Unsupervised Machine Learning Hybrid Approach Integrating Linear Programming in Loss Function: A Robust Optimization Technique

Since its formal introduction by Dantzig in 1947, LP has been widely applied across various fields, including operations research, economics, and engineering, due to its ability to optimize objectives subject to linear constraints (Dantzig, 1951; Bazaraa et al., 2013). However, traditional LP approaches have certain limitations, particularly in dealing with non-linear, high-dimensional, and dynamic environments where relationships among variables are complex and non-linear (Bertsimas & Tsitsiklis, 1997). By contrast, machine learning (ML) methods, especially deep learning, have demonstrated remarkable success in modeling complex patterns and making predictions based on large datasets (LeCun et al., 2015; Goodfellow et al., 2016). Despite these strengths, ML models often lack the explicit interpretability and rigorous constraint satisfaction that LP offers (Rudin, 2019). This has motivated researchers to explore hybrid approaches that combine the strengths of LP and ML, aiming to develop models that are both interpretable and powerful in their predictive capabilities. This paper proposes a novel hybrid method that integrates LP within the loss function of an unsupervised machine learning model. By embedding LP constraints directly into the ML framework, this approach not only maintains the interpretability and constraint satisfaction of LP but also leverages the flexibility and learning capacity of ML. This integration is particularly beneficial in unsupervised or semi-supervised settings, where traditional LP methods may struggle to provide robust solutions due to the lack of labeled data (Amos & Kolter, 2017).