Inverse Optimization with Noisy Data

The goal of inverse optimization is to infer unknown parameters of a forward optimization problem using solution data from the forward problem. Here, we present an approach for estimating parameters of convex forward problems from noisy solution data. In contrast to the noiseless case, which can be solved via convex formulations, we show that the noisy setting generally requires solving a non-convex problem. We prove that an advantage of solving this harder problem is that solutions are risk consistent (predictions are best possible) or parameter estimation consistent (true parameters are recovered) under mild conditions. The novelty of our modeling approach is the use of duality theory to upper bound the value function of the forward problem, which improves smoothness of the nonlinear inverse program in comparison to approaches using the KKT conditions; this also yields a convex subproblem that can be used to solve the inverse problem with enumeration. We present a polynomial-time statistical approximation algorithm for solving the inverse problem when the forward problem is a quadratic program, and show that estimates produced by our approximation algorithm retain the statistical consistency properties achieved by solving the original non-convex problem. Lastly, we present numerical results using synthetic and empirical data that show our approach performs competitively with state-of-the-art inverse optimization techniques.