In this paper we present an active-set method for the solution of $\ell_1$-regularized convex quadratic optimization problems. It is derived by combining a proximal method of multipliers (PMM) strategy with a standard semismooth Newton method (SSN). The resulting linear systems are solved using a Krylov-subspace method, accelerated by certain general-purpose preconditioners which are shown to be optimal with respect to the proximal parameters. Practical efficiency is further improved by warm-starting the algorithm using a proximal alternating direction method of multipliers. We show that the outer PMM achieves global convergence under mere feasibility assumptions. Under additional standard assumptions, the PMM scheme achieves global linear and local superlinear convergence. The SSN scheme is locally superlinearly convergent, assuming that its associated linear systems are solved accurately enough, and globally convergent under certain additional regularity assumptions. We provide numerical evidence to demonstrate the effectiveness of the approach by comparing it against OSQP and IP-PMM (an ADMM and a regularized IPM solver, respectively) on several elastic-net linear regression and $L^1$-regularized PDE-constrained optimization problems.

翻译：本文提出了一种用于求解$\ell_1$正则化凸二次优化问题的活动集方法。该方法通过结合近端乘子法（PMM）策略与标准半光滑牛顿法（SSN）推导得出。所生成的线性系统采用Krylov子空间方法求解，并通过若干通用预条件子加速，这些预条件子被证明在近端参数意义下具有最优性。通过使用近端交替方向乘子法对算法进行热启动，进一步提升了实际计算效率。我们证明了外部PMM算法在仅需可行性假设的条件下即可实现全局收敛。在附加标准假设下，PMM方案可实现全局线性收敛和局部超线性收敛。若关联线性系统求解精度足够，SSN方案具有局部超线性收敛性；而在特定正则性假设下，该方案具有全局收敛性。通过将所提方法与OSQP和IP-PMM（分别为ADMM求解器和正则化内点法求解器）在多个弹性网线性回归及$L^1$正则化偏微分方程约束优化问题上的数值对比，验证了该方法的有效性。