Evolutionary Search for Minimal Elements in Partially Ordered Finite Sets

The task of finding minimal elements of a partially ordered set is a generalization of the task of finding the global minimum of a real-valued function or of finding Pareto-optimal points of a multicriteria optimization problem. It is shown that evolutionary algorithms are able to converge to the set of minimal elements in finite time with probability one, provided that the search space is finite, the time-invariant variation operator is associated with a positive transition probability function and that the selection operator obeys the so-called ‘elite preservation strategy.’