l_p(0<p<=1)正则化问题数值算法研究

l_p (0<p<=1) regularized problems arise from the fields of information sciences, satistics and so on. These problems are a class of nonsmooth minimization. Hence some existing algorithms proposed for smooth minimization cannot be directly applied to l_p (0<p<=1) regularized problems. It is of important practical significance and theoretical value to develop efficient numerical methods for solving these problems and it has attracted great attentions from the field of numerical optimization. The aim of this project is to further study numerical algorithms to solve l_p (0<p<=1) regularized problems: 1、when p=1, we shall present semismooth Newton method to solve the regularized problems and analyze its convergence; 2、when 0<p<1, we shall propose first order algorithms, i.e., iterative thresholding algorithms and projected gradient method, to solve the regularized problems; 3、where 0<p<1, we also present second order algorithms to solve the regularized problems by combining first order algorithms and active set methods.

l_p(0<p<=1)正则化问题来源于信息、统计等学科领域。由于问题是一类非光滑优化问题，现存的一些解光滑优化问题的高效算法已不能直接用来解该问题。发展有效的数值方法解l_p(0<p<=1)正则化问题具有重要的实际意义和理论价值，是近年来优化界尤为关注的一个重要课题。本项目拟进一步研究数值方法解l_p(0<p<=1)正则化问题：1、当p=1时，拟研究半光滑牛顿法求解正则化问题, 分析算法的收敛性；2、当0<p<1时, 拟研究一阶迭代算法(即迭代阈值型算法和投影梯度算法)求解正则化问题；3、当0<p<1时，我们还将结合一阶迭代方法和积极集方法来研究二阶迭代算法求解正则化问题。

正则化优化问题在信息科学和统计学等领域有广泛的应用。近年来，设计迭代算法解正则化优化问题引起了大量的关注。项目组研究了算法解l_{1/2}正则化问题等几类特殊形式的优化问题。具体的研究内容和取得的成果如下：第一、提出了迭代算法解l_p正则化问题，研究了算法的收敛性和迭代复杂度；第二、提出了半光滑牛顿型算法解l_1正则化问题，分析了算法的收敛性；第三、提出了迭代算法解一般形式的正则化问题，分析了算法的收敛性和迭代复杂度；最后、我们还构造了多重分裂迭代算法解一类非线性互补问题，分析了算法的收敛性质和计算效果。