论文标题
一种不精确的增强拉格朗日方法，用于使用应用程序
An Inexact Augmented Lagrangian Method for Second-order Cone Programming with Applications
论文作者
论文摘要
在本文中，我们采用增强的拉格朗日方法（ALM）来解决凸二次二阶编程问题（SECPS）。在文献中已经建立了关于ALM效率的富有成果的结果。最近，它已在[Cui，Sun和Toh，{\ em Math。程序。}，178（2019），pp。381--415]，如果二次生长条件在双重问题的最佳解决方案中成立，则当将ALM应用于原始问题时，KKT残差会收敛到零R-Superlinearear。此外，CUI，Ding和Zhao [{\ Em Siam J. Optim。}，27（2017），第2332-2355页]为二次生长条件提供了足够的条件，可以在度量的次额定性和有限的线性规则条件下持有涉及光谱功能的求解综合矩阵优化问题。在这里，我们采用这些最新想法来分析应用于SECP的ALM的收敛性。据我们所知，到目前为止，还没有为SECPS做类似的工作。在我们的论文中，我们首先提供了足够的条件，以确保SECPS的二次增长条件。凭借这些优雅的理论保证，我们然后设计了一个SOCP求解器，并将其应用于解决各种类别的苏菲省，例如最小的封闭球问题，经典的信任区域子问题，方形 - 根 - 根 - 根套索问题和DIMAC挑战问题。数值结果表明，与现有高度发达的求解器（例如Mosek和SDPT3）相比，所提出的基于ALM的求解器是有效且健壮的。
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, {\em 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 [{\em 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.
