Numerous modeling problems in optimization depend on how the optimal value of a linear or quadratic program changes as a function of its right-hand side. For example, in many stochastic programming formulations, the recourse function is explicitly a function of the second-stage right-hand sides. In bilevel programs, the follower's decisions are constrained to be optimal with respect to their own problem of interest which may be implicitly interpreted as a constraint contingent on how the value of follower optimization problem changes with respect to the leader's decisions.

Computational interest in the wide range of applications stemming from the ability to predict how an optimization problem's optimal values vary with respect to changes in constraint values have motivated the study of the \textit{value function}. The value function of a linear or quadratic program parametrizes the optimal value of an optimization problem by the value of its right-hand side. Previous researchers have shown that value functions are highly structured objects that are amenable for various computational applications.

This thesis develops and analyzes how machine learning techniques used for functional regression problems can be used to learn the value function of continuous linear and quadratic programs. We study and design models to verify the accuracy of neural network representations of the value function, and use the trained network value functions as constraints within various optimization problems.