optimization/steepest_descent.m

   1 function [x, k] = steepest_descent(g, x0, step_size, tolerance, max_iterations)
   2   %
   3   % An implementation of the steepest-descent algorithm, with the
   4   % search direction d_{k} = -\nabla f(x_{k}).
   5   %
   6   % We should terminate when either,
   7   %
   8   %   a) The 2-norm of the gradient at x_{k} is greater than
   9   %   ``tolerance``.
  10   %
  11   %   b) We have performed ``max_iterations`` iterations.
  12   %
  13   % INPUT:
  14   %
  15   %   * ``g`` - the gradient of the function ``f`` we're minimizing.
  16   %
  17   %   * ``x0`` - the starting point for the search.
  18   %
  19   %   * ``step_size`` - a function of x which returns the optimal step
  20   %     size alpha_{k}. This will generally require more information
  21   %     than just x; for example it might need the function ``f`` or the
  22   %     gradient ``g``. However, our caller has this information so it
  23   %     should be incorporated into the step size algorithm that we
  24   %     receive.
  25   %
  26   %   * ``tolerance`` - the stopping tolerance. We stop when the norm
  27   %   of the gradient falls below this value.
  28   %
  29   %   * ``max_iterations`` - a safety net; we return x_{k}
  30   %   unconditionally if we exceed this number of iterations.
  31   %
  32   % OUTPUT:
  33   %
  34   %   * ``x`` - the solution at the final value of x_{k}.
  35   %
  36   %   * ``k`` - the value of k when we stop; i.e. the number of
  37   %   iterations.
  38
  39   % The initial gradient at x_{0} is not supplied, so we compute it
  40   % here and begin the loop at k=1.
  41   x = x0;
  42   g_k = g(x);
  43
  44   if (norm(g_k) < tolerance)
  45     % If x_0 is close enough to a solution, there's nothing for us to
  46     % do! We use g_k (the gradient of f at x_k) instead of d_k because
  47     % their 2-norms will be the same, and g_k is already stored.
  48     return;
  49   end
  50
  51   for k = [1 : max_iterations]
  52     % Loop until either of our stopping conditions are met. If the
  53     % loop finishes, we have implicitly met the second stopping
  54     % condition (number of iterations).
  55     d_k = -g_k;
  56     alpha_k = step_size(x);
  57     x = x + (alpha_k * d_k);
  58     g_k = g(x);
  59
  60     if (norm(g_k) < tolerance)
  61       return;
  62     end
  63   end
  64
  65   % If we make it to the end of the loop, that means we've executed the
  66   % maximum allowed iterations. The caller should be able to examine the
  67   % return value ``k`` to determine what happened.
  68 end