Date: Feb 22, 2014 3:01 AM
Author: Daniel Stariolo
Subject: Some linear constraints seem to be ignored in function NMinimize with
I'm trying to minimize a non-linear function of four variables with some linear constraints. Mathematica 8 is unable to find a good solution giving complex values of the function at some point in the iteration. This implies that one or some contraints are not being enabled in the process. Is this a bug or limitation of the optimization function ?

Function to minimize is

ff[lxw_, lwz_, c_, d_] := - J1 (lxw + lwz) - 2 J2 c +

T (-Log[2] - 1/2 (1 - lxw) Log[(1 - lxw)/4] -

1/2 (1 + lxw) Log[(1 + lxw)/4] -

1/2 (1 - lwz) Log[(1 - lwz)/4] -

1/2 (1 + lwz) Log[(1 + lwz)/4] + 1/2 (1 - d) Log[(1 - d)/16] +

1/8 (1 + 2 c + d - 2 lwz - 2 lxw) Log[

1/16 (1 + 2 c + d - 2 lwz - 2 lxw)])

where

T = 10;

J1 = 1;

J2 = -0.2;

are constant parameters. Then I try

NMinimize[{ff[lxw, lwz, c, d],

2 c + d - 2 lwz - 2 lxw >= -0.999 && -0.999 <= lxw <=

0.999 && -0.999 <= lwz <= 0.999 && -0.999 <= c <= 0.999 &&

d <= 0.9999}, {lxw, lwz, c, d}]

with the result

NMinimize::nrnum: "The function value 5.87777-4.87764 I is not a real number at {c,d,lwz,lxw} = {-0.718817,-1.28595,0.69171,-0.932461}."

I would appreciate if someone can give a hint at what is happening here.