Boolean constraint in an interval for FindMinimum

Question

I have a function with a large number of variables that I am trying to minimize subject to some constraints, but for simplicity let us say that it only has two variables, f[x, y]. The constraint is that another function, g[z], must be positive in the interval given by x and y (in my problem I usually have around 200 variables, and it needs to be monotonical in the interval between the minimum and the maximum)

Edit: Following the comment by Daniel Huber, let's define

f[x_, y_] := x^3 + 2 x y + 3
g[z_] := z^2 - 2 z 

where f is to be minimized in the region {x, y} [Element] Rectangle[{0, 0}, {4, 4}] subject to g[z] >= 0 for all z in the region Interval[{Min[x_solution, y_solution], Max[x_solution, y_solution]}]

If we did not have this constraint we would find that

Minimize[{f[x, y], constraint}, {x, y}]\[Element] Rectangle[{0, 0}, {4, 4}]
(* {3, {x -> 0, y -> 2}} *)

But g[z] < 0 for z ∈ Interval[{0, 2}]

Instead, the solution that I am looking for would be {19, {x -> 2, y -> 2}}

In my problem, the coefficients in the polynomial of g[z] are part of the variables that I am trying to optimize, while the rest of the variables are what defines the interval in which I want this polynomial to be positive (so I will take the minimum and maximum of these variables when defining the interval, similar to taking the minimum and maximum of x and y in this toy model), and it is not easy to find workarounds like studying the behavior of g[z] before optimizing f[x,y].

Additionally, my function involves a lot of terms and Minimize cannot solve it, so I would prefer solutions involving FindMinimum instead.

I hope the question is clearer now!

Answer 1

I hope I understood your question correctly. I assume that by "monotonically increasing" you mean increasing in the x and y direction. Here is an example.

With two example functions:

f[x_, y_] = 0.4 Sin[x y] ;
g[x_, y_] = (x - 1)^2 (y - 1)^2;

Without the condition on g:

sol = Minimize[{f[x, y]}, {x, y} \[Element] Rectangle[{0, 0}, {4, 4}]]
Show[
 Plot3D[f[x, y], {x, y} \[Element] Rectangle[{0, 0}, {4, 4}]],
 Graphics3D[{Green, PointSize[0.03], 
   Point[{x, y, sol[[1]]} /. sol[[2]]]}]]

The minimum is at: {x -> 3.40016, y -> 2.30988}

However, with the condition on g:

sol = Minimize[{f[x, y], D[g[x, y], x] > 0, 
   D[g[x, y], y] > 0}, {x, y} \[Element] Rectangle[{0, 0}, {3, 3}]]
Show[
 Plot3D[{f[x, y], g[x, y]}, {x, y} \[Element] 
   Rectangle[{0, 0}, {4, 4}], PlotStyle -> Opacity[0.5]],
 Graphics3D[{Green, PointSize[0.03], 
   Point[{x, y, sol[[1]]} /. sol[[2]]]}]]

The minimum is now at: {x -> 2.61006, y -> 1.80547}