Yunda Dong In this article, we consider the Krasnosel’ski\u{\i}-Mann iteration for approximating a fixed point of any given non-expansive operator in real Hilbert spaces, and we study an inertial version proposed by Maing\'{e} recently. As a result, we suggest new conditions on the inertial factors to ensure weak convergence. They are free of iterates and depend on … Read more
In this article, we consider the problem of finding zeros of two-operator monotone inclusions in real Hilbert spaces, and the second operator has been linearly composed. We suggest an extended splitting method: At each iteration, it mainly solves one resolvent for each operator, respectively. For these two resolvents, the involved two scaling factors can be … Read more
In this article, we consider monotone inclusions in real Hilbert spaces and suggest a new splitting method. The associated monotone inclusions consist of the sum of one bounded linear monotone operator and one inverse strongly monotone operator and one maximal monotone operator. The new method, at each iteration, first implements one forward-backward step as usual … Read more
In this article, we consider monotone inclusions of two operators in real Hilbert spaces, in which one is further assumed to be Lipschitz continuous, and we suggest adding an inertial term to a splitting method at each iteration. The associated weak convergence is analyzed under standard assumptions. The way of choosing steplength is self-adaptive via … Read more
In this paper, we consider how to directly apply the BFGS method to finding a zero point of any given monotone gradient and thus suggest new conditions to locate the corresponding step lengths. The suggested conditions involve curvature condition and merely use gradients’ computations. Furthermore, they can guarantee convergence without any other restrictions. Finally, preliminary … Read more
In this paper, we consider the Douglas-Rachford splitting method for monotone inclusion in Hilbert spaces. It can be implemented as follows: from the current iterate, first use forward-backward step to get the intermediate point, then to get the new iterate. Generally speaking, the sum operator involved in the Douglas-Rachford splitting takes the value of every … Read more
In this paper, we give two new results on the proximal point algorithm for finding a zero point of any given maximal monotone operator. First, we prove that if the sequence of regularization parameters is non-increasing then so is that of residual norms. Then, we use it to show that the proximal point algorithm can … Read more