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

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 subproblem of the DC algorithm (DCA) to define a descent direction of the objective from that point, and then a monotone line search starting from it is performed in order to find a new point which decreases the objective function when compared with the point generated by the subproblem of DCA. This procedure improves the performance 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 provides 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–Łojasiewicz property of the objective function. Some numerical experiments show that the nmBDCA outperforms the DCA, such as its monotone version.