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 |
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