The Theory of Filled Function Method for Finding Global Minimizers of Nonlinearly Constrained Minimization Problems
Year: 1987
Journal of Computational Mathematics, Vol. 5 (1987), Iss. 1 : pp. 1–9
Abstract
This paper is an extension of [1]. In this paper the descent and ascent segments are introduced to replace respectively the descent and ascent directions in [1] and are used to extend the concepts of S-basin and basin of a minimizer of a function. Lemmas and theorems similar to those in [1] are proved for the filled function $$P(x,r,p)= \frac{1}{r+F(x)}exp(-|x-x^*_1|^2/\rho^2),$$ which is the same as that in [1], where $x^*_1$ is a constrained local minimizer of the problem (0.3) below and $$F(x)=f(x)+\sum^{m'}_{i=1}\mu_i|c_i(x)|+ \sum^m_{i=m'+1}\mu_i max(0, -c_i(x))$$ is the exact penalty function for the constrained minimization problem
$\mathop{\rm min}\limits_x f(x)$,
subject to $$c_i(x) = 0 , i = 1, 2, \cdots, m',$$ $$c_i(x) \ge 0 , i = m'+1, \cdots, m,$$ where $μ_i ＞ 0 \ (i=1, 2, \cdots, m)$ are sufficiently large. When $x^*_1$ has been located, a saddle point or a minimizer $\hat{x}$ of $P(x,r,\rho)$ can be located by using the nonsmooth minimization method with some special termination principles. The $\hat{x}$ is proved to be in a basin of a lower minimizer $x^*_2$ of $F(x)$, provided that the ratio $\rho^2/[r+F(x^*_1)]$ is appropriately small. Thus, starting with $\hat{x}$ to minimize $F(x)$, one can locate $x^*_2$. In this way a constrained global minimizer of (0.3) can finally be found and termination will happen.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/1987-JCM-9526
Journal of Computational Mathematics, Vol. 5 (1987), Iss. 1 : pp. 1–9
Published online: 1987-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 9