🚀 go-pugleaf

It seems these days that calculus of variations is a lost art. I have
talked to a number of top-flight mathematicians about the following
problem and none of them could help in solving it. (I can't solve it
either). Can anyone help?

Let  L(x) be an unknown function, let u be a parameter to be optimized,
let F be an known integer, and let P(a,b) be a known (but messy!) function.

Find L(x) and u to minimize

	u
       ---
       \
	\   L(b)
	/
       /
       ---
       b=1

subject to:


	u	L(b)
       ---     -----
       \       \
	\       \       P(a,b)   >= F
	/       /
       /       /
       ---     -----
       b=1    a= -L(b)

I know how to compute P(a,b), but dealing with it analytically is
intractable. It is the product of two Dickman's rho functions.

Can anyone help? How do I solve the above problem?
 
Notes:

(1) One can, of course, turn the above discrete problem into a continuous
    one by changing the sums into Stieltje's integrals.

(2) P(a,b) is as follows:

Let rho(t) be Dickman's rho function, given by the differential-delay
equation   t rho'(t) = rho(t-1).  Let FB and M be integers.  Then, P(a,b) is:

mu(a,b) mu'(a,b)

where

mu(a,b) = integral from  beta-alpha to beta-1 of  rho(t)/(beta-alpha) dt

where  alpha = log(FB)/log(a + bM)
       beta = 2*alpha

mu'(a,b) = integral from beta'-alpha' to beta'-1 of rho(t)/(beta'-alpha') dt

where	alpha' = log(FB)/log(a^d - k b^d)      d,k known
	beta' = 2 alpha'

mu and mu' are respectively the probability that a+bM  and a^d-kb^d factor
into the product of primes up to FB, with possibly a single additional factor
between FB and FB^2.
 
Basically what is happening is that we want to find a function bounding an
area so as to minimize that area, subject to a (joint) density function 
over that area summing to at least F.
--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"