A Tractable Approach To Inverse Optimization Under Euclidean Norm

Abstract

The conventional optimization assumes that the problem and its parameters are known, and it utilizes this information to determine the optimal solution. Inverse optimization works in reverse by determining different parameters of an optimization model such that a given dataset of observed decisions from the past becomes optimal for the model. The parameters imputed through inverse optimization can be in the objective function and/or the constraints of the model. When inferring the constraint parameters, the choice of objective for the inverse optimization problem can result in different inverse optimal solutions. However, it is unclear which solution provides the best fit to the data. In this study, a goodness-of-fit measure is first introduced to evaluate the fit between the model and data and determine the quality of the inferred feasible region based on the distances of data points from its boundary. Next, employing this measure as the objective function, a multi-point inverse optimization problem under the Euclidean norm is proposed to infer the feasible region of a linear optimization model. Given the nonlinear nature of the Euclidean norm, a relaxation technique using the non-smooth L1 penalty function is proposed for the inverse optimization problem. This reformulates the non-convex mixed-integer quadratically-constrained programming problem into a mixed-integer quadratic programming problem which is more tractable. Then, an exact heuristic method and a greedy heuristic method are introduced to alleviate the computational challenges of the problem. Finally, two application examples to illustrate the practicality and effectiveness of our proposed model and solution approach are presented. In the first application, our model determines the implicit criteria based on which a patient is identified as an outpatient without requiring hospital supervision. The second application focuses on improving the recommended diets by uncovering hidden preferences and suggesting meal plans based on individuals' past food choices.