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An open global optimization problem on the unit sphere. (English) Zbl 0822.90125

Summary: The author poses the following problem: \[ \text{global max}\quad \Phi_ n(x)= \prod_{1\leq i< j\leq n} \| x_ i- x_ j\|\quad\text{s.t.}\quad \| x_ i\|= 1,\quad i= 1,\dots,n\tag{1} \] and \(x_ i\in \mathbb{R}^ 3\), for all \(i= 1,\dots, n\). The points \(x_ 1,\dots, x_ n\) which give the global maximum of (1) are called the elliptic Fekete points (of order \(n\)). Problem (1) has many local maxima and saddle points. There is no known algorithm for computing an exact (or approximate) global maximum for the above problem.

MSC:

90C30 Nonlinear programming
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References:

[1] Michael Shub and Steve Smale (1993), Complexity of Bezout’s Theorem III. Condition Number and Packing,J. of Complexity 9, 4-14. · Zbl 0846.65018 · doi:10.1006/jcom.1993.1002
[2] M. Tsuji (1959),Potential Theory in Modern Function Theory, Maruzen Co., Tokyo. · Zbl 0087.28401
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