On the order of smoothness of the maximal convex continuation of a Boolean function
D. N. Barotova,
R. N. Barotovb a Financial University under the Government of the Russian Federation (Moscow)
b Khujand state university named after academician Bobojon Gafurov (Khujand)
Abstract:
This paper is devoted to establishing the order of smoothness of
$f_{NR}(x_1,x_2,...,x_n)$ — the largest convex continuation to
$[0,1]^n$ of any Boolean function
$f_{B}(x_1,x_2,...,x_n)$. As a result of the study, for each Boolean function
$f_{B}(x_1,x_2,...,x_n)$, the order of differentiability of
$f_{NR}(x_1,x_2,...,x_n)$ — the corresponding greatest convex continuation to
$[0,1]^n$ — was established, namely, firstly, the greatest convex continuation
$f_{NR}(x_1,x_2,...,x_n)$ was estimated from both sides so that, which implies the continuity of
$f_{NR}(x_1,x_2,...,x_n)$ on
$[0,1]^n$ for any natural
$n$, and secondly, it was proved that if the number of essential variables of the Boolean function
$f_{B}(x_1,x_2,...,x_n)$ is less than two, then on
$[0,1]^n$ the greatest convex continuation
$f_{NR}(x_1,x_2,...,x_n)$ is infinite differentiable, and if there are at least two, then on
$[0,1]^n$ the largest convex continuation
$f_{NR}(x_1,x_2,...,x_n)$ is not differentiable, i.e. it is only continuous.
Keywords:
convex continuation of a Boolean function, Boolean function, convex function, global optimization, local extremum.
UDC:
519.716.322,
517.518.244,
512.563 Received: 25.01.2025
Revised: 27.08.2025
DOI:
10.22405/2226-8383-2025-26-3-58-70
Fulltext:
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