A New Accelerated Forward–Backward Splitting Algorithm for Monotone Inclusions with Application to Data Classification

This paper proposes a new accelerated fixed-point algorithm based on a double-inertial extrapolation technique for solving structured variational inclusion and convex bilevel optimization problems. The underlying framework leverages fixed-point theory and operator splitting methods to address inclusion problems of the form 0∈(A+B)(x), where A is a cocoercive operator and B is a maximally monotone operator defined on a real Hilbert space. The algorithm incorporates two inertial terms and a relaxation step via a contractive mapping, resulting in improved convergence properties and numerical stability. Under mild conditions of step sizes and inertial parameters, we establish strong convergence of the proposed algorithm to a point in the solution set that satisfies a variational inequality with respect to a contractive mapping. Beyond theoretical development, we demonstrate the practical effectiveness of the proposed algorithm by applying it to data classification tasks using Deep Extreme Learning Machines (DELMs). In particular, the training processes of Two-Hidden-Layer ELM (TELM) models is reformulated as convex regularized optimization problems, enabling robust learning without requiring direct matrix inversions. Experimental results on benchmark and real-world medical datasets, including breast cancer and hypertension prediction, confirm the superior performance of our approach in terms of evaluation metrics and convergence. This work unifies and extends existing inertial-type forward–backward schemes, offering a versatile and theoretically grounded optimization tool for both fundamental research and practical applications in machine learning and data science.