TY - JOUR
T1 - An inexact augmented lagrangian method for second-order cone programming with applications
AU - Liang, Ling
AU - Sun, Defeng
AU - Toh, Kim Chuan
N1 - Funding Information:
\ast Received by the editors October 19, 2020; accepted for publication (in revised form) April 6, 2021; published electronically July 13, 2021. https://doi.org/10.1137/20M1374262 \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The research of the second author is supported in part by the Hong Kong Research Grant Council under grant PolyU 153014/18P. The third author is supported in part by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant (MOE-2019-T3-1-010). \dagger Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076 ([email protected]). \ddagger Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong ([email protected]). \S Department of Mathematics, and Institute of Operations Research and Analytics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076 ([email protected]).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.
PY - 2021/7/13
Y1 - 2021/7/13
N2 - In this paper, we adopt the augmented Lagrangian method (ALM) to solve convex quadratic second-order cone programming problems (SOCPs). Fruitful results on the efficiency of the ALM have been established in the literature. Recently, it has been shown in [Cui, Sun, and Toh, Math. Program., 178 (2019), pp. 381-415] that if the quadratic growth condition holds at an optimal solution for the dual problem, then the KKT residual converges to zero R-superlinearly when the ALM is applied to the primal problem. Moreover, Cui, Ding, and Zhao [SIAM J. Optim., 27 (2017), pp. 2332-2355] provided sufficient conditions for the quadratic growth condition to hold under the metric subregularity and bounded linear regularity conditions for solving composite matrix optimization problems involving spectral functions. Here, we adopt these recent ideas to analyze the convergence properties of the ALM when applied to SOCPs. To the best of our knowledge, no similar work has been done for SOCPs so far. In our paper, we first provide sufficient conditions to ensure the quadratic growth condition for SOCPs. With these elegant theoretical guarantees, we then design an SOCP solver and apply it to solve various classes of SOCPs, such as minimal enclosing ball problems, classical trust-region subproblems, square-root Lasso problems, and DIMACS Challenge problems. Numerical results show that the proposed ALM based solver is efficient and robust compared to the existing highly developed solvers, such as Mosek and SDPT3.
AB - In this paper, we adopt the augmented Lagrangian method (ALM) to solve convex quadratic second-order cone programming problems (SOCPs). Fruitful results on the efficiency of the ALM have been established in the literature. Recently, it has been shown in [Cui, Sun, and Toh, Math. Program., 178 (2019), pp. 381-415] that if the quadratic growth condition holds at an optimal solution for the dual problem, then the KKT residual converges to zero R-superlinearly when the ALM is applied to the primal problem. Moreover, Cui, Ding, and Zhao [SIAM J. Optim., 27 (2017), pp. 2332-2355] provided sufficient conditions for the quadratic growth condition to hold under the metric subregularity and bounded linear regularity conditions for solving composite matrix optimization problems involving spectral functions. Here, we adopt these recent ideas to analyze the convergence properties of the ALM when applied to SOCPs. To the best of our knowledge, no similar work has been done for SOCPs so far. In our paper, we first provide sufficient conditions to ensure the quadratic growth condition for SOCPs. With these elegant theoretical guarantees, we then design an SOCP solver and apply it to solve various classes of SOCPs, such as minimal enclosing ball problems, classical trust-region subproblems, square-root Lasso problems, and DIMACS Challenge problems. Numerical results show that the proposed ALM based solver is efficient and robust compared to the existing highly developed solvers, such as Mosek and SDPT3.
KW - Augmented Lagrangian method
KW - Minimal enclosing ball problem
KW - Quadratic growth condition
KW - Second-order cone programming
KW - Square-root Lasso problem
KW - Trust-region subproblem
UR - http://www.scopus.com/inward/record.url?scp=85110320379&partnerID=8YFLogxK
U2 - 10.1137/20M1374262
DO - 10.1137/20M1374262
M3 - Journal article
AN - SCOPUS:85110320379
SN - 1052-6234
VL - 31
SP - 1748
EP - 1773
JO - SIAM Journal on Optimization
JF - SIAM Journal on Optimization
IS - 3
ER -