A Perturbation Framework for Convex Minimization and Monotone Inclusion Problems with Nonlinear Compositions
基于Rockafellar扰动理论，提出框架分析并求解涉及非线性复合函数的非光滑凸最小化和单调包含问题，在希尔伯特空间中导出首个能完全分解所有组件的块迭代原始-对偶算法。
We introduce a framework based on Rockafellar’s perturbation theory to analyze and solve general nonsmooth convex minimization and monotone inclusion problems involving nonlinearly composed functions as well as linear compositions. Such problems have been investigated only from a primal perspective and only for nonlinear compositions of smooth functions in finite-dimensional spaces in the absence of linear compositions. In the context of Banach spaces, the proposed perturbation analysis serves as a foundation for the construction of a dual problem and of a maximally monotone Kuhn–Tucker operator, which is decomposable as the sum of simpler monotone operators. In the Hilbertian setting, this decomposition leads to a block-iterative primal-dual algorithm that fully splits all the components of the problem and appears to be the first proximal splitting algorithm for handling nonlinear composite problems. Various applications are discussed. Funding: The work of L. M. Briceño-Arias was supported by Agencia Nacional de Investigación y Desarrollo-Chile [Grant Fondo Nacional de Desarrollo Científico y Tecnológico 1190871, Grant Centro de Modelamiento Matemático ACE210010, Grant Centro de Modelamiento Matemático FB210005, and basal Funds for Centers of Excellence], and the work of P. L. Combettes was supported by the National Science Foundation [Grant DMS-1818946].