Optimization with assumptions

Question

I have the following problem, when I am trying to optimize function with pre-defined assumptions.

I am using Mathematica 8 and I wrote the following simple code

$Assumptions = (m > 0)
Minimize[{x^m, x >= 1}, x]

It is clear, that the answer is 1, since $x^m\ge 1$ if $x\ge 1$ for all positive $m$. However, Mathematica fails to calculate this simple problem.

What do I do wrong? Should I use another function for minimization?

Answer 1

Try This,

NMinimize[{X^m, X >= 1, m > 0, m \[Element] Integers}, {X, m}]

This will give you what you like

{1., {X -> 1., m -> 2}}

Answer 2

Minimization (especially in several dimensions) is in general a tough task, and it is usually better to facilitate the task for Mathematica. So normally in calculations in Mathematica it is easier to replace the minimization procedure by an equivalent one, i.e. to take the derivative. The following example

function[x_, m_] := x^m;
derivative = D[function[x, m], x];
Solve[derivative == 0, x]

gives the desired analytical result

{{x -> 0^(1/(-1 + m))}}

for any parameter m. It tells you that the only extremum point point of the function is 0 (for m<>1). This means that if you take the interval x>=1 for minimization, your function does not have extremum points inside it and hence attains its minimum at the boundary.

You can also directly write

Solve[{derivative == 0,x>1,m>0}, x]

to see that no extremum points exist inside the defined interval.

To conclude, you can analytically minimize complicated expressions in Mathematica, but you should wisely choose the way you formulate the problem for the computer.

Thanks to gwr for pointing out that my previous answer didn't work.