Past exam of the continuous optimisation course of the University of Oxford / 2025 / 1 / c / iii

Interpret boundedness above uniformly: there is an $M<\infty$ such that $B^k⪯MI$ for every $k$. Since $B^k$ is a positive-definite symmetric matrix, the spectral theorem implies $(B^k{)}^{-1}⪰M^{-1}I$. ThereforeCombining the Armijo condition with ${\alpha }^k\ge C>0$ yieldsIf $f\ge L$, a telescoping sum givesThe nonnegative series converges, so its terms tend to zero:This proves first-order global convergence in the sense of stationarity from uniformly bounded quasi-Newton matrices. It does not imply that every nonconvex objective function reaches a global minimum, or that the iterates themselves converge. Every finite limit point is a stationary point, by continuity of the gradient.