多项式优化问题极小值数量及最优值下界分析

Number of minimum value and lower bound of optimal value for polynomial optimization problem

为解决多元多项式的极小值数量及无约束多项式优化问题(POP)的全局最优值,首先给出了关于Liqun,Koklay提出的当n≤2时,具有r个变量的2n或2n+1阶多项式,最多有nr个孤立局部极小值的猜测的证明过程.其次,由于无约束多项式优化问题一般是非凸的,NP难的,其全局最优值不易求解,故利用张量的相关知识,给出了计算二阶无约束多元多项式全局最优值下界的理论估计及证明过程,此理论简单、方便.从而可以更好的计算全局最优值.

In order to solve the minimum number of multivariate polynomials and the global optimal value of the unconstrained polynomial optimization problem(POP),this paper first gives the proof of the conjecture of Liqun and Koklay that a 2n-degree or 2n+1-degree polynomial of r variables has at mostrn isolated local minima when n ≤2.Secondly,since the problem of unconst-rained polynomial optimization is generally nonconvex,NP hard and its global optimal value is not easy to be solved,so by using the relevant knowledge of tensor,the theory and the proof of the lower bound of the global optimal value of the second order unconstrained multivariate polynomials are given.This theory is simple and convenient,which can be used to calculate the global optimal value better.