[Paper Review] Self-Supervised Primal-Dual Learning for Constrained Optimization

Seonho Park, Pascal Van Hentenryck|arXiv (Cornell University)|Aug 18, 2022

Reservoir Engineering and Simulation Methods4 citations

TL;DR

This paper proposes Primal-Dual Learning (PDL), a self-supervised method that jointly trains primal and dual neural networks to approximate solutions of constrained optimization problems by mimicking the trajectory of the Augmented Lagrangian Method (ALM). Unlike supervised methods, PDL eliminates the need for pre-solved instances or optimization solvers during training, achieving negligible constraint violations and minor optimality gaps—comparable to ALM—across nonlinear benchmarks including QP, QCQP, and AC-OPF problems.

ABSTRACT

This paper studies how to train machine-learning models that directly approximate the optimal solutions of constrained optimization problems. This is an empirical risk minimization under constraints, which is challenging as training must balance optimality and feasibility conditions. Supervised learning methods often approach this challenge by training the model on a large collection of pre-solved instances. This paper takes a different route and proposes the idea of Primal-Dual Learning (PDL), a self-supervised training method that does not require a set of pre-solved instances or an optimization solver for training and inference. Instead, PDL mimics the trajectory of an Augmented Lagrangian Method (ALM) and jointly trains primal and dual neural networks. Being a primal-dual method, PDL uses instance-specific penalties of the constraint terms in the loss function used to train the primal network. Experiments show that, on a set of nonlinear optimization benchmarks, PDL typically exhibits negligible constraint violations and minor optimality gaps, and is remarkably close to the ALM optimization. PDL also demonstrated improved or similar performance in terms of the optimality gaps, constraint violations, and training times compared to existing approaches.

Motivation & Objective

• To address the limitations of supervised learning in constrained optimization, which require costly pre-solved instance generation and often fail to ensure feasibility due to aggregated constraint penalties.
• To develop a self-supervised alternative that bypasses the need for pre-solved training data and optimization solvers during training and inference.
• To improve solution quality by enabling instance-specific constraint multipliers through a joint primal-dual network training scheme.
• To demonstrate that self-supervised primal-dual learning can achieve optimality gaps and constraint violations comparable to the Augmented Lagrangian Method (ALM), a standard optimization solver.
• To explore the potential of dual network approximations for downstream tasks such as price estimation or warm-starting solvers.

Proposed method

• PDL formulates a self-supervised learning objective that mimics the iterative trajectory of the Augmented Lagrangian Method (ALM), training both primal and dual neural networks in tandem.
• The method uses instance-specific dual variables (multipliers) in the primal loss function to penalize constraint violations, enabling dynamic and adaptive feasibility enforcement.
• The primal network predicts decision variables, while the dual network estimates Lagrange multipliers, both updated iteratively to converge toward the true primal and dual solutions.
• The training process does not require access to pre-solved instances or an optimization solver; instead, it relies solely on the ALM dynamics and the problem’s mathematical formulation.
• The framework is generic and supports various neural network architectures, including those with implicit layers for feasibility restoration.
• PDL is applied to convex and non-convex quadratic programs, QCQP, and AC-OPF problems, with performance evaluated via optimality gap and constraint violation metrics.

Experimental results

Research questions

• RQ1 Can a self-supervised method eliminate the need for pre-solved training instances in learning input-output mappings for constrained optimization?
• RQ2 Can a primal-dual learning framework with instance-specific multipliers outperform existing supervised and primal-only self-supervised methods in feasibility and optimality?
• RQ3 How does PDL’s performance compare to the Augmented Lagrangian Method (ALM) in terms of solution quality and constraint violation?
• RQ4 What is the training efficiency of PDL relative to supervised baselines, especially in problems requiring extensive data generation?
• RQ5 Can the dual network approximation from PDL support downstream applications such as price estimation or warm-starting optimization solvers?

Key findings

• On convex and non-convex QP problems, PDL achieved an optimality gap of 0.03(0.01) and a maximum constraint violation of 0.08(0.02), demonstrating negligible infeasibility and high accuracy.
• For AC-OPF problems (case56 and case118), PDL achieved an optimality gap of 0.37(0.52) and maximum constraint violation of 0.03(0.01), outperforming supervised penalty and Lagrangian relaxation baselines.
• In QCQP problems, PDL achieved an optimality gap of 0.05(0.01) and constraint violation of 0.02(0.00), showing strong performance with minimal error.
• PDL training times were comparable to those of the supervised baseline LD (e.g., 5553.2s vs. 3572.3s for QCQP), while avoiding the 715-hour data generation cost required by supervised methods.
• For AC-OPF case118, PDL required 7605.1s for training, while data generation for the supervised baseline took 13,120.3s, highlighting PDL’s efficiency in data-scarce settings.
• PDL’s solutions were remarkably close to ALM, with optimality gaps and constraint violations consistently lower than or competitive with existing methods across all benchmarks.