A simplex is defined as a body in
dimensions consisting of vertices. Specifying
the location of each vertex fully defines the simplex. In two dimensions,
the simplex is a triangle. In three dimensions, it is a tetrahedron.
As the algorithm proceeds, the simplex makes its way downward toward
the location of the minimum through a series of steps. These steps can
be divided into reflections, expansions, and contractions. Most steps
are reflections, which consist of moving the vertex of the simplex where
the objective function is largest (worst) through the opposite face of
the simplex to a lower (better) point. Reflections maintain the volume
of the simplex. When possible, an expansion can accompany the reflection
to increase the size of the simplex and speed convergence by allowing
larger steps. Conversely, contractions “shrink” the simplex, allowing
it to settle into a minimum or pass through a small opening like the
neck of an hourglass.
This method has the highest probability of finding the global minimum
when it is started with big initial steps. The initial simplex will span
a greater fraction of the design space, and the chances of getting trapped
in a local minimum are smaller. However, for complex hyper-dimensional
topographies, the method can break down.