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The Cauchy step for a quadratic trust-region model
Definition
Input a model of Quadratic trust-region models and the actual-to-predicted reduction ratio with gradient
, symmetric model matrix , and
radius . If , return with status stationary. If
, set and define
Return with status step.
Depends on
Used by
- A finite-budget trust-region method with ratio-based radius updates Algorithm
- Dogleg steps for positive-definite trust-region models Algorithm
- Truncated conjugate gradients with boundary and negative-curvature exits Algorithm
- An indefinite quadratic has a unique boundary trust-region minimizer Example
- Newton, Cauchy, and truncated-CG steps on one indefinite model Example
- Positive-definite Newton and dogleg steps are well-defined Lemma
- Retraction trust-region ratios are accurate at small radii Lemma
- The actual-to-predicted ratio approaches one at small radii away from stationarity Lemma
- The Cauchy step gives a quantitative model decrease Lemma
- Truncated-CG iterates are feasible and lie in generated Krylov subspaces Lemma
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, second edition (standard reference, not scraped)
- Nick Gould, Continuous Optimization Algorithms, Part 3.2: Trust-Region Methods (standard reference, not scraped)