We consider the problem of minimizing a convex objective which is the sum of a smooth part, with Lipschitz continuous gradient, and a nonsmooth part. Inspired by various applications, we focus on the case when the nonsmooth part is a composition of a proper closed convex function P and a nonzero affine map, with the proximal mappings of τP , τ > 0, easy to compute. In this case, a direct applic...

We present a new Bregman-function-based algorithm which is a modification of the generalized proximal point method for solving the variational inequality problem with a maximal monotone operator. The principal advantage of the presented algorithm is that it allows a more constructive error tolerance criterion in solving the proximal point subproblems. Furthermore, we eliminate the assumption of...

We describe a primal-dual application of the proximal point algorithm to nonconvex minimization problems. Motivated by the work of Spingarn and more recently by the work of Kaplan and Tichatschke about the proximal point methodology in nonconvex optimization. This paper discusses some local results in two directions. The first one concerns the application of the proximal method of multipliers t...

In this paper a new class of proximal-like algorithms for solving monotone inclusions of the form T (x) 3 0 is derived. It is obtained by applying linear multi-step methods (LMM) of numerical integration in order to solve the differential inclusion ẋ(t) ∈ −T (x(t)), which can be viewed as a generalization of the steepest decent method for a convex function. It is proved that under suitable cond...

In this paper, convergence of the sequence generated by the inexact form of the inertial proximal algorithm is studied. This algorithm which is obtained by the discretization of a nonlinear oscillator with damping dynamical system, has been introduced by Alvarez and Attouch (2001) and Jules and Maingé (2002) for the approximation of a zero of a maximal monotone operator. We establish weak and s...

The proximal point methods have been widely used in the last decades to approximate the solutions of nonlinear equations associated with monotone operators. Inspired by the iterative procedure defined by B. Martinet (1970), R.T. Rockafellar introduced in 1976 the so-called proximal point algorithm (PPA) for a general maximal monotone operator. The sequence generated by this iterative method is ...

In this paper we describe a systematic procedure to analyze the convergence of degenerate preconditioned proximal point algorithms. We establish weak results under mild assumptions that can be easily employed in context splitting methods for monotone inclusion and convex minimization problems. Moreover, show degeneracy preconditioner allows reduction variables involved iteration updates. streng...

Based on the A−maximal (m)−relaxed monotonicity frameworks, the approximation solvability of a general class of variational inclusion problems using the relaxed proximal point algorithm is explored, while generalizing most of the investigations, especially of Xu (2002) on strong convergence of modified version of the relaxed proximal point algorithm, Eckstein and Bertsekas (1992) on weak conver...