Finance Notes_Part_15

Finance Notes_Part_15 - Simplices The volume of a simplex...

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Unformatted text preview: Simplices The volume of a simplex of vertices Y = {y 0 , y 1 , . . . , y n } is defined by vol(Y ) = |det(S (Y ))| n! where S (Y ) = y 1 − y 0 · · · y n − y 0 . Note that vol(Y ) > 0 (since the vertices are affinely independent). A measure of geometry for simplices is the normalized volume: von(Y ) = vol Audet and Vicente (SIOPT 2008) 1 Y diam(Y ) Unconstrained optimization . 37/109 Simplicial Direct-Search Methods The Nelder-Mead method: Considers a simplex at each iteration, trying to replace the worst vertex by a new one. For that it performs one of the following simplex operations: reflexion, expansion, outside contraction, inside contraction. −→ Costs 1 or 2 function evaluations (per iteration). If they all the above fail the simplex is shrunk. −→ Additional n function evaluations (per iteration). Audet and Vicente (SIOPT 2008) Unconstrained optimization 38/109 Nelder Mead simplex operations (reflections, expansions, outside contractions, inside contractions) y1 y2 y ic yc y oc yr ye y0 y c is the centroid of the face opposed to the worse vertex y 2 . Audet and Vicente (SIOPT 2008) Unconstrained optimization 39/109 Nelder Mead simplex operations (shrinks) y1 y2 y0 Audet and Vicente (SIOPT 2008) Unconstrained optimization 40/109 McKinnon counter-example: The Nelder-Mead method: Attempts to capture the curvature of the function. Is globally convergent when n = 1. Can fail for n > 1 (e.g. due to repeated inside contractions). Audet and Vicente (SIOPT 2008) Unconstrained optimization 41/109 ...
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Finance Notes_Part_15 - Simplices The volume of a simplex...

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