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I use

NMinimize[f[{a, b, c, d}, 0, 0], {a, b, c, d}]

f[{a,b,c,d}, 0, 0]/.%%[[2]] returns 3.3493*10^-17, while f[P1, 0, 0] returns 0.373542.

What's worse is that f[({a, b, c, d}/.%%%%[[2]]), 0, 0] also returns the inaccurate result 0.373542.

What I wanted to do is find a linear combination that minimizes at the first spot, then define a coefficient vector from that and minimize at the second spot with the constraint that the coefficient vector has to be orthogonal to the first. f[{a, b, c, d}, x, y] normalizes the coefficient vector btw.

Edit: The function f is called LK4 below, I take tau = I, xi1 = xi2 = 0 for now, i.e. f[{a, b, c, d}, x, y] = LK4[{a, b, c, d}, I, 0, 0, x, y]

I use (LK4 is defined below, but its shape should have nothing to do with my problem)

NMinimize[LK4[{a, b, c, d}, I, 0, 0, 0, 0], {a, b, c, d}]

LK4[{a, b, c, d}, I, 0, 0, 0, 0]/.%%[[2]] returns 3.3493*10^-17, while LK4[P1, I, 0, 0, 0, 0] returns 0.373542.

What's worse is that LK4[({a, b, c, d}/.%%%%[[2]]), 0, 0, 0, 0] also returns the inaccurate result 0.373542.

What I wanted to do is find a linear combination that minimizes at the first spot, then define a coefficient vector from that and minimize at the second spot with the constraint that the coefficient vector has to be orthogonal to the first. LK4[{a, b, c, d}, tau, xi1, xi2, x, y] normalizes the coefficient vector btw.

Definition of LK4:

Edit: The function f is called LK4 below, I take tau = I, xi1 = xi2 = 0 for now.

Edit: The function f is called LK4 below, I take tau = I, xi1 = xi2 = 0 for now, i.e. f[{a, b, c, d}, x, y] = LK4[{a, b, c, d}, I, 0, 0, x, y]

Edit: The function f is called LK4 below, I take tau = I, xi1 = xi2 = 0 for now.

    Phi[j_, qM_, x_, y_, tau_, xi1_, xi2_] := 
  Surd[2 qM Im[tau]/Abs[tau]^2, 4] Exp[
    I 2 Pi (xi1 x + xi2 y)]  Exp[- Pi I Conjugate[tau]/Abs[tau]^2 ( 
       j^2/qM + qM x^2) + 
     2 Pi I j Conjugate[tau]/Abs[tau]^2 (x + tau y)] * 
   N[EllipticTheta[3, 
     Pi Conjugate[tau]/Abs[tau]^2 (qM (x + tau y) - j), 
     Exp[ -Pi I Conjugate[tau]/Abs[tau]^2 qM]], 20];
PhiEven[j_, qM_, x_, y_, tau_, xi1_, xi2_] := 
  Phi[j, qM, x, y, tau, xi1, xi2] /; j == 0;
PhiEven[j_, qM_, x_, y_, tau_, xi1_, xi2_] := 
  Phi[j, qM, x, y, tau, xi1, xi2] /;  
   qM/2 \[Element] Integers && j == qM/2;
PhiEven[j_, qM_, x_, y_, tau_, xi1_, xi2_] := 
  1/Sqrt[2] (Phi[j, qM, x, y, tau, xi1, xi2] + 
     Phi[qM - j, qM, x, y, tau, xi1, xi2]);
PhiOdd[j_, qM_, x_, y_, tau_, xi1_, xi2_] := 
  1/Sqrt[2] (Phi[j, qM, x, y, tau, xi1, xi2] - 
      Phi[qM - j, qM, x, y, tau, xi1, xi2]) /; j != 0 && j != qM/2;
LK4[coeff_, tau_, xi1_, xi2_, x_, y_] := 
  Sum[coeff[[a]]/(Sqrt@Total[coeff^2]) PhiEven[a - 1, 6, x, y, tau, 
     xi1, xi2], {a, 1, 4}];