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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
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