On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators (Q929604)

This paper introduces general iterative methods (called Algorithms 2.1 and 2.2) for finding zeros of a maximal monotone operator in a Hilbert space which unify two previously studied iterative methods: the relaxed proximal point algorithm [\textit{H.\,K.\thinspace Xu}, J.~Lond.\ Math.\ Soc., II.\ Ser.\ 66, No.\,1, 240--256 (2002; Zbl 1013.47032)] and the inexact hybrid extragradient proximal point algorithm [\textit{R.\,S.\thinspace Burachik, S.\,Scheimberg} and \textit{B.\,F.\thinspace Svaiter}, J.~Optim.\ Theory Appl.\ 111, No.\,1, 117--136 (2001; Zbl 1054.90088)]. The authors establish both weak and strong convergence of the methods under suitable assumptions. The Algorithm 2.1 introduced in this paper also provides a unified framework for the convergence analysis of the classical proximal point method, the perturbed proximal point method for optimization [\textit{B.\,Lemaire}, Lect.\ Notes Econ.\ Math.\ Syst.\ 382, 39--51 (1992; Zbl 0763.90072)], the perturbed proximal point method [\textit{R.\,S.\thinspace Burachik, A.\,N.\thinspace Iusem} and \textit{B.\,F.\thinspace Svaiter}, Set-Valued Anal.\ 5, No.\,2, 159--180 (1997; Zbl 0882.90105)] and the hybrid extragradient proximal point method [\textit{M.\,V.\thinspace Solodov} and \textit{B.\,F.\thinspace Svaiter}, Set-Valued Anal.\ 7, 323--345 (1999; Zbl 0959.90038)].