论文标题
零 - 正则化PLQ复合优化问题的近端MM方法
A proximal MM method for the zero-norm regularized PLQ composite optimization problem
论文作者
论文摘要
本文涉及一类零正规化的分段线性季度（PLQ）综合最小化问题，该问题涵盖了零 - 正规化的$ \ ell_1 $ -loss最小化问题作为特殊情况。对于这类非洞穴的非平滑问题，我们表明其等效的MPEC重新印象在全球Optima的集合中部分平静，并利用此属性来得出同等的DC代理家族。然后，我们提出了一种近端大型化最小化方法（MM）方法，这是DC算法框架中不使用凸的松弛方法，用于求解一种DC替代物之一，该替代物是一个涉及三个非单词术语的SemiconVex PLQ最小化问题。对于这种方法，我们建立了其全局收敛性和收敛的线性速率，在适当的条件下，生成的序列的极限不仅是局部最优值，而且在统计意义上也是一个良好的关键点。使用双重半齿牛顿方法来确认我们的理论发现，并与融合的无限次无限型AMPM进行了数值比较，并通过综合MM方法进行了合成和真实数据进行数值实验，并进行了数值比较，以验证DC的一部分DC替代质量和计算的质量质量时光，以验证其优越的时间和计算。
This paper is concerned with a class of zero-norm regularized piecewise linear-quadratic (PLQ) composite minimization problems, which covers the zero-norm regularized $\ell_1$-loss minimization problem as a special case. For this class of nonconvex nonsmooth problems, we show that its equivalent MPEC reformulation is partially calm on the set of global optima and make use of this property to derive a family of equivalent DC surrogates. Then, we propose a proximal majorization-minimization (MM) method, a convex relaxation approach not in the DC algorithm framework, for solving one of the DC surrogates which is a semiconvex PLQ minimization problem involving three nonsmooth terms. For this method, we establish its global convergence and linear rate of convergence, and under suitable conditions show that the limit of the generated sequence is not only a local optimum but also a good critical point in a statistical sense. Numerical experiments are conducted with synthetic and real data for the proximal MM method with the subproblems solved by a dual semismooth Newton method to confirm our theoretical findings, and numerical comparisons with a convergent indefinite-proximal ADMM for the partially smoothed DC surrogate verify its superiority in the quality of solutions and computing time.
