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Re: Finding the Local Minima of a somewhat complicated function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116021] Re: Finding the Local Minima of a somewhat complicated function
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Sat, 29 Jan 2011 05:27:56 -0500 (EST)

I believe you can solve it for all a in one go:

Clear[fun]
fun[a_, y_] = (1000 - 5*a)/10000000000 +
   332*(y/10000000000 + (1 - y)*(1/5000000000 + y))
+(1/2)*y*(1000 - 5*a + 1000*(-1/10)) +
   0.695*a*(-(Log[10000000000]/10000000000) + (1 - y)*Log[1 - y] +
      y*Log[y]) // Rationalize;

start = 10^(-20);
end = 1 - 10^(-20);

Minimize[{fun[a, y], start <= y <= end}, y]

{(333000000008300000000829999999917 -
     1250000000000000000000000000000 a)/
   2500000000000000000000000000000000000000, {y ->
    9999999999/
     20000000000 + (\[Sqrt](8300000014980000000083 -
          50000000000 a + (-333000000008300000000829999999917 +
             1250000000000000000000000000000 a)/
           25000000000000000000))/(20000000000 Sqrt[83])}}

Bobby

On Fri, 28 Jan 2011 05:14:32 -0600, Andrew DeYoung  
<adeyoung at andrew.cmu.edu> wrote:

> Hi,
>
> I have a function that consists of two variables, y and a.  I would
> like to find the local minimum of the function in y for various
> constant values of a.
>
> For example, the list of a values is given by:
>
> atable = Range[100, 300, 5];
>
> For each a value in atable, I want to find the local minimum of the
> function in y.  My function is "fun," and I use code like the
> following:
>
> fun=(1000-5*a)/10000000000+332*(y/10000000000+(1-y)*(1/5000000000+y))
> +(1/2)*y*(1000-5*a+1000*(-1/10))+0.695*a*(-(Log[10000000000]/
> 10000000000)+(1-y)*Log[1-y]+y*Log[y]);
>
> startPoint = 10^(-20);
> endPoint = 1-10^(-20);
> minData = Table[FindMinimum[fun /. a -> j, {y, startPoint, endPoint}],
> {j, atable}]
>
> Above, I use startPoint=10^(-20) and endPoint=1-10^(-20) because the
> function is indeterminate at y=0 and at y=1.  When I run the above
> code, I see that for most (but not all) values of a, Mathematica does
> not find a local minimum.  Why is this so?
>
> Of course, it could be that the function does not have a local minimum
> at those values of a where Mathematica does not find one.  But, if I
> plot fun at a=300, for example, the plot shows that there is a local
> minimum at something like y=0.945:
>
> Plot[fun /. a -> 300, {y, 0, 1}]
>
> But if I ask Mathematica to find that local minimum...
>
> FindMinimum[fun /. a -> 300, {y, startPoint, endPoint}]
>
> ...Mathematica will not find it.
>
> Can you please help me understand why I am having trouble finding the
> local minima?
>
> Many thanks in advance,
>
> Andrew DeYoung
> Carnegie Mellon University
>


-- 
DrMajorBob at yahoo.com


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