This article introduces an iterative algorithm that is created by integrating the $S^{*}$-iteration process with the inertial forward-backward algorithm. The algorithm is designed to solve optimization problems formulated as variational inclusions in a real Hilbert space. We establish the weak convergence of the algorithm under conventional assumptions. One of the applications of the algorithm is to solve the extreme learning machine, which can be transformed into the variational inclusion problem. Different algorithms, with all parameters set to be identical, are employed to solve the stroke classification problem in order to evaluate the algorithm’s performance. The results indicate that our algorithm converges faster than others and achieves a precision of 93.90%, a recall of 100%, and an F1-score of 96.58%.

Upper Northern Thailand is a critical region reflecting the severity of seasonal PM2.5 pollution and its transboundary spread, yet accurate spatial monitoring remains difficult due to complex mountainous terrain and a sparse network of physical monitoring stations. To resolve this observational limitation under a fixed sensor budget, this study develops a multi-criteria spatial optimization framework using weighted-sum scalarization to identify optimal locations for network expansion. We integrate set-dependent geostatistical Average Kriging Variance (AKV) reduction, meteorological correlation graphs, and cross-source satellite discrepancy into a unified combinatorial objective. Feature selection reveals that coupling rainfall, relative humidity, and temperature yields the best predictive performance, achieving an $R^2$ of 0.8785 and minimizing RMSE to $33.91\mathsf{\mu }{\mathrm{g}/\mathrm{m}}^3$ under Universal Kriging (UK). While Ordinary Kriging (OK) and Universal Kriging produce consistent spatial patterns, UK systematically secures superior variance reduction by accounting for coordinate-based spatial trends. Comparing optimization algorithms demonstrates distinct search dynamics: Genetic Algorithms (GAs) converge faster for smaller network expansion sizes, whereas Simulated Annealing (SA) becomes more scalable and efficient when adding 20 or more stations. Incorporating graph centrality measures (betweenness centrality and clustering coefficient) actively promotes spatial dispersion and prevents candidate stations from clustering redundantly in high-uncertainty areas. Furthermore, retrospective ground-station masking experiments demonstrate that the framework reliably targets genuine observational blind spots, reducing out-of-sample prediction RMSE by up to 40.98% upon network expansion. The proposed framework provides a practical, data-driven decision-support screening tool for environmental agencies seeking cost-effective air quality infrastructure expansion in data-sparse regions.