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Dual Interior Point Optimization Learning

Michael Klamkin, Mathieu Tanneau, Pascal Van Hentenryck

TL;DR

This work tackles real-time constrained optimization by learning dual-feasible proxies that guarantee valid dual bounds. It introduces Smoothed Self-Supervised Learning (S3L), a barrier-regularized loss that smooths dual training, and a dual completion layer offering closed-form forward and backward computations, removing the need for slow implicit layers. The method is validated on large parametric DCOPF problems, where S3L and a closed-form dual completion achieve sub-1% dual gaps on average with substantial speedups (hundreds to thousands of times faster than commercial solvers). Compared to prior dual-learning approaches, S3L yields tighter dual bounds and faster training/inference, demonstrating strong practical potential for large-scale, real-time optimization tasks.

Abstract

In many practical applications of constrained optimization, scale and solving time limits make traditional optimization solvers prohibitively slow. Thus, the research question of how to design optimization proxies -- machine learning models that produce high-quality solutions -- has recently received significant attention. Orthogonal to this research thread which focuses on learning primal solutions, this paper studies how to learn dual feasible solutions that complement primal approaches and provide quality guarantees. The paper makes two distinct contributions. First, to train dual linear optimization proxies, the paper proposes a smoothed self-supervised loss function that augments the objective function with a dual penalty term. Second, the paper proposes a novel dual completion strategy that guarantees dual feasibility by solving a convex optimization problem. Moreover, the paper derives closed-form solutions to this completion optimization for several classes of dual penalties, eliminating the need for computationally-heavy implicit layers. Numerical results are presented on large linear optimization problems and demonstrate the effectiveness of the proposed approach. The proposed dual completion outperforms methods for learning optimization proxies which do not exploit the structure of the dual problem. Compared to commercial optimization solvers, the learned dual proxies achieve optimality gaps below $1\%$ and several orders of magnitude speedups.

Dual Interior Point Optimization Learning

TL;DR

This work tackles real-time constrained optimization by learning dual-feasible proxies that guarantee valid dual bounds. It introduces Smoothed Self-Supervised Learning (S3L), a barrier-regularized loss that smooths dual training, and a dual completion layer offering closed-form forward and backward computations, removing the need for slow implicit layers. The method is validated on large parametric DCOPF problems, where S3L and a closed-form dual completion achieve sub-1% dual gaps on average with substantial speedups (hundreds to thousands of times faster than commercial solvers). Compared to prior dual-learning approaches, S3L yields tighter dual bounds and faster training/inference, demonstrating strong practical potential for large-scale, real-time optimization tasks.

Abstract

In many practical applications of constrained optimization, scale and solving time limits make traditional optimization solvers prohibitively slow. Thus, the research question of how to design optimization proxies -- machine learning models that produce high-quality solutions -- has recently received significant attention. Orthogonal to this research thread which focuses on learning primal solutions, this paper studies how to learn dual feasible solutions that complement primal approaches and provide quality guarantees. The paper makes two distinct contributions. First, to train dual linear optimization proxies, the paper proposes a smoothed self-supervised loss function that augments the objective function with a dual penalty term. Second, the paper proposes a novel dual completion strategy that guarantees dual feasibility by solving a convex optimization problem. Moreover, the paper derives closed-form solutions to this completion optimization for several classes of dual penalties, eliminating the need for computationally-heavy implicit layers. Numerical results are presented on large linear optimization problems and demonstrate the effectiveness of the proposed approach. The proposed dual completion outperforms methods for learning optimization proxies which do not exploit the structure of the dual problem. Compared to commercial optimization solvers, the learned dual proxies achieve optimality gaps below and several orders of magnitude speedups.
Paper Structure (11 sections, 10 equations, 2 figures, 1 table)

This paper contains 11 sections, 10 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions and Outline
  3. Notations
  4. Smoothed Self-Supervised Learning for Dual Optimization Proxies
  5. S3L: the Forward Pass
  6. S3L: the Backward Pass
  7. Numerical Experiments
  8. Optimal Power Flow Formulation
  9. Experiment details
  10. Results
  11. Conclusion

Figures (2)

  • Figure 1: Illustration of the proposed S3L method. From left to right: given input data $(\mathbf{A}, \mathbf{b}, \mathbf{c}, \mathbf{l}, \mathbf{u}$), a neural network first predicts $\mathbf{y}$. A generalized dual completion layer then recovers $(\mathbf{z}^{l}_{\mu}, \mathbf{z}^{u}_{\mu})$ in closed form (Section \ref{['sec:forward']}), and the loss $\mathcal{L}_\mu(\mathbf{y},\mathbf{z}^l_\mu,\mathbf{z}^u_\mu)$ is evaluated. Gradient information $\partial_{\mathbf{y}} \mathcal{L}_\mu$ can be computed either by automatic differentiation (following operations marked with dashed arrows) or in closed-form from $\mathbf{y}$ directly (Section \ref{['sec:backward']}).
  • Figure 2: Training curves for DLL and S3L on the 2869_pegase case. Left: evolution of mean objective gap. Right: evolution of maximum objective gap. The shaded area indicates the range between best and worst seed.