An inexact proximal decomposition method for variational inequalities with separable structure

This paper presents an inexact proximal method for solving monotone variational inequality problems with a given separable structure. The proposed algorithm is a natural extension of the Proximal Multiplier Algorithm with Proximal Distances (PMAPD) proposed by Sarmiento et al. [Optimization, 65(2), (2016), pp. 501-537], which unified the works of Chen and Teboulle (PCPM method) and Kyono and Fukushima (NPCPMM) developed for solving convex programs with a particular separable structure. The resulting method combines the recent proximal distances theory introduced by Auslender and Teboulle [SIAM J. Optim., 16 (2006), pp. 697-725] with a decomposition method given by Chen and Teboulle for convex problems and extends the results of the Entropic Proximal Decomposition Method proposed by Auslender and Teboulle, which used to Logarithmic Quadratic proximal distances. Under some mild assumptions on the problem we prove a global convergence of the primal-dual sequences produced by the algorithm.

Belief Propagation Network for Hard Inductive Semi-Supervised Learning

Given graph-structured data, how can we train a robust classifier in a semi-supervised setting that performs well without neighborhood information? In this work, we propose belief propagation networks (BPN), a novel approach to train a deep neural network in a hard inductive setting, where the test data are given without neighborhood information. BPN uses a differentiable classifier to compute the prior distributions of nodes, and then diffuses the priors through the graphical structure, independently from the prior computation. This separable structure improves the generalization performance of BPN for isolated test instances, compared with previous approaches that jointly use the feature and neighborhood without distinction. As a result, BPN outperforms state-of-the-art methods in four datasets with an average margin of 2.4% points in accuracy.

An inexact proximal algorithm for variational inequalities

An inexact proximal algorithm for variational inequalities O. Sarmiento, E. A. Papa Quiroz and P. R. Oliveira Programa de Ingeniería de Sistemas y Computación - COPPE, Universidad Federal de Rio de Janeiro, 68511 CEP: 21941-972, Rio de Janeiro, Brasil. DOI: https://doi.org/10.33017/RevECIPeru2015.0018/ Abstract This paper presents a new inexact proximal method for solving monotone variational inequality problems with a given separable structure. The resulting method combines the recent proximal distances theory introduced by Auslender and Teboulle (2006) with a decomposition method given by Chen and Teboulle that was proposed to solve convex optimization problems. This method extends and generalizes proximal methods using Bregman, Phi-divergences and Quadratic logarithmic distances. Taking mild assumptions we prove that the primal-dual sequences produced by algorithm is well-defined and converge to optimal solution of the variational inequality problem. Furthermore, we show some numerical experiments, for the particular case to solve convex optimization problem, showing that the algorithm is perfectly implementable. Keywords: Inexact proximal method, variational inequality, separable structure, proximal distances. Resumen En este artículo presentamos un nuevo método proximal inexacto para resolver problemas de desigualdad variacional monótono con una estructura separable. El método resultante combina la reciente teoria de distancias proximales introducidas por Auslender y Teboulle (2006) con un método de descomposición proximal dado por Chen y Teboulle que fue propuesto para resolver problemas de optimización convexa. Este método extiende y generaliza métodos proximales usando distancias de Bregman, Phi-divergencias y logaritmo cuadrático, Asumiendo hipotesis adecuadas probamos que la sucesión primal-dual generada por el algoritmo está bien definido y converge a la solución óptima de un problema de desigualdad variacional. Además presentamos algunos resultados computacionales para el caso particular de resolver problemas de optimización convexa, mostrando asi que el algoritmo es perfectamente implementable. Descriptores: Método proximal inexacto, desigualdad variacional, estructura separable, distancias proximales.

SQP alternating direction method with a new optimal step size for solving variational inequality problems with separable structure

In this paper, we suggest and analyze a new alternating direction scheme for the separable constrained convex programming problem. The theme of this paper is twofold. First, we consider the square-quadratic proximal (SQP) method. Next, by combining the alternating direction method with SQP method, we propose a descent SQP alternating direction method by using the same descent direction as in [6] with a new step size ?k. Under appropriate conditions, the global convergence of the proposed method is proved. We show the O(1/t) convergence rate for the SQP alternating direction method. Some preliminary computational results are given to illustrate the efficiency of the proposed method.