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Finance Notes_Part_20

# Finance Notes_Part_20 - TR methods for DFO(global...

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TR methods for DFO (global convergence) Theorem (Global convergence (2nd order) — TRM) If f has Lipschitz continuous second derivatives then max ± k∇ f ( x k ) k , - λ min [ 2 f ( x k )] ² -→ 0 . -→ Compactness of L ( x 0 ) is not necessary. -→ True for simple decrease. -→ Use of fully-quadratic models when necessary. Audet and Vicente (SIOPT 2008) Unconstrained optimization 58/109

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Given a sample set Y = { y 0 ,y 1 ,...,y p } and a polynomial basis φ , one considers a system of linear equations: M ( φ,Y ) α = f ( Y ) , where M ( φ,Y ) = φ 0 ( y 0 ) φ 1 ( y 0 ) ··· φ p ( y 0 ) φ 0 ( y 1 ) φ 1 ( y 1 ) ··· φ p ( y 1 ) . . . . . . . . . . . . φ 0 ( y p ) φ 1 ( y p ) ··· φ p ( y p ) f ( Y ) = f ( y 0 ) f ( y 1 ) . . . f ( y p ) . Example: φ = { 1 ,x 1 ,x 2 ,x 2 1 / 2 ,x 2 2 / 2 ,x 1 x 2 } . Audet and Vicente (SIOPT 2008)
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Finance Notes_Part_20 - TR methods for DFO(global...

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