无限闭凸集族凸可行性问题中投影算法的线性收敛

In this project, we aim to investigate the convex feasibility problem in Hilbert space with an infinite family of closed convex sets with nonempty intersection, and give the convergence analysis of projection-based methods by means of bounded linear regularity. First, for a possibly infinite family of closed convex sets, we will provide several sufficient conditions for ensuring the bounded linear regularity in terms of the relaxed interior-point conditions. Moreover, by exploiting the constructive technique and numerical analysis methods, we will establish some quantitative results on linear regularity for an infinite family of polyhedrons. Then, we propose a general projection-based method for solving convex feasibility problem. By using bounded linear regularity, we will establish some linear convergence results of this method under a new control strategy introduced here. Finally, we will study the convergence of the subgradient algorithm. For achieving this goal, we will introduce some new Slater type conditions. By using these constraint qualifications, we will establish the linear convergence results for the subgradient algorithm considered in this proposal.

本项目将研究Hilbert空间中带无限闭凸集族的凸可行性问题，并利用线性正则给出投影算法的收敛性分析。首先，利用内点条件，建立了保证无限闭凸集系统是有界线性正则的充分条件；并且针对无限个多面体的情形，运用构造性思想和数值分析方法，建立了线性正则的定量估计。然后，提出一种一般的解决凸可行性问题的投影算法。通过构造新的控制策略，我们利用有界线性正则条件建立了其线性收敛的结果。最后，针对次梯度算法，通过对函数系统引进Slater类条件，保证函数系统对应的水平集系统是有界线性正则的，进而建立次梯度算法的线性收敛结果。

本项目主要研究Hilbert空间中带无限闭凸集族的凸可行性问题，以及局部凸拓扑向量空间中无限非凸优化问题的稳定性。首先，建立了保证无限闭凸集系统是有界线性正则的充分条件；针对无限凸可行性问题，提出一种广义的投影算法，并利用有界线性正则条件建立了其线性收敛的结果。然后，我们考虑了Hilbert空间中非光滑变分不等式问题。为此，我们提出一种Halpern类迭代算法，并建立了算法的强收敛结果。最后，针对目标函数和约束函数都是两个函数差的无限优化问题，我们给出了解集映射、可行解集映射以及最优值函数在优化问题扰动下的连续依赖性。