I have $8$ independent variables, out of which:
$6$ variables have $63$ states
one variable has $200$ states
one variable has $3$ states
Thus, in total, they create a search space of $63^6 \cdot 200 \cdot 3 \approx 32 \text{ trillion}$. How can I find the optimal rows that minimize a custom objective function? Can we use combinatorial optimization algorithms? If so, what categories of algorithm, I need to look into for problems with such huge search spaces?
I am not much familiar in the field of optimization, but with some preliminary googling, I came across, google OR-tools, particularly constraints optimization. But, from my initial thoughts, I think it is bit hard to analytically describe my objective function. But, I haven't tried it.
EDITED
This is how my objective function will look like.
$$\min y = x_1 - \frac{x_2-1}{x_1} + \frac{x_3-1}{x_1 x_2} - \frac{x_4-1}{x_1 x_2 x_3} + \frac{x_5-1}{x_1 x_2 x_3 x_4} - \frac{x_6-1}{x_1 x_2 x_3 x_4 x_5} + \frac{x_7-1}{x_1 x_2 x_3 x_4 x_5 x_6} - \frac{x_8-1}{x_1 x_2 x_3 x_4 x_5 x_6 x_7} $$
Constraints:
$$1\leq x_1\leq 200$$ $$1\leq x_2,x_3,x_4,x_5,x_6,x_7\leq 63$$ $$0\leq x_8\leq 2$$
$$0.01x_1+0.5(x_2+x_3+x_4+x_5+x_6+x_7)+15x_8=C$$
where $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8$ are integers and $C$ is a constant.
So, from the above, I am guessing it mostly falls under MINLP or Constraints Optimization ? But, in most of the online examples the objective functions are much simpler. Hence, what kind of algorithms is suitable for such complex objective functions ?