New inertial factors of the Krasnoselskii-Mann iteration

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

Weak convergence of an extended splitting method for monotone inclusions

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

A new splitting method for monotone inclusions of three operators

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

New inertial factors of a splitting method for monotone inclusions

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

Step lengths in BFGS method for monotone gradients

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

An asymptotic inclusion speed for the Douglas-Rachford splitting method in Hilbert spaces

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