Volume 5, Issue 1
The Theory of Filled Function Method for Finding Global Minimizers of Nonlinearly Constrained Minimization Problems
DOI:

J. Comp. Math., 5 (1987), pp. 1-9

Published online: 1987-05

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• Abstract

This paper is an extension of [1]. In this paper the descent and ascent segements are introduced to replace respectively the descent and ascent diretions 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 funciton $$P(x,r,p)= \frac{1}{r+F(x)}exp(-|x-x^*_1|^2/\rho^2),$$ which 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 \min_x f(x) (0.3), subject to $$c_i(x) = 0 , i = 1, 2, \cdots, m',$$ $$c_i(x) \ge 0 , i = m'+1, \cdots, m,$$ where $mu › 0$ (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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@Article{JCM-5-1, author = {Ren-Pu Ge}, title = {The Theory of Filled Function Method for Finding Global Minimizers of Nonlinearly Constrained Minimization Problems}, journal = {Journal of Computational Mathematics}, year = {1987}, volume = {5}, number = {1}, pages = {1--9}, abstract = { This paper is an extension of [1]. In this paper the descent and ascent segements are introduced to replace respectively the descent and ascent diretions 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 funciton $$P(x,r,p)= \frac{1}{r+F(x)}exp(-|x-x^*_1|^2/\rho^2),$$ which 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 \min_x f(x) (0.3), subject to $$c_i(x) = 0 , i = 1, 2, \cdots, m',$$ $$c_i(x) \ge 0 , i = m'+1, \cdots, m,$$ where $mu › 0$ (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. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9526.html} }
TY - JOUR T1 - The Theory of Filled Function Method for Finding Global Minimizers of Nonlinearly Constrained Minimization Problems AU - Ren-Pu Ge JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 9 PY - 1987 DA - 1987/05 SN - 5 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/jcm/9526.html KW - AB - This paper is an extension of [1]. In this paper the descent and ascent segements are introduced to replace respectively the descent and ascent diretions 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 funciton $$P(x,r,p)= \frac{1}{r+F(x)}exp(-|x-x^*_1|^2/\rho^2),$$ which 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 \min_x f(x) (0.3), subject to $$c_i(x) = 0 , i = 1, 2, \cdots, m',$$ $$c_i(x) \ge 0 , i = m'+1, \cdots, m,$$ where $mu › 0$ (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.