Strongly convergent inertial proximal point method with correction terms: Theory, applications in machine learning for Air Quality prediction

This paper introduces a novel Halpern-type proximal point algorithm that incorporates both inertial and correction terms for solving monotone inclusion problems in real Hilbert spaces. The inertial component accelerates convergence, while the correction term enhances numerical stability and dampens oscillations. Under standard assumptions, we establish strong convergence to a solution. Beyond qualitative convergence, we derive explicit convergence rates: with αk=1/(k+1), we prove ∑i=1k‖yi−xi−1‖2=O(logk) and min1≤i≤k‖yi−wi−1‖2=O(logk/k), providing quantitative performance guarantees. To demonstrate practical utility, we implement the algorithm within an extreme learning machine (ELM) framework to forecast urban benzene (C6H6) concentrations using multisensor air-quality data. The numerical experiments confirm the algorithm’s effectiveness in high-dimensional machine learning applications, highlighting its robustness, stability, and computational efficiency.