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An algorithm for finding the global maximum of a multimodal, multivariate function. (English) Zbl 0598.90075
A method is presented for finding the global maximum of a Lipschitzian function f(x) (with Lipschitz constant K) over a cube $$A_ i\leq x^ i\leq B_ i$$, $$i=1,2,...,N$$ in $$R^ N$$ $$(x^ i$$ is the i-th coordinate of x). Like a number of previously known methods, this one constructs a sequence of points $$x_ 0$$, $$x_ 1,...,x_ n,...$$, starting from an arbitrary point $$x^ 0$$ in the cube, such that $$x_{n+1}$$ is a global maximizer of $$F_ n(x)=\min \{Kx-x_ j| +f(x_ j): j=0,1,...,n\}$$ over the cube (it is proved in the paper that any cluster point of such a sequence is a global maximizer of f, though the fact has been known previously). The main difficulty with this approach is how to find a global maximizer of a function of the form $$F(x)=\min \{K| x-x_ j| +h_ i:$$ $$i=0,1,...,m\}$$. Starting from the observation that the graph of F is the intersection of a number of conical surfaces, the author establishes that any point of the graph which corresponds to a relative maximum must be a solution to a system of N linear equations (in $$N+1$$ variables $$x^ 1,...,x^ N,z)$$ and a quadratic equation. She then proposes to compute the solutions of all such systems of equations in order to chose the global maximizer.
The algorithm seems to be efficient only for N very small. Actually, the computational experience is reported only for functions of two variables.
Reviewer: H.Tuy

##### MSC:
 90C30 Nonlinear programming 65K05 Numerical mathematical programming methods
##### Keywords:
Lipschitzian function; global maximizer; conical surfaces
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##### References:
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