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

This paper proposes a generalized Hopfield network for solving general constrained convex optimization problems. First, the existence and the uniqueness of solutions to the generalized Hopfield network in the Filippov sense are proved. Then, the Lie derivative is introduced to analyze the stability of the network using a differential inclusion. The optimality of the solution to the nonsmooth constrained optimization problems is shown to be guaranteed by the enhanced Fritz John conditions. The convergence rate of the generalized Hopfield network can be estimated by the second-order derivative of the energy function. The effectiveness of the proposed network is evaluated on several typical nonsmooth optimization problems and used to solve the hierarchical and distributed model predictive control four-tank benchmark.