Inverse Optimization Latent Variable Models for Learning Costs Applied to Route Problems

2025-09-19

AI Summary

For constrained optimization problems (COPs) with unknown cost functions, this paper proposes an inverse optimization latent variable model that jointly learns the latent distribution of cost functions and structured, constraint-satisfying decodings. Methodologically, it is the first to integrate latent variable modeling with inverse optimization, employing the Fenchel–Young loss to enable gradient backpropagation through non-differentiable, deterministic solvers—thus supporting end-to-end optimization in latent space. Key contributions include: (1) modeling the *distribution*—rather than a point estimate—of cost functions, enabling behavior diversity modeling and generalizable inference across multi-agent or multi-scenario settings; and (2) guaranteeing strictly feasible and interpretable decoding paths. Evaluated on real-world vessel and taxi trajectory datasets, as well as synthetic graph-path data, the method achieves significant improvements in path reconstruction accuracy, predictive distribution quality, and interpretability of latent representations.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Learning representations for solutions of constrained optimization problems (COPs) with unknown cost functions is challenging, as models like (Variational) Autoencoders struggle to enforce constraints when decoding structured outputs. We propose an Inverse Optimization Latent Variable Model (IO-LVM) that learns a latent space of COP cost functions from observed solutions and reconstructs feasible outputs by solving a COP with a solver in the loop. Our approach leverages estimated gradients of a Fenchel-Young loss through a non-differentiable deterministic solver to shape the latent space. Unlike standard Inverse Optimization or Inverse Reinforcement Learning methods, which typically recover a single or context-specific cost function, IO-LVM captures a distribution over cost functions, enabling the identification of diverse solution behaviors arising from different agents or conditions not available during the training process. We validate our method on real-world datasets of ship and taxi routes, as well as paths in synthetic graphs, demonstrating its ability to reconstruct paths and cycles, predict their distributions, and yield interpretable latent representations.

Problem

• Learning cost functions from observed solutions
• Reconstructing feasible outputs using solver
• Capturing distribution over cost functions

Innovation

• Latent space learning for cost functions
• Fenchel-Young loss with non-differentiable solver
• Distribution over cost functions captures diversity