Maximize this function

Question

For the following matrix

$$M = \begin{pmatrix} f1[x,y,z] & f2[x,y,z]\\ g1[x,y,z] & g2[x,y,z]\\ h1[x,y,z] & h2[x,y,z] \end{pmatrix} $$

with

f1[x_,y_,z_] := E^(-E^(2 Im[x]) Abs[y Sqrt[1 - z] + 0.1 Sqrt[z]]^2)
g1[x_,y_,z_] := E^(-E^(2 Im[x]) Abs[y Sqrt[1 - z] + 0.1 I Sqrt[z]]^2)
h1[x_,y_,z_] := E^(-E^(2 Im[x]) Abs[y Sqrt[1 - z] + Sqrt[z]]^2)

f2[x_,y_,z_] := E^(-E^(2 Im[x]) Abs[y Sqrt[1 - z]])
g2[x_,y_,z_] := E^(-E^(2 Im[x]) Abs[y Sqrt[1 - x]])
h2[x_,y_,z_] := E^(-E^(2 Im[x]) Abs[y Sqrt[1 - y]])

Now I want to optimise the value of:

P = f1[x,y,z] pos[1] + f2[x,y,z] pos[2]

where pos[j] returns the row corresponding to the highest value in column j . However, I don't know how to define pos[j]. I currently have

MWE

pos[1] = Position[{f1[x,y,z], g1[x,y,z], h1[x,y,z]]}, Max[f1[x,y,z], g1[x,y,z], h1[x,y,z]]]

pos[2] = Position[{f2[x,y,z], g2[x,y,z], h2[x,y,z]]}, Max[f2[x,y,z], g2[x,y,z], h2[x,y,z]]]

Poptimized = Maximize[P, {x, y, z}]

But this is not working. This is since pos[1] and pos[2] return empty lists. I thought that wrapping it inside the Maximize would have it evaluate. I want something structured as follows:

Maximize[f1[x,y,z] pos[1] + f2[x,y,z] pos[2],{x,y,z}]

where pos[1] and pos[2] both also depend on x,y,z and so need to be optimised too.

Answer 1

Not a complete answer but try this. Since all function has the same max value, how do you decide which function you choose? Mathematica chose the last one.

M = {{-x - y^2 - 2 z , x - y^2 + 2 z },
{-x - y - z^2, x + 2 y + z^2},
{-x - y - z^3, 2 x - y + z}};

$M=\left( \begin{array}{cc} -x-y^2-2 z & x-y^2+2 z \\ -x-y-z^2 & x+2 y+z^2 \\ -x-y-z^3 & 2 x-y+z \\ \end{array} \right)$

pos1 = First@Ordering[MaxValue[#, {x, y, z}] & /@ M[[All, 1]], -1]
pos2 = First@Ordering[MaxValue[#, {x, y, z}] & /@ M[[All, 2]], -1]

P = NMaximize[M[[1,1]] pos1 + M[[1,2]] pos2,{x, y, z}]