Point Methods and Nonconvex Optimization , 1998, J. Glob. Optim..

[6]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[7]  Chris C. N. Chu,et al.  FastPlace 3.0: A Fast Multilevel Quadratic Placement Algorithm with Placement Congestion Control , 2007, 2007 Asia and South Pacific Design Automation Conference.

[8]  Xiaoming Yuan,et al.  Matrix completion via an alternating direction method , 2012 .

[9]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[10]  Jarrod A. Roy,et al.  Capo: robust and scalable open-source min-cut floorplacer , 2005, ISPD '05.

[11]  Junfeng Yang,et al.  A Proximal Point Algorithm for Log-Determinant Optimization with Group Lasso Regularization , 2013, SIAM J. Optim..

[12]  M. H. Xu.  Proximal Alternating Directions Method for Structured Variational Inequalities , 2007 .

[13]  Yao-Wen Chang,et al.  Essential Issues in Analytical Placement Algorithms , 2009, IPSJ Trans. Syst. LSI Des. Methodol..

[14]  Jonathan Eckstein.  Some Saddle-function splitting methods for convex programming , 1994 .

[15]  Xiaoming Yuan.  An improved proximal alternating direction method for monotone variational inequalities with separable structure , 2011, Comput. Optim. Appl..

[16]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[17]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[18]  S. Karamardian,et al.  Seven kinds of monotone maps , 1990 .

[19]  Patrick H. Madden,et al.  Recursive bisection placement: feng shui 5.0 implementation details , 2005, ISPD '05.

[20]  Anatoly S. Antipin,et al.  Extra-proximal methods for solving two-person nonzero-sum games , 2009, Math. Program..

[21]  Joseph R. Shinnerl,et al.  mPL6: enhanced multilevel mixed-size placement , 2006, ISPD '06.

[22]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[23]  Bingsheng He,et al.  A new inexact alternating directions method for monotone variational inequalities , 2002, Math. Program..

[24]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[25]  Yao-Wen Chang,et al.  NTUplace3: An Analytical Placer for Large-Scale Mixed-Size Designs With Preplaced Blocks and Density Constraints , 2008, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[26]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..