Polynomial fitting, regularization, Kobon triangle, Fujimura's triangle problem
using Optim
n = [3 4 5 6 7 8 9 13 15 17]
k = [1 2 5 7 11 15 21 47 65 85]
lambda = 2.5
f(lam, x) = 1/size(k)[2] * sum((x[1] .+ x[2].*(n) .+ x[3].*(n).^2 .- (k)) .^ 2) - lam * sum(abs.(x))
#f(lam, x) = 1/size(k)[2] * sum((x[1] .+ x[2].*(n) .+ x[3].*(n).^2 .+ x[4].*(n).^3 .+ x[5].*(n).^4 .+ x[6].*(n).^5 .+ x[7].*(n).^6 .+ x[8].*(n).^7 .- (k)) .^ 2) - lam * sum(abs.(x).^2)
#x0 = [0.0, 0.0, 0.0, 0.0, 0.0]
x0 = [0.0, 0.0, 0.0]
for i in 1:100
lam = 0. + 0.002 * i
g(x) = f(lam, x)
res = optimize(g, x0).minimum
println(abs(res),"###", lam)
end
lam1 = 0.1
costl1(x) = f(lam1, x)
l1res = optimize(costl1, x0).minimizer
3-element Vector{Float64}:
-0.682241878486419
-0.658912921983581
0.3350440666781199
Kobon Triangle Problem
If we draw k lines, what is the largest number of triangles without overlapping?
This is we called Kobon triangle problem, or Fujimura's triangle problem. Actually, this is unsolved problem (2024 now)
We know some upper bound such as floor of (k-2)k/3
I conducted quadratic fitting, and obtain similar coefficients.
for polynomial fitting and its regularization, please see the following documents.
https://iopscience.iop.org/article/10.1088/1742-6596/1213/4/042054/pdf
Comment : Looks very interesting, but I don't fully understand how to solve that. Anyway this is just practice of programming.
