A boosted DC algorithm for non-differentiable DC components with non-monotone line search

TL;DR

This paper proposes a non-monotone line search variant of the boosted DC algorithm (nmBDCA) for non-convex, non-differentiable problems, improving convergence and outperforming traditional DCA methods.

Contribution

It introduces a non-monotone line search strategy for BDCA, enabling handling of non-differentiable DC components and providing convergence guarantees.

Findings

nmBDCA outperforms traditional DCA in numerical tests.

The method guarantees convergence to critical points under certain conditions.

Provides iteration-complexity bounds and full convergence results.

Abstract

We introduce a new approach to apply the boosted difference of convex functions algorithm (BDCA) for solving non-convex and non-differentiable problems involving difference of two convex functions (DC functions). Supposing the first DC component differentiable and the second one possibly non-differentiable, the main idea of BDCA is to use the point computed by the DC algorithm (DCA) to define a descent direction and perform a monotone line search to improve the decreasing the objetive function accelerating the convergence of the DCA. However, if the first DC component is non-differentiable, then the direction computed by BDCA can be an ascent direction and a monotone line search cannot be performed. Our approach uses a non-monotone line search in the BDCA (nmBDCA) to enable a possible growth in the objective function values controlled by a parameter. Under suitable assumptions, we show that any cluster point of the sequence generated by the nmBDCA is a critical point of the problem under consideration and provide some iteration-complexity bounds. Furthermore, if the first DC component is differentiable, we present different iteration-complexity bounds and prove the full convergence of the sequence under the Kurdyka-\L{}ojasiewicz property of the objective function. Some numerical experiments show that the nmBDCA outperforms the DCA such as its monotone version.