Deterministic Global Optimization by Yaroslav D. Sergeyev & Dmitri E. Kvasov

Author:Yaroslav D. Sergeyev & Dmitri E. Kvasov
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

In this context, we should recall (see Fig. 1.​7 in the introductory Chap. 1) that the efficient diagonal strategy can be also viewed as a procedure generating a series of curves similar to traditional space-filling curves (see, e.g., [238, 263, 315, 323])—adaptive diagonal curves (see [176, 279, 288]). They join the vertices a and b of the initial hyperinterval D and remain continuous during the whole process of partitioning. Each of these curves is constructed on the main diagonals of hyperintervals obtained during the current partition of D. Particularly, the initial adaptive diagonal curve coincides with the main diagonal [a, b] of the hyperinterval D. Then, every partitioning of a hyperinterval substitutes the segment of the curve, corresponding to the main diagonal of , with the polygonal line connecting the points , v, u, and (see the right picture in Fig. 3.8). Thus, a new curve is generated, which differs from the previous one only within the subdomain . This reflects the idea of adaptive local partition of the curve within the selected subdomain (without modifying other segments) at the current iteration. In Fig. 3.10, a partition (and obtained diagonals) of a two-dimensional interval D after ten subdivisions executed by the non-redundant diagonal partition strategy (left picture) and the corresponding adaptive diagonal curve (right picture) are presented. Trial points of f(x) are indicated by black dots. In the left picture (corresponding to the partition from Fig. 3.6), the numbers around the dots indicate iterations at which the objective function is evaluated at the corresponding points (if enclosed in brackets, the corresponding function value has been read from the vertex database). The curve starts at the point , goes through the points , , , and finishes at the point .

Fig. 3.11Adaptive diagonal curves become denser in the vicinity of the global minimizers of f(x) if the selection of hyperintervals for partitioning is realized appropriately

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