0 down vote favorite I'm trying to solve something like this:
find the smallest n, s.t. $x_1 + f(x_1)\times x_2 + f(x_1,x_2)\times x_3 + ... + f(x_1,...,x_n-1)\times x_n > 100$, under the constraint that $|x_i| < 1$, for all $i = 1, 2, ..., n$.
The actual problem is about some iterative algorithm. I want to find the appropriate parameter s.t. the algorithm terminates in minimal iterations.
I want to know if there is a name for this kind of problem. Pointing to any fields, papers, theorems, or any thoughts would be helpful.
Ok, it's hard to come up with a concrete example because they are either too hard or too trivial, but here is one:
I want to travel from (0,0) to (100, 100). For each step I'm allowed to move along a vector of length determined by the function f. But each step size has to be <= 1.
Suppose I'm at step t. Then $f(x_1, x_2, ..., ) = (1-\eta x_1^2)(1-\eta x_2^2)...(1-\eta x_{t-1}^2)$ is the longest distance I can travel at step t, where $x_1,x_2,...,x_{t-1}$ are the distance I moved at step $1$ to $t-1$, and $\eta$ is some parameter, say here we set $\eta = 0.05$.
Question is how to design the move of each step (direction and length), s.t. the number of steps to reach $(100,100)$ is minimized.
Does that help clarify what I mean?
You clearly should travel along the diagonal, so you are asking to cover a distance of $100 \sqrt 2$. I can do it in $4993$ steps, but I do not know if it is minimal. A little playing suggested I should start with a constant step until $f$ forced the step smaller, then take the maximum allowed step. I made an Excel spreadsheet to implement this, then used Goal Seek trying to set the distance to $141.421\approx 100 \sqrt 2$. I got there with a starting step of $0.031498243$ but could not get there at all with $4992$ steps.