Nemirovski's analysis (SIAM J. Optim. 15:229---251, 2005) indicates that the extragradient method has the O(1/t) convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, a class of Fejer monotone projection and contraction methods is developed. Until now, only convergence results are available to these projection and contraction methods, though the numerical experiments indicate that they always outperform the extragradient method. The reason is that the former benefits from the `optimal' step size in the contraction sense. In this paper, we prove the convergence rate under a unified conceptual framework, which includes the projection and contraction methods as special cases and thus perfects the theory of the existing projection and contraction methods. Preliminary numerical results demonstrate that the projection and contraction methods converge twice faster than the extragradient method.

http://www.optimization-online.org/DB_FILE/2011/07/3097.pdf

On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators

In this paper, a self-adaptive three-term conjugate gradient method is proposed for solving monotone nonlinear equations with convex constraints. Under milder conditions, the global convergence of the method is proved. Numerical experiments reported in this paper illustrate that the method is stable and efficient for monotone nonlinear equations, especially for the large-scale problems with convex constraints.

A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints

Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. To solve the subproblems of these algorithms, the projection method takes the iteration in form of u k+1=P ? [u k ?? k d k ]. Interestingly, many of them can be paired such that $\tilde{u}^{k} = P_{\varOmega}[u^{k} - \beta_{k}F(v^{k})] = P_{\varOmega}[\tilde {u}^{k} - (d_{2}^{k} - G d_{1}^{k})]$ , where inf?{β k }>0 and G is a symmetric positive definite matrix. In other words, this projection equation offers a pair of directions, i.e., $d_{1}^{k}$ and $d_{2}^{k}$ for each step. In this paper, for various APPAs we present a unified framework involving the above equations. Unified characterization is investigated for the contraction and convergence properties under the framework. This shows some essential views behind various outlooks. To study and pair various APPAs for different types of variational inequalities, we thus construct the above equations in different expressions according to the framework. Based on our constructed frameworks, it is interesting to see that, by choosing one of the directions ( $d_{1}^{k}$ and $d_{2}^{k}$ ) those studied proximal-like methods always utilize the unit step size namely ? k ?1.

Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods

Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. In Part I of this paper (He et al. in Proximal-like contraction methods for monotone variational inequalities in a unified framework I: effective quadruplet and primary methods, 2010), we proposed a unified framework consisting of an effective quadruplet and a corresponding accepting rule. Under the framework, various existing APPAs can be grouped in the same class of methods (called primary or elementary methods) which adopt one of the geminate directions in the effective quadruplet and take the unit step size. In this paper, we extend the primary methods by using the same effective quadruplet and the accepting rule. The extended (general) contraction methods need only minor extra even negligible costs in each iteration, whereas having better properties than the primary methods in sense of the distance to the solution set. A set of matrix approximation examples as well as six other groups of numerical experiments are constructed to compare the performance between the primary (elementary) and extended (general) methods. As expected, the numerical results show the efficiency of the extended (general) methods are much better than that of the primary (elementary) ones.

Proximal-like contraction methods for monotone variational inequalities in a unified framework II: general methods and numerical experiments

This article examines the application of a path-based algorithm to the static and fixed demand asymmetric traffic assignment problem. The algorithm is of the simplicial decomposition type and it solves the equilibration or master problem step by means of five existing projection methods for variational inequality problems to evaluate their performance on real traffic networks. The projection methods evaluated are: (1) a cost approximation-based method for minimizing the Fukushima's gap function, (2) the modified descent method of Zhu and Marcotte (1988), (3) the double projection method of Khobotov (1987) and three of its recently developed variants (Nadezhkina and Takahashi, 2006; Wang et al., 2010; and He et al., 2012); (4) the method of Solodov and Svaiter (1999); and (5) the method of Solodov and Tseng (1996). These projection methods do not require evaluation of the Jacobians of the path cost functions. The source for asymmetries are link costs with interactions, as in the case of priority ruled junctions. The path-based algorithm has been computationally tested using the previous projection methods on three medium to large networks under different levels of congestion and the computational results are presented and discussed. Comparisons are also made with the basic projection algorithm for the fixed demand asymmetric traffic assignment problem. Despite the lack of monotonicity properties of the test problems, the only method that failed to converge under heavy congestion levels was the basic projection algorithm. The fastest convergence was obtained in all cases solving the master problem step using the method of He et al. (2012), which is a variant of Khobotov's method.

/pdf/applying-projection-based-methods-to-the-asymmetric-traffic-151xv8o6w4.pdf

Xiang Wang

Papers

Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods

Proximal-like contraction methods for monotone variational inequalities in a unified framework II: general methods and numerical experiments

Steplengths in the extragradient type methods