Convergence of Two-Step Inertial Mann-Type Forward–Backward–Forward with an Application to Breast Ultrasound Image Classification

In this paper, we propose a two-step inertial Mann-type forward–backward–forward method with a self-adaptive step-size strategy for solving variational inclusion problems in real Hilbert spaces. The proposed method modifies two inertial terms to exploit information from previous iterates, while the self-adaptive step-size rule avoids the need for prior knowledge of the Lipschitz constant of the underlying operator. Under suitable conditions on the operators and control parameters, we establish the weak convergence of the generated sequence to a solution of the variational inclusion problem. The proposed method is then applied to train ELM classifiers in a two-stage breast ultrasound classification framework using the BUSI dataset. Stage 1 distinguishes normal images from lesion-containing images, while Stage 2 classifies lesions as benign or malignant using predicted mask regions of interest. Numerical results and comparisons with other optimization methods and standard machine learning classifiers demonstrate the applicability and classification performance of the proposed method.