# Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Minimal Polynomials

## 3. Estimation of the Constant $\mathit{C}$ in a Markov-Type Inequality

## 4. Optimization Problem in a Markov-Type Inequality

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Baran, M.; Bialas-Ciez, L. On the behaviour of constants in some polynomial inequalities. Ann. Pol. Math.
**2019**, 123, 43–60. [Google Scholar] [CrossRef] - Baran, M.; Bialas-Ciez, L.; Milowka, B. On the best exponent in Markov’s inequality. Potential Anal.
**2013**, 38, 635–651. [Google Scholar] [CrossRef] [Green Version] - Baran, M. Markov inequality on sets with polynomial parametrization. Ann. Polon. Math.
**1994**, 60, 69–79. [Google Scholar] [CrossRef] - Bialas-Ciez, L.; Goetgheluck, P. Constants in Markov’s inequality on convex sets. East J. Approx.
**1995**, 1, 379–389. [Google Scholar] - Bun, M.; Thaler, J. Dual lower bounds for approximate degree and Markov-Bernstein inequalities. Inf. Comput.
**2015**, 243, 2–25. [Google Scholar] [CrossRef] - Nadzhmiddinov, D.; Subbotin, Y.N. Markov Inequalities for Polynomials on Triangles. Math. Notes Acad. Sci. USSR
**1989**, 46, 627–631. [Google Scholar] [CrossRef] - Baran, M. Bernstein type theorems for compact sets in ${\mathbb{R}}^{N}$ revisited. J. Approx. Theory
**1994**, 79, 190–198. [Google Scholar] [CrossRef] [Green Version] - Baran, M. Complex equilibrium measure and Bernstein type theorems for compact sets in ${\mathbb{R}}^{n}$. Proc. Amer. Math. Soc.
**1995**, 123, 485–494. [Google Scholar] - Sofonea, F.; Tincu, I. On an Inequality for Legendre Polynomials. Mathematics
**2020**, 8, 2044. [Google Scholar] [CrossRef] - Ozisik, S.; Riverce, B.; Warburon, T. On the Constants in Inverse Inequalities in L
_{2}. Technical Report. 2010. Available online: https://hdl.handle.net/1911/102161 (accessed on 5 December 2020). - Piazzon, F.; Vianello, M. A note on total degree polynomial optimization by Chebyshev grids. Optim. Lett.
**2018**, 12, 63–71. [Google Scholar] - Piazzon, F.; Vianello, M. Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies. Optim. Lett.
**2019**, 13, 1325–1343. [Google Scholar] [CrossRef] - Sommariva, A.; Vianello, M. Discrete norming inequalities on sections of sphere, ball and torus. arXiv
**2018**, arXiv:1802.01711. [Google Scholar] - Wilhelmsen, D.R. A Markov inequality in several dimensions. J. Approx. Theory
**1974**, 11, 216–220. [Google Scholar] [CrossRef] [Green Version] - Sarantopoulos, Y. Bounds on the derivatives of polynomials on Banach spaces. Math. Proc. Camb. Philos. Soc.
**1991**, 110, 307–312. [Google Scholar] [CrossRef] - Kroó, A. Révész, S. On Bernstein and Markov-type inequalities for multivariate polynomials on convex bodies. J. Approx. Theory
**1999**, 99, 134–152. [Google Scholar] [CrossRef] [Green Version] - Kroó, A. On the existence of optimal meshes in every convex domain on the plane. J. Approx. Theory
**2019**, 238, 26–37. [Google Scholar] [CrossRef] [Green Version] - Davydov, O. Smooth Finite Elements and Stable Splitting; Berichte “Reihe Mathe-matik” der Philipps-Universit at Marburg: Marburg, Germany, 2007. [Google Scholar]
- Angermann, L.; Heneke, C. Interpolation, Projection and Hierarchical Bases in Discontinuous Galerkin Methods. Numer. Math. Theory Methods Appl.
