Maximum Optimality Margin: A Unified Approach for Contextual Linear Programming and Inverse Linear Programming

TL;DR

This paper introduces the maximum optimality margin approach for predictive analytics in linear programming, offering a unified, efficient, and theoretically sound method that requires only optimal solutions for training, applicable to both contextual and inverse linear programming.

Contribution

It proposes a novel max-margin loss function based on optimality conditions, improving computational efficiency and theoretical guarantees, and enabling inverse linear programming with minimal data requirements.

Findings

The method achieves better generalization bounds.

It is computationally efficient and scalable.

Numerical experiments validate its effectiveness.

Abstract

In this paper, we study the predict-then-optimize problem where the output of a machine learning prediction task is used as the input of some downstream optimization problem, say, the objective coefficient vector of a linear program. The problem is also known as predictive analytics or contextual linear programming. The existing approaches largely suffer from either (i) optimization intractability (a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii) requiring strong condition(s) such as no constraint or loss calibration. We develop a new approach to the problem called \textit{maximum optimality margin} which designs the machine learning loss function by the optimality condition of the downstream optimization. The max-margin formulation enjoys both computational efficiency and good theoretical properties for the learning procedure. More importantly, our new approach only needs the observations of the optimal solution in the training data rather than the objective function, which makes it a new and natural approach to the inverse linear programming problem under both contextual and context-free settings; we also analyze the proposed method under both offline and online settings, and demonstrate its performance using numerical experiments.