对偶理论在一类多项式全局优化中的应用

Application of Canonical Duality Theory to Global Optimization with Polynomials

摘要:

用Canonical对偶理论,讨论一类高阶多项式全局最优化问题的求解.首先将无约束多项式全局优化问题转换成箱体约束下的多项式全局优化问题,之后通过构造非线性变换对偶函数及相应的共轭函数,得到原问题的Canonical对偶问题.进一步通过求解对偶问题的最优解,导出原多项式全局优化问题的最优解,并给出对偶问题是凹函数的证明.最后应用所得方法,计算一个二元6次多项式全局最优化实例.

关键词:Canonical对偶理论; 全局优化; 高阶多元多项式

Abstract:

A class of global optimization problem with polynomial is investigated with canonical duality theory.The unconstrained polynomial optimization problem is transformed into box constrained global optimization.The canonical dual function is defied for a solution to the original global optimization with polynomial problem by solving the dual problem.In addition,the dual problem proves to be a concave optimization.Finally,an example about binary six-order polynomial global optimization is illustrated.

Key words:Canonical duality theory; global optimization; high-order multivariate polynomials