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A uniform descent angle makes the Wolfe gradient norms vanish

Statement

Under the hypotheses of Wolfe steps imply the Zoutendijk summability condition, suppose there is a constant $δ > 0$ such that $\cos θ_{k} \geq δ$ for every $k$ . Then $∥ \nabla f (x_{k}) ∥_{2} \to 0$ . This is convergence of gradient norms to zero; it does not assert convergence of $(x_{k})$ or that a limit point is a minimizer.

Facts & Assumptions

Given: The hypotheses and angle convention of Wolfe steps imply the Zoutendijk summability condition, and $δ > 0$ with $\cos θ_{k} \geq δ$ for all $k$ .

[F1]

Wolfe steps imply the Zoutendijk summability condition: the nonnegative series $\sum_{k \geq 0} \cos^{2} θ_{k} ∥ \nabla f (x_{k}) ∥_{2}^{2}$ is finite.

Proof

Given: The hypotheses and notation in the Statement.

1.1givenF1algebra

Since $\cos^{2} θ_{k} \geq δ^{2}$ , [F1] implies $\sum_{k \geq 0} ∥ \nabla f (x_{k}) ∥_{2}^{2} \leq δ^{- 2} \sum_{k \geq 0} \cos^{2} θ_{k} ∥ \nabla f (x_{k}) ∥_{2}^{2} < \infty$ .

2.1step 1.1choosealgebra∎

Fix any $ε > 0$ . There can be only finitely many $k$ with $∥ \nabla f (x_{k}) ∥_{2} \geq ε$ , since $N$ such indices would contribute at least $N ε^{2}$ to the finite series in step 1.1, for every $N$ . Thus the gradient norms are eventually below each positive $ε$ , which is exactly $∥ \nabla f (x_{k}) ∥_{2} \to 0$ .

Depends on

Wolfe steps imply the Zoutendijk summability condition

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, 2nd ed. (standard reference, not scraped)