Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications

In this work, we construct four efficient multi-step inertial relaxed algorithms based on the monotonic step-length criterion which does not require any information about the norm of the underlying operator or the use of a line search procedure for split feasibility problems in infinite-dimensional Hilbert spaces. The first and the third are the general multi-step inertial-type methods, which unify two steps of the multi-step inertial terms with the golden ratio-based and an alternating golden ratio-based extrapolation steps, respectively, to improve the speed of convergence of their sequences of iterates to a solution of the problem, while the second and the fourth are the three-term conjugate gradient-like and multi-step inertial-type methods, which integrate both the three-term conjugate gradient-like direction and a multi-step inertial term with the golden ratio-based and an alternating golden ratio-based extrapolation steps, respectively, to accelerate their sequences of iterates toward a solution of the problem. Under some simple and weaker assumptions, we formulate and prove some strong convergence theorems for each of these algorithms based on the convergence theorem of a golden ratio-based relaxed algorithm with perturbations and the alternating golden ratio-based relaxed algorithm with perturbations in infinite-dimensional real Hilbert spaces. Moreover, we analyze their applications in classification problems for an interesting real-world dataset based on the extreme learning machine (ELM) with the $\ell_{1}-\ell_{2}$ hybrid regularization approach and in solving constrained minimization problems in infinite-dimensional Hilbert spaces. In all the experiments, our proposed algorithms, which generalizes several algorithms in the literature, comparatively achieve better performance than some related algorithms.