Finance Notes_Part_19

Finance Notes_Part_19 - Fully-linear models For a class of...

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Fully-linear models For a class of fully-linear models, one must also guarantee that: There exists a model-improvement algorithm , that in a finite, uniformly bounded (with respect to x and Δ ) number of steps can: certificate that a given model is fully linear on B ( x ; Δ) , — or (if the above fails), find a model that is fully linear on B ( x ; Δ) . Audet and Vicente (SIOPT 2008) Unconstrained optimization 53/109
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Fully-quadratic models Given a point x and a trust-region radius Δ , a model m ( y ) around x is called fully quadratic if It is twice continuous differentiable with Lipschitz continuous second derivatives. The following error bounds hold (. ..): k∇ 2 f ( y ) - ∇ 2 m ( y ) k ≤ κ eh Δ y B ( x ; Δ) k∇ f ( y ) - ∇ m ( y ) k ≤ κ eg Δ 2 y B ( x ; Δ) and | f ( y ) - m ( y ) | ≤ κ ef Δ 3 y B ( x ; Δ) . Audet and Vicente (SIOPT 2008) Unconstrained optimization 54/109
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TR methods for DFO (basics) Set x k +1 to x k + Δ x k (success)
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This document was uploaded on 10/30/2011 for the course FIN 3403 at University of Florida.

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Finance Notes_Part_19 - Fully-linear models For a class of...

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