GPS-X Technical Reference

Optimizer 372
CHAPTER 14
Optimizer
INTRODUCTION
This chapter describes the GPS-X optimizer and its associated forms. More detail on the
maximum likelihood objective function and the statistical tests provided in the solution report can
be found in Appendix A and Appendix B, and the Optimizer Solution Report. The complete list of
nomenclature used is found in Appendix C: Nomenclature. The references cited in this chapter
are listed in Appendix D: References.
When using the optimizer in GPS-X it is recommended that layouts be built in double precision.
Switch to double precision by selecting the Double Precision option in the
Options > Preferences > Build tab.
OPTIMIZER DESCRIPTION
The optimizer is a module designed to minimize the value of a user-selected objective function by
adjusting the free variables in this function. In the case of parameter estimation, these free
variables are the unknown process parameters. The optimizer uses the Nelder-Mead simplex
method (Press et al., 1986) for minimization. The algorithm has been modified to handle bounds
on the optimization (i.e. free) variables.
The simplex method is a multi-dimensional procedure that does not rely on gradient information.
The algorithm searches through the multidimensional "surface" using a direct search method to
find a local minimum of the objective function.
The procedure starts with an initial point in the multi-dimensional parameter space and then
generates new points in space by perturbing the initial point a scaled amount along each
parameter direction. This leads to p + 1 points in space that define a polyhedron (the simplex)
where p is the number of optimization variables. The points are called the vertices of the simplex.
At each iteration, the simplex method reflects the vertex with the highest function value (worst
point) through the centroid of the remaining p points of the polyhedron. The amount of reflection
is controlled by a reflection constant. If the reflected vertex is the new best point (lowest
function value) then the polyhedron is expanded along the direction of reflection. The amount of
expansion is controlled by an expansion constant. If the expanded vertex is better than the
reflected vertex it is taken as the new best point.
GPS-X Technical Reference