DCA based Algorithm with Extrapolation for Nonconvex Nonsmooth Optimization

TL;DR

This paper introduces a novel DCA-based algorithm with extrapolation for nonconvex nonsmooth optimization, extending convergence analysis without requiring Lipschitz continuity, and demonstrates its effectiveness through numerical experiments.

Contribution

It develops a new accelerated DCA algorithm for nonconvex nonsmooth problems without Lipschitz assumptions, with proven convergence and superior performance.

Findings

Converges to critical points under broad conditions

Global convergence of the entire sequence is established

Numerical results show improved efficiency over existing methods

Abstract

In this paper, we focus on the problem of minimizing the sum of a nonconvex differentiable function and a DC (Difference of Convex functions) function, where the differentiable function is not restricted to the global Lipschitz gradient continuity assumption. This problem covers a broad range of applications in machine learning and statistics such as compressed sensing, signal recovery, sparse dictionary learning, and matrix factorization, etc. We take inspiration from the Nesterov's acceleration technique and the DC algorithm to develop a novel algorithm for the considered problem. Analyzing the convergence, we study the subsequential convergence of our algorithm to a critical point. Furthermore, we justify the global convergence of the whole sequence generated by our algorithm to a critical point and establish its convergence rate under the Kurdyka-Lojasiewicz condition. Numerical experiments on the nonnegative matrix completion problem are performed to demonstrate the efficiency of our algorithm and its superiority over well-known methods.