Nelder Mead Method

NelderMead(M, F [, p])

perform a nelder mead minimization problem for the cost funciton F on the manifold M. If the initial population p is not given, a random set of points is chosen.

This algorithm is adapted from the Euclidean Nelder-Mead method, see https://en.wikipedia.org/wiki/Nelder–Mead_method and http://www.optimization-online.org/DB_FILE/2007/08/1742.pdf.

Input

• M – a manifold $\mathcal M$
• F – a cost function $F\colon\mathcal M\to\mathbb R$ to minimize
• population – (n+1 random_point(M)) an initial population of $n+1$ points, where $n$ is the dimension of the manifold M.

Optional

• stopping_criterion – (StopAfterIteration(2000)) a StoppingCriterion
• α – (1.) reflection parameter ($\alpha > 0$)
• γ – (2.) expansion parameter ($\gamma$)
• ρ – (1/2) contraction parameter, $0 < \rho \leq \frac{1}{2}$,
• σ – (1/2) shrink coefficient, $0 < \sigma \leq 1$
• retraction_method – (ExponentialRetraction) the rectraction to use
• inverse_retraction_method - (LogarithmicInverseRetraction) an inverse retraction to use.

and the ones that are passed to decorate_options for decorators.

Output

• either x the last iterate or the complete options depending on the optional keyword return_options, which is false by default (hence then only x is returned).

NelderMead(M, F [, p])

perform a nelder mead minimization problem for the cost funciton F on the manifold M. If the initial population p is not given, a random set of points is chosen. If it is given, the computation is done in place of p.

For more options see NelderMead.

NelderMeadOptions <: Options

Describes all parameters and the state of a Nealer-Mead heuristic based optimization algorithm.

Fields

The naming of these parameters follows the Wikipedia article of the Euclidean case. The default is given in brackets, the required value range after the description

• population – an Array{point,1} of $n+1$ points $x_i$, $i=1,\ldots,n+1$, where $n$ is the dimension of the manifold.
• stopping_criterion – (StopAfterIteration(2000)) a StoppingCriterion
• α – (1.) reflection parameter ($\alpha > 0$)
• γ – (2.) expansion parameter ($\gamma>0$)
• ρ – (1/2) contraction parameter, $0 < \rho \leq \frac{1}{2}$,
• σ – (1/2) shrink coefficient, $0 < \sigma \leq 1$
• x – (p[1]) - a field to collect the current best value
• retraction_method – ExponentialRetraction() the rectraction to use, defaults to the exponential map
• inverse_retraction_method - LogarithmicInverseRetraction an inverse_retraction(M,x,y) to use.

Constructors

NelderMead(M,stop, retr; α=1. , γ=2., ρ=1/2, σ=1/2)

construct a Nelder-Mead Option with a set of dimension(M)+1 random points.

NelderMead(p, stop retr; α=1. , γ=2., ρ=1/2, σ=1/2)

construct a Nelder-Mead Option with a set p of points