An efficient inexact symmetric Gauss–Seidel based majorized ADMM for high-dimensional convex composite conic programming

In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the i...

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Vydané v:	Mathematical programming Ročník 161; číslo 1-2; s. 237 - 270
Hlavní autori:	Chen, Liang, Sun, Defeng, Toh, Kim-Chuan
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2017 Springer Nature B.V
Predmet:	Accuracy Calculus of Variations and Optimal Control; Optimization Combinatorics Conics Convergence Convex analysis Cost engineering Criteria Full Length Paper Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Principal components analysis Quadratic programming Semidefinite programming Studies Symmetry Theoretical Convex quadratic semidefinite programming Alternating direction method of multipliers 65K05 90C25 90C22 Convex conic programming Majorization 90C06 Symmetric Gauss–Seidel
ISSN:	0025-5610, 1436-4646
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Abstract	In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the inexact symmetric Gauss–Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block-variable. One distinctive feature of our proposed method (the sGS-imsPADMM) is that it only needs one cycle of an inexact sGS method, instead of an unknown number of cycles, to solve each of the subproblems involved. With some simple and implementable error tolerance criteria, the cost for solving the subproblems can be greatly reduced, and many steps in the forward sweep of each sGS cycle can often be skipped, which further contributes to the efficiency of the proposed method. Global convergence as well as the iteration complexity in the non-ergodic sense is established. Preliminary numerical experiments on some high-dimensional linear and convex quadratic SDP problems with a large number of linear equality and inequality constraints are also provided. The results show that for the vast majority of the tested problems, the sGS-imsPADMM is 2–3 times faster than the directly extended multi-block ADMM with the aggressive step-length of 1.618, which is currently the benchmark among first-order methods for solving multi-block linear and quadratic SDP problems though its convergence is not guaranteed.
AbstractList	In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block-variable. One distinctive feature of our proposed method (the sGS-imsPADMM) is that it only needs one cycle of an inexact sGS method, instead of an unknown number of cycles, to solve each of the subproblems involved. With some simple and implementable error tolerance criteria, the cost for solving the subproblems can be greatly reduced, and many steps in the forward sweep of each sGS cycle can often be skipped, which further contributes to the efficiency of the proposed method. Global convergence as well as the iteration complexity in the non-ergodic sense is established. Preliminary numerical experiments on some high-dimensional linear and convex quadratic SDP problems with a large number of linear equality and inequality constraints are also provided. The results show that for the vast majority of the tested problems, the sGS-imsPADMM is 2-3 times faster than the directly extended multi-block ADMM with the aggressive step-length of 1.618, which is currently the benchmark among first-order methods for solving multi-block linear and quadratic SDP problems though its convergence is not guaranteed. In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the inexact symmetric Gauss–Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block-variable. One distinctive feature of our proposed method (the sGS-imsPADMM) is that it only needs one cycle of an inexact sGS method, instead of an unknown number of cycles, to solve each of the subproblems involved. With some simple and implementable error tolerance criteria, the cost for solving the subproblems can be greatly reduced, and many steps in the forward sweep of each sGS cycle can often be skipped, which further contributes to the efficiency of the proposed method. Global convergence as well as the iteration complexity in the non-ergodic sense is established. Preliminary numerical experiments on some high-dimensional linear and convex quadratic SDP problems with a large number of linear equality and inequality constraints are also provided. The results show that for the vast majority of the tested problems, the sGS-imsPADMM is 2–3 times faster than the directly extended multi-block ADMM with the aggressive step-length of 1.618, which is currently the benchmark among first-order methods for solving multi-block linear and quadratic SDP problems though its convergence is not guaranteed.
Author	Chen, Liang Toh, Kim-Chuan Sun, Defeng
Author_xml	– sequence: 1 givenname: Liang surname: Chen fullname: Chen, Liang organization: College of Mathematics and Econometrics, Hunan University – sequence: 2 givenname: Defeng surname: Sun fullname: Sun, Defeng organization: Department of Mathematics and Risk Management Institute, National University of Singapore – sequence: 3 givenname: Kim-Chuan surname: Toh fullname: Toh, Kim-Chuan email: mattohkc@nus.edu.sg organization: Department of Mathematics, National University of Singapore
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Snippet	In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization... In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization...
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SubjectTerms	Accuracy Calculus of Variations and Optimal Control; Optimization Combinatorics Conics Convergence Convex analysis Cost engineering Criteria Full Length Paper Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Principal components analysis Quadratic programming Semidefinite programming Studies Symmetry Theoretical
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Title	An efficient inexact symmetric Gauss–Seidel based majorized ADMM for high-dimensional convex composite conic programming
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