**2015**, 8, 425–450. [Google Scholar] [CrossRef] [Green Version] - Baran, M.; Kowalska, A.; Ozorka, P. Optimal factors in Vladimir Markov’s inequality in L
^{2}Norm. Sci. Tech. Innov.**2018**, 2, 64–73. [Google Scholar] - Bialas-Ciez, L.; Jedrzejowski, M. Transfinite Diameter of Bernstein Sets in ${\mathbb{C}}^{N}$. J. Inequal. Appl.
**2002**, 7, 393–404. [Google Scholar] - Bloom, T.; Calvi, J.P. On multivariate minimal polynomials. Math. Proc. Camb. Phil. Soc.
**2000**, 129, 417–432. [Google Scholar] [CrossRef] - Bialas-Ciez, L.; Calvi, J.P. Homogeneous minimal polynomials with prescribed interpolation conditions. Proc. Am. Math. Soc.
**2016**, 368, 8383–8402. [Google Scholar] [CrossRef] - Newman, D.J.; Xu, Y. Tchebycheff polynomials on a triangular region. Constr. Approx
**1993**, 9, 543–546. [Google Scholar] [CrossRef] - Peherstorfer, F. Minimal polynomials for compact sets of the complex plane. Constr. Approx
**1996**, 12, 481–488. [Google Scholar] [CrossRef] - Davis, P.J. Approximation and Interpolation; Dover Publication: New York, NY, USA, 1975. [Google Scholar]
- Aron, R.M.; Klimek, M. Supremum norms for quadratic polynomials. Arch. Math.
**2001**, 76, 73–80. [Google Scholar] [CrossRef] - Sinha, A.; Malo, P.; Deb, K. Evolutionary algorithm for bilevel optimization using approximations of the lower level optimal solution mapping. Eur. J. Oper. Res.
**2017**, 257, 395–411. [Google Scholar] [CrossRef] - Byrd, R.H.; Gilbert, J.C.; Nocedal, J. A Trust Region Method Based on Interior Point Techniques for Nonlinear Programming. Math. Prog.
**2000**, 89, 149–185. [Google Scholar] [CrossRef] [Green Version] - Byrd, R.H.; Hribar, M.E.; Nocedal, J. An Interior Point Algorithm for Large-Scale Nonlinear Programming. SIAM J. Optim.
**1999**, 9, 877–900. [Google Scholar] - Gao, Y.; Zhang, G.; Lu, J.; Wee, H.-M. Particle swarm optimization for bi-level pricing problems in supply chains. J. Glob. Optim.
**2011**, 51, 245–254. [Google Scholar] [CrossRef] [Green Version] - Pedersen, M.E. Good Parameters for Particle Swarm Optimization; Hvass Laboratories: Luxembourg, 2010. [Google Scholar]
- Zhang, Y.; Wang, S.; Ji, G. A Comprehensive Survey on Particle Swarm Optimization Algorithm and Its Applications. Math. Probl. Eng.
**2015**, 2015, 38. [Google Scholar] [CrossRef] [Green Version] - Goldberg, D.E. Genetic Algorithms in Search. In Optimization & Machine Learning; Addison-Wesley: Boston, MA, USA, 1989. [Google Scholar]
- Yang, J.; Zhang, M.; He, B.; Yang, C. Bi-level programming model and hybrid genetic algorithm for flow interception problem with customer choice. Comput. Math. Appl.
**2009**, 57, 1985–1994. [Google Scholar] - Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. Optimization by Simulated Annealing. Science
**1983**, 220, 671–680. [Google Scholar] [PubMed] - Sahin, K.H.; Ciric, A.R. A dual temperature simulated annealing approach for solving bilevel programming problems. Comput. Chem. Eng.
**1998**, 23, 11–25. [Google Scholar] [CrossRef]

