计算机工程与应用 ›› 2011, Vol. 47 ›› Issue (30): 26-28.

• 研究、探讨 • 上一篇    下一篇

结合权余量和粒子群方法求解变分优化问题

高雷阜,齐 微   

  1. 辽宁工程技术大学 理学院,系统科学研究所,辽宁 阜新 123000
  • 收稿日期:1900-01-01 修回日期:1900-01-01 出版日期:2011-10-21 发布日期:2011-10-21

Combining right surplus with particle swarm method to solve variational optimization problems

GAO Leifu,QI Wei   

  1. Institute of Mathematics and Systems Science,Liaoning Technical University,Fuxin,Liaoning 123000,China
  • Received:1900-01-01 Revised:1900-01-01 Online:2011-10-21 Published:2011-10-21

摘要: 鉴于对变分问题的研究,发现了可以将微分方程与变分问题进行转化的原理,将求解微分方程的权余量方法与标准粒子群算法进行结合,旨在找到一种求解变分优化问题的较好算法。在权余量方法的计算中,把方程组转化为求近似目标函数的极小值的变分问题,接着用标准粒子群算法对其求解。对于因变量的幂为高次时的变分问题,可以直接用改进的算法对其求解。对于因变量的幂为一次时的变分问题,需要对等式左右分别平方后,再转化为求极小值的变分问题并求解。用一个变分问题的实例进行检验,发现改进后的算法容易,计算过程简单,结果较好。对近似目标函数的基的个数取三四个进行比较,对于具体问题,应该选取较好的适应实际问题的次数,得到较好的结果。从理论上分析,该算法具有可行性与一定的实际意义。

关键词: 变分优化, 微分方程, 权余量方法, 粒子群算法

Abstract: In view of studying variational problems and finding the principle of transformation in the issue of differential equations and variational problems,the solution of combining right surplus method of differential equations with standard particle swarm algorithm to find a better algorithm for variational optimization problems.In the right surplus calculation method,the equations are transformed into the approximate objective function and the minimum variational problem,the standard particle swarm algorithm is used to solve them.On the dependent the powers of variable are high times in the variational problem,it can directly use the modified algorithm to solve it.On the dependent the powers of variable are one time,need to respectively square on the left and right equation,then it is converted to minimum variational problem and solving.And then the algorithm is tested with an variational problem example and founding that the improved algorithm is easy,calculation process is simple,the result is good.And the number of the base of the approximate objective function takes three or four times to compare,for specific issues,it should be better adapted to practical problem of selecting the number of times,and then getting better results.From the theoretical analysis,the algorithm is also feasible and it has practical significance.

Key words: variational optimization, differential equations, right surplus method, Particle Swarm Optimization algorithm