I was analyzing a bunch of games and often I wanted to find the integer(s) such that $f(i)=\max_{j\in Z\cap[a,b]} f(j)$ where $f:[a,b]\rightarrow R$ is concave. I proved a few simple results myself, and I imagine that my theorems are trivial consequences of published theorems. Has anybody here seen theorems proven that are similar to the simple theorems shown below? The Discrete Convex Analysis papers seem to be much more focused on finding the minimum value of $f(x)$ where $f(x)$ is convex and $x$ is confined to a subset of $Z^n$ with $n>1$.
I would appreciate it if anyone can give me a reference for convex analysis that has theorems similar to the ones below, or better yet a reference that contains theorems which are more general than the ones I give below and that I could be use to prove the theorems below.
Here is a list of the simple theorems that I found useful for maximizing concave $f(x)$ over integers. There are also bounds that you can put on the integer solutions if the function is thrice differentiable. I haven't worked those out yet.
Theorem 1: Suppose that $f:[a,b]\rightarrow R$ is strictly concave, $a$ and $b$ are real numbers, $a<b$, and there exists a real $x$ satisfying $a\leq x<x+1 \leq b$ and $f(x)=f(x+1)$. Then, for any integer $z\in [a,b]$,
$x<z<x+1$ if and only if $ f(z)>f(x)$ and
$x\leq z\leq x+1$ if and only if $ f(z)\geq f(x)$.
Furthermore, $\max_{i\in Z\cap [a,b]} f(i)$ is well defined and for any integer $j$, and $\max_{i\in Z\cap [a,b]} f(i) = f(j)$ if and only if $j\in\{\mathrm{ceil}(x), \mathrm{floor}(x+1)\}$.
Corollary 1: Suppose that $f(x) = a x^2 +b x + c$ with $a<0$ and $\{b,c\}\subset R$. Then for any integer $j$, $\max_{i\in Z\cap [a,b]} f(i) = f(j)$ if and only if $j\in\{\mathrm{ceil}(x)), \mathrm{floor}(x+1)\}$ with $x=-\frac{b}{2 a}-1/2.$
Theorem 2: Assume that $f:[a,b]\rightarrow R$ is strictly concave with $a$ and $b$ real satisfying $a+1<b$. Let $x\in[a,b]$ such that $f(x) = \max_{y\in[a,b]} f(y)$ and let $m =\max_{j\in[a,b]\cap Z} f(j)$. Then exactly one of the following is true:
- there exists a unique integer $i$ such that $f(i) = m$ and for that unique integer $i$, $|i-x|<1$, or
- there exists a unique integer $i$ such that $\{j\in Z |\ f(j)=m\}=\{i,i+1\}$ and that $i$ satisfies $x-1 \leq i \leq x$.
Theorem 1 until 'Furthermore' is well known and does not need $z$ to be integer: the 'only if' follows directly from the definition of a concave function, and the 'if' is, e.g., (2) in this paper.
These theorems remind me of the news vendor problem with discrete demand, and marginal analysis.