A Perturbation Framework for Convex Minimization and Monotone Inclusion Problems with Nonlinear Compositions

TL;DR

This paper develops a perturbation framework based on Rockafellar's theory to analyze and solve complex nonsmooth convex and monotone problems involving nonlinear and linear compositions, leading to new duality and primal-dual algorithms.

Contribution

It introduces a novel perturbation analysis for nonsmooth convex and monotone problems with nonlinear compositions, enabling dual problem construction and a block-iterative primal-dual algorithm.

Findings

First proximal splitting algorithm for nonlinear composite problems.

Framework applies to Banach and Hilbert spaces.

Provides duality and decomposition results for complex problems.

Abstract

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.