An Inexact Boosted Difference of Convex Algorithm for Nondifferentiable Functions

TL;DR

This paper introduces an inexact boosted difference of convex algorithm (InmBDCA) for nonconvex, nondifferentiable problems, improving convergence and efficiency over existing methods through approximate subproblem solutions and nonmonotone linesearch.

Contribution

It proposes an inexact, nonmonotone variant of BDCA that handles nondifferentiability and provides theoretical convergence guarantees and complexity bounds.

Findings

InmBDCA outperforms nmBDCA and DCA in numerical tests.

The algorithm converges to critical points under certain conditions.

Provides iteration complexity bounds for the proposed method.

Abstract

In this paper, we introduce an inexact approach to the Boosted Difference of Convex Functions Algorithm (BDCA) for solving nonconvex and nondifferentiable problems involving the difference of two convex functions (DC functions). Specifically, when the first DC component is differentiable and the second may be nondifferentiable, BDCA utilizes the solution from the subproblem of the DC Algorithm (DCA) to define a descent direction for the objective function. A monotone linesearch is then performed to find a new point that improves the objective function relative to the subproblem solution. This approach enhances the performance of DCA. However, if the first DC component is nondifferentiable, the BDCA direction may become an ascent direction, rendering the monotone linesearch ineffective. To address this, we propose an Inexact nonmonotone Boosted Difference of Convex Algorithm (InmBDCA). This algorithm incorporates two main features of inexactness: First, the subproblem therein is solved approximately allowing us for a controlled relative error tolerance in defining the linesearch direction. Second, an inexact nonmonotone linesearch scheme is used to determine the step size for the next iteration. Under suitable assumptions, we demonstrate that InmBDCA is well-defined, with any accumulation point of the sequence generated by InmBDCA being a critical point of the problem. We also provide iteration-complexity bounds for the algorithm. Numerical experiments show that InmBDCA outperforms both the nonsmooth BDCA (nmBDCA) and the monotone version of DCA in practical scenarios.