稀疏约束优化问题的二阶最优性条件及稳定性研究

Sparsity constrained optimization has been applied widely in signal and image processing, machine learning, pattern recognition and computer vision, and so on. Recently, the models and algorithms of sparsity optimization have been rapidly developed, while the correlation theory is lack. In this project, we consider the sparsity optimization with set inclusion constraint by utilizing the theory in convex analysis, variational analysis and perturbation analysis. We mainly study the theory of first- and second-order optimality conditions, sensitivity analysis and error bound, and construct the algorithm, discuss the local convergence and rate of convergence of this algorithm.

稀疏约束优化问题被广泛应用于信号回收、图像处理、机器学习、图像识别、计算机显像等诸多领域，倍受学者关注. 目前，有关稀疏优化问题的模型、算法发展迅速，而有关稀疏优化问题的最优性条件，稳定性分析等理论结果却相对匮乏. 本项目以带有集合包含约束的稀疏优化问题为对象，利用凸分析、变分分析和扰动分析等知识，研究一阶、二阶最优性条件，稳定性分析与误差界，设计最优化算法，并讨论算法的局部收敛性，计算收敛速度.