The Extended Second APG Method for Constrained DC Problems

In this paper, we develop the extended proximal gradient algorithm with Nesterov’s second acceleration (EAPGs) for constrained difference-of-convex (DC) optimization problems. EAPGs has two key links to existing methods: it extends APGs (for unconstrained DC problems) by adopting the constraint handling idea from Auslender’s ESQM, and serves as a variant of ESQMe with extrapolation replaced by Nesterov’s second acceleration. Under basic assumptions, we establish the subsequential convergence of EAPGs. By introducing a restart technique and leveraging the Kurdyka–Łojasiewicz (KL) property of a suitable potential function, we further prove its global convergence, analyze its convergence rate, and do so under weaker conditions than those for APGs. Additionally, we propose EAPGsr by adding practical restart criteria to EAPGs. Numerical experiments verify the criteria’s efficiency and show that EAPGsr performs well against state-of-the-art methods for constrained and unconstrained DC problems.