Many engineering optimization problems involve a special kind of discrete v
ariable that can be represented by a number, but this representation has no
significance. Such variables arise when a decision involves some situation
like a choice from an unordered list of categories. This has two implicati
ons: The standard approach of solving problems with continuous relaxations
of discrete variables is not available, and the notion of local optimality
must be defined through a user-specified set of neighboring points. We pres
ent a class of direct search algorithms to provide limit points that satisf
y some appropriate necessary conditions for local optimality for such probl
ems. We give a more expensive version of the algorithm that guarantees addi
tional necessary optimality conditions. A small example illustrates the dif
ferences between the two versions. A real thermal insulation system design
problem illustrates the efficacy of the user controls for this class of alg
orithms.