About NMinimize with an array of variable

Question

I was trying to test NMinimize on $$ \int_0^1 (a+b(cx+d))^2 dx $$, then I guess it should be minimized when the integrand equals 0. It works fine when the integrand is only (a+bx)^2, I use the code:

z1[x_, y_] := y[[1]]*x + y[[2]] ;
Y = Table[y[i], {i, 1, 2}];
h[Y_] := Integrate[(z1[x, Y])^2, {x, 0, 1}]
NMinimize[h[Y], Y, AccuracyGoal -> 20, PrecisionGoal -> 18, 
 WorkingPrecision -> 40]

And I get the output :

{3.550736290059893672842701884381905283185*10^-144, {y[1] -> 
   6.527544368345474836762769419275328411033*10^-72, 
  y[2] -> -3.263772184172737418381384709637664205517*10^-72}}

But when I do the composition, I was confused, what I wrote is:

z1[x_, y_] := y[[1]]*x + y[[2]] ;
z2[x_, y_] := y[[3]]*x + y[[4]];
z[x_, y_] := z1@z2@x;
Y = Table[y[i], {i, 1, 4}];
h[Y_] := Integrate[(z[x, Y])^2, {x, 0, 1}];
NMinimize[h[Y], Y, AccuracyGoal -> 20, PrecisionGoal -> 18, 
 WorkingPrecision -> 40]

Where the $z[x,y]$ gives

y[[2]] + y[[1]] (x y[[3]] + y[[4]])

and $h[Y]$ gives

y[[2]]^2 + y[[1]] y[[2]] y[[3]] + 1/3 y[[1]]^2 y[[3]]^2 + 
 2 y[[1]] y[[2]] y[[4]] + y[[1]]^2 y[[3]] y[[4]] + y[[1]]^2 y[[4]]^2

which seem fine, but the result of NMinimize is :

NMinimize[
 y[[2]]^2 + y[[1]] y[[2]] y[[3]] + 1/3 y[[1]]^2 y[[3]]^2 + 
  2 y[[1]] y[[2]] y[[4]] + y[[1]]^2 y[[3]] y[[4]] + 
  y[[1]]^2 y[[4]]^2, {y[1], y[2], y[3], y[4]}, AccuracyGoal -> 20, 
 PrecisionGoal -> 18, WorkingPrecision -> 40]

I dont know why I cannot get an answer from it, thanks for any help !

Answer 1

$Version

(* "14.0.0 for Mac OS X ARM (64-bit) (December 13, 2023)" *)

Clear["Global`*"]

z1[x_, y_List] := {x, 1} . y

Y = Array[y, 2];

h[y_List] := Integrate[z1[x, y]^2, {x, 0, 1}]

NMinimize[h[Y], Y]

(* {0., {y[1] -> 0., y[2] -> 0.}} *)

or

Minimize[h[Y], Y]

(* {0, {y[1] -> 0, y[2] -> 0}} *)

Y = Array[y, 4];

z1 must be given two arguments.

z[x_, y_List] := z1[z1[x, y[[{3, 4}]]], y[[{1, 2}]]]

h2[y_List] := Integrate[(z[x, y])^2, {x, 0, 1}];

NMinimize[h2[Y], Y]

(* {0., {y[1] -> -0.0000139272, y[2] -> -8.43686*10^-6, 
  y[3] -> -1.11022*10^-16, y[4] -> -0.605783}} *)

or

Minimize[h2[Y], Y]

(* {0, {y[1] -> -1, y[2] -> 0, y[3] -> 0, y[4] -> 0}} *)