Stability Analysis And Applications Of Neural Networks For Non-smooth Pseudoconvex Optimization

Neural network has been taken a great concern on solving smooth optimization andconvex optimization problems. In recent years, with the deep study of the theory ofconvex analysis, non-smooth analysis, set-valued map and differential inclusion, theapplication of neural network for solving non-smooth optimization problems andnon-convex optimization problem has been taken great attention. As a special case ofnon-convex optimization problem, the pseudo-convex optimization problem has beenfound to be more prevalent with respect to the other non-convex optimization.Pseudo-convex optimization has a large number of applications in practice, such as:fractional programming, machine vision, production plans.Based on the optimization theory, convex analysis, non-smooth analysis, set-valuedmap, differential inclusion and the Lyapunov stability theorem, this paper solved twotypes of pseudo-convex optimization by establishing suitable non-linear neural networksdescribed by differential inclusion. Correspond the equilibrium point of the network to theoptimal optimization problem, we obtained the conditions of the existence, uniqueness andstability of the equilibrium point of the neural network by constructing suitable Lyapunovfunctions. The main work is as follows:Firstly, we introduced the research background and the developments of neuralnetworks, showed that it is necessary to investigate the stability of optimization problems.Besides, in order to present the neural network for solving optimization problems, weintroduced the basic theory on the optimization problem and relative stability theoremSecondly, we developed a one-layer recurrent neural network for solvingpseudoconvex optimization with box set constraints. Based on the projection theory, theequilibrium point of the proposed neural network is proved to be equivalent to the optimalsolution of the optimization problem; Based on Lyapunov stability theorem, we provedthat the proposed neural network is stable in the sense of Lyapunov; Using the techniqueof Clarke nonsmooth analysis, the finite-time state convergence to the feasible regiondefined by the constraint conditions is also addressed. Lastly, we presented a projection neural network for nonsmooth pseudoconvexoptimization with general constraints. Based on Lyapunov stability theorem and thetechnique of Clarke nonsmooth analysis, the proposed neural network is proved to beglobally asymptotically stable, the state trajectory of the proposed neural network globallyconvergent to an equilibrium point of the network is also addressed.