NMinimize with assumptions gives result that does not satisfy assumptions

Question

I am trying to determine if a function is always positive given a set of assumptions. I am using NMinimize following the advice of previous posts in this site. However, I am getting a global minimum that does not satisfy the assumptions.

I use the following code:

assumptions = {w3 > 0 && w2 > w3 && w1 > 2 w2 - w3 && 0 < b < w3 && 0 < l < 1};
NMinimize[
  {1/6 
     ((2 b l + 2 w1 + 5 w2 - 3 l w2 - 7 w3 + 3 l w3) / (w2 - l w2 - w3 + l w3) - 
       Sqrt[
         (4 b^2 l^2 + 4 b l (2 w1 + 5 w2 - 3 l w2 - 7 w3 + 3 l w3) + 
         (-2 w1 + w2 - 3 l w2 + w3 + 3 l w3)^2) / ((-1 + l)^2 (w2 - w3)^2)]) - 
   (2 w1 - w2 - w3)/(b*2 l + 2 w1 - w2 - w3), 
   assumptions}, 
  {b, l, w1, w2, w3}]

However, Mathematica returns

{-5.32491,
{b -> 0.0796325, l -> 0.0617359, w1 -> 0.52956, w2 -> 0.14148, w3 -> 0.189851}}

where w2 < w3.

I have tried adding the assumptions directly to NMinimize or to write a function for the expression, but the results are the same.

Does someone know a better way to solve this problem?

Answer 1

Reduce the function to be minimized to only one Denominator with Together.

assumptions = {w3 > 0 && w2 > w3 && w1 > 2 w2 - w3 && 0 < b < w3 && 
               0 < l < 1};

si[b_, l_, w1_, w2_, w3_] = 
    1/6 ((2 b l + 2 w1 + 5 w2 - 3 l w2 - 7 w3 + 3 l w3)/(w2 - l w2 - w3 +
       l w3) - 
   Sqrt[(4 b^2 l^2 + 
       4 b l (2 w1 + 5 w2 - 3 l w2 - 7 w3 + 3 l w3) + (-2 w1 + 
          w2 - 3 l w2 + w3 + 3 l w3)^2)/((-1 + l)^2 (w2 - 
          w3)^2)]) - (2 w1 - w2 - w3)/(b*2 l + 2 w1 - w2 - w3) // 
Together // Simplify[#, assumptions] & 

(*   -(-4 b^2 l^2 + 
2 b l (-4 w1 - 4 w2 + 3 l w2 + 8 w3 - 3 l w3 + Sqrt[
   4 b^2 l^2 - 
    4 b l (-2 w1 - 5 w2 + 3 l w2 + 7 w3 - 3 l w3) + (-2 w1 + w2 - 
      3 l w2 + w3 + 3 l w3)^2]) + (2 w1 - w2 - w3) (-2 w1 + w2 - 
   3 l w2 + w3 + 3 l w3 + Sqrt[
   4 b^2 l^2 - 
    4 b l (-2 w1 - 5 w2 + 3 l w2 + 7 w3 - 3 l w3) + (-2 w1 + w2 - 
      3 l w2 + w3 + 3 l w3)^2]))/(6 (-1 + l) (w2 - w3) (-2 b l - 
  2 w1 + w2 + w3))   *)

NMinimize[{si[b,l,w1,w2,w3], assumptions}, {b, l, w1, 
w2, w3}]

(*   {-3.58991*10^-15, {b -> 0.0547977, l -> 2.47803*10^-7, w1 -> 0.69935, 
       w2 -> 0.0934508, w3 -> 0.0864639}}   *)

Edit

A plot of the function shows, the whole lines with b==0 and l==0 reach the minimum zero.

Manipulate[
  Plot3D[si[b, l, w1, w2, w3], {b, 10^-3, 10}, {l, 10^-3, 1}, 
RegionFunction -> Function[{b}, 0 < b < w3], PlotPoints -> 50, 
AxesLabel -> {"b", "l", "si"}, ImageSize -> 400], {{w1, 9}, 
2 w2 - w3, 2 w2 - w3 + 10, Appearance -> "Labeled"}, {{w2, 7}, w3, 
w3 + 10, Appearance -> "Labeled"}, {{w3, 6}, 10^-5, 10, 
Appearance -> "Labeled"}, ContinuousAction -> False]

Attention. If the sliders show a red point, you are outside the allowed variable range.