In this paper we introduce an inexact proximal point algorithm using proximal distances for solving variational inequality problems when the mapping is pseudomonotone or quasimonotone. Under some natural assumptions we prove that the sequence generates by the algorithm is convergent for the pseudomonotone case and weakly convergent for the quasimonotone ones. This approach unifies the results o...

In this paper, we consider a framework of inexact proximal point methods for convex optimization that allows a relative error tolerance in the approximate solution of each proximal subproblem and establish its convergence rate. We then show that the well-known forward-backward splitting algorithm for convex optimization belongs to this framework. Finally, we propose and establish the iteration-...

An implementable proximal point algorithm is established for the smooth nonconvex unconstrained minimization problem. At each iteration, the algorithm minimizes approximately a general quadratic by a truncated strategy with step length control. The main contributions are: (i) a framework for updating the proximal parameter; (ii) inexact criteria for approximately solving the subproblems; (iii) ...

In this paper, we analyze the celebrated EM algorithm from the point of view of proximal point algorithms. More precisely, we study a new type of generalization of the EM procedure introduced in [Chretien and Hero (1998)] and called Kullback-proximal algorithms. The proximal framework allows us to prove new results concerning the cluster points. An essential contribution is a detailed analysis ...

We propose a proximal Newton method for solving nondiieren-tiable convex optimization. This method combines the generalized Newton method with Rockafellar's proximal point algorithm. At each step, the proximal point is found approximately and the regu-larization matrix is preconditioned to overcome inexactness of this approximation. We show that such a preconditioning is possible within some ac...

We consider an algorithm that successively samples and minimizes stochastic models of the objective function. We show that under weak-convexity and Lipschitz conditions, the algorithm drives the expected norm of the gradient of the Moreau envelope to zero at the rate O(k−1/4). Our result yields the first complexity guarantees for the stochastic proximal point algorithm on weakly convex problems...

We analyze worst-case complexity of a Proximal augmented Lagrangian (Proximal AL) framework for nonconvex optimization with nonlinear equality constraints. When an approximate first-order (second-order) optimal point is obtained in the subproblem, $$\epsilon $$ original problem can be guaranteed within $${\mathcal {O}}(1/ \epsilon ^{2 - \eta })$$ outer iterations (where $$\eta user-defined para...

In this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set-valued operators in a reflexive Banach space. This algorithm, which we call the proximalprojection method is, essentially, a fixed point procedure and our convergence results are based on new generalizations of Lemma Opial. We show how the proximal-projection meth...

The proximal point algorithm (PPA) is classical, and the resulting proximal subproblems may be as difficult as the original problem. In this paper, we show that with appropriate choices of proximal parameters, the application of PPA to the linearly constrained convex programming can result in easy proximal subproblems. In particular, under some practical assumptions on the objective function, t...

An algorithm is developped for minimizing nonsmooth convex functions. This algortithm extends Elzinga-Moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to Elzinga-Moore’s algorithm what a proximal bundle method is to Kelley’s algorithm. As in proximal bundle methods, a quadratic pr...