[Paper Review] Learning Linear Programs from Optimal Decisions

Yingcong Tan, Daria Terekhov+1 more|arXiv (Cornell University)|Jun 16, 2020

Machine Learning and Algorithms53 references8 citations

TL;DR

This paper proposes a gradient-based framework for learning linear programs from optimal decisions, enabling joint optimization of costs, constraints, and loss functions via implicit differentiation. It successfully learns synthetic linear programs and minimum-cost multi-commodity flow instances, outperforming non-gradient methods in high-dimensional parameter spaces using a PyTorch-compatible interior point solver with backpropagation support.

ABSTRACT

We propose a flexible gradient-based framework for learning linear programs from optimal decisions. Linear programs are often specified by hand, using prior knowledge of relevant costs and constraints. In some applications, linear programs must instead be learned from observations of optimal decisions. Learning from optimal decisions is a particularly challenging bi-level problem, and much of the related inverse optimization literature is dedicated to special cases. We tackle the general problem, learning all parameters jointly while allowing flexible parametrizations of costs, constraints, and loss functions. We also address challenges specific to learning linear programs, such as empty feasible regions and non-unique optimal decisions. Experiments show that our method successfully learns synthetic linear programs and minimum-cost multi-commodity flow instances for which previous methods are not directly applicable. We also provide a fast batch-mode PyTorch implementation of the homogeneous interior point algorithm, which supports gradients by implicit differentiation or backpropagation.

Motivation & Objective

• To address the challenge of learning linear programs when only optimal decisions are observed, rather than costs or constraints.
• To develop a flexible, end-to-end differentiable framework that jointly learns all LP parameters—costs, constraints, and loss functions—under a parametric formulation.
• To overcome key difficulties in inverse optimization, including infeasible solutions, empty feasible regions, and non-unique optimal decisions.
• To enable generalization to novel conditions by learning a parametric linear program (PLP) that reproduces observed optimal decisions.

Proposed method

• Formulates the learning problem as a bi-level optimization where the outer loop optimizes parameters w to make observed decisions feasible and optimal.
• Uses a parametric linear program (FOP) with coefficients dependent on features u and parameters w, enabling generalization to new conditions.
• Employs a homogeneous interior point algorithm implemented in PyTorch that supports gradients via implicit differentiation or backpropagation.
• Applies sequential quadratic programming (SQP) with implicit gradients to efficiently optimize w in high-dimensional spaces.
• Introduces a differentiable solver that enables end-to-end training by backpropagating through the solution of the inner optimization problem.
• Handles non-unique optimal solutions and empty feasible regions by incorporating feasibility and optimality constraints into the loss function.

Experimental results

Research questions

• RQ1 Can a gradient-based method effectively learn all parameters of a linear program—including costs, inequality and equality constraints—from observed optimal decisions?
• RQ2 How does the proposed method handle cases where the observed decision is infeasible or where the feasible region is empty under certain parameter settings?
• RQ3 What is the performance of gradient-based optimization (e.g., SQP) compared to non-gradient methods (e.g., COBYLA, SLSQP) in high-dimensional parameter spaces?
• RQ4 Can the method generalize to unseen conditions by learning a parametric linear program that consistently reproduces optimal decisions?
• RQ5 How robust is the method to variations in numerical tolerances and solver configurations?

Key findings

• Gradient-based SQP methods consistently outperform non-gradient methods like COBYLA and SLSQP, especially in high-dimensional parameter spaces.
• The method successfully learns synthetic linear programs and minimum-cost multi-commodity flow instances, even when prior methods are not directly applicable.
• The homogeneous interior point solver in PyTorch enables differentiable optimization, supporting backpropagation through the solution of the inner LP.
• The approach achieves zero training loss on 100% of tested instances with D=10, M1=80, and M2=2, demonstrating strong convergence.
• The method remains robust to changes in numerical tolerance and PyTorch version, indicating stable performance across configurations.
• Non-gradient methods like RS fail to find feasible parameters due to infeasibility of equality constraints, highlighting the necessity of gradient-based optimization.