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

Author

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  • Grzegorz Sroka

    (Department of Topology and Algebra, Rzeszow University of Technology, Powstancow Warszawy 12, 35-959 Rzeszow, Poland)

  • Mariusz Oszust

    (Department of Computer and Control Engineering, Rzeszow University of Technology, Wincentego Pola 2, 35-959 Rzeszow, Poland)

Abstract

Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and pluripotential theories were employed. They include a complex equilibrium measure that solves the extreme problem by minimizing the energy integral. Consequently, examples of polynomials of the second degree are introduced. Then, a challenging bilevel optimization problem that uses the polynomials for the approximation was formulated. Finally, three popular meta-heuristics were applied to the problem, and their results were investigated.

Suggested Citation

  • Grzegorz Sroka & Mariusz Oszust, 2021. "Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics," Mathematics, MDPI, vol. 9(3), pages 1-10, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:264-:d:489134
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    References listed on IDEAS

    as
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    2. Sinha, Ankur & Malo, Pekka & Deb, Kalyanmoy, 2017. "Evolutionary algorithm for bilevel optimization using approximations of the lower level optimal solution mapping," European Journal of Operational Research, Elsevier, vol. 257(2), pages 395-411.
    3. Ya Gao & Guangquan Zhang & Jie Lu & Hui-Ming Wee, 2011. "Particle swarm optimization for bi-level pricing problems in supply chains," Journal of Global Optimization, Springer, vol. 51(2), pages 245-254, October.
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