In this paper, we study a modification of the Golden Ratio Algorithm (GRAAL) for solving monotone variational inequalities. We present an inertial Bregman modification of GRAAL for solving the aforementioned problem. Our proposed algorithm contains the inertial technique, the Bregman distance and a fully adaptive stepsize. We present a convergence result when the cost operator is monotone and locally Lipschitz continuous. Furthermore, we obtain the sublinear rate of convergence of our proposed method. Finally, we present numerical experiments to illustrate the applicability of our proposed method.

This paper studies new accelerated optimization algorithms and applied the algorithms to prediction of breast cancer through machine learning approach. To do this, we first introduce new fast CQ algorithms and obtain weak convergence results. In one of our proposed algorithms (inertial-type CQ Algorithm), the inertial choice could be negative and even greater than 1 with no on-line rule imposed in order to obtain convergence results. This is a major improvement over other inertial-type algorithms in the literature where inertial choices are restrictive to [0 ,1) and on-line rule is imposed. Then we validate the applicability of the proposed CQ algorithms to real-life applications by predicting breast cancer by updating the optimal weight in machine learning. We use the mammographic mass dataset from the UC Irvine machine learning repository that is available on the UCI website as a training set to show the superiority of our algorithms over existing ones in the literature.