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How to define piecewise functions of two variables for NMinimize?

Posted 12 days ago
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Hey,

I'm new, so sorry if I somehow messed up by posting in the wrong place. Anyway, onto my question: I defined two functions

\[Beta][w_, x_] := 
 Piecewise[{{0, 
    w < 150 && x < 150}, {1/
     3, (w < 150 && 150 <= x < 300) || (150 <= w < 300 && 
       x < 150)}, {1/
     2, (w < 150 && x >= 300) || (x < 150 && w >= 300)}, {2/
     3, (150 <= w < 300 && 150 <= x < 300)}, {5/
     6, (150 <= w < 300 && x >= 300 ) || (150 <= x < 300 && 
       w >= 300 )}, {1, w >= 300 && x >= 300}}]

and

\[Gamma][w_, x_, y_] := 
 Piecewise[{{0, 
    w < 150 && x < 150 && y < 150}, {1/
     3, (150 <= w < 300 && x < 150 && y < 150) || (150 <= x < 300 && 
       w < 150 && y < 150) || (150 <= y < 300 && w < 150 && 
       x < 150)}, {1/
     2, (w >= 300 && x < 150 && y < 150 ) || (x >= 300 && w < 150 && 
       y < 150 ) || (y >= 300 && x < 150 && w < 150 )}, {2/
     3, (150 <= w < 300 && 150 <= x < 300 && 
       y < 150) || (150 <= w < 300 && 150 <= y < 300 && 
       x < 150) || (150 <= y < 300 && 150 <= x < 300 && w < 150)}, {5/
     6, (w >= 300 && 150 <= x < 300 && y < 150) || (w >= 300 && 
       150 <= y < 300 && x < 150) || (x >= 300 && 150 <= w < 300 && 
       y < 150) || (x >= 300 && 150 <= y < 300 && 
       w < 150) || (y >= 300 && 150 <= w < 300 && 
       x < 150) || (y >= 300 && 150 <= x < 300 && 
       w < 150)}, {1, (w >= 150 && x >= 150 && y >= 150) || (w < 150 && 
       x >= 300 && y >= 300) || (x < 150 && w >= 300 && 
       y >= 300) || (y < 150 && x >= 300 && w >= 300)}}]

and then use them along with a third piecewise function in one variable

\[Alpha][w_] := 
 Piecewise[{{0, w < 150}, {1/3, 150 <= w < 300}, {1/2, w >= 300}}]

The piecewise functions themselves all work, but I get an error when evaluating:

NMinimize[{w + x + y + z, 
  w + \[Alpha][w] x + \[Beta][w, x] y + \[Gamma][w, x, y] z >= 1/3 && 
   w \[Element] PositiveIntegers && x \[Element] PositiveIntegers && 
   y \[Element] PositiveIntegers && 
   z \[Element] PositiveIntegers }, {w, x, y, z}]

Can anyone tell me why this doesn't work and if there is some way to do this correctly? Thank you very much!

2 Replies

Jesco,

I do not really know why your approach using Piecewise is not working. But you could try with interpolated functions instead:

alpha = Quiet@FunctionInterpolation[\[Alpha][w], {w, 0, 500}, InterpolationOrder -> 0];
beta = FunctionInterpolation[\[Beta][w, x], {w, 0, 500}, {x, 0, 500}, InterpolationOrder -> 0];
gamma = FunctionInterpolation[\[Gamma][w, x, y], {w, 0, 500}, {x, 0, 500}, {y, 0, 500}, InterpolationOrder -> 0];

And then:

NMinimize[{w + x + y + z, 
  w + alpha[w] x + beta[w, x] y + gamma[w, x, y] z >= 1/3 && 
   w \[Element] PositiveIntegers && x \[Element] PositiveIntegers && 
   y \[Element] PositiveIntegers && 
   z \[Element] PositiveIntegers}, {w, x, y, z}]

(*   Out:    {4.`,{w\[Rule]1,x\[Rule]1,y\[Rule]1,z\[Rule]1}}   *)

Given your definitions this result is not really surprising - well, at least it seems to be correct.

Does that help? Regards -- Henrik

POSTED BY: Henrik Schachner

Hey Henrik,

that does help, thank you! I'll have to look into interpolated functions in Mathematica sometime.

I actually got it to work by using Minimize instead of NMinimize and simplifying the definitions of beta and gamma by reusing alpha. I was actually interested in a slightly different problem (with another piecewise function on the RHS instead of just 1/3 and some conditions to remove the trivial w=x=y=z=1 solution), but that change didn't pose any problems.

Kind regards,

Jesco

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