Contractive difference-of-convex algorithms

TL;DR

This paper introduces a contractive difference-of-convex algorithm (cDCA) that improves convergence speed by leveraging contraction properties and adaptive termination rules, addressing slow convergence issues in traditional DCA methods.

Contribution

The paper demonstrates that the subproblem in DCA can be viewed as a fixed point of a contraction and proposes cDCA with an adaptive termination rule for faster convergence.

Findings

cDCA has global subsequential convergence

cDCA exhibits global convergence of the entire sequence

Preliminary numerical results are promising

Abstract

The difference-of-convex algorithm (DCA) and its variants are the most popular methods to solve the difference-of-convex optimization problem. Each iteration of them is reduced to a convex optimization problem, which generally needs to be solved by iterative methods such as proximal gradient algorithm. However, these algorithms essentially belong to some iterative methods of fixed point problems of averaged mappings, and their convergence speed is generally slow. Furthermore, there is seldom research on the termination rule of these iterative algorithms solving the subproblem of DCA. To overcome these defects, we ffrstly show that the subproblem of the linearized proximal method (LPM) in each iteration is equal to the ffxed point problem of a contraction. Secondly, by using Picard iteration to approximately solve the subproblem of LPM in each iteration, we propose a contractive difference-ofconvex algorithm (cDCA) where an adaptive termination rule is presented. Both global subsequential convergence and global convergence of the whole sequence of cDCA are established. Finally, preliminary results from numerical experiments are promising.