# 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:
 [1] M. Andreatta, A. Bezdek and Jan P. Boroski, The problem of Malfatti: Two centuries of debate, The Mathematical Intelligencer, 33 (2011), 72-76. doi: 10.1007/s00283-010-9154-7. Google Scholar [2] N. Andrei, Hybrid conjugate gradient algorithm for unconstrained optimization}, JOTA, 141 (2009), 249-264. doi: 10.1007/s10957-008-9505-0. Google Scholar [3] A. Anikin, A. Gornov and A. Andrianov, Computational technologies for Morse potential optimization, Abstracts of IV Internetional conference ''Optimization and applications'' (OPTIMA-2013), (2013), 22-23. Google Scholar [4] J. W. David and J. P. K. Doye, Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms, The Journal of Physical Chemistry A, 28 (1997), 5111-5116. Google Scholar [5] R. Enkhbat, An algorithm for maximizing a convex function over a simple set}, Journal of Global Optimization, 8 (1996), 379-391. doi: 10.1007/BF02403999. Google Scholar [6] R. Enkhbat, Global optimization approach to Malfatti's problem}, Journal of Global Optimization, 65 (2016), 33-39. doi: 10.1007/s10898-015-0372-6. Google Scholar [7] R. Enkhbat, M. V. Barkova and M. V. Strekalovsky, Solving Malfatti's high dimensional problem by global optimization}, Numerical Algebra, Control and Optimization, 6 (2016), 153-160. doi: 10.3934/naco.2016005. Google Scholar [8] R. P. Fedorenko, Approximate solution of some optimal control problems}, USSR Computational Mathematics and Mathematical Physics, 4 (1964), 89-116. Google Scholar [9] H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem}, Math. Mag., 41 (1967), 251-252. Google Scholar [10] N. Gernet, The Fundamental Problem of the Calculus of Variations, St. Petersburg, Erlich, (1913), [in Russian].Google Scholar [11] M. Goldberg, On the original Malfatti problem, Math. Mag., 40 (1967), 241-247. Google Scholar [12] H. Lob and H. W. Richmond, On the solutions of the Malfatti problem for a triangle, Proc. London Math. Soc., 2 (1930), 287-301. doi: 10.1112/plms/s2-30.1.287. Google Scholar [13] G. A. Los, Malfatti's optimization problem, Dep. Ukr. NIINTI, July 5,1988, in Russian.Google Scholar [14] G. Malfatti, Memoria sopra una problema stereotomico, Memoria di Matematica e di Fisica della Societa italiana della Scienze, 10 (1803), 235-244. Google Scholar [15] H. Melissen, Packing and Covering with Circles, Thesis, Univ. Utrecht, 1997.Google Scholar [16] B. T. Polyak, The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112. Google Scholar [17] M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2016), 21-50. doi: 10.3934/naco.2017002. Google Scholar [18] M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, (2016), DOI: 10. 1007/s10479-016-2355-5. doi: 10.1007/s10479-016-2355-5. Google Scholar [19] S. K. Roy, G. Maity, G. W. Weber and S. Z. Alparslan Gok, Conic Scalarization approach to solve multi-choice multi-objective transportation problem with interval goal, Annals of Operations Research, (2016), DOI: 10. 1007/s10479-016-2283-4. doi: 10.1007/s10479-016-2283-4. Google Scholar [20] S. K. Roy, G. Maity and G. W. Weber, Multi-objective two-stage grey transportation problem using utility function with goals, Central European Journal of Operations Research, 25 (2017), 417-439. doi: 10.1007/s10100-016-0464-5. Google Scholar [21] A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066. Google Scholar [22] K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems, Pitman Monographs and Surveys in Pure and Applied Mathematics. New York, Longman Scientific & Technical, 1991. Google Scholar [23] F. A. Valentine, The problem of Lagrange with differential inequalities as added side conditions, Dissertation Univ. of Chicago, 1937. Google Scholar [24] V. A. Zalgaller and G. A. Los, The solution of Malfatti's problem, Journal of Mathematical Sciences, 72 (1994), 3163-3177. doi: 10.1007/BF01249514. Google Scholar

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##### References:
