Optimization algorithms for finding the optimal settings in large search space I have a 8 independent variables, out of 8 variables, 6 variables have 63 states, one variable have 200 states and last one have 3 states. So, in total, they create a search space of $63^6 \cdot  200 \cdot 3$, approximately 32 trillion rows.
How can I find the optimal rows, that minimizes a custom objective function? Can we use combinatorics optimization algorithms ? If so, what categories of algorithm, I need to look into for problem such huge search space

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 ?
 A: "There is no free lunch": this applies also to optimization. You have to characterize your multi-variable objective function in order to get better algorithms than random picking.
You may want to look at:

*

*Linear or quadratic formulation, possibly after a change of variables: very effective softwares exist to deal with these cases. A typical combinatorial optimization problem such as TSP is actually best resolved by linear programming (with very elaborate models).

*Convexity, to get unicity of minimum, and allow methods such as interior-point.

*Symetries, to prune the search space.

*Monotone variation of function with regard to some variable.

*Simple lower bounds to the objective function, so that finding points with good (= low) objective value, enables to cut entire branches of the search tree, when it can be proven that they cannot contain a better result (branch-and-bound).

*etc.

Your search space is actually not that big: with a few orders of magnitude simplification, it may be reachable by exhaustive search. That depends also upon the time it takes to compute the objective function, and the computation hardware you have.
Another thing to consider is the problem context: usually some heuristics come to mind, relying on "physical" properties of the domain.
If the problem can be expressed as a sequence of choices, dynamic programming may be possible. You need to define a state that includes all necessary data to make further choices, and have a limited number of states. See knapsack problem as a typical example.
If there is no simplification, there are still elaborate algorithms to deal with. The general idea is to try a few combinations at random, then balance between exploration (of new zones in the search space) and exploitation (of the zones that have already given good results); MCTS (Monte-Carlo Tree Search) is an example.
If nothing goes or you do not have time to code algorithms, you may look at Black-Box optimization softwares: they apply all tricks of the trade for you. Well, not really: a good home made algorithm is usually vastly better, but it also takes vastly more time to code it, so that's a trade-off.
