Optimisation of continuous function in [0,1]
I've been struggling for a while with the following problem. I'd be very grateful if any of you have advice or pointers.
Basically there's a set of variables whose domain is [0,1] (extremes included) x1, ..., xn.
The function to maximise looks like this
It's a sum of products where variables can appear or not and the coefficient is 1 or -1.
x1*x2 - x1*x3*x4 + x2*x4
more in general the function looks like
C11*x2*...*xn + ... + C1n*x1*...*xn-1 +
CN1*x1 + ... CNN*x2 +
with the constants being 1, -1 or 0
My intuition is that there the maximum of this function always belongs to the corners e.g. [1,0,0,0,1] or [1,1,1,0,0] but I haven't managed to prove it yet.
Thanks for the help.