A Dual-Projective Double Inertial Forward-Backward Splitting Algorithm for Variational Inclusion Problems with Applications to External Validation in Breast Cancer Diagnosis

This study proposes a novel algorithm that combines a double inertial technique with a two step projection-based forward-backward splitting method to solve variational inclusion problems in real Hilbert spaces. A weak convergence theorem is established under suitable conditions, and its validity is demonstrated in an infinite-dimensional setting. The model is evaluated in a machine learning context using external val idation—training on public data and testing on separate real-world clinical data for breast cancer prediction. Experimental results show promising classification performance: accuracy of 83.16%, precision of 86.56%, recall of 85.82%, and F1-score of 86.19%. The proposed method achieves superior computational efficiency, requiring only 47 iterations and 0.2998 seconds, compared to over 200 iterations and 2.3 seconds for baseline algorithms. Confusion matrix and ROC analysis confirm robust multi-class classification, with notable separability in Class 2 and Class 4. Training and loss curves further demonstrate model stability and generalizability, with no signs of overfitting. Overall, the algorithm presents a fast, accurate, and practical solution for real-world medical applications