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:在本文件中,我们提出了一个解决$ell_1美元正规化的共振方形优化问题的主动设定方法,其方法是将一种最接近的倍增法战略与标准的半线性牛顿法相结合,由此产生的线性系统使用Krylov-子空间方法加以解决,由某些普通用途先决条件人加速,这些先决条件人显示,与准准参数相比是最佳的。通过使用一种准氧化交替的乘数方向法来温暖启动算法,实际效率得到进一步提高。我们表明,外部PMMM在仅仅可行的假设下实现了全球趋同。在额外的标准假设下,PMM计划实现了全球线性和局部超线性趋同。SNSN计划是局部超线性趋同,假设其相关的线性系统在一定的定期性假设下得到足够准确的解决,全球趋同。我们提供了数字证据,通过将这种方法与OSQP和IP-PMM(ADM和固定化的IPMSSS)分别与若干弹性和磁性磁性磁性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬</s>