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 Incoming links: Stationarity from uniformly bounded quasi-Newton matrices

Past exam of the continuous optimisation course of the University of Oxford / 2025 / 1 / c / iii / Solution 2026-10-25

Interpret boundedness above uniformly: there is an

M < \infty

such that

B^{k} ⪯ M I

for every

k

. Since

B^{k}

is a positive-definite symmetric matrix, the spectral theorem implies

(B^{k})^{- 1} ⪰ M^{- 1} I

. Therefore

- \nabla f (x^{k})^{T} s^{k} = \nabla f (x^{k})^{T} (B^{k})^{- 1} \nabla f (x^{k}) \geq \frac{1}{M} ∥\nabla f (x^{k}) ∥^{2} .

Combining the Armijo condition with

α^{k} \geq C > 0

yields

f (x^{k}) - f (x^{k + 1}) \geq \frac{β _{1} C}{M} ∥\nabla f (x^{k}) ∥^{2} .

If

f \geq L

, a telescoping sum gives

\frac{β _{1} C}{M} \sum_{k = 0}^{N} ∥\nabla f (x^{k}) ∥^{2} \leq f (x^{0}) - f (x^{N + 1}) \leq f (x^{0}) - L .

The nonnegative series converges, so its terms tend to zero:

k \to \infty lim ∥\nabla f (x^{k}) ∥ = 0 .

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.

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