Research On Two Optimization Problems Based On Neural Dynamics

Neural networks have been widely studied by the researchers due to its parallel and quick property.Various neural networks have been proposed to solve various optimization problems.Generalized eigenvalue problem and quadratic zero-one programming problem are two important optimization problems in real life,and have a wide range of applications in Aerospace Engineering,location decision and quadratic assignment.Based on neural dynamics optimization method,this paper is to solve these two optimization problems.For the generalized eigenvalue problem,the traditional methods are numerical algorithms.However,the traditional numerical algorithm suffers from some disadvantages,such as higer computational complexity and longer computational time.In general,it is unable to solve generalized eigenvalue problem in real time,especially when solving large-scale optimization problems.To overcome these drawbacks,in this paper,based on Lyapunov functional method and generalized eigen-decomposition theorem,a recurrent neural network is presented for solving the generalized eigenvalue problem.It is proved that any state of the neural network is convergent to the generalized eigenvector corresponding to the largest generalized eigenvalue.This conclusion depends on more general assumptions,because it removes the restrictions on the algebraic multiplicity and geometric multiplicity of generalized eigenvalues,and generalizes the existing relevant conclusions.Finally,some comparisons and numerical examples are introduced to illustrate the effectiveness of the proposed neural network.For the quadratic zero-one programming problem,the neural network models for solving this problem mostly depend on penalty parameters.However,it is not easy to choose the exact penalty parameters in practical applications.In o rder to avoid this difficulty,in this paper,based on the basic idea of the Scholtes' relaxation scheme,the original quadratic zero-one programming problem is approximated by a parameterized nonlinear programming problem.Then,an artificial neural network without penalty parameters is proposed to solve the related parameterized nonlinear programming problem.It is proved that the equilibrium point of the presented neural network is stable in the sense of Lyapunov.Finally,some numerical examples are presented to substantiate the effectiveness of the proposed neural network.