On the diagonal proximal point algorithms

Abstract

We study the asymptotic behaviour of the diagonal proximal point algorithms governed by a sequence of maximally monotone operators. With a summability condition on the Brézis–Haraux functions of the sequence of operators, we obtain several ergodic and weak convergence results in the monotone and subgradient cases. This approach is of relevance to and motivated by Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790]. Some strong convergence results are presented as well. Our results in this paper include discrete versions of the results in Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790], and also unify and improve many existing results in Attouch et al. [Prox-penalization and splitting methods for constrained variational problems. SIAM J Optim. 2011;21:149–173], Cabot [Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization. SIAM J Optim. 2005;15:555–572], Lehdili and Moudafi [Combining the proximal algorithm and Tikhonov regularization. Optimization. 1996;37:239–252], Xu [A regularization method for the proximal point algorithm. J Glob Optim. 2006;36:115–125].