Introduction to Global Optimization Exploiting Space-Filling Curves by Yaroslav D. Sergeyev Roman G. Strongin & Daniela Lera

Author:Yaroslav D. Sergeyev, Roman G. Strongin & Daniela Lera
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

100/3

15,757

8,671

5,399

9,458

6,783

4,699

15,982

15,617

10,295

GA2

4/3

1,053

1,484

649

1,025

278

1,664

473

378

875

53/2

2,972

4,215

2,207

3,073

725

4,491

103 ∗

94 ∗

2,235

100/3

2,108

4,090

2,023

2,828

667

4,196

154 ∗

153 ∗

2,027

These methods have been chosen because they can be easily found by a final user. Unfortunately, our experience with both algorithms has shown that solving the system (3.3.11) can be a problem itself. Particularly, we note that, when N increases, the two curves and from (3.3.12) tend to flatten (see Figs. 3.5 and 3.6) and if the intersection point is close to the boundaries of the subinterval , then the system (3.3.11) can be difficult to solve. In some cases the methods looking for the roots of the system do not converge to the solution.

For example, Fig. 3.6 presents the case when the point (denoted by “*”) which approximates the root is obtained out of the search interval . Thus, the system (3.3.11) is not solved and, as a consequence, the algorithm GJE does not find the global minima of the objective function. These cases are shown in Table 3.4 by “–.”

Numerical experiments described in Table 3.4 have been executed with the following parameters. The exact constants H ≥ h have been used in the methods GJE and GA1. Parameters and r = 1. 5 have been used in the GA2. All global minimizers have been found by the algorithms GJE and GA1. Note that the parameter r influences the reliability of the method GA2. For example, the algorithm GA2 has found only one global minimizer in the experiments marked by “*.” The value r = 3. 5 allows one to find all global minimizers.

Fig. 3.5The two curves and

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