# zbMATH — the first resource for mathematics

A new method for global optimisation (Alienor). (English) Zbl 0701.90083
The Alienor technique is a new deterministic method for global optimization. It is based on a reducing transformation allowing to express n variables as a function of a single variable $$\theta\geq 0$$. To do that generalized Archimedes spirals are used. For two variables (x,y) written in polar coordinates $$x=r \cos \theta$$, $$y=r \sin \theta$$, the Archimedes spiral, $$r=a\theta$$, is introduced and leads to $$x=a\theta \cos \theta$$, $$y=a\theta \sin \theta.$$
Then a general global optimization problem Glob.Min f(x$${}_ 1,...,x_ n)$$, f continuous, is transformed into an approximated problem Glob.Min $$f^*(\theta)$$ depending on the single variable $$\theta$$, where $$f^*(\theta)$$ is the restriction of f to the generalized Archimedes spiral defined by the Alienor transformation $$f^*(\theta)=f(h_ 1(\theta)$$, $$h_ 2(\theta),...,h_ n(\theta))$$. The functions $$h_ i(\theta)$$ coming from $$x_ i=h_ i(\theta)$$, $$i=1,...,n$$ are obtained explicitly from the reducing transformation. In other words, the Alienor transformation allows to draw dense paths in $$R^ n$$ (Peano curves).
The method may be improved by introducing a parallel structure and a deterministic chaos. This technique has a lot of interesting applications and specially (1) the identification of models coming from physics, chemistry, biology,...; (2) the optimization of industrial processes; (3) the numerical resolution of optimal control problems; (4) the numerical resolution of nonlinear functional equations.
The Alienor technique works quickly because its calculation time is proportional to n, the number of variables. It has been compared to Monte-Carlo techniques for global optimization.
Reviewer: Y.Cherruault

##### MSC:
 90C30 Nonlinear programming 90-08 Computational methods for problems pertaining to operations research and mathematical programming 90C90 Applications of mathematical programming
Full Text:
##### References:
 [1] DOI: 10.1108/eb005642 · Zbl 0503.49020 [2] Cherruault Y., C.R. Acad. Sc. Paris, t. 296 pp 178– (1983) [3] Cherruault Y., No. 24 pp 33– (1985) [4] Mandelbrot B., Flammarion (1975) [5] Cherruault Y., No. 10 pp 879– (1987) [6] Cea J., Proceedings of the 7th IFIP Conference (1975) [7] Cherruault Y., Reidel (1986) [8] DOI: 10.1016/0378-4754(84)90105-8 [9] Cherruault Y., R.A.I.R.O. Série Recherche Opérationnelle 21 (1) pp 51– (1987) [10] Cherruault Y., P. U. F. (1983) [11] Davis P.J., Blaisdell (1963) [12] Rice J.R., The Approximation of Functions 1 (1965) · Zbl 0154.14903 [13] Rice J.R., The Approximation of Functions 2 (1969) · Zbl 0185.30601 [14] DOI: 10.1073/pnas.51.1.24 · Zbl 0136.36201 [15] Godfrey K., Compartmental Models and their Application (1983) [16] Norton J.P., Introduction to Identification (1986) · Zbl 0617.93064 [17] Bellman R., Dynamic Programming and Modern Control Theory (1965) · Zbl 0245.49015 [18] Karpouzas I., Optimisation et Contrôle Optimal des Systèmes Linéaires et Non-Linéaires, Thèse d’Etat (1987) [19] Adomian G., Non-linear Stochastic Systems Theory and Applications to Physics (1988)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.