NMinimize: How to avoid solutions that do not satisfy constraints within a certain tolerance?

This NMinimize::nosat documentation page explains the there are no solutions found if this warning message is displayed.

NMinimize[{((e*(1 - Sqrt[(g - e)^2 + (f - h)^2]) + (g - e)*(1 - 
          Sqrt[f^2 + e^2])) + (h*(1 - 
          Sqrt[(g - e)^2 + (f - h)^2]) + (f - h)*(1 - 
          Sqrt[g^2 + h^2])))/((g + f)*
     Max[1 - Sqrt[(g - e)^2 + (f - h)^2], 1 - Sqrt[g^2 + h^2]]), 
  0 <= e <= 1, 0 <= f <= 1, e^2 + f^2 == 1, e <= g <= 1, 0 <= h <= f, 
  Sqrt[(g - e)^2 + (f - h)^2] <= 1, g^2 + h^2 <= 1}, {e, f, g, h}]

{0.404445, {e -> 0.00756254, f -> 0.999323, g -> 0.868352, 
  h -> 0.490688}}

0.00756254^2 + .999323^2 <1

and all the rest is not very valid in a probe too.

Most probable this is ill-conditioned or overly constrained. The is a chance that the 4-dimensionality causes some strong divergence and NMinimize to finish the search for a minimum somewhere near or at the borders, so on the circle on in the circles.

Since {e,f}, and {g,h} are unit circles around zero in 2 dimensions there is a chance to use the constraints for visual control of the solution.

objective = {((e*(1 - Sqrt[(g - e)^2 + (f - h)^2]) + (g - e)*(1 - 
          Sqrt[f^2 + e^2])) + (h*(1 - 
          Sqrt[(g - e)^2 + (f - h)^2]) + (f - h)*(1 - 
          Sqrt[g^2 + h^2])))/((g + f)*
     Max[1 - Sqrt[(g - e)^2 + (f - h)^2], 1 - Sqrt[g^2 + h^2]])}

constraints = {0 <= e <= 1, 0 <= f <= 1, e^2 + f^2 == 1, e <= g <= 1, 
  0 <= h <= f, Sqrt[(g - e)^2 + (f - h)^2] <= 1, g^2 + h^2 <= 1}

Solve[constraints, {e, f, g, h}]

{{f -> ConditionalExpression[Sqrt[
    1 - e^2], (0 < g <= Sqrt[1 - h^2] && 0 <= e <= g && 
       1/2 < h <= 1) || (0 < g <= Sqrt[2 h - h^2] && 0 <= e <= g && 
       0 <= h <= 1/2) || (0 <= h <= 1/2 && 
       Sqrt[2 h - h^2] < g <= Sqrt[1 - h^2] && 
       g/2 - 1/2 Sqrt[(4 h^2 - g^2 h^2 - h^4)/(g^2 + h^2)] <= e <= 
        g)]}, {e -> ConditionalExpression[0, 0 <= h <= 1], 
  f -> ConditionalExpression[1, 0 <= h <= 1], 
  g -> ConditionalExpression[0, 0 <= h <= 1]}}

The solutions on the border of the unit circle are irrelevant.

This can be used in a

RegionPlot3D[(0 < g <= Sqrt[1 - h^2] && 0 <= e <= g && 
    1/2 < h <= 1) || (0 < g <= Sqrt[2 h - h^2] && 0 <= e <= g && 
    0 <= h <= 1/2) || (0 <= h <= 1/2 && 
    Sqrt[2 h - h^2] < g <= Sqrt[1 - h^2] && 
    g/2 - 1/2 Sqrt[(4 h^2 - g^2 h^2 - h^4)/(g^2 + h^2)] <= e <= 
     g), {e, 0, 1}, {g, 0, 1}, {h, 0, 1}]

NMinimize[{((e*(1 - Sqrt[(g - e)^2 + (f - h)^2]) + (g - e)*(1 - 
          Sqrt[f^2 + e^2])) + (h*(1 - 
          Sqrt[(g - e)^2 + (f - h)^2]) + (f - h)*(1 - 
          Sqrt[g^2 + h^2])))/((g + f)*
     Max[1 - Sqrt[(g - e)^2 + (f - h)^2], 1 - Sqrt[g^2 + h^2]]), 
  f == Sqrt[
   1 - e^2], (0 < g <= Sqrt[1 - h^2] && 0 <= e <= g && 
     1/2 < h <= 1) || (0 < g <= Sqrt[2 h - h^2] && 0 <= e <= g && 
     0 <= h <= 1/2) || (0 <= h <= 1/2 && 
     Sqrt[2 h - h^2] < g <= Sqrt[1 - h^2] && 
     g/2 - 1/2 Sqrt[(4 h^2 - g^2 h^2 - h^4)/(g^2 + h^2)] <= e <= 
      g)}, {e, g, h}]

{-1.38043*10^13, {e -> 9.30374*10^-6, g -> 0.887035, h -> 0.53828}}

This again is very marginal. Might be the complete plane for e near zero is degenerate a solution.

That the problem is ill-posed might be due to an error in the question.

Or it is an situation like this

Plot[{1/(1 + g), Sqrt[1/4 + g^2], 1/4 + g^2}, {g, 0, 1}]