Learning Generalized Linear Programming Value Functions

We develop a theoretically-grounded learning method for the Generalized Linear Programming Value Function (GVF), which models the optimal value of a linear programming (LP) problem as its objective and constraint bounds vary. This function plays a fundamental role in algorithmic techniques for large-scale optimization, particularly in decomposition for two-stage mixed-integer linear programs (MILPs). This paper establishes a structural characterization of the GVF that enables it to be modeled as a particular neural network architecture, which we then use to learn the GVF in a way that benefits from three notable properties. First, our method produces a true under-approximation of the value function with respect to the constraint bounds. Second, the model is input-convex in the constraint bounds, which not only matches the structure of the GVF but also enables the trained model to be efficiently optimized over using LP.