Methods to Minimize Functions and Integrals over $\mathbb{N}$. In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable function $f_\mu (n)$. Moreover, I have a couple messy integral expressions involving floor and ceiling functions, say $$\int_{n\mu}^{\left \lceil{n \mu}\right \rceil} f(t) \, dt,$$ again I need to minimize this for $n \in \mathbb{N}$ given $\mu$.  
My question is,:   
What are some methods for minimizing functions and integrals over subsets of $\mathbb{R}$, specifically $\mathbb{N}$? 
Say, given a messy quadratic in $n$ and $\mu$, what is the best way to minimize such a function over $n \in \mathbb{N}$? To specify, I'm looking for methods that avoid using computational resources, such as Matlab, these should be purely analytical. 
Also, any references would be remarkably useful. 
 A: I think there is no simple method to minimize over $\mathbb{N}$. This is a very complex topic in math. Using the computation it will be much more simple to find minimum (and sometimes even more faster). Anyway, I am familiar with some problems of this type: 
For example find minimum of $f(n) = (x-0.25)^2$ we use $f'(n) =0$ and find that on $\mathbb{R}$ $x_{min} = 0.25$ in this case on $\mathbb{N}$ $x_{min} = [0.25]=0$, but this is not true to any function. You can find several families of functions that this method is working on them, and check if your function in one of them.
More complex example is
$f(n) = \frac{n}{n-1}-1.1$ the minimum is at $n=0$. One way to obtain this analyticaly is to use taylor series and than the previous method or use the fact that the function is monotonically decreasing...
It's get really though if your function has sin or cos. For example to find minimum on $\mathbb{N}$ of $f(n)=sin(2n)$.
Using derivative we can find that the minimum is on the points $n=3\pi/4+2\pi k$, It can be shown that for any $\epsilon>0$ there exists $k$ such that $f(3\pi/4+2\pi k)+1<\epsilon$. Therefore  $f(n)$ has a horizontal asymptote on $\mathbb{N}$

Sometimes parts of the function can be dissmised, for example
$g(n)=h(n)sin(\pi n)$
has no maximum on $\mathbb{R}$, but $g(n)=0$ on $\mathbb{N}$
 To find minimum on $\mathbb{N}$ analytically, you must study carefully your function and be creative and open minded. Most of the time I'm just using integer programming.
