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

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.

Automated summary

This paper examines the upper approximation of the constant in a Markov-type inequality on a simplex using minimal polynomial and pluripotential theories. The complex equilibrium measure that solves the extreme problem by minimizing the energy integral is included. Examples of second degree polynomials are introduced, followed by formulating a challenging bilevel optimization problem using the polynomials for approximation. Three popular meta-heuristics were then applied to investigate the results.