Finance Notes_Part_16

# Finance Notes_Part_16 - Let Y k be the sequence of...

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McKinnon counter-example Audet and Vicente (SIOPT 2008) Unconstrained optimization 42/109

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Nelder-Mead on McKinnon example (two diﬀerent starting simplices) Audet and Vicente (SIOPT 2008) Unconstrained optimization 43/109
Modiﬁed Nelder-Mead methods For Nelder-Mead to globally converge one must: Control the internal angles ( normalized volume ) in all simplex operations but shrinks. CAUTION (very counterintuitive) : Isometric reﬂections only preserve internal angles when n = 2 or the simplices are equilateral . -→ Need for a back-up polling. Impose suﬃcient decrease instead of simple decrease for accepting new iterates: f ( new point ) f ( previous point ) - o ( simplex diameter ) . Audet and Vicente (SIOPT 2008) Unconstrained optimization 44/109

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Unformatted text preview: Let { Y k } be the sequence of simplices generated. Let f be bounded from below and uniformly continuous in R n . Theorem (Step size going to zero) The diameters of the simplices converge to zero: lim k-→ + ∞ diam ( Y k ) = 0 . Theorem (Global convergence) If f is continuously diﬀerentiable in R n and { Y k } lies in a compact set then { Y k } has at least one stationary limit point x * . Audet and Vicente (SIOPT 2008) Unconstrained optimization 45/109 Simplex gradients It is possible to build a simplex gradient: y y 1 y 2 ∇ s f ( y ) = ± y 1-y y 2-y ²-> ³ f ( y 1 )-f ( y ) f ( y 2 )-f ( y ) ´ . Audet and Vicente (SIOPT 2008) Unconstrained optimization 46/109...
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Finance Notes_Part_16 - Let Y k be the sequence of...

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