NL constrained optimization with generalized function

Question

I have a objective function H, which follows following constrains: p - c - a > 0 && p > c > a && 0 < p < 1 && 0 < c < 1 && 0 < a < 1 && 0 < f[s] && Derivative[1][f][s] > 0 && Derivative[2][f][s] < 0 To have unique global maxima, I am assuming hessian of H to be negative definite. So, one more condition I got from here. But when I am using Maximize[{obj, const}, {var}] I am not able to get answer. Please help!

Edit 1: After a bit research on necessary conditions for KKT conditions I came to know that inequalities should be convex. Now I wonder how to tell Mathematica that these constrains are convex 0 < f[s] && Derivative[1][f][s] > 0 && Derivative[2][f][s] < 0

I can see the following code does not work.

    H = (-a - c + p) (1 - (2 p - f[s] Subscript[k, 2])/(
    f[s] (2 Subscript[k, 1] + Subscript[k, 2])))
D[H, {{p, s}, 2}]
(*{{-(4/(f[s]*(2*Subscript[k, 1] + 
       Subscript[k, 2]))), 
   (2*(-a - c + p)*Derivative[1][f][
       s])/(f[s]^2*
      (2*Subscript[k, 1] + 
       Subscript[k, 2])) + 
    (Subscript[k, 2]*
      Derivative[1][f][s])/
     (f[s]*(2*Subscript[k, 1] + 
       Subscript[k, 2])) + 
    ((2*p - f[s]*Subscript[k, 2])*
      Derivative[1][f][s])/
     (f[s]^2*(2*Subscript[k, 1] + 
       Subscript[k, 2]))}, 
  {(2*(-a - c + p)*Derivative[1][f][
       s])/(f[s]^2*
      (2*Subscript[k, 1] + 
       Subscript[k, 2])) + 
    (Subscript[k, 2]*
      Derivative[1][f][s])/
     (f[s]*(2*Subscript[k, 1] + 
       Subscript[k, 2])) + 
    ((2*p - f[s]*Subscript[k, 2])*
      Derivative[1][f][s])/
     (f[s]^2*(2*Subscript[k, 1] + 
       Subscript[k, 2])), 
   (-a - c + p)*
    (-((2*Subscript[k, 2]*
        Derivative[1][f][s]^2)/
       (f[s]^2*(2*Subscript[k, 1] + 
         Subscript[k, 2]))) - 
     (2*(2*p - f[s]*Subscript[k, 2])*
       Derivative[1][f][s]^2)/
      (f[s]^3*(2*Subscript[k, 1] + 
        Subscript[k, 2])) + 
     (Subscript[k, 2]*
       Derivative[2][f][s])/
      (f[s]*(2*Subscript[k, 1] + 
        Subscript[k, 2])) + 
     ((2*p - f[s]*Subscript[k, 2])*
       Derivative[2][f][s])/
      (f[s]^2*(2*Subscript[k, 1] + 
        Subscript[k, 2])))}}*)

For Hessian matrix [A B, C D] to be negative definite require A<0 and A D-BC>0 Here as f[s], k1, k2 >0 hence $-(4/(f[s] (2 Subscript[k, 1]+Subscript[k, 2])))<0$ for sure Checking AD- BC

