The multi-directional search (MDS)^{[1]} algorithm is a direct search simplex method that is closely related to
Nelder-Mead^{[2]} method, in which non-degenerate simplex of dimension *n+1*
is updated (for an *n*-dimensional parameter vector) at each step. The
volume enclosed by the simplex reduces until it encloses an extremum of the objective
function.

A step in the process begins with stored values of the simplex vertices (parameter
vectors) and the associated objective function values. The vertex with the best (minimum)
value of the objective function is identified, and a set of *n* search
directions is defined by the edges that connect the best vertex to the remaining vertices
in the simplex. The length of each edge defines the length of the associated step in that
direction. i.e., the new sample points are obtained by "reflecting" each vertex about the
best vertex, with the connecting edge defining the reflection plane normal. The new sample
points, together with the previous best vertex, form a new simplex that is accepted for
the next iteration if at least one of its vertices has a better objective function value
than the previous best vertex. If the simplex is not accepted there is a simple set of
expansion and/or contraction steps (wherein the search directions are maintained but the
step-size is changed) that are performed to find an acceptable new simplex. The process
terminates when the simplex nodes, and the corresponding goal function values, become
"close enough". More precisely, the two criteria are:

(4.1) |

where δ is a user defined parameter, and NES is the number of experiments in one iteration.

One advantage this method has over Nelder-Mead and some other direct search approaches is that it is inherently parallel, because the processing and function evaluations associated with the different search directions are independent and can be performed simultaneously on different processors.

Analyst exposes several parameters for MDS:

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