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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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