Inferring Linear Feasible Regions using Inverse Optimization

TL;DR

This paper introduces an inverse optimization framework to infer linear feasible regions from observed data, enabling better understanding and prediction of feasible solutions in linear programming problems.

Contribution

It proposes a novel methodology to recover the complete constraint matrix from multiple observations, including a tractable reformulation and generalized loss functions.

Findings

Validated approach with numerical examples

Demonstrated application in diet recommendation problem

Highlighted differences among proposed loss functions

Abstract

Consider a problem where a set of feasible observations are provided by an expert and a cost function is defined that characterizes which of the observations dominate the others and are hence, preferred. Our goal is to find a set of linear constraints that would render all the given observations feasible while making the preferred ones optimal for the cost (objective) function. By doing so, we infer the implicit feasible region of the linear programming problem. Providing such feasible regions (i) builds a baseline for categorizing future observations as feasible or infeasible, and (ii) allows for using sensitivity analysis to discern changes in optimal solutions if the objective function changes in the future. In this paper, we propose an inverse optimization framework to recover the constraints of a forward optimization problem using multiple past observations as input. We focus on linear models in which the objective function is known but the constraint matrix is partially or fully unknown. We propose a general inverse optimization methodology that recovers the complete constraint matrix and then introduce a tractable equivalent reformulation. Furthermore, we provide and discuss several generalized loss functions to inform the desirable properties of the feasible region based on user preference and historical data. Our numerical examples verify the validity of our approach, emphasize the differences among the proposed measures, and provide intuition for large-scale implementations. We further demonstrate our approach using a diet recommendation problem to show how the proposed models can help impute personalized constraints for each dieter.