Fundamentals - Preventive Biomechanics - page 116

Fig. 3.27 Simplex shape modifications in 2-dimensional Euclidian space R 2 : a Reflection,
b Expansion, c Partial Contraction (inside or outside) and d Shrinking. Dashed triangle: current
simplex, shaded triangle: potentially new simplex
as well as the independent variables of the parameter vector are scalar values. To
perform parameter optimization under the condition that the model function is not
explicitly accessible and only function values are available, an appropriate opti-
mization algorithm must be used. This condition is present in the case of foam and
tissue parameter identification employing the finite element method to solve the
boundary value problem. One possible approach is to employ the downhill simplex
strategy (Nelder-Mead method or amoeba algorithm) devised by Nelder and Mead
(1965), which is subsequently discussed in more detail. The basic procedure
leading to the modified version of Nelder and Mead was first introduced by
Spendley et al. (1962).
Further information about the downhill simplex method can be found in Ha-
slinger and Mäkinen (2003), Kolda et al. (2003), Schwefel (1994) and Walter
(1991).
The simplex strategy of Nelder and Mead should not to be confused with the
simplex method of linear programming, introduced by Wood and Dantzig (1949)
and Dantzig (1949), where an optimum of a linear objective function, subject to a
number of linear equality or inequality constraints is sought. Although both
strategies make use of a geometric figure called a simplex, linear programming
uses the simplex differently.
The downhill simplex method is a local unconstrained minimizing algorithm
for general nonlinear functions in multi dimensions, not relying on analytical or
numerical derivative information. It thus can be classed as a heuristic method that
begins with (n ? 1) vertices (parameter vectors) in the n-dimensional space R n ,
instead of one initial starting point. During optimization, it decides on further
procedures following a scheme, using information gathered in previous iterations.
Starting with an initial (random) guess, n further vertices of a non-degenerate
simplex are generated. This may be accomplished through multiplication of each
variable of the initial parameter vector with a factor to generate, in general, a