**Figure 1.**Influence of the coefficients u on the objective function ${C}_{4}$. Three coefficients are selected: ${a}_{0}$ (first row), ${a}_{1}$ (second row), and ${a}_{2}$ (third row).

**Figure 2.**Convergence plots for the values of ${C}_{4}$ for sample runs of the hybrid approach with the Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Genetic Algorithm (GA). The plots contain the best result at each function call and the mean value of ${C}_{4}$ for the processed solutions.

SYMBOLS | ${\mathit{x}}_{0,0}$ | ${\mathit{x}}_{1,0}$ | ${\mathit{x}}_{0,1}$ | ${\mathit{x}}_{1,1}$ | ${\mathit{x}}_{1,2}$ | ${\mathit{x}}_{2,1}$ | ${\mathit{x}}_{2,0}$ | ${\mathit{x}}_{0,2}$ | ${\mathit{x}}_{3,0}$ | ${\mathit{x}}_{0,3}$ |
---|---|---|---|---|---|---|---|---|---|---|

VALUES | $2\pi $ | $2\pi /3$ | $2\pi /3$ | $2\pi /15$ | $2\pi /35$ | $2\pi /35$ | $2\pi /5$ | $2\pi /5$ | $2\pi /7$ | $2\pi /7$ |

Example | Minimal Polynomial $\mathit{P}(\mathit{x},\mathit{y})$ | ${\parallel \mathit{P}\parallel}_{{\mathit{S}}_{2}}$ | ${\parallel \mathit{gradP}\parallel}_{{\mathit{S}}_{2}}$ | ${\parallel \mathit{gradP}\parallel}_{{\mathit{S}}_{2}}/4{\parallel \mathit{P}\parallel}_{{\mathit{S}}_{2}}$ |
---|---|---|---|---|

1 | ${y}^{2}-y+\frac{1}{9}+xy-\frac{4}{9}{x}^{2}+\frac{2}{9}x$ | 5/36 | $\sqrt{202}$/9 | C = 2.842534081 |

2 | ${y}^{2}-y+\frac{1}{8}+\frac{1}{8}{x}^{2}+\frac{33}{24}xy-\frac{19}{56}$ | 99/784 | 10$\sqrt{585}$/168 | C = 2.850293143 |

3 | ${y}^{2}-y+\frac{1}{8}+\frac{61}{40}xy+\frac{1}{4}{x}^{2}-\frac{1}{2}x$ | 221/1760 | $\sqrt{3281}$/40 | C = 2.851041419 |

4 | ${y}^{2}-y+\frac{1}{8}+\frac{3}{2}xy+\frac{1}{4}{x}^{2}-\frac{1}{2}x$ | 1/8 | $\sqrt{2}$ | C = 2.828427125 |

**Table 3.**Optimization results obtained by the compared algorithms. The best values of C obtained in the first experiment are written in bold.

Algorithm | ${\mathit{C}}_{1},{\mathit{C}}_{2}$ | ${\mathit{C}}_{3},{\mathit{C}}_{4}$ |
---|---|---|

GA | 2.861879141 | 2.852924202 |

SA | 2.917638439 | 2.854084285 |

PSO | 2.925401560 | 2.854206134 |

${C}_{1},{C}_{2}$ | 2.925415050 |

${a}_{0}$ | −2765.096993 |

${a}_{1}$ | −1383.048545 |

${a}_{2}$ | −15148.04101 |

${a}_{3}$ | −10633.59420 |

${a}_{4}$ | 10846.85282 |

${C}_{3},{C}_{4}$ | 2.854206170 |

${a}_{0}$ | 1.169292858 |

${a}_{1}$ | −0.121651344 |

${a}_{2}$ | −0.129695921 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sroka, G.; Oszust, M.
Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics. *Mathematics* **2021**, *9*, 264.
https://doi.org/10.3390/math9030264

**AMA Style**

Sroka G, Oszust M.
Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics. *Mathematics*. 2021; 9(3):264.
https://doi.org/10.3390/math9030264

**Chicago/Turabian Style**

Sroka, Grzegorz, and Mariusz Oszust.
2021. "Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics" *Mathematics* 9, no. 3: 264.
https://doi.org/10.3390/math9030264