Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dual Lagrangian Learning for Conic Optimization

Mathieu Tanneau, Pascal Van Hentenryck

TL;DR

DLL tackles the challenge of obtaining certified dual bounds for parametric conic optimization by learning dual proxies through Dual Lagrangian Learning. It introduces a dual conic completion procedure, differentiable conic projection layers, and a self-supervised training objective, with closed-form completions to avoid implicit layers. The approach yields strong dual bounds and large speedups over DC3 and interior-point solvers on both linear and nonlinear problems, demonstrating practical efficiency and scalability. By providing dual feasibility and optimality guarantees in a broad conic setting, DLL enables fast, certifiable optimization that can be integrated into existing algorithms and extended via graph-based architectures.

Abstract

This paper presents Dual Lagrangian Learning (DLL), a principled learning methodology for dual conic optimization proxies. DLL leverages conic duality and the representation power of ML models to provide high-duality, dual-feasible solutions, and therefore valid Lagrangian dual bounds, for linear and nonlinear conic optimization problems. The paper introduces a systematic dual completion procedure, differentiable conic projection layers, and a self-supervised learning framework based on Lagrangian duality. It also provides closed-form dual completion formulae for broad classes of conic problems, which eliminate the need for costly implicit layers. The effectiveness of DLL is demonstrated on linear and nonlinear conic optimization problems. The proposed methodology significantly outperforms a state-of-the-art learning-based method, and achieves 1000x speedups over commercial interior-point solvers with optimality gaps under 0.5\% on average.

Dual Lagrangian Learning for Conic Optimization

TL;DR

DLL tackles the challenge of obtaining certified dual bounds for parametric conic optimization by learning dual proxies through Dual Lagrangian Learning. It introduces a dual conic completion procedure, differentiable conic projection layers, and a self-supervised training objective, with closed-form completions to avoid implicit layers. The approach yields strong dual bounds and large speedups over DC3 and interior-point solvers on both linear and nonlinear problems, demonstrating practical efficiency and scalability. By providing dual feasibility and optimality guarantees in a broad conic setting, DLL enables fast, certifiable optimization that can be integrated into existing algorithms and extended via graph-based architectures.

Abstract

This paper presents Dual Lagrangian Learning (DLL), a principled learning methodology for dual conic optimization proxies. DLL leverages conic duality and the representation power of ML models to provide high-duality, dual-feasible solutions, and therefore valid Lagrangian dual bounds, for linear and nonlinear conic optimization problems. The paper introduces a systematic dual completion procedure, differentiable conic projection layers, and a self-supervised learning framework based on Lagrangian duality. It also provides closed-form dual completion formulae for broad classes of conic problems, which eliminate the need for costly implicit layers. The effectiveness of DLL is demonstrated on linear and nonlinear conic optimization problems. The proposed methodology significantly outperforms a state-of-the-art learning-based method, and achieves 1000x speedups over commercial interior-point solvers with optimality gaps under 0.5\% on average.
Paper Structure (55 sections, 2 theorems, 53 equations, 1 figure, 5 tables)

This paper contains 55 sections, 2 theorems, 53 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Contributions and outline
  3. Related works
  4. Constrained Optimization Learning
  5. Dual Optimization Learning
  6. Background
  7. Notations
  8. Conic optimization
  9. Dual Lagrangian Learning (DLL)
  10. Dual Conic Completion
  11. Conic Projections
  12. Euclidean projection
  13. Radial projection
  14. Self-Supervised Dual Lagrangian Training
  15. Numerical experiments
  16. ...and 40 more sections

Key Result

theorem 1

Assume that $\forall \hat{y} \in \mathcal{K}^{*}, \exists x : Hx \succ_{\mathcal{C}} h$ and the problem is bounded. Then, $\forall \hat{y} \in \mathcal{K}^{*}, \exists \hat{z} \, {\in} \, \mathcal{C}^{*}: A^{\top} \hat{y} + H^{\top} \hat{z} = c$, i.e., $(\hat{y}, \hat{z})$ is feasible for eq:completion:conic:dual.

Figures (1)

  • Figure 1: Illustration of the proposed DLL scheme. Given input data $(A, b, H, h, c)$, a neural network first predicts $\bar{y} \in \mathbb{R}^{n}$. Next, a conic projection layer computes a conic-feasible $\hat{y} \in \mathcal{K}^{*}$, which is then completed into a full dual-feasible solution $(\hat{y}, \hat{z})$. The model is trained in a self-supervised fashion, by updating the weights $\theta$ to maximize the Lagrangian dual bound $\mathcal{L}(\hat{y})$.

Theorems & Definitions (7)

  • theorem 1: Dual conic completion
  • proof
  • theorem 2: Optimal dual completion
  • proof
  • proof
  • proof
  • proof