Re: Maximum of a rational function of many variables in the unit cube

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Wed, Jun 06, 2018 at 12:50:34AM +0000, Jacques Gélinas wrote:
> Now I have certain rational functions of the coefficients (a,b,c,...) of a real polynomial of degree n>1
> (expressions which look like the convergents of continued fractions) such as
> 
> f1(a,b) = 1/2 * (n-2)/(n-1) * (5-a*b)  /  (3-a);
> 
> f2(a,b,c) = b  *  (3-a)  /  (5-a*b)  * (1 - 1/3*(n-3)/(n-1)*(1-a*b*c/7)/(1-a/3))   \
>           / (1 - 1/2*(n-2)/(n-1)*(1-a*b/5)/(1-a/3) );
> 
> The coefficients are in the unit cube, 0<=a<=1, 0<=b<=1, ..., and I would like to 
> use Pari/GP to test the conjecture that the maxima over the cube is in (0,1] for all n.
> 
> 1. Could I determine the maxima over the vertices, say (0,0),(0,1),(1,0),(1,1) of f1(a,b),
> without using a variable number of "for loops" or constructors like "vector" ?
> 
> 2. What functions are available in GP for a Monte-Carlo search inside the cube for the maxima ?

Given you have a rationale function of real variables, I would suggest
to use a differential geometry method, like Lagrange multipliers:

<https://en.wikipedia.org/wiki/Lagrange_multiplier>

Cheers,
Bill.