AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite-budget retraction Armijo backtracking

Definition

Let $M$ be $St (n, r)$ or $Gr (r, n)$ , let $f : M \to R$ be smooth, and choose one retraction $R$ from QR and polar retractions on Stiefel and Grassmann. At $x \in M$ , let $g = grad f (x)$ . Fix $a \in (0, 1)$ , $β \in (0, 1)$ and an integer budget $J \geq 0$ .

If $g = 0$ , return (x, stationary). Otherwise test, in order, $t_{j} = β^{j}$ for $j = 0, \dots, J$ . At each trial form $y_{j} = R_{x} (- t_{j} g)$ ; if the chosen retraction's full-rank condition fails, reject that trial without evaluating $f (y_{j})$ . Otherwise accept and return $(y_{j}, t_{j})$ at the first index with $f (y_{j}) \leq f (x) - a t_{j} ∥ g ∥^{2} .$ If no trial passes, return (x, budget-exhausted).

Depends on

Used by

Dependency tree · two levels

18 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

P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Chapter 4 (2008) (standard reference, not scraped)
Nicolas Boumal, An Introduction to Optimization on Smooth Manifolds (2023 prepublication) (standard reference, not scraped)
Nicolas Boumal, P.-A. Absil, and Coralia Cartis, Global rates of convergence for nonconvex optimization on manifolds (2018), arXiv v2 (standard reference, not scraped)