Convex duality contracts for production-grade mathematical optimization

TL;DR

This paper introduces a unified theoretical framework for duality contracts in mathematical optimization, enhancing clarity and consistency across different solvers and problem classes within the MathOpt system.

Contribution

It develops an abstract primal-dual scheme based on Fenchel duality, enabling automatic derivation of dual problems and contracts for diverse optimization problem classes.

Findings

Unified duality contracts improve solver interoperability.

Enhanced clarity of optimality conditions for certain problem classes.

Framework implemented in Google’s MathOpt system.

Abstract

Deploying mathematical optimization in autonomous production systems requires precise contracts for objects returned by an optimization solver. Unfortunately, conventions on dual solution and infeasibility certificates (rays) vary widely across solvers and classes of problems. This paper presents the theoretical framework used by MathOpt (a domain-specific language developed and used at Google) to unify these notions. We propose an abstract primal-dual pair based on a simplified Fenchel duality scheme that allows for the mechanical derivation of dual problems and associated contracts for all classes of problems currently supported by MathOpt (including those with linear and quadratic objectives plus linear, conic, quadratic, and two-sided linear constraints). We also show how these contracts can improve clarity of complementary-slackness based optimality conditions for certain classes of problems.