In this paper we present an efficient active-set method for the solution of convex quadratic programming problems with general piecewise-linear terms in the objective, with applications to sparse approximations and risk-minimization. The algorithm is derived by combining a proximal method of multipliers (PMM) with a standard semismooth Newton method (SSN), and is shown to be globally convergent under minimal assumptions. Further local linear (and potentially superlinear) convergence is shown under standard additional conditions. The major computational bottleneck of the proposed approach arises from the solution of the associated SSN linear systems. These are solved using a Krylov-subspace method, accelerated by certain novel general-purpose preconditioners which are shown to be optimal with respect to the proximal penalty parameters. The preconditioners are easy to store and invert, since they exploit the structure of the nonsmooth terms appearing in the problem's objective to significantly reduce their memory requirements. We showcase the efficiency, robustness, and scalability of the proposed solver on a variety of problems arising in risk-averse portfolio selection, $L^1$-regularized partial differential equation constrained optimization, quantile regression, and binary classification via linear support vector machines. We provide computational evidence, on real-world datasets, to demonstrate the ability of the solver to efficiently and competitively handle a diverse set of medium- and large-scale optimization instances.

翻译：本文提出了一种高效的活跃集方法，用于求解目标函数中包含一般分段线性项的凸二次规划问题，并应用于稀疏逼近和风险最小化。该算法通过结合近端乘子法（PMM）与标准半光滑牛顿法（SSN）推导得出，在最小假设下被证明具有全局收敛性。在标准附加条件下，进一步证明了其局部线性（以及潜在超线性）收敛性。该方法主要计算瓶颈在于求解相关SSN线性系统。我们采用Krylov子空间方法求解该系统，并通过某些新型通用预处理子加速，这些预处理子被证明对于近端惩罚参数是最优的。预处理子易于存储和求逆，因为它们利用问题目标函数中非光滑项的结构，显著降低了内存需求。我们通过在风险厌恶投资组合选择、$L^1$正则化偏微分方程约束优化、分位数回归以及基于线性支持向量机的二分类等一系列问题中，展示了所提求解器的效率、鲁棒性和可扩展性。基于真实数据集的计算证据表明，该求解器能够高效且具有竞争力地处理各种中大规模优化实例。