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Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control
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 |
In this paper, we generalize Malfatti's problem as a continuation of works [
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. |
[2] |
N. Andrei,
Hybrid conjugate gradient algorithm for unconstrained optimization}, JOTA, 141 (2009), 249-264.
doi: 10.1007/s10957-008-9505-0. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[8] |
R. P. Fedorenko,
Approximate solution of some optimal control problems}, USSR Computational Mathematics and Mathematical Physics, 4 (1964), 89-116.
|
[9] |
H. Gabai and E. Liban,
On Goldberg's inequality associated with the Malfatti problem}, Math. Mag., 41 (1967), 251-252.
|
[10] |
N. Gernet,
The Fundamental Problem of the Calculus of Variations,
St. Petersburg, Erlich, (1913), [in Russian]. |
[11] |
M. Goldberg,
On the original Malfatti problem, Math. Mag., 40 (1967), 241-247.
|
[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. |
[13] |
G. A. Los,
Malfatti's optimization problem, Dep. Ukr. NIINTI, July 5,1988, in Russian. |
[14] |
G. Malfatti,
Memoria sopra una problema stereotomico, Memoria di Matematica e di Fisica della Societa italiana della Scienze, 10 (1803), 235-244.
|
[15] |
H. Melissen,
Packing and Covering with Circles, Thesis, Univ. Utrecht, 1997. |
[16] |
B. T. Polyak,
The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112.
|
[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. |
[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. |
[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. |
[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. |
[21] |
A. S. Strekalovsky,
On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066.
|
[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. |
[23] |
F. A. Valentine,
The problem of Lagrange with differential inequalities as added side conditions, Dissertation Univ. of Chicago, 1937. |
[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. |
show all references
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. |
[2] |
N. Andrei,
Hybrid conjugate gradient algorithm for unconstrained optimization}, JOTA, 141 (2009), 249-264.
doi: 10.1007/s10957-008-9505-0. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[8] |
R. P. Fedorenko,
Approximate solution of some optimal control problems}, USSR Computational Mathematics and Mathematical Physics, 4 (1964), 89-116.
|
[9] |
H. Gabai and E. Liban,
On Goldberg's inequality associated with the Malfatti problem}, Math. Mag., 41 (1967), 251-252.
|
[10] |
N. Gernet,
The Fundamental Problem of the Calculus of Variations,
St. Petersburg, Erlich, (1913), [in Russian]. |
[11] |
M. Goldberg,
On the original Malfatti problem, Math. Mag., 40 (1967), 241-247.
|
[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. |
[13] |
G. A. Los,
Malfatti's optimization problem, Dep. Ukr. NIINTI, July 5,1988, in Russian. |
[14] |
G. Malfatti,
Memoria sopra una problema stereotomico, Memoria di Matematica e di Fisica della Societa italiana della Scienze, 10 (1803), 235-244.
|
[15] |
H. Melissen,
Packing and Covering with Circles, Thesis, Univ. Utrecht, 1997. |
[16] |
B. T. Polyak,
The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112.
|
[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. |
[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. |
[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. |
[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. |
[21] |
A. S. Strekalovsky,
On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066.
|
[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. |
[23] |
F. A. Valentine,
The problem of Lagrange with differential inequalities as added side conditions, Dissertation Univ. of Chicago, 1937. |
[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. |



| ||
1.9011 | -0.2129 | 3.6336 |
6.7104 | -0.0751 | 1.1775 |
0.4961 | -4.5530 | 0.9282 |
| ||
1.9011 | -0.2129 | 3.6336 |
6.7104 | -0.0751 | 1.1775 |
0.4961 | -4.5530 | 0.9282 |
| ||
1.9609 | -0.2849 | 3.6675 |
6.7807 | -0.0898 | 1.1563 |
0.4795 | -4.6023 | 0.8969 |
0.3978 | 3.8828 | 0.7834 |
| ||
1.9609 | -0.2849 | 3.6675 |
6.7807 | -0.0898 | 1.1563 |
0.4795 | -4.6023 | 0.8969 |
0.3978 | 3.8828 | 0.7834 |
| ||
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 |
| ||
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 |
K | |||
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 | |||
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 |
| ||
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 |
| ||
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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