Appendix A — Algorithms

Algorithm A.1: Golden section method

Input: Optimization problem (1.33), initial interval \(I = [a,b]\), number of evaluations \(N \in \N\)
Output: Approximation of minimum \(\bar{x}_k\)
begin
  set \([a_0, b_0] \gets [a,b]\), \(l_0 \gets b_0 - a_0\);
  calculate \(x_2^- \gets a_0 + \frac 1 {\goldRatio^2} l_0\) & \(x_2^+ \gets a_0 + \frac 1 \goldRatio l_0\);
  evaluate \(f_2^- \gets f(x_2^-)\), \(f_2^+ \gets f(x_2^+)\)
  if \(f_2^- < f_2^+\) do
    set \([a_2, b_2] \gets [a_0, x_2^+]\);
    set \(\mathrm{Increasing} \gets 1\);
  else do
    set \([a_2, b_2] \gets [x_2^-, b_0]\);
    set \(\mathrm{Increasing} \gets 0\);
  end
  set \(l_2 \gets b_2 - a_2\)
  for \(i = 3\) to \(N\) do
    if \(\mathrm{Increasing}\) equals \(1\) do
      set \(x_i^+ \gets x_{i-1}^-\), \(f_i^+ = f_{i-1}^-\);
      calculate \(x_i^- \gets a_{i-1} + \frac 1 {\goldRatio^2} l_{i-1}\);
      evaluate \(f_i^- \gets f(x_i^-)\)
    else do
      set \(x_i^- \gets x_{i-1}^+\), \(f_i^- = f_{i-1}^+\);
      calculate \(x_i^+ \gets a_{i-1} + \frac 1 {\goldRatio} l_{i-1}\);
      evaluate \(f_i^+ \gets f(x_i^+)\)
    end
    if \(f_i^- < f_i^+\) do
      set \([a_i, b_i] \gets [a_{i-1}, x_i^+]\);
      set \(\mathrm{Increasing} \gets 1\);
    else do
      set \([a_i, b_i] \gets [x_i^-, b_{i-1}]\);
      set \(\mathrm{Increasing} \gets 0\);
    end
    set \(l_i \gets b_i - a_i\);
    set \(\bar{x}_i \gets \frac {a_i + b_i} 2\)
  end
end

Algorithm A.2: Fibonacci method

Input: Optimization problem (1.33), initial interval \(I = [a,b]\), number of evaluations \(N \in \N\), tolerance \(\ve > 0\) small
Output: Approximation of minimum \(\bar{x}\)
begin
  set \(A \gets a\), \(B \gets b\), \(n \gets N\);
  define \(s \letDef \frac {1 - \sqrt{5}} {1 + \sqrt{5}}\);
  let \(\rho \gets \frac 1 {\goldRatio} \frac {1 - s^n}{1 - s^{n+1}}\), \(r \gets (1 - \rho) A + \rho B\);
  set \(f^r \gets f(r)\) and \(n \gets n - 1\)
  while \(n > 0\) do
    if \(n > 1\) do
      set \(l \gets \rho A + (1 - \rho) B\)
    else do
      set \(l \gets \ve A + (1 - \ve) r\)
    end
    update \(\rho \gets \frac 1 {\goldRatio} \frac {1 - s^n}{1 - s^{n+1}}\);
    set \(f^l \gets f(l)\) and \(n \gets n - 1\)
    if \(f^l < f^r\) do
      set \([r, B, f^r] \gets [l, r, f^l]\)
    else do
      set \([A, B] \gets [B, l]\)
    end
  end
  set \(\bar{x} = \frac {A + B} 2\)
end