A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set. (English) Zbl 1465.90074
The author considers the minimization problem
\[ F(x) \rightarrow \min \text{ subject to } x \in D, \] where \(F : \mathbb R^m \rightarrow \mathbb R\) is a lower bounded real valued continuous function and \(D\) is a noncompact set in \(\mathbb R^m\). The aim of the paper is to provide a new necessary and sufficient criterion for the existence of a global minimizer of this problem. The application of the crierion is illustrated by examples. The results extend and generalize sufficient condition by E. Demidenko [Comput. Stat. Data Anal. 51, No. 3, 1739–1753 (2006; Zbl 1157.62456)], which is not necessary, which is showed by an appropriate numerical example.
\[ F(x) \rightarrow \min \text{ subject to } x \in D, \] where \(F : \mathbb R^m \rightarrow \mathbb R\) is a lower bounded real valued continuous function and \(D\) is a noncompact set in \(\mathbb R^m\). The aim of the paper is to provide a new necessary and sufficient criterion for the existence of a global minimizer of this problem. The application of the crierion is illustrated by examples. The results extend and generalize sufficient condition by E. Demidenko [Comput. Stat. Data Anal. 51, No. 3, 1739–1753 (2006; Zbl 1157.62456)], which is not necessary, which is showed by an appropriate numerical example.
Reviewer: Karel Zimmermann (Praha)
MSC:
| 90C26 | Nonconvex programming, global optimization |
| 65K10 | Numerical optimization and variational techniques |
| 62J02 | General nonlinear regression |
Citations:
Zbl 1157.62456References:
| [1] | Demidenko, E., Optimization and Regression (1989), Nauka: Nauka Moscow, (in Russian) · Zbl 0675.65060 |
| [2] | Nakamura, T., Existence theorems of a maximum likelihood estimate from a generalized censored data sample, Ann. Inst. Statist. Math., 36, 375-393 (1984) · Zbl 0573.62025 |
| [3] | Nakamura, T.; Lee, C., On the existence of minimum contrast estimates in binary response model, Ann. Inst. Statist. Math., 45, 741-758 (1993) · Zbl 0801.62026 |
| [4] | Demidenko, E., Criteria for global minimum of sum of squares in nonlinear regression, Comput. Statist. Data Anal., 51, 1739-1753 (2006) · Zbl 1157.62456 |
| [5] | Himmelblau, D. M., Applied Nonlinear Programming (1972), McGraw-Hill: McGraw-Hill NY · Zbl 0241.90051 |
| [6] | Bates, D. M.; Watts, D. G., Nonlinear Regression Analysis and Its Applications (1988), Wiley: Wiley New York · Zbl 0728.62062 |
| [7] | Gill, P. E.; Murray, W.; Wright, M. H., Practical Optimization (1981), Academic Press: Academic Press London · Zbl 0503.90062 |
| [8] | Seber, G. A.F.; Wild, C. J., Nonlinear Regression (1989), John Wiley: John Wiley NewYork · Zbl 0721.62062 |
| [9] | Chena, H.; Lv, X.; Qiao, Y., Application of gradient descent method to the sedimentary grain-size distribution fitting, J. Comput. Appl. Math., 233, 1128-1138 (2009) · Zbl 1173.86310 |
| [10] | Demidenko, E., Criteria for unconstrained global optimization, J. Optim. Theory Appl., 136, 375-395 (2008) · Zbl 1145.90053 |
| [11] | Dubeau, F.; Mir, Y., Existence of optimal weighted least squares estimate for three-parametric exponential model, Comm. Statist. Theory Methods, 37, 1383-1398 (2008) · Zbl 1175.65016 |
| [12] | Jukić, D.; Scitovski, R., The existence of optimal parameters of the generalized logistic function, Appl. Math. Comput., 77, 281-294 (1996) · Zbl 0853.65151 |
| [13] | Nievergelt, Y., On the existence of best Mitscherlich, Verhulst, and West growth curves for generalized least-squares regression, J. Comput. Appl. Math., 248, 31-46 (2013) · Zbl 1316.62102 |
| [14] | Nivergelt, Y., Least-squares logistic curves with initial conditions exist, Comm. Statist. Theory Methods, 46, 4016-4030 (2017) · Zbl 1369.41027 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
