optimization/step_length_positive_definite.m

   1 function alpha = step_length_positive_definite(g, Q, p)
   2   ##
   3   ## Find the minimizer alpha of,
   4   ##
   5   ##   phi(alpha) = f(x + alpha*p)
   6   ##
   7   ## where ``p`` is a descent direction,
   8   ##
   9   ##   f(x) = (1/2)<Qx,x> - <b,x>
  10   ##
  11   ## and ``Q`` is positive-definite.
  12   ##
  13   ## The closed-form solution to this problem is given in Nocedal and
  14   ## Wright, (3.55).
  15   ##
  16   ## INPUT:
  17   ##
  18   ##   - ``g`` -- The gradient of f.
  19   ##
  20   ##   - ``Q`` -- The positive-definite matrix in the definition of
  21   ##     ``f`` above.
  22   ##
  23   ##   - ``p`` -- The direction in which ``f`` decreases. The line
  24   ##     along which we minimize f(x + alpha*p).
  25   ##
  26   ## OUTPUT:
  27   ##
  28   ##   - ``alpha`` -- The value which causes ``f`` to decrease the
  29   ##     most.
  30   ##
  31   ## NOTES:
  32   ##
  33   ## All vectors are assumed to be *column* vectors.
  34   ##
  35   alpha = -(g' * p)/(p' * Q * p);
  36 end