We consider a data-driven framework for learning to generate decisions for continuous optimization problems containing constraints that: (i) are not a priori specified, and (ii) vary with an instance-specific input. Our approach uses two machine learning models. The first is a feasibility classifier; we use it as a barrier function in an interior point method (IPM) to train the second model to generate decisions. An oracle is used in training to evaluate the generative model and improve the barrier. In this work, we first develop a theory of optimality for IPMs when given a barrier that approximates the feasible set; we use this to motivate our algorithm and derive several properties, including a generalization bound on producing optimal solutions. Finally, we present preliminary results on an application in predicting personalized radiation therapy treatment plans for head-and-neck cancer.
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