This article uses technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural logarithm, also commonly written as ln(x) or loge(x).
The Chebyshev function , with x < 50The function , for x < 104The function , for x < 107
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev functionϑ (x) or θ (x) is given by
where denotes the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x.
The second Chebyshev functionψ (x) is defined similarly, with the sum extending over all prime powers not exceeding x
where Λ is the von Mangoldt function. The Chebyshev functions, especially the second one ψ (x), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, π (x) (see the exact formula below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem.
Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function:
[1]
By minimizing this function for different values of , one obtains every point on a Pareto front, even in the nonconvex parts.[1] Often the functions to be minimized are not but for some scalars . Then [2]
All three functions are named in honour of Pafnuty Chebyshev.
^ abJoshua Knowles (2 May 2014). "Multiobjective Optimization Concepts, Algorithms and Performance Measures" (PDF). The University of Manchester. p. 34.
^Ho-Huu, V.; Hartjes, S.; Visser, H. G.; Curran, R. (2018). "An improved MOEA/D algorithm for bi-objective optimization problems with complex Pareto fronts and its application to structural optimization" (PDF). Expert Systems with Applications. Delft University of Technology. Page 6 equation (2). doi:10.1016/j.eswa.2017.09.051.
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