A Proximal Neurodynamic Network With Fixed-Time Convergence for Equilibrium Problems and Its Applications.

Area of Science:

• Neuroscience
• Optimization Theory
• Applied Mathematics

Background:

• Equilibrium problems (EPs) are fundamental in game theory and economics.
• Existing neurodynamic networks often exhibit asymptotic or exponential convergence, lacking precise time bounds.
• Transient performance and settling time are critical for real-world applications of optimization algorithms.

Purpose of the Study:

• To introduce a novel fixed-time converging proximal neurodynamic network (FXPNN).
• To analyze the convergence properties and transient performance of the proposed FXPNN.
• To demonstrate the applicability of FXPNN to various optimization and inequality problems.

Main Methods:

• Development of a proximal operator-based neurodynamic network architecture.
• Theoretical analysis of fixed-time convergence under mild conditions.
• Application of the FXPNN to solve composition optimization problems (COPs), l1-regularized least-squares, mixed variational inequalities (MVIs), and variational inequalities (VIs).

Main Results:

• The proposed FXPNN achieves fixed-time convergence to solutions of EPs.
• FXPNN demonstrates superior transient performance compared to existing methods.
• Settling time is independent of initial conditions and can be prescribed.
• Fixed-time convergence for COPs is established using the Polyak-Lojasiewicz condition, relaxing convexity requirements.

Conclusions:

• The FXPNN provides a robust and efficient framework for solving EPs and related problems.
• The fixed-time convergence property offers significant advantages in terms of predictability and performance.
• Numerical validation confirms the effectiveness and superiority of the FXPNN approach.