Parallel LQP alternating direction method for solving variational inequality problems with separable structure

In this paper, we propose a logarithmic-quadratic proximal alternating direction method for structured variational inequalities. The predictor is obtained by solving series of related systems of nonlinear equations, and the new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along a new descent direction. Global convergence of the new method is proved under certain assumptions. Preliminary numerical experiments are included to verify the theoretical assertions of the proposed method.Qatar University Start-Up Grant: QUSG-CAS-DMSP-13/14-8.Scopu

On the Optimal Linear Convergence Rate of a Generalized Proximal Point Algorithm

The proximal point algorithm (PPA) has been well studied in the literature. In particular, its linear convergence rate has been studied by Rockafellar in 1976 under certain condition. We consider a generalized PPA in the generic setting of finding a zero point of a maximal monotone operator, and show that the condition proposed by Rockafellar can also sufficiently ensure the linear convergence rate for this generalized PPA. Indeed we show that these linear convergence rates are optimal. Both the exact and inexact versions of this generalized PPA are discussed. The motivation to consider this generalized PPA is that it includes as special cases the relaxed versions of some splitting methods that are originated from PPA. Thus, linear convergence results of this generalized PPA can be used to better understand the convergence of some widely used algorithms in the literature. We focus on the particular convex minimization context and specify Rockafellar's condition to see how to ensure the linear convergence rate for some efficient numerical schemes, including the classical augmented Lagrangian method proposed by Hensen and Powell in 1969 and its relaxed version, the original alternating direction method of multipliers (ADMM) by Glowinski and Marrocco in 1975 and its relaxed version (i.e., the generalized ADMM by Eckstein and Bertsekas in 1992). Some refined conditions weaker than existing ones are proposed in these particular contexts.Comment: 22 pages, 1 figur

Outer approximation algorithms for pseudomonotone equilibrium problems

AbstractWe propose a new method for solving equilibrium problems on a convex subset C, where the underlying function is continuous and pseudomonotone which is called an outer approximation algorithm. The algorithm is to define new approximating subproblems on the convex domains Ck⊇C,k=0,1,… , which forms a generalized iteration scheme for finding a global equilibrium point. Finally we present some numerical experiments to illustrate the behavior of the proposed algorithms

An Asymmetric Proximal Decomposition Method for Convex Programming with Linearly Coupling Constraints

The problems studied are the separable variational inequalities with linearly coupling constraints. Some existing decomposition methods are very problem specific, and the computation load is quite costly. Combining the ideas of proximal point algorithm (PPA) and augmented Lagrangian method (ALM), we propose an asymmetric proximal decomposition method (AsPDM) to solve a wide variety separable problems. By adding an auxiliary quadratic term to the general Lagrangian function, our method can take advantage of the separable feature. We also present an inexact version of AsPDM to reduce the computation load of each iteration. In the computation process, the inexact version only uses the function values. Moreover, the inexact criterion and the step size can be implemented in parallel. The convergence of the proposed method is proved, and numerical experiments are employed to show the advantage of AsPDM