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2016, 6(2): 153-160. doi: 10.3934/naco.2016005

Solving Malfatti's high dimensional problem by global optimization

1. 

Institute for Systems Dynamics and Control Theory, SB of RAS, 664033 Irkutsk, Russian Federation, Russian Federation, Russian Federation

Received  October 2015 Revised  April 2016 Published  June 2016

We generalize Malfatti's problem which dates back to 200 years ago as a global optimization problem in a high dimensional space. The problem has been formulated as the convex maximization problem over a nonconvex set. Global optimality condition by Strekalovsky [11] has been applied to this problem. For solving numerically Malfatti's problem, we propose the algorithm in [3] which converges globally. Some computational results are provided.
Citation: Rentsen Enkhbat, M. V. Barkova, A. S. Strekalovsky. Solving Malfatti's high dimensional problem by global optimization. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 153-160. doi: 10.3934/naco.2016005
References:
[1]

M. Andreatta, A. Bezdek and Jan P. Boroński., 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]

R. Enkhbat, Global optimization approach to Malfatti's problem, Accepted for JOGO and to appear in 2016. Google Scholar

[3]

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

[4]

H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem, Math. Mag., 41 (1967), 251-252.  Google Scholar

[5]

M. Goldberg, On the original Malfatti problem, Math. Mag., 40 (1967), 241-247.  Google Scholar

[6]

G. A. Los, Malfatti's Optimization Problem [in Russian], Dep. Ukr. NIINTI, July 5, 1988. Google Scholar

[7]

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

[8]

C. Malfatti, Memoria sopra una problema stereotomico,, Memoria di Matematica e di Fisica della Societa italiana della Scienze, ().   Google Scholar

[9]

V. N. Nefedov, Finding the global maximum of a function of several variables on a set given by inequality constraints, Journal of Numerical Mathematics and Mathematical Physics, 27 (1987), 35-51.  Google Scholar

[10]

T. Saaty, Integer optimization methods and related extremal problems [Russian translation], Nauka, Moscow, 1973. 10 (1803), 235-244. Google Scholar

[11]

A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066. Google Scholar

[12]

H. Tverberg, A generalization of Radon's theorem, Journal of the London Mathematical Society, 41 (1966), 123-128.  Google Scholar

[13]

V. A. Zalgaller, An inequality for acute triangles, Ukr. Geom. Sb., 34 (1991), 10-25. Google Scholar

[14]

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

show all references

References:
[1]

M. Andreatta, A. Bezdek and Jan P. Boroński., 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]

R. Enkhbat, Global optimization approach to Malfatti's problem, Accepted for JOGO and to appear in 2016. Google Scholar

[3]

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

[4]

H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem, Math. Mag., 41 (1967), 251-252.  Google Scholar

[5]

M. Goldberg, On the original Malfatti problem, Math. Mag., 40 (1967), 241-247.  Google Scholar

[6]

G. A. Los, Malfatti's Optimization Problem [in Russian], Dep. Ukr. NIINTI, July 5, 1988. Google Scholar

[7]

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

[8]

C. Malfatti, Memoria sopra una problema stereotomico,, Memoria di Matematica e di Fisica della Societa italiana della Scienze, ().   Google Scholar

[9]

V. N. Nefedov, Finding the global maximum of a function of several variables on a set given by inequality constraints, Journal of Numerical Mathematics and Mathematical Physics, 27 (1987), 35-51.  Google Scholar

[10]

T. Saaty, Integer optimization methods and related extremal problems [Russian translation], Nauka, Moscow, 1973. 10 (1803), 235-244. Google Scholar

[11]

A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066. Google Scholar

[12]

H. Tverberg, A generalization of Radon's theorem, Journal of the London Mathematical Society, 41 (1966), 123-128.  Google Scholar

[13]

V. A. Zalgaller, An inequality for acute triangles, Ukr. Geom. Sb., 34 (1991), 10-25. Google Scholar

[14]

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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