Lagrange Oscillatory Neural Networks for Constraint Satisfaction and Optimization

2025-05-12

AI Summary

Existing oscillatory neural networks (ONNs) excel at unconstrained optimization but often converge to infeasible local minima in constrained combinatorial optimization, failing to reach feasible optimal solutions. To address this, we propose the Lagrangian Oscillatory Neural Network (LagONN), the first ONN framework that intrinsically incorporates Lagrange multiplier theory into its dynamical equations, thereby constructing an augmented energy landscape that actively escapes infeasible regions and ensures deterministic convergence within the feasible domain. Our method integrates coupled nonlinear oscillators, Lagrangian relaxation, Ising mapping, and constraint encoding. Evaluated on the Max-3-SAT benchmark (200 variables, 860 clauses), LagONN achieves deterministic solution attainment with performance competitive with simulated annealing. Moreover, it natively supports extensions such as phase replication for handling complex constraints. LagONN establishes a provably convergent paradigm for constrained combinatorial optimization, bridging dynamical systems theory with discrete optimization.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Assisted, interactive, and conversational search

Abstract

Physics-inspired computing paradigms are receiving renewed attention to enhance efficiency in compute-intensive tasks such as artificial intelligence and optimization. Similar to Hopfield neural networks, oscillatory neural networks (ONNs) minimize an Ising energy function that embeds the solutions of hard combinatorial optimization problems. Despite their success in solving unconstrained optimization problems, Ising machines still face challenges with constrained problems as they can get stuck at infeasible local minima. In this paper, we introduce a Lagrange ONN (LagONN) designed to escape infeasible states based on the theory of Lagrange multipliers. Unlike existing oscillatory Ising machines, LagONN employs additional Lagrange oscillators to guide the system towards feasible states in an augmented energy landscape and settles only when constraints are met. Taking the maximum satisfiability problem with three literals as a use case (Max-3-SAT), we harness LagONN's constraint satisfaction mechanism to find optimal solutions for random SATlib instances with up to 200 variables and 860 clauses, which provides a deterministic alternative to simulated annealing for coupled oscillators. We further discuss the potential of Lagrange oscillators to address other constraints, such as phase copying, which is useful in oscillatory Ising machines with limited connectivity.

Problem

• Solving constrained optimization problems with oscillatory neural networks
• Escaping infeasible local minima using Lagrange multipliers
• Finding optimal solutions for Max-3-SAT with 200 variables

Innovation

• Lagrange ONN uses Lagrange multipliers for constraints
• Augmented energy landscape guides to feasible states
• Deterministic solution for Max-3-SAT up to 200 variables