Convergence results for proximal point algorithm with inertial and correction terms

This paper designs and studies a numerically fast proximal point algorithm to solve the monotone inclusion problem in Hilbert spaces. Our first proposed algorithm combines the proximal point algorithm, inertial term, and two correction terms. The addition of two correction terms is considered to further numerically accelerate the convergence speed of the inertial proximal point algorithm already studied in the literature. In our convergence results, we obtain both weak and linear convergence of the first proposed algorithm under some standard assumptions. Furthermore, we modify the first algorithm to obtain a strongly convergent algorithm. Applications of our proposed algorithm to mixed variational inequalities, strongly quasi-convex minimization problems, and Douglas–Rachford splitting algorithm are given. Numerical results show that our proposed algorithm outperforms other related algorithms in the literature.