 
Summary: WEAK CONVERGENCE OF A RELAXED AND INERTIAL HYBRID
PROJECTIONPROXIMAL POINT ALGORITHM FOR MAXIMAL
MONOTONE OPERATORS IN HILBERT SPACE
FELIPE ALVAREZ
SIAM J. OPTIM. c 2004 Society for Industrial and Applied Mathematics
Vol. 14, No. 3, pp. 773782
Abstract. This paper introduces a general implicit iterative method for finding zeros of a
maximal monotone operator in a Hilbert space which unifies three previously studied strategies:
relaxation, inertial type extrapolation and projection step. The first two strategies are intended to
speed up the convergence of the standard proximal point algorithm, while the third permits one
to perform inexact proximal iterations with fixed relative error tolerance. The paper establishes
the global convergence of the method for the weak topology under appropriate assumptions on the
algorithm parameters.
Key words. Hilbert space, maximal monotone operator, proximal point, inexact iteration,
relative error, separating hyperplane, orthogonal projection, relaxation, weak convergence
AMS subject classifications. 90C25, 65K05, 47J25
DOI. 10.1137/S1052623403427859
1. Introduction. From now on, (H, ·, · ) is a real Hilbert space and the set
valued mapping A : H H is a maximal monotone operator, that is, A is monotone,
i.e., x, y H, v A(x), w A(y), v  w, x  y 0, and the graph GrA =
