Here’s the trivial Convex.jl code
using Convex, SCS
N = 50
x = randn(N) .+ LinRange(0, 1, N)
y = Convex.Variable(N)
p = minimize(sumsquares(y - x))
p.constraints += y[2:end] >= y[1:end-1]
Convex.solve!(p, SCS.Optimizer)
plot([x y.value])
With convex optimization, you can easily tune the fit as well, here’s your example and with some smoothing realized by penalizing sumsquares(diff(y)))
t = -10:0.1:10
x = atan.(t) + 0.1*rand(length(t))
Base.diff(y::Convex.Variable) = y[2:end] - y[1:end-1]
y = Convex.Variable(length(t))
p = minimize(sumsquares(y - x) + 10sumsquares(diff(y)))
p.constraints += y[2:end] >= y[1:end-1]
Convex.solve!(p, SCS.Optimizer)
plot([x y.value])
