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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

90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
Full Text: DOI
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