Global optimization reduction of generalized Malfatti's problem

Rentsen Enkhbat; Evgeniya A. Finkelstein; Anton S. Anikin; Alexandr Yu. Gornov

doi:10.3934/naco.2017015

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2017, Volume 7, Issue 2: 211-221. Doi: 10.3934/naco.2017015

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Global optimization reduction of generalized Malfatti's problem

1.
Institute of Mathematics, National University of Mongolia, 210646, Ulaanbaatar, Mongolia
2.
Matrosov Institute for Systems Dynamics and Control Theory SB RAS, 664033, Irkutsk, Russia

^* Corresponding author: R.Enkhbat
^* Corresponding author: R.Enkhbat

Received: December 2016

Revised: May 2017

Published: October 2017

This paper was prepared at the occasion of The 10th International Conference on Optimization: Techniques and Applications (ICOTA 2016), Ulaanbaatar, Mongolia, July 23-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Prof. Dr. Zhiyou Wu, School of Mathematical Sciences, Chongqing Normal University, Chongqing, China, Prof. Dr. Changjun Yu, Department of Mathematics and Statistics, Curtin University, Perth, Australia, and Shanghai University, China, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.

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Abstract

In this paper, we generalize Malfatti's problem as a continuation of works [6,7]. The problem has been formulated as a global optimization problem. To solve Malfatti's problem numerically, we propose the co-called ''Hill method'' which is based on a heuristic approach. Some computational results for two and three-dimensional test problems are provided.

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Mathematics Subject Classification: Primary: 49K, 65K10; Secondary: 90C26.

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Figure 1. Three, four, and five circles inscribed in the set $D$ of Test 1

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Figure 2. Circles for $K=3$ and $K=5$ for test problem 2

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Figure 3. Circles placed into the test polygon 3 for $K=3, 5$

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Figure 4. Spheres placed into polyhedron for $K=3$ and $K=5$

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Figure 5. Spheres placed into test polyhedron 5

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Table 1. Test Problem 1 for $K=3$

$x^*_1$ $x^*_2$ $r^*$

1.9011 -0.2129 3.6336
6.7104 -0.0751 1.1775
0.4961 -4.5530 0.9282

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Table 2. Test Problem 1 for $K=4$

$x^*_1$ $x^*_2$ $r^*$

1.9609 -0.2849 3.6675
6.7807 -0.0898 1.1563
0.4795 -4.6023 0.8969
0.3978 3.8828 0.7834

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Table 3. Test Problem 1 for $K=5$

$x^*_1$ $x^*_2$ $r^*$

1.9607 -0.2849 3.6677
6.7799 -0.0899 1.1567
0.4796 -4.6020 0.8972
0.3973 3.8822 0.7841
-0.3701 -3.6201 0.4016

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Table 4. Test Problem 2 for $K=3, 4, 5$

K $x^*_1$ $x^*_2$ $r^*$

1 1.2601 3.4685 3.4685
2 5.4923 1.2905 1.2905
3 -2.475 1.0056 1.0056
4 5.2051 3.0559 0.4981
5 3.8888 0.4980 0.4981

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Table 5. Test Problem 3 for $K=3, 4, 5$

$x^*_1$ $x^*_2$ $r^*$

0.7187 3.8509 3.8509
5.7749 1.6597 1.6597
-3.8282 1.3422 1.3422
5.2807 3.9803 0.7129
7.7998 0.6176 0.6176

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