Solve for s, and t1…tn that minimizes the following summation:
Σnk = 1 (1 – min(s·tk,Ck)/max(s·tk,Ck)),
where
C1…Cn > 1,
s > 0,
t1…tn ∈ ℤ+
edit to clarify to problem description:
“how fast of an algorithm you require.” Not super fast (But not multiple seconds). n will be around 5-10 or so.
As far as the actual problem, I have a number of “elements” of different sizes on a “page” and this page needs to be translated to a format in which there is a maximum base size for an element of X, and the base size of an element has to be an integer. However in the new format any element can be scaled up by a single scaling factor set for the page.
So C1…Cn were the sizes of the elements on the original page. t1…tn are the new integer sizes in the new page format. (And t1…tn need to be less than X.) The scaling factor for the new page format is s.
More:
As far as what I’ve done previously, I find the largest element on the original page, and if its smaller than X, I just use the existing element sizes on the new page, but rounding each one to an integer. However, If the largest element on the original page is greater than X, I divide its size by X to get the scaling factor s for the new page, and divide C1…Cn by s to get t1…tn. But this results in size discrepancies of something like 1-3% on average for every element on the new page but the largest. Not really all that noticeable, but I’m a perfectionist.
You should read on linear programming with integer unknowns as well.
Even though this is not linear programming it might give you an idea on what to look for.
Also you might head over to https://mathoverflow.net/ to get some help remodelling the problem (you have an optimization problem on an integer domain with discontinuities in goal function and my math is a bit rusty to place it properly; check combinatorial optimization as well)
(for the right solution jump to EDIT4 below)
EDIT:
Regarding linearity
Looking for maximum of
Σnk = 1 (1 – min(s·tk,Ck)/max(s·tk,Ck))
might be the same as looking at the maximum for
Σnk = 1 (max(s·tk,Ck) – min(s·tk,Ck))
providing that
Σnk = 1 max(s·tk,Ck) > 0
(which is always ?true? given your conditions)
And term
Σnk = 1 (max(s·tk,Ck) – min(s·tk,Ck))
can be written as
Σnk = 1 Abs(s·tk – Ck)
which, if the question mark above hold gives the following
So all C = 1, and all t and s → ∞ for which your original expression approaches n.
Ok, so originally I was wrong with my suggestion, because I assumed a question would not degenerate into trivial case, which actually it quite obviously does.
EDIT2:
My math is rusty, the procedure above is not correct (first step), but the conclusion/solution seems to validate so I will not correct (it gets a bit complicated)
EDIT3 (Ck are constants and other corrections):
Maybe I should clean up the answer, I believe the following reasoning is enough as a solution:
The fact that Ck are constant and not equal 1 does not matter. To maximize original expression
Σnk = 1 (1 – min(s·tk,Ck)/max(s·tk,Ck))
you should minimize
Σnk = 1 min(s·tk,Ck)/max(s·tk,Ck)
since domain of everything is positive to make this ratio minimal you have to make numerator as small as possible and denominator as big as possible.
The ratio is zero if
It also approaches zero if
(the above equalities should have used limit notation…)
(each condition is enough on its own and represents the solution, when combining them don’t let s approach 0 while tk approach infinity or vice versa; in such case it matters which one approaches it faster)
EDIT4: (actual solution)
Well basically all of the above is giving answer to a wrong question, because I was looking for maximum of the original goal function, not the minimum.
As for minimum it is a bit more interesting, the minimum is reached if each term
min(s·tk,Ck)/max(s·tk,Ck) = 1
This is the maximum of this term given the domain of the parameters. If we assume (for now) that Ck is integer then the solution can be found for
s = 1
tk=Ck
However Ck is not integer in general case, so we need to find s for which Ck/s is integer.
If we can write Ck as
Nk/Dk where N, D ∈ ℤ+ (in another words if Ck is rational)
then a solution can be
s = 1/∏nk = 1Dk
tk = Nk/Dk · ∏nk = 1Dk
(which is ∈ ℤ+)
Note:
Instead of choosing s to be a product of all denominators it could be set to biggest common denominator, and then tk can be calculated appropriately.
Note2:
Plotting diagrams of the functions in question helped me catch my error of misreading the question (realizing that minimum is much more interesting). Also I realized that the functions are continuous (but not smooth, so derivations are discontinuous).
Note3:
The above solution works for rational numbers, but I imagine that irrational numbers would not make the solution useless as decimal or other rational representations of irrational numbers would give an approximate solution proportionally close to real solution as the representation is close to actual value of irrational number.