I've got a question whether there is a mathematical expression for a grid search? I have two parameters a and b in [0;1]. Depending on the values of a and b, I get a value for my function (the value is called MAPE for Mean Absolute Percentage Error).
What I do is the following: Let a run from 0.1, 0.2, ... , 0.9 and b from 0.1, 0.2, ..., 0.9, I get 81 different MAPE values for each combination of a and b. The "optimal" value for a and b is those where the MAPE value is the lowest of the computed 81 different values. My question is: How can this grid search expressed mathematically?
I think you are looking for arg min.
Let $f$ be your function that gives you the MAPE based on $a$ and $b$, and let $S = \{0.1, 0.2, ... , 0.9\}$. Then you are computing $$\text{arg min}_{a,b \in S} f(a,b),$$
where $\text{arg min}_{a,b \in S} f(a,b)$ is defined as
$$ \left\{ (a,b) \in S \; | \; \forall \; (x,y) \in S: f(a,b) \le f(x,y) \right\}.$$
Not that the result is a set of pairs, since there may be many minimal pairs.