# American Institute of Mathematical Sciences

June  2017, 7(2): 211-221. doi: 10.3934/naco.2017015

## 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

Received  December 2016 Revised  May 2017 Published  June 2017

Fund Project: 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.

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.

Citation: Rentsen Enkhbat, Evgeniya A. Finkelstein, Anton S. Anikin, Alexandr Yu. Gornov. Global optimization reduction of generalized Malfatti's problem. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 211-221. doi: 10.3934/naco.2017015
##### References:

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##### References:
Three, four, and five circles inscribed in the set $D$ of Test 1
Circles for $K=3$ and $K=5$ for test problem 2
Circles placed into the test polygon 3 for $K=3, 5$
Spheres placed into polyhedron for $K=3$ and $K=5$
Spheres placed into test polyhedron 5
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
 $x^*_1$ $x^*_2$ $r^*$ 1.9011 -0.2129 3.6336 6.7104 -0.0751 1.1775 0.4961 -4.5530 0.9282
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
 $x^*_1$ $x^*_2$ $r^*$ 1.9609 -0.2849 3.6675 6.7807 -0.0898 1.1563 0.4795 -4.6023 0.8969 0.3978 3.8828 0.7834
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
 $x^*_1$ $x^*_2$ $r^*$ 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
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
 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
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
 $x^*_1$ $x^*_2$ $r^*$ 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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