 Solution

ID: past-exam-of-the-numerical-linear-algebra-and-continuous-optimisation-course-of-the-university-of-oxford/2007/b/7/v/solution

p_{k} = - g (x_{k})

gives

f (x_{k}) - f (x_{k + 1}) \geq β α_{k} ∥ g (x_{k}) ∥_{2}^{2}

f (x_{0}) - f (x_{j + 1}) \geq β k = 0 \sum j α_{k} ∥ g (x_{k}) ∥_{2}^{2} .

f \geq f_{i n f}

, these partial sums of nonnegative terms are bounded above by

(f (x_{0}) - f_{i n f}) / β

. The resulting series converges, so its terms tend to zero:

α_{k} ∥ g (x_{k}) ∥_{2}^{2} \to 0

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