Hamiltonian Descent Algorithm Improves Solutions for Large-Scale Nonlinear Programming Problems

Nonlinear programming forms the backbone of optimisation challenges across diverse fields, including power networks, chemical processes and financial modelling, yet solving these problems at scale remains notoriously difficult when they involve complex, nonlinear constraints. Mingze Li, Lei Fan, and Zhu Han, all from the University of Houston, present a new approach to tackle this issue, developing a method that combines Hamiltonian Descent with an Augmented Lagrangian framework. This innovative technique transforms challenging constrained problems into a form suitable for efficient solution using a dynamic process simulated by a novel algorithm, the Simulated Bifurcation method. The team demonstrates the effectiveness of their method through application to a Power-to-Hydrogen System, offering a promising advance for optimisation in complex real-world scenarios.

Nonlinear programming (NLP) finds extensive application in diverse fields including naval engineering, communication networks, and financial engineering. However, solving large-scale, nonconvex NLP problems remains a significant challenge due to the complexity of the solution landscape and the presence of nonlinear nonconvex constraints. Researchers have now developed a Quantum Hamiltonian Descent based Augmented Lagrange Method (QHD-ALM) framework to address these large-scale, constrained nonconvex NLP problems. These problems are common in fields like power systems, chemical engineering, and finance, and are notoriously difficult to solve, often trapping conventional algorithms in suboptimal solutions. The team’s work centres on bridging the gap between quantum algorithms and classical computation. While QHD offers a potentially powerful way to navigate complex optimisation landscapes, current quantum hardware is limited in scale and prone to errors.

To overcome these limitations, the researchers implemented a classical algorithm called Simulated Bifurcation (SB) that mimics the behaviour of quantum systems. SB effectively simulates the core principles of quantum annealing, allowing it to explore multiple solutions simultaneously using a network of classical oscillators. This allows the benefits of quantum-inspired optimisation to be realised on conventional computers. The newly developed framework, called QHD-ALM, tackles constrained NLP problems by reformulating them into a form suitable for the quantum-inspired optimisation process. ALM converts the original problem with complex constraints into a simpler, unconstrained one, allowing QHD to efficiently search for optimal solutions. By combining these two methods, the researchers have created a robust and versatile algorithm capable of handling a wider range of real-world problems.

QHD-ALM Optimisation for Complex Systems

This research presents a novel optimisation framework, QHD-ALM, which combines Quantum Hamiltonian Descent (QHD) with the Augmented Lagrange Method (ALM) to address challenging large-scale, non-convex nonlinear programming problems. The integration of QHD into the ALM framework improves convergence, offering a potentially more efficient approach to solving complex optimisation tasks. Application of the algorithm to a hydrogen production management system demonstrates both its efficiency and ability to achieve optimal solutions. The results indicate that QHD-ALM achieves comparable or better objective values than existing methods, such as IPOPT, while requiring significantly less computational effort. This improved performance is particularly notable when solving non-convex problems, where traditional methods often require multiple restarts to find high-quality solutions. QHD offers a way to overcome limitations of classical methods in dealing with non-convexity, while ALM converts constrained optimisation problems into a series of unconstrained subproblems. The core innovation lies in combining these techniques to enhance the global exploration capabilities of ALM and improve convergence speed. To demonstrate the effectiveness of QHD-ALM, the team applied it to a hydrogen energy production management problem.

This application highlights the potential of the framework to optimise complex energy systems, improving efficiency and reducing costs. The results demonstrate that the combined approach can effectively navigate the challenges of large-scale, non-convex optimisation, offering a promising alternative to existing methods. This advancement could lead to significant improvements in various industrial applications, enabling more efficient and sustainable solutions to complex optimisation challenges.