Strong Convergence of an Inexact Proximal Point Algorithm for Equilibrium Problems in Banach Spaces

We develop an inexact proximal point algorithm for solving equilibrium problems in Banach spaces which consists of two principal steps and admits an interesting geometric interpretation. At a certain iterate, first we solve an inexact regularized equilibrium problem with a flexible error criterion to obtain an axillary point. Using this axillary point and the inexact solution of the previous iterate, we construct two appropriate hyperplanes which separate the current iterate from the solution set of the given problem. Then the next iterate is defined as the Bregman projection of the initial point onto the intersection of two halfspaces obtained from the two constructed hyperplanes containing the solution set of the original problem. Assuming standard hypotheses, we present a convergence analysis for our algorithm, establishing that the generated sequence strongly and globally converges to a solution of the problem which is the closest one to the starting point of the algorithm.     

Modified descent-projection method for solving variational inequalities

In this paper, we propose a modified descent-projection method for solving variational inequalities. The method makes use of a descent direction to produce the new iterate and can be viewed as an improvement of the descent-projection method by using a new step size. Under certain conditions, the global convergence of the proposed method is proved. In order to demonstrate the efficiency of the proposed method, we provide numerical results for a traffic equilibrium problems.     

An improved two-step method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is a noncooperative game in which the strategy set of each player, as well as his payoff function, depend on the rival players strategies. As a generalization of the standard Nash equilibrium problem (NEP), the GNEP has recently drawn much attention due to its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. In this paper, we propose an improved two-step (a prediction step and a correction step) method for solving the quasi-variational inequality (QVI) formulation of the GNEP. Per iteration, we first do a projection onto the feasible set defined by the current iterate (prediction) to get a trial point; then, we perform another projection step (correction) to obtain the new iterate. Under certain assumptions, we prove the global convergence of the new algorithm. We also present some numerical results to illustrate the ability of our method, which indicate that our method outperforms the most recent projection-like methods of Zhang et al. (2010).     

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.     

On finite termination of an iterative method for linear complementarity problems

Based on a well-known reformulation of the linear complementarity problem (LCP) as a nondifferentiable system of nonlinear equations, a Newton-type method will be described for the solution of LCPs. Under certain assumptions, it will be shown that this method has a finite termination property, i.e., if an iterate is sufficiently close to a solution of LCP, the method finds this solution in one step. This result will be applied to a recently proposed algorithm by Harker and Pang in order to prove that their algorithm also has the finite termination property.     

Approximating a Zero of Sum of Two Monotone Operators Which Solves a Fixed Point Problem in Reflexive Banach Spaces

Abstract

The purpose of this paper is to introduce an iterative method for approximating a point in the set of zeros of the sum of two monotone mappings, which is also a solution of a fixed point problem for a Bregman strongly nonexpansive mapping in a real reflexive Banach space. With our iterative technique, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a variational inclusion problem for sum of two monotone mappings and the set of solutions of a fixed point problem for Bregman strongly nonexpansive mapping. We give applications of our result to convex minimization problem, convex feasibility problem, variational inequality problem, and equilibrium problem. Our result complements and extends some recent results in literature.     

A Strongly Convergent Method for Nonsmooth Convex Minimization in Hilbert Spaces

In this article, we propose a strongly convergent variant on the projected subgradient method for constrained convex minimization problems in Hilbert spaces. The advantage of the proposed method is that it converges strongly when the problem has solutions, without additional assumptions. The method also has the following desirable property: the sequence converges to the solution of the problem which lies closest to the initial iterate.     

多目标规划求解中修正权系数的方法

我们利用 p级数方法求解多目标规划问题 MOP,并用分层法的思想确定权系数 .求解多目标规划问题 MOP就相当于求解分层的多目标规划问题 L SP.这样 ,我们就可以确定这个函数的目标函数解 ,如果这个解不是满足决策者要求的 Pareto有效解 ,就改变原 MOP问题的权系数。我们就用这个迭代的方法求解多目标规划问题 MOP。     

A half-space projection method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is an n-person noncooperative game in which each player’s strategy set depends on the rivals’ strategy set. In this paper, we presented a half-space projection method for solving the quasi-variational inequality problem which is a formulation of the GNEP. The difference from the known projection methods is due to the next iterate point in this method is obtained by directly projecting a point onto a half-space. Thus, our next iterate point can be represented explicitly. The global convergence is proved under the minimal assumptions. Compared with the known methods, this method can reduce one projection of a vector onto the strategy set per iteration. Numerical results show that this method not only outperforms the known method but is also less dependent on the initial value than the known method.     

