Finance Notes_Part_30

Finance Notes_Part_30 - From Coordinate Search to OrthoMads...

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Unformatted text preview: From Coordinate Search to OrthoMads OrthoMads – 2008 u xk Audet and Vicente (SIOPT 2008) Optimization under general constraints 76/109 From Coordinate Search to OrthoMads OrthoMads – 2008 u   u  k xu u  u  Audet and Vicente (SIOPT 2008) Optimization under general constraints 76/109 From Coordinate Search to OrthoMads Union of all normalized directions grows dense in the unit sphere OrthoMads – 2008 u   u  k xu u  u  #¡ uy e gg T££ ¡ ! e s d eg  £¡ d B ¨ ‰ r d eg £¡ ¨ r ¨$ yr ˆ ˆrdeg £¡ ¨$$ X ˆˆ rˆ£ $ ˆ$ˆ¨$ g¡ r$ˆ e¨ dr ' E $¨ £d ¡e $$$¨ g rˆˆˆ W $ z ¨ ¡£ ge rr ˆ d ¨¨ j r % ¨ ¡£ ge d r ge d © ¡£ ‚ d ¡£  ¡ £ cgg e … ‡e infinite number of directions Audet and Vicente (SIOPT 2008) Optimization under general constraints 76/109 From Coordinate Search to OrthoMads Union of all normalized directions grows dense in the unit sphere OrthoMads – 2008 u   u  k xu u  u  #¡ uy e gg T££ ¡ ! e s d eg  £¡ d B ¨ ‰ r d eg £¡ ¨ r ¨$ yr ˆ ˆrdeg £¡ ¨$$ X ˆˆ rˆ£ $ ˆ$ˆ¨$ g¡ r$ˆ e¨ dr ' E $¨ £d ¡e $$$¨ g rˆˆˆ W $ z ¨ ¡£ ge rr ˆ d ¨¨ j r % ¨ ¡£ ge d r ge d © ¡£ ‚ d ¡£  ¡ £ cgg e … ‡e infinite number of directions OrthoMads is deterministic. Results are reproducible on any machine. At any given iteration, the directions are orthogonal. Audet and Vicente (SIOPT 2008) Optimization under general constraints 76/109 OrthoMads: Halton Pseudo-random sequence OrthoMads uses the pseudo-random Halton sequence (1960) to generate a sequence {ut }∞ of vectors in Rn that is dense in the t=1 hypercube [0, 1]n . The Mads convergence analysis requires that all trial points belong to a mesh that gets finer and finer as the algorithm unfolds. The direction ut is translated and scaled to 2ut −e so that it belongs to 2 ut − e the unit sphere (e is the vector of ones). The direction is rounded to the nearest mesh direction q . Audet and Vicente (SIOPT 2008) Optimization under general constraints 77/109 ...
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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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