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Mathematica

Minimize Usage

MaiLin Shi
Posted 12 years ago
Hi guys, I'm a beginer in mathematica ,and now want to use it to check my homework answer, but the software result seems to be wrong.
_______________
code:
Minimize[{9.0 n (m - 1.0)^4 + (m n - 2.0)^2, -1.0 <= m <=
    4.0 && -1.0 <= n <= 4.0}, {m, n}]
 picture:
POSTED BY: MaiLin Shi
7 Replies
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I think that the issue is that NMinimize is more likely to get "stuck" in a local minimum and never get to the global one.  So it is less clever with local minima.
POSTED BY: Karl Isensee
@Bruce, I think you would like to say "NMinimize is less clever about dealing with global minima" ;-)
POSTED BY: Shenghui Yang
The explanation for the poor result from your Minimize run is that when Minimize is given
approximate numbers (floating-point), it switches to NMinimize which handles floating-point
numbers better. NMinimize is less clever about dealing with local minima.  When Minimize
is given an equation and constraints that use only exact (Integer or Rational) numbers,
it can use more mathematical techniques.
In[1]:= Minimize[{9.0 n (m - 1.0)^4 + (m n - 2.0)^2, -1.0 <= m <= 4.0 && -1.0 <= n <= 4.0}, {m, n}]
Out[1]= {-143., {m -> -1., n -> -1.}}

In[2]:= Minimize[{9 n (m - 1)^4 + (m n - 2)^2, -1 <= m <= 4 && -1 <= n <= 4}, {m, n}, Integers]
Out[2]= {-693, {m -> 4, n -> -1}}
POSTED BY: Bruce Miller
Just to reemphasize something from above:

Mathematica supports both exact numbers and approximate numbers (with finite numerical precision - aka floating point numbers):

http://reference.wolfram.com/mathematica/tutorial/ExactAndApproximateResults.html

Essentially, a floating point number is a bit different than the kind of numbers that you learn about in a math class.

 When using a symbolic function like Minimize, it is best to use exact numbers only.  Using approximate numbers may result in unexpected numerical issues since symbolic functions may not be as numerically stable as numerical functions.

In your specific case, simply switching to exact numbers gets you the answer that you expect.  Note that all I have done is removed all the decimal points - in Mathematica, approximate numbers have decimal points while exact numbers do not:
In[28]:= Minimize[{9 n (m - 1)^4 + (m n - 2)^2, -1 <= m <= 4 && -1 <=
    n <= 4}, {m, n}]

Out[28]= {-693, {m -> 4, n -> -1}}
POSTED BY: Karl Isensee
As mentioned in the documentation, NMinimize only guarantees a global minimum value only if the cost function and its constraint are linear. You may need to try different functions and also plot the reuslt to help you find the global miminum. Another function you can try is: 
FindMinValue[{9.0 n (m - 1.0)^4 + (m n - 2.0)^2, -1.0 <= m <= 4.0 && -1.0 <= n <= 4.0}, {m, n}]
POSTED BY: Shenghui Yang
Minimize[{9 n (m - 1)^4 + (m n - 2)^2, -1 <= m <= 4 && -1 <= n <= 4}, {m, n}, Integers]
seems to give the answer you seek.
{-693, {m -> 4, n -> -1}}
POSTED BY: C ormullion
FindMinimum[{9.0 n (m - 1.0)^4 + (m n - 2.0)^2, -1.0 <= m <= 4.0 && -1.0 <= n <= 4.0}, {m, n}]
 gives your answer.

{-693., {m -> 4., n -> -1.}}

Your subject title has been edited to be more informative.
POSTED BY: Steven Christensen
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