# American Institute of Mathematical Sciences

• Previous Article
A new smoothing approach to exact penalty functions for inequality constrained optimization problems
• NACO Home
• This Issue
• Next Article
Output feedback overlapping control design of interconnected systems with input saturation
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 and 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. [2] R. Enkhbat, Global optimization approach to Malfatti's problem, Accepted for JOGO and to appear in 2016. [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. [4] H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem, Math. Mag., 41 (1967), 251-252. [5] M. Goldberg, On the original Malfatti problem, Math. Mag., 40 (1967), 241-247. [6] G. A. Los, Malfatti's Optimization Problem [in Russian], Dep. Ukr. NIINTI, July 5, 1988. [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. [8] C. Malfatti, Memoria sopra una problema stereotomico,, Memoria di Matematica e di Fisica della Societa italiana della Scienze, (). [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. [10] T. Saaty, Integer optimization methods and related extremal problems [Russian translation], Nauka, Moscow, 1973. 10 (1803), 235-244. [11] A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066. [12] H. Tverberg, A generalization of Radon's theorem, Journal of the London Mathematical Society, 41 (1966), 123-128. [13] V. A. Zalgaller, An inequality for acute triangles, Ukr. Geom. Sb., 34 (1991), 10-25. [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.

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. [2] R. Enkhbat, Global optimization approach to Malfatti's problem, Accepted for JOGO and to appear in 2016. [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. [4] H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem, Math. Mag., 41 (1967), 251-252. [5] M. Goldberg, On the original Malfatti problem, Math. Mag., 40 (1967), 241-247. [6] G. A. Los, Malfatti's Optimization Problem [in Russian], Dep. Ukr. NIINTI, July 5, 1988. [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. [8] C. Malfatti, Memoria sopra una problema stereotomico,, Memoria di Matematica e di Fisica della Societa italiana della Scienze, (). [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. [10] T. Saaty, Integer optimization methods and related extremal problems [Russian translation], Nauka, Moscow, 1973. 10 (1803), 235-244. [11] A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066. [12] H. Tverberg, A generalization of Radon's theorem, Journal of the London Mathematical Society, 41 (1966), 123-128. [13] V. A. Zalgaller, An inequality for acute triangles, Ukr. Geom. Sb., 34 (1991), 10-25. [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.
 [1] Rentsen Enkhbat, Evgeniya A. Finkelstein, Anton S. Anikin, Alexandr Yu. Gornov. Global optimization reduction of generalized Malfatti's problem. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 211-221. doi: 10.3934/naco.2017015 [2] Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial and Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789 [3] Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 [4] Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087 [5] Yong Xia. New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems. Journal of Industrial and Management Optimization, 2009, 5 (4) : 881-892. doi: 10.3934/jimo.2009.5.881 [6] Jutamas Kerdkaew, Rabian Wangkeeree, Rattanaporn Wangkeeree. Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 93-107. doi: 10.3934/naco.2021053 [7] Nazih Abderrazzak Gadhi, Fatima Zahra Rahou. Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021216 [8] Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial and Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483 [9] Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231 [10] Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial and Management Optimization, 2020, 16 (1) : 55-70. doi: 10.3934/jimo.2018140 [11] Xiaoqing Ou, Suliman Al-Homidan, Qamrul Hasan Ansari, Jiawei Chen. Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021199 [12] Tiago Carvalho, Luiz Fernando Gonçalves. A flow on $S^2$ presenting the ball as its minimal set. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4263-4280. doi: 10.3934/dcdsb.2020287 [13] Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047 [14] Tran Ngoc Thang, Nguyen Thi Bach Kim. Outcome space algorithm for generalized multiplicative problems and optimization over the efficient set. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1417-1433. doi: 10.3934/jimo.2016.12.1417 [15] Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67 [16] Ye Tian, Qingwei Jin, Zhibin Deng. Quadratic optimization over a polyhedral cone. Journal of Industrial and Management Optimization, 2016, 12 (1) : 269-283. doi: 10.3934/jimo.2016.12.269 [17] Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089 [18] Bhawna Kohli. Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3209-3221. doi: 10.3934/jimo.2020114 [19] Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial and Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1 [20] Saeid Ansary Karbasy, Maziar Salahi. Quadratic optimization with two ball constraints. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 165-175. doi: 10.3934/naco.2019046

Impact Factor: