Fast and efficient optimization algorithms for split variational inequality problem with applications in data classification problems

Split variational inequality problems (SVIPs) have become a comprehensive framework for representing intricate coupled systems in fields such as optimization, machine learning, signal processing, and inverse problems. In this study, we present three innovative Mann-type iterative methods that integrate self-adjusting step-size selection and inertial dynamics to address SVIPs. Theoretically, we proved weak convergence theorems for each of these algorithms under mild and standard conditions, expanding the traditional results by permitting inertial parameters to surpass the usual range of [0, 1) without necessitating restrictive on-line rules. The algorithms were developed to overcome significant computational hurdles in large-scale nonlinear models, and were particularly effective for data-driven applications. Inspired by practical biomedical classification challenges, we incorporated these algorithms into the training process of the Extreme Learning Machine (ELM) to determine optimal output weights with enhanced numerical stability and precision. This improvement significantly boosted the predictive capabilities of ELM models for early disease detection. Empirical tests on four standard medical datasets, heart disease, lung cancer, heart failure, and prostate cancer, showed notable improvements in classification accuracy and robustness. The blend of relaxed inertial conditions, adaptive step-size updating, and Mann-type structures provided a versatile and efficient framework for solving SVIPs and advancing machine learning-based disease diagnostics.