 Solution

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

Use the uniform lower step bound from part (iv):

0 \leq ∥ g (x_{k}) ∥_{2}^{2} \leq \frac{α _{k} ∥ g ( x _{k} ) ∥ _{2}^{2}}{α _{m i n}} ⟶ 0.

Thus

g (x_{k}) \to 0

. This is convergence to stationarity in the usual global convergence theorem for the steepest descent method. It does not itself prove that the iterates converge, or that any stationary limit is a global minimizer of a nonconvex objective.

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