The Research Of Several Kinds Of Non-linear Optimization Problems Based On Neural Network

Nonlinear optimizaion problems are the approximations of the facts in reality.We could give an accurate solution using precise langugage of mathematics throughthis kind of approximation. Since the1980s, the introduction of neural network inthis area made people feel gradually that it could provide an almost perfect solutionfor the optimization problems because it could meet all the needs in many aspects.As the research continues, many research results are achieved and applied to realityby the researchers in this field. In recent years, the application of recurrent neuralnetwork greatly shortens the distance between theory and application.In this paper, we investigate two types of nonlinear optimization problems: onetype is the pseudoconvex optimization problems with linear equality constraints,from the properties of variational inequalities, we propose a recurrent neuralnetwork to solve the this type of optimizaiton problems; the other type is the convexoptimizaiton problems subject to linear equality and nonlinear inequality constraints,we construct a recurrent neural network to solve the this type of optimizationproblem based on the KKT condition of them. Then, we prove that the equilibriumpoints of the neural networks are equivalent to the optimal solutions of optimizationproblems. We define two different Lyapunov functions and prove the equilibrium’sstability of constructed neural network in the sense of Lyapunov throughanalysizing the properties of the Lyapunov functions. And by redifining another twoLyapunov functions, we prove that the solutions of neural network from specificinitial points are convergent to the optimal solutions of the nonlinear optimizationproblems. In addition, for the first type of optimization problem, we provev itsexponential convergence under specific conditions. Through some examples, we usethe numerical computation to verify the stability of the neural networks.