有理单变元表示在优化问题上的应用

利用零维多项式系统的有理单变元表示,给出了求多项式在有限点集上的正性判定算法.同时,结合不等式证明,呈现了目标函数在零维系统约束下最优化的一个纯代数算法,从而将多元函数约束优化问题转化为单变元函数在单变元多项式约束下的优化问题.新算法不仅能处理目标函数为多项式的最优化问题,而且还能处理目标函数为有理分式函数和根式函数的的最优化问题,并且给出了目标函数最优值的精确区间表示,使得能任意精度地逼近最优值.

Rational Univariate Representation and Its Application in Optimization Problem

Based on rational univariate representation, an algorithm to verify the polynomial positive definiteness on a finite number of points is presented. Also, a pure algebraic method is given to obtain the optimum value of a objective function with zero-dimensional constraints. Compared with SOSTOOLS, experimental results obtained by the algebraic approach is more exact.