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多项式的全局极小值的下界

Lower bounds on the global minimum of a polynomial

Mehdi Ghasemi, Jean Bernard Lasserre, Murray Marshall

发表信息：
Comput. Optim. Appl. 56(1) (2013)
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摘要：

我们推广了Ghasemi和Marshall [SIAM. J. Opt.22（2）（2012），pp 460-473]，以获得对于任何实数$M$，可通过几何编程计算的、在闭球${x \in \mathbb{R}^n : \sum x_i^{2d} \le M}$上的$n$变量$x = (x_1,...,x_n)$中的次数为$ \le 2d$的多元多项式$f(x) \in \mathbb{R}[x]$的下界$f_{{\rm gp},M}$。我们将这个界与Ghasemi和Marshall得到的（全局）下界$f_{\rm gp}$进行比较，也与Lasserre得到的可通过半定规划计算的下界的层次结构进行比较[SIAM J. Opt. 11（3）（2001）pp 796-816]。我们的计算结果表明，约束$f_{{\rm gp},M}$提高了约束$f_{\rm gp}$和计算的$f_{{\rm gp},M}$，像$f_{\rm gp}$，可以进行快速和容易的多项式具有大量的变量和/或大的程度，假设一个合理的稀疏系数，情况下，相应的计算使用半定规划故障。

英文摘要：

We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\rm gp},M}$ for a multivariate polynomial $f(x) \in \mathbb{R}[x]$ of degree $ \le 2d$ in $n$ variables $x = (x_1,...,x_n)$ on the closed ball ${x \in \mathbb{R}^n : \sum x_i^{2d} \le M}$, computable by geometric programming, for any real $M$. We compare this bound with the (global) lower bound $f_{\rm gp}$ obtained by Ghasemi and Marshall, and also with the hierarchy of lower bounds, computable by semidefinite programming, obtained by Lasserre [SIAM J. Opt. 11(3) (2001) pp 796-816]. Our computations show that the bound $f_{{\rm gp},M}$ improves on the bound $f_{\rm gp}$ and that the computation of $f_{{\rm gp},M}$, like that of $f_{\rm gp}$, can be carried out quickly and easily for polynomials having of large number of variables and/or large degree, assuming a reasonable sparsity of coefficients, cases where the corresponding computation using semidefinite programming breaks down.

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