  In this paper we introduce the Boosted Double-proximal Subgradient Algorithm
(BDSA), a novel splitting algorithm designed to address general structured
nonsmooth and nonconvex mathematical programs expressed as sums and differences
of composite functions. BDSA exploits the combined nature of subgradients from
the data and proximal steps, and integrates a line-search procedure to enhance
its performance. While BDSA encompasses existing schemes proposed in the
literature, it extends its applicability to more diverse problem domains. We
establish the convergence of BDSA under the Kurdyka--Lojasiewicz property and
provide an analysis of its convergence rate. To evaluate the effectiveness of
BDSA, we introduce a novel family of challenging test functions with an
abundance of critical points. We conduct comparative evaluations demonstrating
its ability to effectively escape non-optimal critical points. Additionally, we
present two practical applications of BDSA for testing its efficacy, namely, a
constrained minimum-sum-of-squares clustering problem and a nonconvex
generalization of Heron's problem.
