Generalized Neural Network Methods For Solving A Class Of Nonsmoth Optimization Problems

A plenty of optimization problems can be originated from military science, management, economics, engineering science and so on. And the methods for solving those problems all have more and more popular applications in the fields of image, communication, process of designing and operation, analysis of manufacturing plants, economical decision and so on. With the development of the science and technique, the dimensions and structures of the optimization problems, which occur in a wealth of fields, become more and more complex. These all lead to motivate us to find more and more effective mathematical models to solve those problems. Especially in the process of solving real-time solutions, when we solve the problems with higher dimensions and deeper structures, we always have to take the computational time into account. As we know, the time depends greatly on the dimensions and structures of the problems, and the complexity of the algorithms. Generally speaking, it is less effective for the traditional numerical algorithms to solve the real-time solutions of such optimization problems. Because their core is iterative method, it is hard to deal with the complex problems. The appearance of the neural network gives us the way, which explores its merits rather than solve the problems under the ristriction of the iterative method. The merits of neural network in dealing with the high dimensions and deep structures are the neural network's adaptive and parallel property. These can improve the computation and training time. As a result, it is of great development to explore the neural network for solving optimization problems. And some mathmatical models were proposed. However, these methods were partly paid attention to the anlysis of smooth optimization problem, and partly to approximate the nonsmooth object functions into smooth ones. To the decades, neural network for solving nonsmooth optimization problems is gradually taking its road, which gets benefits of the further analysis of set-value mapping and nonsmooth theory. Consequently, the investigation of such minimax optimization problems does not merely stay the level of applying numerical algorithms. The neural network method gradually plays an important role. Especially, based on the differential inclusions theory and convex analysis theory, neural network for solving nonsmooth optimization is thought highly of its value.On the basis of the analysis above, at first, this paper gives the development of neural network in the fields of solving the optimization problems, the associated definitions and lemmas on the nonsmooth theory and convex analysis. Then, this paper introduces three kinds of generalized neural network models. And they are mainly used to solve a class of unconstrained minimax optimization problems and a kind of minimax optimization problems with linear equation constraits respectively. More details are as follows:(1) A generalized neural network is proposed to solve a class of minimax optimization problems with unconstrained nonsmooth cost functions. It mainly takes on the account of the differential inclusions theory, the stability theory, and the extended Lojasiewicz inequality to solve a class of unconstrained minimax optimization problems with subanalytic and convex cost functions. And we give the effenciency of the theoretical results of this generalized neural network. (2) A generalized neural network is proposed to solve a class of minimax optimization problems with linear equation constraints. On the condition of linear equation constraints, we mainly depends on the projection neural network, the differential inclusions theory and the stability theory to construct a generalized nerual network. It is used to investigate how to slove a class of minimax optimization problems with linear equation constraints more effectively. And we give the illustrations of the theoretical results of this generalized neural network.(3) We study the convergence of a projection-based generalized neural network and its application to nonsmooth optimization problems. The projection-based generalized neural network plays an important role in the smooth optimization problems. And it hasn't attained to its maturity. Considering of the nonsmoothness of the minimax problems, we mainly study the application of a projection-based generalized neural network into the case of nonsmooth optimization on the analysis of the above two sections. And we also show the effenciency of the theoretical results of this generalized neural network.In addition, after the theoretical analysis of the convergence of each generalized neural network, we give some illustrative examples to show the effenciency of the theoretical results.