Mathematica NMinimize runs into memory problems

Question

I'm trying to minimize my function "FunctionToMinimize", which is defined as follows:

FunctionToMinimize[a_, b_, c_, d_] := (2.35*Sqrt[
   Variance[1/2*
     (a*#1 + b*#2 + c*#3 + d*#4)
    ]
   ]
  /Mean[1/2*(a*#1 + b*#2 + c*#3 + d*#4)]) 
&[DataList1[[1 ;; 1000]],DataList2[[1 ;; 1000]],
DataList3[[1 ;; 1000]], DataList4[[1 ;; 1000]]]

The four parameters a,b,c and d are restricted to be somewhere between 0.5 and 1.5. My Problem is now, that if I call

NMinimize[{Funktion[w, x, y, z],
0.75 < w < 1.25 && 0.75 < y < 1.25 && 0.75 < x < 1.25 && 0.75 < z < 1.25}, 
{w, x, y, z}]

the Mathematica kernel shuts down because it has not enough memory. If I use only the first 100 entries in my DataLists, it will find me results (in 4.1 sec), but if I use DataList[[1;;1000]] or even more entries, the kernel crashes.

Has anybody an idea, why the NMinimize function uses so much memory? I would need to have the minimization for 150'000 events in each list...

Thanks for your answer, Cheers, Andreas

Answer 1

I would guess (but haven't in any way checked) that the problem is that on each call to your function, Mathematica is trying to construct a symbolic expression derived from all your data and that occupies much more memory than you'd expect.

Regardless, the good news -- if you haven't long since moved on and forgotten about this problem -- is that you can turn the function into something much simpler.

So, first of all, the 2.35 and the 1/2s just change your function by a constant factor and don't affect where the minimum is, so let's ignore them. Next, your function is always non-negative, so minimizing it is the same as minimizing its square, so let's do that.

So now you're trying to minimize var(aw+bx+cy+dz)/mean(aw+bx+cy+dz)^2 where w,x,y,z are (perhaps quite long) vectors.

Now your numerator and denominator are both just quadratic forms in a,b,c,d whose coefficients depend (in fixed ways) on those vectors. Specifically, suppose your vectors have length N. Then your function is just

[sum(aw+bx+cy+dz)^2/N - sum(aw+bx+cy+dz)^2/N^2] / (sum(aw+bx+cy+dz)^2/N^2)

which you might prefer to write as N sum(aw+bx+cy+dz)^2 / sum(aw+bx+cy+dz)^2 - 1

and in that fraction, e.g., the coefficient of bc in the numerator is 2 sum(xy), and the coefficient in the denominator is 2 sum(x) sum(y).

So you can take your big vectors, compute the relevant coefficients once, and then just ask Mathematica to optimize a function of the form (quadratic / quadratic), which should be pretty painless.