Some generalized forward reflected backward methods for solving convex minimization problems

In this paper, we are concerned with the iterative approximation of a solution to the unconstrained convex minimization problem in a real Hilbert space. Our first method is motivated by the recent advancement of the reflected backward method and the generalized reflected backward method. We also introduce a Halpern generalized reflected backward method. The method employs an increasing step size, eliminating the need to know the Lipschitz constant of the gradient of one of the functions. We establish the weak convergence of the core generalized forward reflected backward method (GFRB) and prove the strong convergence of the proposed Halpern generalized forward reflected backward method (HGFRB) to a solution of the convex minimization problem. The practicability and efficiency of the method are demonstrated through several numerical examples, including one application to the regression problem.