Banach空间中极大单调算子的一个邻近点算法

设E是Banach空间,T∶E→2E*是极大单调算子,T-10≠ф.令x0∈E,yn=(J λnT)-1xn en,xn 1=J-1(αnJxn (1-αn)Jyn),n0,λn>0,αn∈[0,1],文章研究了{xn}收敛性.     

Banach空间有限个极大单调算子公共零点的迭代强收敛定理

令E为实光滑、一致凸Banach空间,E为其对偶空间.令Ai E×E,i=1,2,…,m,为极大单调算子且∩mi=1Ai-10≠.将引进一个新定义、给出一种新迭代算法,并利用Lyapunov泛函与广义投影算子等技巧,证明迭代序列强收敛于Ai的公共零点,i=1,2,…,m.去掉了以往结论中过强的限定条件,是对笔者以往工作的延续。     

Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space

In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim. 14 (1976), 877–898] and Kamimura and Takahashi [J. Approx. Theory 106 (2000), 226–240]. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem.     

极大单调算子和非扩展映射的迭代收敛定理及其应用

In this paper, two iterative schemes for approximating common element of the set of zero points of maximal monotone operators and the set of fixed points of a kind of generalized nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Two strong convergence theorems are obtained and their applications on finding the minimizer of a kind of convex functional are discussed, which extend some previous work.     

Convergence of over-relaxed contraction-proximal point algorithm in Hilbert spaces

The proximal point algorithm (PPA) is a classical method for finding zeros of maximal monotone operators. It is known that the algorithm only has weak convergence in a general Hilbert space. Recently, Wang, Wang and Xu proposed two modifications of the PPA and established strong convergence theorems on these two algorithms. However, these two convergence theorems exclude an important case, namely, the over-relaxed case. In this paper, we extend the above convergence theorems from under-relaxed case to the over-relaxed case, which in turn improve the performance of these two algorithms. Preliminary numerical experiments show that the algorithm with over-relaxed parameter performs better than that with under-relaxed parameter.     

Regular Maximal Monotone Operators

The purpose of this paper is to introduce a class of maximal monotone operators on Banach spaces that contains all maximal monotone operators on reflexive spaces, all subdifferential operators of proper, lsc, convex functions, and, more generally, all maximal monotone operators that verify the simplest possible sum theorem. Dually strongly maximal monotone operators are also contained in this class. We shall prove that if T is an operator in this class, then (the norm closure of its domain) is convex, the interior of co(dom(T)) (the convex hull of the domain of T) is exactly the set of all points of at which T is locally bounded, and T is maximal monotone locally, as well as other results.     

Banach空间中极大单调算子零点的投影算法

本文设计了一种极大单调算子零点的带误差项的新投影迭代算法,并在Banach空间中,利用Lyapunov泛函与广义投影映射等技巧,证明了迭代序列强收敛于极大单调算子零点的结论.     

Banach空间中有限个极大单调算子公共零点的投影算法

设计了一种带误差项的新投影迭代算法,利用Lyapunov泛函与广义投影映射等技巧,在Banach空间中,证明了迭代序列强收敛于有限个极大单调算子公共零点的结论.     

论强单调算子下的邻点算法

考虑强单调算子下的邻点算法.证明了它容许弱于可和性条件的绝对误差以及每个松弛因子不超过某个较小正数的相对误差.     

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.     

Modified Proximal-Point Algorithm for Maximal Monotone Operators in Banach Spaces

We introduce an iterative sequence for finding the solution to 0∈T(v), where T : E⇉E ^* is a maximal monotone operator in a smooth and uniformly convex Banach space E. This iterative procedure is a combination of iterative algorithms proposed by Kohsaka and Takahashi (Abstr. Appl. Anal. 3:239–249, 2004) and Kamamura, Kohsaka and Takahashi (Set-Valued Anal. 12:417–429, 2004). We prove a strong convergence theorem and a weak convergence theorem under different conditions respectively and give an estimate of the convergence rate of the algorithm. An application to minimization problems is given. This work was partially supported by the National Natural Sciences Grant 10671050 and the Heilongjiang Province Natural Sciences Grant A200607. The authors thank the referees for useful comments improving the presentation and Professor K. Kohsaka for pointing out Ref. 7.     

Banach空间有限个极大单调算子公共零点的迭代注记

令E为实光滑、一致凸Banach空间,E*为其对偶空间.令AiE×E*,i=1,2,…,m,为极大单调算子且∩mi=1Ai-10≠φ.将给出一种计算量较小的新迭代算法,并利用Lyapunov泛函与广义投影算子等技巧,证明迭代序列弱收敛于A的公共零点,i=1,2,…,m.     

A Regularization Method for the Proximal Point Algorithm

A regularization method for the proximal point algorithm of finding a zero for a maximal monotone operator in a Hilbert space is proposed. Strong convergence of this algorithm is proved.Hong-Kun Xu: Supported in part by NRF     

Hilbert空间中Lipschitz单调映像变分不等式解的另一个逼近定理

首先给出了Hilbert空间中Lipschitz单调映像变分不等式解的迭代格式,并证明了其收敛性.作为应用,证明了Hilbert空间中Lipschitz伪压缩映像的强收敛定理,扩展了已知的相关结果.     

Weak and Strong Convergence of Solutions to Accretive Operator Inclusions and Applications

Our purpose in this paper is to approximate solutions of accretive operators in Banach spaces. Motivated by Halpern's iteration and Mann's iteration, we prove weak and strong convergence theorems for resolvents of accretive operators. Using these results, we consider the convex minimization problem of finding a minimizer of a proper lower semicontinuous convex function and the variational problem of finding a solution of a variational inequality.     

极大单调算子的一个邻近点算法

设E是自反的Banach空间,T∶E→2E是极大单调算子.T-10≠.令x0∈E,yn=(J λnT)-1xn en,xn 1=J-1(αnJxn (1-αn)Jyn),n≥0,λn>0,αn∈[0,1],本文研究了{xn}收敛性.     

The Sum and Chain Rules for Maximal Monotone Operators

This paper is primarily concerned with the problem of maximality for the sum A + B and composition L* ML in non-reflexive Banach space settings under qualifications constraints involving the domains of A, B, M. Here X, Y are Banach spaces with duals X*, Y*, A, B: X ⇉ X*, M: Y ⇉ Y* are multi-valued maximal monotone operators, and L: X → Y is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results on the topic are presented.      

Banach空间中m-d-增生算子零点迭代算法的构造

研究了Banach空间中m-d-增生算子零点的迭代算法的构造问题,获得了一个强收敛定理.