Inexact cuts in SDDP applied to multistage stochastic nondifferentiable problems

In [13], an Inexact variant of Stochastic Dual Dynamic Programming (SDDP) called ISDDP was introduced which uses approximate (instead of exact with SDDP) primal dual solutions of the problems solved in the forward and backward passes of the method. That variant of SDDP was studied in [13] for linear and for differentiable nonlinear Multistage Stochastic … Read more

A proximal-Newton method for unconstrained convex optimization in Hilbert spaces

We propose and study the iteration-complexity of a proximal-Newton method for finding approximate solutions of the problem of minimizing a twice continuously differentiable convex function on a (possibly infinite dimensional) Hilbert space. We prove global convergence rates for obtaining approximate solutions in terms of function/gradient values. Our main results follow from an iteration-complexity study of … Read more

Iteration-complexity of a Rockafellar’s proximal method of multipliers for convex programming based on second-order approximations

This paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar’s proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar’s PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm … Read more

Regularized HPE-type methods for solving monotone inclusions with improved pointwise iteration-complexity bounds

This paper studies the iteration-complexity of new regularized hybrid proximal extragradient (HPE)-type methods for solving monotone inclusion problems (MIPs). The new (regularized HPE-type) methods essentially consist of instances of the standard HPE method applied to regularizations of the original MIP. It is shown that its pointwise iteration-complexity considerably improves the one of the HPE method … Read more

Diffusion Methods for Classification with Pairwise Relationships

We define two algorithms for propagating information in classification problems with pairwise relationships. The algorithms involve contraction maps and are related to non-linear diffusion and random walks on graphs. The approach is also related to message passing and mean field methods. The algorithms we describe are guaranteed to converge on graphs with arbitrary topology. Moreover … Read more

A dynamic approach to a proximal-Newton method for monotone inclusions in Hilbert spaces, with complexity $\bigo(1/n^2)$

In a Hilbert setting, we introduce a new dynamic system and associated algorithms aimed at solving by rapid methods, monotone inclusions. Given a maximal monotone operator $A$, the evolution is governed by the time dependent operator $I -(I + \lambda(t) {A})^{-1}$, where, in the resolvent, the positive control parameter $\lambda(t)$ tends to infinity as $t … Read more

Primal-dual regularized SQP and SQCQP type methods for convex programming and their complexity analysis

This paper presents and studies the iteration-complexity of two new inexact variants of Rockafellar’s proximal method of multipliers (PMM) for solving convex programming (CP) problems with a fi nite number of functional inequality constraints. In contrast to the first variant which solves convex quadratic programming (QP) subproblems at every iteration, the second one solves convex constrained … Read more

A note on Fejér-monotone sequences in product spaces and its applications to the dual convergence of augmented Lagrangian methods

In a recent Math. Program. paper, Eckstein and Silva proposed a new error criterion for the approximate solutions of augmented Lagrangian subproblems. Based on a saddle-point formulation of the primal and dual problems, they proved that dual sequences generated by augmented Lagrangians under this error criterion are bounded and that theirs limit points are dual … Read more

An inexact block-decomposition method for extra large-scale conic semidefinite programming

In this paper, we present an inexact block-decomposition (BD) first-order method for solving standard form conic semidefinite programming (SDP) which avoids computations of exact projections onto the manifold defined by the affine constraints and, as a result, is able to handle extra large SDP instances. The method is based on a two-block reformulation of the … Read more

A hybrid proximal extragradient self-concordant primal barrier method for monotone variational inequalities

In this paper we present a primal interior-point hybrid proximal extragradient (HPE) method for solving a monotone variational inequality over a closed convex set endowed with a self-concordant barrier and whose underlying map has Lipschitz continuous derivative. In contrast to the method of “R. D. C. Monteiro and B. F. Svaiter, Iteration-Complexity of a Newton … Read more