Decomposition and Successive Decomposition Methods and Algorithms for Nonconvex Optimization

TL;DR

This paper introduces novel primal, dual, and successive decomposition algorithms for nonconvex optimization problems with complex structures, enabling parallel implementation and improved convergence.

Contribution

It develops generalized decomposition methods for nonconvex problems that exploit problem structure and extend existing convex decomposition techniques.

Findings

Proposed methods produce stationary points of original nonconvex problems.

Algorithms enable parallel and distributed implementations.

Numerical results demonstrate effectiveness and advantages.

Abstract

Existing results on decomposition methods and algorithms for nonconvex problems are minimal. Parallel decomposition algorithms do not exist for nonconvex problems with coupling nonlinear equality constraints. Besides, decomposition structures (i.e., coupling variables and constraints) are not fully exploited in designing decomposition methods and algorithms. In this paper, we consider nonconvex problems with decomposition structures that are more general than those handled in the existing literature. We propose primal and dual decomposition and successive primal and successive dual decomposition methods and algorithms for these nonconvex problems, which exploit decomposition structures, allow for parallel and distributed implementations, produce the original nonconvex problems' stationary points, and offer good opportunities to achieve superior tradeoff between convergence performance and computation time. Finally, we compare the proposed methods and algorithms, extend them to indirect and two-level decomposition methods and algorithms, and provide examples and numerical results to demonstrate their respective values. Notably, the proposed decomposition and successive decomposition methods and algorithms generalize the basic ones for convex problems, and the proposed successive decomposition methods and algorithms extend the existing ones for nonconvex problems, together enriching the decomposition theory.