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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Vydáno v:Mathematical programming Ročník 161; číslo 1-2; s. 237 - 270
Hlavní autoři: Chen, Liang, Sun, Defeng, Toh, Kim-Chuan
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2017
Springer Nature B.V
Témata:
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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Shrnutí: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-016-1007-5