Evolutionary Global Optimization, Manifolds and Applications by Hime Aguiar e Oliveira Junior

Author:Hime Aguiar e Oliveira Junior
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Final step

Obtain , being the output of previous steps, expected to be the global minimizer of . Therefore, is expected to be a global minimizer of f. As before, the index i corresponds to the minimum value for which is defined.

Considering that it is not so easy to construct atlases for arbitrary submanifolds, it seems helpful to present some hints about the synthesis of charts for , or the way to properly choose the independent variables to be generated by the auxiliary optimization algorithm. The material presented in [3, 10] is taken as the basis for the following considerations:

Let P be an element of . Then it is possible to build a slice chart from a centered chart of and another one of P.

Assuming that and that the full rank hypothesis is satisfied, it is true that has rank p, and takes the canonical basis of onto a spanning set of —when , it is possible to extract a basis from it.

Let us represent such a basis by , in which is a strictly increasing function. Moreover, representing the extra unit vectors as , where is also a strictly increasing function, we can set as

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