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.

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);

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

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);

60 if (norm(g_k) < tolerance)

61 return;

62 end

63 end

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