Investigation On Neural Network Theory Related To Convex Optimization Problem

This paper develops a neural network modelled by a differential inclusion for solving the general nonsmooth convex optimization problems. Compared with the existed neural networks for solving nonsmooth convex optimization problems, this neural network has a wider domain for application. Under a suitable assumption on the constraint set, it is proved that for a given nonsmooth convex optimization problem under the conditions of Lipschitz and sufficiently large penalty parameters, any trajectory of the neural network can reach the feasible region in finite time and stays there thereafter. Moreover, we can prove that the trajectory of the neural network constructed by a differential inclusion and with arbitrarily given initial value, converges to the set of the equilibrium points of the neural network, whose elements are all the optimal solutions of the primary constrained optimization problem. Similarly, we give the result that for a given nonsmooth convex optimization problem under the condition of convex only, any trajectory of the neural network can reach the feasible region in finite time and stays there thereafter with large enough penalty parameters. Furthermore, some illustrative examples are given.