Simplify[
  (-(4/(f[s]*(2*Subscript[k, 1] + 
        Subscript[k, 2]))))*
    ((-a - c + p)*
     (-((2*Subscript[k, 2]*
         Derivative[1][f][s]^2)/
        (f[s]^2*(2*Subscript[k, 1] + 
          Subscript[k, 2]))) - 
      (2*(2*p - f[s]*Subscript[k, 
           2])*Derivative[1][f][s]^
         2)/(f[s]^3*
        (2*Subscript[k, 1] + 
         Subscript[k, 2])) + 
      (Subscript[k, 2]*
        Derivative[2][f][s])/
       (f[s]*(2*Subscript[k, 1] + 
         Subscript[k, 2])) + 
      ((2*p - f[s]*Subscript[k, 2])*
        Derivative[2][f][s])/
       (f[s]^2*(2*Subscript[k, 1] + 
         Subscript[k, 2])))) - 
   ((2*(-a - c + p)*Derivative[1][f][
        s])/(f[s]^2*
       (2*Subscript[k, 1] + 
        Subscript[k, 2])) + 
     (Subscript[k, 2]*
       Derivative[1][f][s])/
      (f[s]*(2*Subscript[k, 1] + 
        Subscript[k, 2])) + 
     ((2*p - f[s]*Subscript[k, 2])*
       Derivative[1][f][s])/
      (f[s]^2*(2*Subscript[k, 1] + 
        Subscript[k, 2])))*
    ((2*(-a - c + p)*Derivative[1][
         f][s])/(f[s]^2*
       (2*Subscript[k, 1] + 
        Subscript[k, 2])) + 
     (Subscript[k, 2]*
       Derivative[1][f][s])/
      (f[s]*(2*Subscript[k, 1] + 
        Subscript[k, 2])) + 
     ((2*p - f[s]*Subscript[k, 2])*
       Derivative[1][f][s])/
      (f[s]^2*(2*Subscript[k, 1] + 
        Subscript[k, 2])))]
(*-((4*((a + c)^2*Derivative[1][f][s]^
                2 + 2*p*(-a - c + p)*f[s]*
              Derivative[2][f][s]))/
      (f[s]^4*(2*Subscript[k, 1] + 
              Subscript[k, 2])^2))*)

Solve[
  -((4*((a + c)^2*Derivative[1][f][
          s]^2 + 2*p*(-a - c + p)*
        f[s]*Derivative[2][f][s]))/
     (f[s]^4*(2*Subscript[k, 1] + 
        Subscript[k, 2])^2)) == 0, 
  f[s]]
(*{{f[s] -> ((a + c)^2*
      Derivative[1][f][s]^2)/
     (2*(a + c - p)*p*
      Derivative[2][f][s])}}*)

\text{So, for Hessian to be negative definite we have to have }

\frac{f[s]$>$(a+c)^2 f'(s)^2}{2 p (a+c-p) f''(s)}

Maximizing the objective function

Maximize[{H, 
   f[s] > ((a + c)^2*
       Derivative[1][f][s]^2)/
      (2*(a + c - p)*p*
       Derivative[2][f][s]) && 
    p - c - a > 0 && 0 < p < 1 && 
    0 < c < 1 && 0 < a < 1 && 
    0 < f[s] && Derivative[1][f][
      s] > 0 && Derivative[2][f][
      s] < 0}, {p, s}]

But it does not work

Following is the drive link to my code if anyone wants to view it: https://drive.google.com/file/d/1M9kE4GeOborZ1l8NQhrVmjNDYhaurYOZ/view?usp=sharing

Thanks in advance

Answer 1

Not an answer, but some observations.

Solve[-((4*((a + c)^2*Derivative[1][f][s]^2 + 
         2*p*(-a - c + p)*f[s]*Derivative[2][f][s]))/(f[
         s]^4*(2*Subscript[k, 1] + Subscript[k, 2])^2)) == 0, f[s]]

Produces a solvable ODE:

fs = DSolveValue[
  f[s] == ((a + c)^2 Derivative[1][f][s]^2)/(
   2 (a + c - p) p (f^\[Prime]\[Prime])[s]), f[s], s]

The derivatives can be computed:

dfs = D[fs, s]
dfs2 = D[fs, {s, 2}]

This can be inserted into your constraints:

constraints = {f[
      s] > ((a + c)^2*Derivative[1][f][s]^2)/(2*(a + c - p)*p*
        Derivative[2][f][s]) && p - c - a > 0 && 0 < p < 1 && 
    0 < c < 1 && 0 < a < 1 && 0 < f[s] && Derivative[1][f][s] > 0 && 
    Derivative[2][f][s] < 0} /. {f[s] -> fs, f'[s] -> dfs, 
   Derivative[2][f][s] -> dfs2}

But, the constraints appear to be inconsistent?

Simplify[constraints] (*False*)