Minimizing a sum of exponential functions I want to minimize this function:
$$ g_0(\psi)= \sum_{m=0}^{M-1}e^{j(am^2+bm)}e^{jm\psi} $$
where $a$ and $b$ are constants for which I want to minimize the function.
Can anyone help me regarding this. Will some optimization technique work?

In your expression:
$$g_0(\psi)= \sum_{m=0}^{M-1}e^{j(am^2+bm)}e^{jm\psi}$$
why dont you express the exponent (or more precisely the coefficient of $j$) in this way:
$$am^2+m(b+\psi)=a\left(m+\dfrac{b+\psi}{2a}\right)^2-\dfrac{(b+\psi)^2}{4a}$$
In this way, you could factor out $e^{-j\frac{(b+\psi)^2}{4a}}$ and concentrate on a simplified sum (that has something to do with a certain $\Theta$ function if $M$ is large https://en.wikipedia.org/wiki/Theta_function).
Regarding optimization, to you want to minimize $|g_0(\psi)|$ (note the $||$ I have placed). Could you give a little hint about the context of appearance of this function? It might help.
