董云达的个人主页

This is the main part of Dong’s Ph.D. dissertation, completed in April of 2003. It can be found in Dong’s Researchgate as well.

A Complete List of Articles

A family of operator splitting methods for maximal monotone operators is investigated.It generalizes the Douglas–Peaceman–Rachford–Varga class of methods in the way that it allows the scaling parameters to vary from iteration to iteration non-monotonically.Conditions for convergence of methods within this family and for obtaining a linear rate of convergence are given. These conditions cover more general cases than the existing ones.

In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and the operator’s value at each iterate, we put forth a new idea to achieve a new result on the speed at which the distance sequence tends to zero globally, provided that the problem’s solution set is nonempty and the sequence of squares of the regularization parameters is nonsummable. We show that it is comparable to a classical result of Brézis and Lions in general and becomes better whenever the proximal point algorithm does converge strongly.

库恩-塔克条件是同时含有等式与不等式约束函数的优化问题的最优性条件,已成为优化与控制、运筹与管理的一个核心内容。笔者通过多年的教学实践,发现了两个容易忽略的问题。第一个,对比经典的拉格朗日乘子法(针对于仅含等式约束函数的优化问题)，它的推导方法并不是库恩一塔克条件的推导方法一个推广。第二个,库恩-塔克条件中的拉格朗日乘子(存在的话) 未必是唯一的。

In this article, we consider the problem of finding zeros of monotone inclusions of three operators in real Hilbert spaces, where the first operator’s inverse is strongly monotone and the third is linearly composed, and we suggest an extended splitting method.

In this article, we consider monotone inclusions of three operators in real Hilbert spaces and suggest an inertial version of a generalized Douglas-Rachford splitting. Under standard assumptions, we prove its weak and strong convergence properties. The newly-developed proof techniques are based on the characteristic operator and thus are more self-contained and less convoluted. Rudimentary experiments demonstrated that our suggested inertial splitting method can efficiently solve some large-scale test problems.

In this article, we consider the problem of finding a zero of systems of monotone inclusions in real Hilbert spaces.Furthermore, each monotone inclusion consists of three operators and the third is linearly composed. We suggest a splitting method for solving them, At each iteration, for each monotone inclusion, it mainly needs computations of three resolvents for individual operator. This method can be viewed as a powerful extension of the classical Douglas–Rachford splitting. Under the weakest possible assumptions, by introducing and using the characteristic operator, we analyze its weak convergence.

In this article, we propose two new splitting methods for solving systems of three-operator monotone inclusions in real Hilbert spaces, where the first operator is continuous monotone, the second is maximal monotone and the third is maximal monotone and is linearly composed.

Here are two preprints. One is my manuscript entitled “Directed Krasnosel’skii-Mann Iteration”. The other is my manuscript entitled “On an accelerated Krasnosel’skii-Mann iteration”, and I ever submitted it to JOTA in August, 2023. The former is a revised version of the latter.