The Neural Network For Solving Optimization Problems

Optimization problems are commonly encountered in modern science and technology , such as optimal control, signal processing, pattern recognition, and so on. Over years, a Variety of nummerical algorithms have been developed for solving linear and nonlinear programming problems, such as the gradient projection method, the projrction method and so on. In many engineering applications, the real-time solution of optimization problems is required. However, traditional algorithms may not be efficient since the computing time required for a solution greatly depends on dimension ,the structure of the problems and the complexity of algorithms. One possible and very promising approach to real-time optimization is to apply artificial neural network. The neural network's adaptive and parallel property can improve the computation and training time. In 1985 Tank and Hopfield first proposed the neural network for linear programming problems. Their work has inspired many researchers to investigate other neural network models for solving programming problems, such as Lagrange neural network , dual neural network , feedback neural network, projection neural network. In the past three decades, neural networks for solving optimization problem have been rather extensively studied and some important results have also been reported. Some of them have been applied in engineering control and optimization. In real applications, neural networks with simple architecture and good performance are desired. However, most existing neural networks have some limitations and disadvantages in the convergence conditions or architecture complexity.On the basis of the analysis above, this paper first gives the results of neural network for solving the nonlinear programming problems, the associated definitions and lemmas on the smooth theory and convex analysis. Then, this paper study necessary and sufficient conditions of the optimal solutiona ,which solve the convex quadratic programming problems with equality and inequality constraints ,the convex optimization problems with linear constraints and the nonlinear convex programming problems and we propose neural networks for solving the above three problems. Comparing with the existing neural networks, the proposed neural networks have fewer state variables and simpler architecture. More details are as follows:(1) In first section, a new neural network for solving convex quadratic programming problems with equality and inequality constrains has been proposed. Comparing with other neural networks for convex quadratic optimization, the proposed neural network is designed with fewer neurons and simpler architecture. It is shown here that the proposed neural network is stable in sence of Lyapunov and globally convergent to an optimal solution. Simulation results demonstrate the superior performance of the proposed neural network.(2) In second section, a new neural network was presented for solving convex programs with linear constrains. Under the condition that the objective function is convex, the proposed neural network is shown to be stable in the sence of Lyapunov and globally converges to the optimal solution of problem. Comparing with the existing neural networks for solving such problems, the proposed neural network dose not require the condition that the objective function is strictly convex, which expanded its scope of appliccation. (3) In third section, a new neural network is proposed to solve nonlinear convex programming problems. The proposed neural network is shown to be asymptotically stable in the sence of Lyapunov. Comparing with the existing neural networks,the proposed neural network has fewer state variables and simpler architure. Numerial examples show that the proposed network is feasible and efficient.