A Neural Network Based on the Metric Projector for Solving SOCCVI Problem

Area of Science:

• Optimization
• Numerical Analysis
• Computational Science

Background:

• Variational inequalities (VI) are fundamental in optimization and game theory.
• Second-order cone constrained variational inequality (SOCCVI) problems present significant computational challenges.
• Existing neural network approaches for SOCCVI have limitations in stability and convergence.

Purpose of the Study:

• To propose an efficient neural network for solving SOCCVI problems.
• To enhance the stability and convergence properties of neural network solutions for SOCCVI.
• To establish theoretical guarantees for the network's performance.

Main Methods:

• Formulating the SOCCVI using Karush-Kuhn-Tucker (KKT) conditions.
• Employing a smoothing function for the metric projection to handle complementarity.
• Analyzing network stability using standard results and second-order sufficient conditions.
• Proving the nonsingularity of the KKT system's Jacobian.

Main Results:

• The proposed neural network efficiently solves SOCCVI problems.
• Second-order sufficient conditions ensure exponential stability.
• The Jacobian of the KKT system is proven to be nonsingular.
• Numerical experiments confirm the network's superior performance over existing methods.

Conclusions:

• The novel neural network offers an efficient and stable approach to SOCCVI.
• The theoretical analysis supports the observed superior convergence and stability.
• This work advances the state-of-the-art in neural network applications for variational inequalities.