Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convex duality contracts for production-grade mathematical optimization

Juan Pablo Vielma, Ross Anderson, Joey Huchette

TL;DR

This work presents MathOpt's unified dual-contract framework, a modular primal–dual scheme built on a simplified Fenchel view that operates on $opt_P = \inf_x \{ c\cdot x + \sum_i g_i(F^i(x)) \}$ with a corresponding abstract dual $opt_D$. By decomposing constraints into modules via $F^i$ and $g_i$, and leveraging $F^i_\nabla$ and $g_i^*$, the authors derive mechanical dual contracts that cover LP, conic, quadratic objectives, and two-sided linear constraints, while preserving standard dual properties such as weak duality and complementary slackness. The paper then demonstrates how these contracts specialize to classical forms (conic, Lagrangian, Fenchel) and discusses connections to Domain Driven Dual, GNLP, and Knotro-like two-sided optimality conditions, highlighting improved clarity for optimality certificates and infeasibility/ray certificates. The contributions yield a precise, extensible framework that unifies solver-specific dual forms, enabling robust user guidance, testing, and cross-solver interoperability in production optimization pipelines. Overall, this framework provides a rigorous, mechanical method to generate dual contracts across a broad class of convex problems, with practical implications for solver interfaces in production-critical systems.

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.

Convex duality contracts for production-grade mathematical optimization

TL;DR

This work presents MathOpt's unified dual-contract framework, a modular primal–dual scheme built on a simplified Fenchel view that operates on with a corresponding abstract dual . By decomposing constraints into modules via and , and leveraging and , the authors derive mechanical dual contracts that cover LP, conic, quadratic objectives, and two-sided linear constraints, while preserving standard dual properties such as weak duality and complementary slackness. The paper then demonstrates how these contracts specialize to classical forms (conic, Lagrangian, Fenchel) and discusses connections to Domain Driven Dual, GNLP, and Knotro-like two-sided optimality conditions, highlighting improved clarity for optimality certificates and infeasibility/ray certificates. The contributions yield a precise, extensible framework that unifies solver-specific dual forms, enabling robust user guidance, testing, and cross-solver interoperability in production optimization pipelines. Overall, this framework provides a rigorous, mechanical method to generate dual contracts across a broad class of convex problems, with practical implications for solver interfaces in production-critical systems.

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.
Paper Structure (30 sections, 32 theorems, 108 equations, 1 table)

This paper contains 30 sections, 32 theorems, 108 equations, 1 table.

Table of Contents

  1. Introduction
  2. Duality contracts in Linear Programming
  3. Existing dual forms
  4. Conic dual and its extensions
  5. Lagrangian duality and its extensions
  6. Fenchel dual
  7. Contributions and outline
  8. Notation
  9. Primal problem definitions and assumptions
  10. Dual problem definitions and properties
  11. Rays and infeasibility certificates
  12. Specific constraints
  13. Conic constraints (including one-sided linear inequalities)
  14. Two-sided (or ranged) linear inequalities
  15. Quadratic Objective
  16. ...and 15 more sections

Key Result

Theorem 1

If $\bar{x}\in \mathbb{R}^J$ and $(\bar{y},\bar{w})\in \mathbb{R}^{M}\times \mathbb{R}^{I\times J}$ are such that $(\bar{y},\bar{w})$ satisfies stationarity condition stationarity, then In particular, if $\bar{x}$ and $(\bar{y},\bar{w})$ are feasible, then

Theorems & Definitions (58)

  • Definition 1: Primal Feasibility
  • Definition 2: Primal objective value
  • Definition 3: Dual Feasibility
  • Definition 4: Dual objective function and value
  • Theorem 1: Weak Duality
  • Theorem 2: Optimality Conditions
  • Definition 5: Complementary Slackness
  • Corollary 1
  • Lemma 1
  • Definition 6
  • ...and 48 more