Constrained Quaternion-Variable Convex Optimization: A Quaternion-Valued Recurrent Neural Network Approach.

Area of Science:

• Computational mathematics
• Neural networks
• Optimization theory

Background:

• Constrained convex optimization problems are fundamental in many scientific and engineering fields.
• Existing methods often decompose quaternion problems into real or complex domains, limiting direct applicability.
• Quaternion variables offer advantages in representing multidimensional data and systems.

Purpose of the Study:

• To propose a novel quaternion-valued one-layer recurrent neural network (RNN) for constrained convex function optimization.
• To develop optimization algorithms directly within the quaternion field using generalized Hamilton-real (GHR) calculus.
• To rigorously analyze the stability and convergence properties of the proposed network.

Main Methods:

• Development of a quaternion-valued one-layer RNN architecture.
• Application of generalized Hamilton-real (GHR) calculus for quaternion gradient computation.
• Utilization of chain rules and Lyapunov theorem for system analysis.
• Design of optimization algorithms directly in the quaternion field.

Main Results:

• The proposed quaternion RNN effectively resolves constrained convex function optimization problems.
• The network demonstrates finite-time stabilization of system dynamics.
• The states converge to the optimal solution of the optimization problems.
• Numerical simulations validate the theoretical findings.

Conclusions:

• The quaternion-valued RNN provides a direct and efficient approach to quaternion optimization.
• The GHR calculus enables novel gradient-based optimization techniques in the quaternion field.
• The study establishes the theoretical foundation and practical viability of quaternion neural networks for complex optimization tasks.