一类含有双正则项的大规模凸约束优化问题的两阶段增广拉格朗日法

In recent years, the field of optimization has undergone tremendous growth with its applications being found in many areas where more and more large-scale optimization models have been proposed by researchers. In this proposal, we will consider solving an important class of such problems, i.e., the large-scale convex composite optimization, including a few nonlinear semidefinite programming problems and many regularized statistical regression problems, whose objective function is the sum of a smooth term and two nonsmooth regularization terms. With the rapid growth of the dimension sizes, especially with the advent of the “big data” analytics, it has become more and more challenging to solve these problems efficiently. Consequently, how to handle such problems has been an important issue in the field of optimization. This proposal aims to solving such problems by conducting theoretical and algorithmic research as well as numerical experiments. We plan to design an efficient two-phase augmented Lagrangian based method, which combines both the first-order approaches and the second-order nonsmooth Newton method to exploit the merits they possess, respectively. We will analyze the corresponding constraint non-degenerate property and the error bound conditions, and properly leverage the second-order sparsity of specific problem classes to enhance the efficiency of the designed algorithm. This proposal will further develop and complete the theory and methods of solving large-scale optimization problems, as well as provide theoretical guarantee and practical algorithms for researchers from related application areas.

随着最优化学科近些年来的快速发展及其在多个领域中应用的不断深化，越来越多的大规模最优化问的题模型被这些领域的学者们所提出。本项目计划研究求解这其中的一类重要的大规模凸合成优化问题，包括几种非线性半定规划问题以及多种正则化的统计回归问题。这类问题的目标函数通常是一个光滑函数和两个非光滑函数的和。随着实际应用中所需求解的问题的规模越来越大，尤其是在“大数据”的背景下，如何有效地求解它们是优化学科发展中所需研究的一个重点。本项目拟从理论分析、算法设计及数值试验三个方面来研究这类问题的求解方法，有针对性地将基于增广拉格朗日函数的一阶方法与二阶的半光滑牛顿法方法相结合，充分发挥这两类方法各自的长处，分析相关的约束非退化性质以及误差界，研究并利用问题本身的二阶稀疏性来提高算法的效率。本项目的研究将会进一步发展并完善大规模最优化问题求解的理论和方法，也为从事相关问题应用研究的人员提供理论依据和实用算法。

随着最优化学科近些年来的快速发展及其在多个领域中应用的不断深化，越来越多的大规模最优化问的题模型被这些领域的学者们所提出。本项目计划研究求解这其中的一类重要的大规模凸合成优化问题，包括几种非线性半定规划问题以及多种正则化的统计回归问题。这类问题的目标函数通常是一个光滑函数和两个非光滑函数的和。随着实际应用中所需求解的问题的规模越来越大，尤其是在“大数据”的背景下，如何有效地求解它们是优化学科发展中所需研究的一个重点。本项目从理论分析、算法设计及数值试验三个方面来研究这类问题的求解方法，有针对性地将将基于增广拉格朗日函数的一阶方法与二阶的半光滑牛顿法方法相结合，充分发挥这两类方法各自的长处，分析相关的约束非退化性质以及误差界，研究并利用问题本身的二阶稀疏性来提高算法的效率。项目课题依托本项目在项目执行期内发表论文5篇，从理论、算法及数值试验三个方面对大规模凸合成优化问题数值计算进行了深入的研究。本项目的研究进一步发展并完善大规模最优化问题求解的理论和方法，也为从事相关问题应用研究的人员提供理论依据和实用算法。