 [1] M. Andreatta, A. Bezdek and Jan P. Boroski, The problem of Malfatti: Two centuries of debate, The Mathematical Intelligencer, 33 (2011), 72-76. doi: 10.1007/s00283-010-9154-7. Google Scholar [2] N. Andrei, Hybrid conjugate gradient algorithm for unconstrained optimization}, JOTA, 141 (2009), 249-264. doi: 10.1007/s10957-008-9505-0. Google Scholar [3] A. Anikin, A. Gornov and A. Andrianov, Computational technologies for Morse potential optimization, Abstracts of IV Internetional conference ''Optimization and applications'' (OPTIMA-2013), (2013), 22-23. Google Scholar [4] J. W. David and J. P. K. Doye, Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms, The Journal of Physical Chemistry A, 28 (1997), 5111-5116. Google Scholar [5] R. Enkhbat, An algorithm for maximizing a convex function over a simple set}, Journal of Global Optimization, 8 (1996), 379-391. doi: 10.1007/BF02403999. Google Scholar [6] R. Enkhbat, Global optimization approach to Malfatti's problem}, Journal of Global Optimization, 65 (2016), 33-39. doi: 10.1007/s10898-015-0372-6. Google Scholar [7] R. Enkhbat, M. V. Barkova and M. V. Strekalovsky, Solving Malfatti's high dimensional problem by global optimization}, Numerical Algebra, Control and Optimization, 6 (2016), 153-160. doi: 10.3934/naco.2016005. Google Scholar [8] R. P. Fedorenko, Approximate solution of some optimal control problems}, USSR Computational Mathematics and Mathematical Physics, 4 (1964), 89-116. Google Scholar [9] H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem}, Math. Mag., 41 (1967), 251-252. Google Scholar [10] N. Gernet, The Fundamental Problem of the Calculus of Variations, St. Petersburg, Erlich, (1913), [in Russian].Google Scholar [11] M. Goldberg, On the original Malfatti problem, Math. Mag., 40 (1967), 241-247. Google Scholar [12] H. Lob and H. W. Richmond, On the solutions of the Malfatti problem for a triangle, Proc. London Math. Soc., 2 (1930), 287-301. doi: 10.1112/plms/s2-30.1.287. Google Scholar [13] G. A. Los, Malfatti's optimization problem, Dep. Ukr. NIINTI, July 5,1988, in Russian.Google Scholar [14] G. Malfatti, Memoria sopra una problema stereotomico, Memoria di Matematica e di Fisica della Societa italiana della Scienze, 10 (1803), 235-244. Google Scholar [15] H. Melissen, Packing and Covering with Circles, Thesis, Univ. Utrecht, 1997.Google Scholar [16] B. T. Polyak, The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112. Google Scholar [17] M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2016), 21-50. doi: 10.3934/naco.2017002. Google Scholar [18] M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, (2016), DOI: 10. 1007/s10479-016-2355-5. doi: 10.1007/s10479-016-2355-5. Google Scholar [19] S. K. Roy, G. Maity, G. W. Weber and S. Z. Alparslan Gok, Conic Scalarization approach to solve multi-choice multi-objective transportation problem with interval goal, Annals of Operations Research, (2016), DOI: 10. 1007/s10479-016-2283-4. doi: 10.1007/s10479-016-2283-4. Google Scholar [20] S. K. Roy, G. Maity and G. W. Weber, Multi-objective two-stage grey transportation problem using utility function with goals, Central European Journal of Operations Research, 25 (2017), 417-439. doi: 10.1007/s10100-016-0464-5. Google Scholar [21] A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066. Google Scholar [22] K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems, Pitman Monographs and Surveys in Pure and Applied Mathematics. New York, Longman Scientific & Technical, 1991. Google Scholar [23] F. A. Valentine, The problem of Lagrange with differential inequalities as added side conditions, Dissertation Univ. of Chicago, 1937. Google Scholar [24] V. A. Zalgaller and G. A. Los, The solution of Malfatti's problem, Journal of Mathematical Sciences, 72 (1994), 3163-3177. doi: 10.1007/BF01249514. Google Scholar
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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