The Boosted Double-proximal Subgradient Algorithm for nonconvex optimization
提出一种新的分裂算法BDSA，用于求解结构化的非光滑非凸数学规划问题，通过结合次梯度和近端步骤并集成线搜索，在Kurdyka–Łojasiewicz性质下证明收敛性，并在聚类和Heron问题中验证有效性。
Abstract 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 linesearch 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–Łojasiewicz property and provide an analysis of its convergence rate. To evaluate the effectiveness of BDSA, we introduce two novel test functions with an abundance of critical points. We conduct comparative evaluations, including algorithms with inertial terms, that illustrate 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.