# Minimizes a function (variance over 1mln data-points)? **URL:** \ **Category:** Questions\ **Tags:** calculus, optimization\ **Created:** [October 24, 2016, 10:53am UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505 "2016-10-24T10:53:30Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![saur12ab](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/saur12ab/32/15904_2.png) [@saur12ab](https://community.wolfram.com/u/saur12ab)\ **Post date:** [October 24, 2016, 10:53am UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505/1 "2016-10-24T10:53:30Z")
I have attached below one Mathematica notebook and a MATLAB file where I try to find the \theta which minimizes a function (variance over 1mln data-points). The MATLAB code runs pretty fast, while the execution of the Mathematica code freezes my machine. Any suggestions how to make the Mathematica code executable (at least)? ---
**Author:** ![sghuisman](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/sghuisman/32/27850_2.png) [@sghuisman](https://community.wolfram.com/u/sghuisman)\ **Post date:** [October 24, 2016, 12:07pm UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505/2 "2016-10-24T12:07:11Z")
The problem is that: ``` $HistoryLength=1; S=100;K=70;\[Sigma]=0.2;T=1;r=0.01;\[Alpha]=r-\[Sigma]^2/2; n=100; Z=RandomVariate[NormalDistribution[],n]; fVarPut[\[Theta]_]:=Variance[E^(-r T) Ramp[K-S E^(\[Alpha] T+\[Sigma] Sqrt[T](Z+\[Theta]))]E^(-\[Theta](Z+\[Theta])+\[Theta]^2/2)]; LogPlot[fVarPut[x],{x,-2,4},PlotRange->All] ``` looks like something like: ![ScreenShot2016-10-24at14.04.50.png](https://global.discourse-cdn.com/wolfram/original/2X/7/7e146a01a69fdf87174b7dddb37d00ce3cd8ff94.png) so fVarPut[100] will always give 0. it is ‘flat’ on the right (equal to 0, because of the Ramp function), so there is not _one_ minimum… ---
**Author:** ![fkampas](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/fkampas/32/41217_2.png) [@fkampas](https://community.wolfram.com/u/fkampas)\ **Post date:** [October 24, 2016, 12:11pm UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505/3 "2016-10-24T12:11:36Z")
The definition of pVarPut needs to signal that the function only works with a numeric argument. In[1]:= S = 100; K = 70; [Sigma] = 0.2; T = 1; r = 0.01; [Alpha] = r - [Sigma]^2/2; ``` In[2]:= n = 100; Z = RandomVariate[NormalDistribution[], n]; In[4]:= fVarPut[\[Theta]_?NumberQ] := Variance[E^(-r T) Ramp[K - S E^(\[Alpha] T + \[Sigma] Sqrt[ T] (Z + \[Theta]))] E^(-\[Theta] (Z + \[Theta]) + \[Theta]^2/ 2)]; In[5]:= (* freezes even with WorkingPrecision\[Rule]5 *) NMinimize[fVarPut[x], x, WorkingPrecision -> 5] Out[5]= {0, {x -> 3.3686}} In[6]:= FindMinimum[fVarPut[x], {x, -2}, WorkingPrecision -> 5] Out[6]= {1.0248*10^-382, {x -> -32.000}} ``` ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [October 24, 2016, 12:43pm UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505/4 "2016-10-24T12:43:01Z")
I don’t get any freeze with your code. However, it may run faster if you use the immediate assignment ``` fVarPut[\[Theta]_?NumberQ] = Variance[E^(-r T) Ramp[ K - S E^(\[Alpha] T + \[Sigma] Sqrt[ T] (Z + \[Theta]))] E^(-\[Theta] (Z + \[Theta]) + \ \[Theta]^2/2)]; ``` instead of the delayed assignment :=, because with delayed assignment Mathematica has to recalculate the `Variance` formula every time `fVarPut` is called. ---
**Author:** ![saur12ab](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/saur12ab/32/15904_2.png) [@saur12ab](https://community.wolfram.com/u/saur12ab)\ **Post date:** [October 24, 2016, 12:49pm UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505/5 "2016-10-24T12:49:43Z")
Thank you very much all of you! I was about to write that it freezes either way even if I ask for the minimum for the interval (-4,-1), when I read Frank’s paramount remark. So much better now! 🙂 ---
**Author:** ![sghuisman](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/sghuisman/32/27850_2.png) [@sghuisman](https://community.wolfram.com/u/sghuisman)\ **Post date:** [October 24, 2016, 12:49pm UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505/6 "2016-10-24T12:49:51Z")
This will calculate the variance in a symbolic way once; that might be or not be a smart idea depending on the number of points. Definitely not a smart move if n is large, it will do a lot of (comparatively) slow symbolic processing. [Frank Kampas](http://community.wolfram.com/web/fkampas)’s solution is a good idea; to turn it into a numeric computation, which is very fast. It will avoid Mathematica trying to do any symbolic manipulation on it. ---
**Author:** ![sghuisman](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/sghuisman/32/27850_2.png) [@sghuisman](https://community.wolfram.com/u/sghuisman)\ **Post date:** [October 24, 2016, 12:52pm UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505/7 "2016-10-24T12:52:37Z")
You will need to constrain it using something like: ``` FindMinimum[f,{x,x0,xmin,xmax}] ``` As this problem will have a minimum for x → -infinity and for x\>a for some ‘a’ it is constant (kind of a minimum) (a depends on the random values). ---
**Author:** ![saur12ab](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/saur12ab/32/15904_2.png) [@saur12ab](https://community.wolfram.com/u/saur12ab)\ **Post date:** [October 24, 2016, 1:14pm UTC](https://community.wolfram.com/t/minimizes-a-function-variance-over-1mln-data-points/9505/8 "2016-10-24T13:14:47Z")
I have a poor machine, Gianluca.. and a bit weird I guess, because it gives me that the implementation with “=” is slower than the one with “:=”