Suppose an unknown function of real variables, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times ...\times \mathbb{R}\rightarrow\mathbb{R}$, where it takes multiple inputs $x_1,x_2,...,x_{35}$ and return a value $y$.
Suppose that $|f|$ is bounded by an unknown large number $M$ on interval $x_i\in [0,1]$ for $i=1,2,...35$.
We thus investigate $f$ in $[0,1]^{35}$. Suppose that $f$ has large but finite discontinuous in $[0,1]^{35}$, i.e. such as $x_1$ has discontinuous at $0.1,0.001,0.005,0.5,...,0.9$ when $x_j\in [0,0.000001]$ for $j=2,...,35$.
Question:
What's the quickest and most efficient way to estimate $M$ and optimize $f$ in $[0,1]^{35}$?
Because we don't know the exact form of the function, simple analytical approach does not apply. However, because the function is bounded and continuous in very small regions across $[0,1]^{35}$, this may created some sort of dependency. Thus a analytical computational solution may be possible.
However, because of those discontinuous, approach such as newton's does not work well. Further, because of the high dimension in domain, simple trust zone methods does not work well.
I thought about grouping, i.e. splits 35 dimension into 15 initial one(region I), 15 in the middle(region II), and 15 at the end(region III), then select 10 points of largest outcome from region I while fix region II, III, then select 10 points of largest outcome from region II while fix region I, III, then select 10 points of largest outcome from region II while fix region I, III, then I, II, III,... and optimize them in a circle.
I also thought about using assumed statistical distribution to select the points to further accelerate the this process. Further, because the discontinuous is finite, say $N$, we may be able to develop a statistics of $M$ much faster than we optimized the value.
What's the quickest and most efficient way to estimate $M$ and optimize $f$ in $[0,1]^{35}$?