An Analysis of a Neural Dynamical Approach to Solving Optimization Problems

Recently, a neural dynamical approach to solving linearly constrained variational inequality problems is presented, and its stability and convergence are conjectured by simulation. This technical note analyzes the global stability and convergence of the neural dynamical approach. Theoretically, it is shown that the neural dynamical approach is convergent globally to a solution when the nonlinear mapping is monotone at the solution. Unlike existing convergence results of neural dynamical methods for solving linearly or nonlinearly variational inequalities, our main results don´t assume the differentiability condition of the nonlinear mapping. Therefore, the neural dynamical approach can be further guaranteed to solve linearly constrained monotone variational inequality problems with a non-smooth mapping. Comparsions and examples illustrative significance of the obtained results on non-smooth mapping.