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.
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