2 (d) exhibits the successive simplexes of NMSA and Fig.

Comparison of Performance on Han's Function: NS-NMSA found the optimal point (0, -1.3623898059) costing only 161 and 157 function-evaluations when started with simplexes H 1 and H 2 respectively.

Iteration-wise the progress of NMSA and NS-NMSA for Han's function for two initial simplexes H 1 and H 2 have been shown in Fig.

If L is a subcollection of simplexes in K closed under containment and intersection, then L is a complex in its own right, called a subcomplex of K.

The set of simplexes of K of dimension at most l is a subcomplex of K, called the l-skeleton of K, denoted [skel.sup.l](K).

Let S = ([s.sub.0], ..., [s.sub.p]) and T = ([t.sub.0], ..., [t.sub.q]) be simplexes whose combined sets of vertexes are affinely independent.

If K and L are simplicial complexes, not necessarily of the same dimension, then their join, denoted K [multiplied by] L, is the collection of simplexes K [union] L [union] {S [multiplied by] T|S [element of] K, T [element of] L}.

Let |K| be the subset [[union].sub.S [element of] K] S of a high-dimensional Euclidean space [R.sup.l], that is, the union of the simplexes of K.

Geometric and abstract simplexes are closely related: any affinely independent set of vectors {[v.sub.0], ..., [v.sub.n]} span both a geometric and abstract simplex.

An abstract complex K is a collection of abstract simplexes closed under containment, that is, if S is in K, so is any face of S.

The notions of dimension, join, and subcomplex are defined for abstract simplexes and complexes in the obvious way.

In the rest of this paper, it is convenient to use abstract and geometric simplexes and complexes more or less interchangeably.

The map is a simplicial map if it also carries simplexes of K to simplexes of L.

As seen in Figure 5, the simplicial map may "collapse" some simplexes.

--Each simplex of K is the union of finitely many simplexes in [Sigma](K).