NMaximize problem

Question

Could anybody explain to me why the following code fails ?

 Reap[
 NMaximize[{x1^2 + x2^2 + x3^2, 0.001 <= x1 <= 0.002, 
   0 <= x2 <= 0.002, 1 <= x3 <= 10, x3 \[Element] Integers }, {x1, x2,
    x3}, Method -> "SimulatedAnnealing", 
  EvaluationMonitor :> Sow[{x1, x2, x3}]]
 ]

In fact it outputs :

{{-\[Infinity], {x1 -> Indeterminate, x2 -> Indeterminate, 
   x3 -> Indeterminate}}, {}}

If i put x1's lower bound to 0 then it works but it uses only 0 values for x1

I suppose it has something to do with the integer x3.

Thanks in advance.

Answer 1

By now, Mathematica is unable to do exact optimization with mixed integer and real variables.

Luckily, in your case, removing the constraint about x3 being an integer but rewriting the problem to fit into the exact optimization approach gives an optimum in which x3 is an exact integer.

Maximize[{x1^2 + x2^2 + x3^2, 
          1/1000 <= x1 <= 2/1000 && 0 <= x2 <= 2/1000 && 1 <= x3 <= 10},
          {x1, x2, x3}]

And this gives {12500001/125000, {x1 -> 1/500, x2 -> 1/500, x3 -> 10}} as a solution.

Trying

Maximize[{x1^2 + x2^2 + x3^2, 
          1/1000 <= x1 <= 2/1000 && 0 <= x2 <= 2/1000 && 1 <= x3 <= 10 &&
                  Element[x3, Integers]},
          {x1, x2, x3}]

Results in error:

Maximize::mixdom: Exact optimization with mixed real and integer variables is not yet implemented.

I think that the "N" functions (NMaximize, NMinimize, NSolve, etc) only deal with finite precision approximate numbers so they wouldn't "respect" integers (they are supposed to replace them with an approximate number).