An Inertial Projection Neural Network for Solving Variational Inequalities

Area of Science:

• Optimization Theory
• Computational Neuroscience
• Applied Mathematics

Background:

• Projection neural networks (PNNs) have emerged as effective tools for solving monotone variational inequalities (VIs) and convex optimization problems.
• Existing neural network (NN) models often rely on the steepest descent method, presenting limitations in addressing complex optimization challenges.

Purpose of the Study:

• To introduce an Inertial Projection Neural Network (IPNN) model by incorporating an inertial term into first-order PNNs.
• To enhance the capability of neural networks for solving variational inequalities and a wider range of constrained optimization problems.
• To explore the potential of IPNNs in finding Karush-Kuhn-Tucker (KKT) optimal solutions for nonconvex optimization problems.

Main Methods:

• Development of a novel Inertial Projection Neural Network (IPNN) model.
• Theoretical analysis to establish the stability conditions of the proposed IPNN.
• Application of the IPNN to solve constrained optimization problems related to variational inequalities.

Main Results:

• The proposed IPNN model demonstrates stability under specific conditions.
• The IPNN is shown to be applicable to a broader class of constrained optimization problems than traditional PNNs.
• The inertial term in IPNNs helps overcome drawbacks associated with steepest descent-based NNs, facilitating exploration of KKT solutions for nonconvex problems.
• Simulation results confirm the effectiveness and performance of the IPNN on three numerical examples.

Conclusions:

• The Inertial Projection Neural Network (IPNN) offers a stable and effective approach for solving monotone variational inequalities.
• IPNNs provide a more versatile framework for tackling complex optimization problems, including nonconvex ones, compared to existing neural network models.
• The integration of inertial dynamics enhances the performance and applicability of projection neural networks in optimization research.