neldermead.algo: Nelder-Mead Algorithm
In sbihorel/neldermead: R Port of the 'Scilab' Neldermead Module

Description Usage Arguments Details Value Author(s)

neldermead.algo performs an optimization without restart using the method associated with the method element of the neldermead object; neldermead.fixed, neldermead.variable, neldermead.box, boxlinesearch, neldermead.storehistory, neldermead.termination, and neldermead.interpolate are utility functions for neldermead.algo.

  neldermead.algo(this = NULL)
  neldermead.fixed(this = NULL)
  neldermead.variable(this = NULL)
  neldermead.box(this = this)
  boxlinesearch(this = NULL, n = NULL, xbar = NULL, xhigh = NULL, fhigh = NULL,
                rho = NULL)
  neldermead.storehistory(this = NULL, n = NULL, fopt = NULL, xopt = NULL,
                          fv = NULL, xcoords = NULL)
  neldermead.termination(this = NULL, fvinitial = NULL, oldfvmean = NULL,
                         newfvmean = NULL, previousxopt = NULL,
                         currentxopt = NULL, simplex = NULL)
  neldermead.interpolate(x1 = NULL, x2 = NULL, fac = NULL)

`this`	A neldermead object.
`n`	Number of variables.
`xbar`	The centroid.
`xhigh`	The high point.
`fhigh`	The value of the cost function at `xhigh`.
`rho`	The reflection factor.
`fopt`	The current value of the function at the current optimum point estimate.
`xopt`	The current optimum point estimate.
`fv`	The function values, with size nbve x 1.
`xcoords`	Matrix of size n x n+1, coordinates of the n+1 vertices
`fvinitial`	The initial cost function value.
`oldfvmean`	The old cost function value average on the simplex.
`newfvmean`	The new cost function value average on the simplex.
`previousxopt`	The previous point estimate.
`currentxopt`	The current point estimate.
`simplex`	The simplex. The best point estimate in the simplex is expected to be stored at 1, while the worst point estimate in the simplex is expected to be stored at n+1.
`x1`	The first reference point estimate to perform the interpolation.
`x2`	The second reference point estimate to perform the interpolation.
`fac`	A factor to perform the interpolation.

neldermead.fixed

The simplex algorithm with fixed size simplex. We implement the following 'rules' of the method of Spendley et al.

Rule 1 is strictly applied, but the reflection is done by reflection of the high point, since we minimize a function instead of maximizing it, like Spendley.
Rule 2 is NOT implemented, as we expect that the function evaluation is not subject to errors.
Rule 3 is applied, i.e. reflection with respect to next to high point. A shrink step is included, with shrinkage factor sigma.

Rule 1. Ascertain the lowest reading y, of yi ... Yk+1 Complete a new simplex Sp by excluding the point Vp corresponding to y, and replacing it by V* defined as above.

Rule 2. If a result has occurred in (k + 1) successive simplexes, and is not then eliminated by application of Rule 1, do not move in the direction indicated by Rule 1, or at all, but discard the result and replace it by a new observation at the same point.

Rule 3. If y is the lowest reading in So , and if the next observation made, y* , is the lowest reading in the new simplex S , do not apply Rule 1 and return to So from Sp . Move out of S, by rejecting the second lowest reading (which is also the second lowest reading in So).

neldermead.variable

The original Nelder-Mead algorithm, with variable-size simplex.

neldermead.box

The Nelder-Mead algorithm, with variable-size simplex and modifications by Box for bounds and inequality constraints.

boxlinesearch

Called by neldermead.box, i.e. Box's method. Perform a line search from xbar, on the line (xhigh,xbar). The reflected point estimate satisfies the following constraints:

fr < fhigh
xr satisfies the bounds constraints
xr satisfies the nonlinear positive inequality constraints
xr satisfies the linear positive inequality constraints

The method is based on projection and scaling toward the centroid.

neldermead.storehistory

Store the optimization history into the neldermead object.

neldermead.termination

Determine if the algorithm must continue or terminate. The function uses the cost function average in the simplex instead of the best cost function value. This is because the function average changes at each iteration. Instead, the best function value has a step-by-step evolution and may not change between two successive iterations, leading to a stop of the algorithm.

neldermead.interpolate

Compute the point estimate xi as an interpolation between x1 and x2, as follows: xi = (1+fac)x1 - fac*x2

neldermead.fixed, neldermead.variable, and neldermead.box

Return the updated neldermead object, containing the optimum point estimate.

boxlinesearch

Return a list with the following elements:

this: The updated neldermead object.
status: TRUE if the search is successful, FALSE otherwise.
xr: The reflected point estimate.
fr: The value of the cost function at xr.

neldermead.storehistory

Return the updated neldermead object.

neldermead.termination

Return a list with the following elements:

this: The updated neldermead object
terminate: TRUE if the algorithm terminates, FALSE if the algorithm must continue.
status: The termination status: 'continue', 'maxiter', 'maxfuneval', 'tolf', 'tolx', 'tolsize', 'tolsizedeltafv', 'kelleystagnation', 'tolboxf', 'tolvariance' or the user-defined termination status.

neldermead.interpolate