Inverse Optimization: Closed-form Solutions, Geometry and Goodness of fit

TL;DR

This paper introduces a unified inverse optimization framework with a closed-form solution and a goodness-of-fit metric, enabling better estimation and evaluation of linear models in noisy real-world data.

Contribution

It presents a nonconvex inverse optimization model with a closed-form solution and a novel goodness-of-fit metric analogous to R^2, with applications in production planning and cancer therapy.

Findings

Closed-form solution for inverse optimization model

Goodness-of-fit metric ρ similar to R^2 and polynomial-time computable

Lower bound for ρ that is tight and easier to compute

Abstract

In classical inverse linear optimization, one assumes a given solution is a candidate to be optimal. Real data is imperfect and noisy, so there is no guarantee this assumption is satisfied. Inspired by regression, this paper presents a unified framework for cost function estimation in linear optimization comprising a general inverse optimization model and a corresponding goodness-of-fit metric. Although our inverse optimization model is nonconvex, we derive a closed-form solution and present the geometric intuition. Our goodness-of-fit metric, $\rho$, 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. While $\rho$ is computable in polynomial-time, we derive a lower bound that possesses the same properties, is tight for several important model variations, and is even easier to compute. We demonstrate the application of our framework for model estimation and evaluation in production planning and cancer therapy.