Finance Notes_Part_31

# Finance Notes_Part_31 - meshes that get inﬁnitely ﬁne...

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OrthoMads : Householder orthogonal matrix The Householder transformation is then applied to the integer direction q : H = k q k 2 I n - 2 qq T , where I n is the identity matrix. By construction, H is an integer orthogonal basis of R n . The poll directions for OrthoMads are deﬁned to be the columns of H and - H . A lower bound on the cosine of the maximum angle between any arbitrary nonzero vector v R n and the set of directions in D is deﬁned as κ ( D ) = min 0 6 = v R n max d D v T d k v kk d k . With OrthoMads the measure κ ( D ) = 1 n is maximized over all positive bases. Audet and Vicente (SIOPT 2008) Optimization under general constraints 78/109

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Convergence analysis - OrthoMads with extreme barrier Theorem As k → ∞ , the set of OrthoMads normalized poll directions is dense in the unit sphere. Audet and Vicente (SIOPT 2008) Optimization under general constraints 79/109
Convergence analysis - OrthoMads with extreme barrier Theorem As k → ∞ , the set of OrthoMads normalized poll directions is dense in the unit sphere. Theorem Let ˆ x be the the limit of a subsequence of mesh local optimizers on

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Unformatted text preview: meshes that get inﬁnitely ﬁne. If f is Lipschitz near ˆ x , then f ◦ (ˆ x,v ) ≥ for all v ∈ T H Ω (ˆ x ) . Assuming more smoothness, Abramson studies second order convergence. Audet and Vicente (SIOPT 2008) Optimization under general constraints 79/109 Open, closed and hidden constraints Consider the toy problem: min x ∈ R 2 x 2 1-√ x 2 s.t.-x 2 1 + x 2 2 ≤ 1 x 2 ≥ Audet and Vicente (SIOPT 2008) Optimization under general constraints 80/109 Open, closed and hidden constraints Consider the toy problem: min x ∈ R 2 x 2 1-√ x 2 s.t.-x 2 1 + x 2 2 ≤ 1 x 2 ≥ Closed constraints must be satisﬁed at every trial vector of decision variables in order for the functions to evaluate. Here x 2 ≥ is a closed constraint, because if it is violated, the objective function will fail. Audet and Vicente (SIOPT 2008) Optimization under general constraints 80/109...
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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_31 - meshes that get inﬁnitely ﬁne...

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