A novel neural network for variational inequalities with linear and nonlinear constraints

Area of Science:

• Optimization Theory
• Computational Neuroscience
• Applied Mathematics

Background:

• Variational inequality provides a unified framework for diverse optimization and equilibrium problems.
• Existing neural network models face limitations in handling nonmonotone problems and often require parameter tuning.

Purpose of the Study:

• To propose a novel neural network model for solving variational inequalities with linear and nonlinear constraints.
• To ensure Lyapunov stability and convergence to exact solutions for the proposed network.
• To demonstrate the model's applicability to nonmonotone problems and its reduced complexity.

Main Methods:

• Development of a neural network model based on sufficient and necessary conditions for variational inequality solutions.
• Analysis of network stability using Lyapunov stability criteria for both asymmetric and gradient mappings.
• Introduction of a new energy function to prove finite-time convergence for the gradient mapping variant.

Main Results:

• The proposed neural network guarantees Lyapunov stability and convergence to exact solutions.
• The gradient mapping variant achieves finite-time convergence under mild conditions.
• The model effectively solves nonmonotone variational inequality problems without adjustable parameters and with lower complexity.

Conclusions:

• The novel neural network offers a simple, efficient, and broadly applicable solution for variational inequalities.
• Its ability to handle nonmonotone problems and reduced complexity present significant advantages over existing methods.
• Numerical examples validate the network's performance and transient behavior, highlighting its application potential.