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Safeguarded PR+ converges in gradient norm under strong Wolfe

Statement

Let $D, f, x_{0}, L_{0}, N, L$ satisfy the hypotheses of A uniform descent angle makes the Wolfe gradient norms vanish. Consider an infinite sequence following the PR+ coefficient recurrence and candidate-or-restart direction selection of Fletcher–Reeves and safeguarded Polak–Ribière nonlinear conjugate gradients, with its fixed $σ, C > 0$ , $d_{0} = - g_{0}$ , every gradient nonzero, feasible updates $x_{k + 1} = x_{k} + t_{k} d_{k}$ , and every $t_{k}$ a strong Wolfe step with $0 < c_{1} < c_{2} < 1 / 2$ . Then $∥ \nabla f (x_{k}) ∥_{2} ⟶ 0.$ This claim concerns the infinite recurrence with successful updates at every index, not an infinite execution of the finite- $K$ algorithm. Raw PR+ without the stated safeguard is not covered.

Facts & Assumptions

Given: The smoothness and lower-bound hypotheses of A uniform descent angle makes the Wolfe gradient norms vanish, an infinite successful safeguarded PR+ recurrence sequence, and $0 < c_{1} < c_{2} < 1 / 2$ .

[F1]

The safeguarded PR+ direction has a uniform angle bound: every nonzero safeguarded PR+ direction is descending and has $\cos θ_{k} \geq \min {1, σ / C} > 0$ .

[F2]

A uniform descent angle makes the Wolfe gradient norms vanish: under the stated lower-bound and Lipschitz-gradient hypotheses, Wolfe steps and a uniform positive descent-angle bound imply $∥ \nabla f (x_{k}) ∥_{2} \to 0$ .

Proof

Given: The hypotheses and notation in the Statement.

1.1givenF1algebra

Set $δ_{P R} = \min {1, σ / C} > 0$ . By [F1], every search direction has descent-angle cosine at least $δ_{P R}$ , including a direction reset to $- g_{k}$ .

2.1givenF2step 1.1∎

Strong Wolfe implies Wolfe by Wolfe and strong Wolfe line-search conditions. All hypotheses of [F2] therefore hold with this $δ_{P R}$ , and [F2] gives $∥ \nabla f (x_{k}) ∥_{2} \to 0$ .

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, second edition (standard reference, not scraped)
Jean Charles Gilbert and Jorge Nocedal, Global Convergence Properties of Conjugate Gradient Methods for Optimization, SIAM Journal on Optimization 2(1), 21–42 (1992), DOI 10.1137/0802003 (standard reference, not scraped)