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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 DirectSearch Methods
The NelderMead 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 counterexample:
The NelderMead 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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 Spring '06
 Tapley
 Finance, Tetrahedron, Unconstrained Optimization, Polytope, VICENTE, Audet

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