A superlinearly convergent projection method for constrained systems of nonlinear equations

In this paper, a new projection method for solving a system of nonlinear equations with convex constraints is presented. Compared with the existing projection method for solving the problem, the projection region in this new algorithm is modified which makes an optimal stepsize available at each iteration and hence guarantees that the next iterate is more closer to the solution set. Under mild conditions, we show that the method is globally convergent, and if an error bound assumption holds in addition, it is shown to be superlinearly convergent. Preliminary numerical experiments also show that this method is more efficient and promising than the existing projection method. This work was done when Yiju Wang visited Chongqing Normal University.     

On the linear convergence of the circumcentered-reflection method

In order to accelerate the Douglas–Rachford method we recently developed the circumcentered-reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.     

Nonlinear Proximal Decomposition Method for Convex Programming

In this paper, we propose a new decomposition method for solving convex programming problems with separable structure. The proposed method is based on the decomposition method proposed by Chen and Teboulle and the nonlinear proximal point algorithm using the Bregman function. An advantage of the proposed method is that, by a suitable choice of the Bregman function, each subproblem becomes essentially the unconstrained minimization of a finite-valued convex function. Under appropriate assumptions, the method is globally convergent to a solution of the problem.     

单障碍问题区域分解法的单调收敛性与收敛速度估计

本文我们考虑一类典型的椭圆型算子的障碍问题的区域分解算法,分析算法的单调收敛性并给出相应的收敛速度估计.障碍问题有着重要的物理背景（参见[3,9]）.近些年来,有关障碍问题的区域分解法方面的研究已经有一些成果.关于线性算子情形,读者可参看[1,2,5,7,8,10,12,13,14,15,17]等文献,而对于非线性算子情形,读者可参看[4,6,16,18].在这些文献中,已经有部分涉及到算法的收敛速度估计.例如,文[15,16]给出了有限元区域分解算法的迭代误差的渐近最大模估计,文[13]给出了求解具M-阵的有限维互补问题     

Inexact Proximal Point Methods for Equilibrium Problems in Banach Spaces

We introduce two inexact proximal-like methods for solving equilibrium problems in reflexive Banach spaces and establish their convergence properties, proving that the sequence generated by each one of them converges to a solution of the equilibrium problem under reasonable assumptions.     

Inexact Iterative Bregman Method for Optimal Control Problems

In this article, we investigate an inexact iterative regularization method based on generalized Bregman distances of an optimal control problem with control constraints. We show robustness and convergence of the inexact Bregman method under a regularity assumption, which is a combination of a source condition and a regularity assumption on the active sets. We also take the discretization error into account. Numerical results are presented to demonstrate the algorithm.     

Selectable Set Randomized Kaczmarz

The Randomized Kaczmarz method (RK) is a stochastic iterative method for solving linear systems that has recently grown in popularity due to its speed and low memory requirement. Selectable Set Randomized Kaczmarz is a variant of RK that leverages existing information about the Kaczmarz iterate to identify an adaptive “selectable set” and thus yields an improved convergence guarantee. In this article, we propose a general perspective for selectable set approaches and prove a convergence result for that framework. In addition, we define two specific selectable set sampling strategies that have competitive convergence guarantees to those of other variants of RK. One selectable set sampling strategy leverages information about the previous iterate, while the other leverages the orthogonality structure of the problem via the Gramian matrix. We complement our theoretical results with numerical experiments that compare our proposed rules with those existing in the literature.     

An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem

In this paper, we are concerned with finding the least solution to the tensor complementarity problem. When the involved tensor is strongly monotone, we present a way to estimate the nonzero elements of the solution in a successive manner. The procedure for identifying the nonzero elements of the solution gives rise to an iterative method of solving the tensor complementarity problem. In each iteration, we obtain an iterate by solving a lower-dimensional tensor equation. After finitely many iterations, the method terminates with a solution to the problem. Moreover, the sequence generated by the method is monotonically convergent to the least solution to the problem. We then extend this idea for general case and propose a sequential mathematical programming method for finding the least solution to the problem. Since the least solution to the tensor complementarity problem is the sparsest solution to the problem, the method can be regarded as an extension of a recent result by Luo et al. (Optim Lett 11:471–482, 2017). Our limited numerical results show that the method can be used to solve the tensor complementarity problem efficiently.     

On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Real Hilbert Spaces

This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problems in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo line search converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.     

Implicit iterative algorithms of the split common fixed point problem for Bregman quasi-nonexpansive mapping in Banach spaces

In this paper, we study a modified implicit rule for finding a solution of split common fixed point problem of a Bregman quasi-nonexpansive mapping in Banach spaces. We propose a new iterative algorithm and prove the strong convergence theorem under appropriate conditions. As an application, the results are applied to solving the zero problem and the equilibrium problem.     

Extragradient Methods for Vector Equilibrium Problems in Banach Spaces

We consider the vector equilibrium problem, which extends the scalar equilibrium problem to vector valued bifunctions, in a Banach space setting. We propose an extragradient method for solving this problem. Under suitable assumptions on the bifunction, we prove that the generated sequence is weakly convergent to a solution of the problem. Then, we propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the problem.