Goodness of Fit in Inverse Optimizaiton

The classical inverse optimization methodology for linear optimization assumes a given solution is a candidate to be optimal. Real data, however, is imperfect and noisy: there is no guarantee that a given solution is optimal for any cost vector. Inspired by regression, this paper presents a unified framework for cost function estimation in linear optimization consisting of a general inverse optimization model and a corresponding goodness-of-fit metric. Although our inverse optimization model is in general nonconvex, we derive a closed-form solution and present the corresponding geometric intuition. Our goodness-of-fit metric, $\rho$, termed the coefficient of complementarity, has similar properties to $R^2$ from regression and is quasiconvex in the input data, leading to an intuitive geometric interpretation. We derive a lower bound for $\rho$ that possesses the same properties but is more tractable. We demonstrate the application of our framework for model estimation and evaluation in production planning and cancer therapy.