A Generalized Hopfield Network for Nonsmooth Constrained Convex Optimization: Lie Derivative Approach

Area of Science:

• Computational Mathematics
• Optimization Theory
• Neural Networks

Background:

• Constrained convex optimization problems are prevalent in various scientific and engineering fields.
• Traditional methods may struggle with nonsmoothness and complex constraints.
• Hopfield networks offer a dynamical systems approach to optimization.

Purpose of the Study:

• To propose a generalized Hopfield network capable of solving general constrained convex optimization problems.
• To rigorously analyze the theoretical properties of the proposed network.
• To demonstrate its practical applicability on complex control tasks.

Main Methods:

• Formulation of a generalized Hopfield network for constrained optimization.
• Proof of existence and uniqueness of solutions using Filippov's theory.
• Stability analysis via Lie derivative and differential inclusions.
• Application of enhanced Fritz John conditions for optimality verification.
• Estimation of convergence rate using the second-order derivative of the energy function.

Main Results:

• Existence and uniqueness of solutions for the generalized Hopfield network are proven.
• The enhanced Fritz John conditions guarantee the optimality of solutions for nonsmooth problems.
• The convergence rate is theoretically estimated.
• The network's effectiveness is demonstrated on benchmark problems.

Conclusions:

• The generalized Hopfield network provides a robust framework for constrained convex optimization.
• The theoretical analysis confirms its stability and optimality guarantees.
• The network is effective for complex control applications like model predictive control.