The Bregman Modified Second APG Method for DC Optimization Problems

The Difference of Convex (DC) optimization problem holds significant importance in both optimization theory and practical applications. In recent years, numerous algorithms have been developed to solve DC problems, demonstrating strong performance across various applications. However, most of these methods typically require the L-smoothness property of the objective function; without this condition, the optimization process may exhibit slow convergence or even fail to converge. To address this limitation, researchers often employ Bregman distance as an alternative to Euclidean distance in existing DC algorithms. While this substitution relaxes the requirements on DC functions, it simultaneously introduces greater complexity in theoretical analysis. In this article, we present the Bregman modified second Accelerated Proximal Gradient method (BAPGs), which incorporates Bregman distance into the modified second APG framework (APGs) introduced in Lin and Liu (2019). We establish fundamental decreasing and convergence properties of the sequence generated by BAPGs under standard assumptions. Moreover, with the additional Kurdyka-Łojasiewicz property, we provide a rigorous proof of global convergence and derive explicit convergence rate estimates. Our numerical experiments on two practical problems demonstrate the theoretical validity of BAPGs while showcasing its performance compared to: 1) the original APGs method, 2) existing Bregman-type algorithms, and 3) classical DC approaches. The results confirm that BAPGs effectively balances computational efficiency with relaxed smoothness requirements.