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

Some Theoretical Properties of Evolutionary Algorithms under Partially Ordered Fitness Values

Presently, the limit theory of evolutionary algorithms (EA) for mono-criterion optimization under certainty is well developed. The situation is different for the fields of evolutionary optimization under complete or partial uncertainty, multiple criteria and so forth. Since these problem classes may be seen as special cases of the task of finding the set of minimal (or maximal) elements in partially ordered sets, a limit theory for EAs that can cope with this kind of problem passes all properties and results on its special cases mentioned above

A framework of quantum-inspired multi-objective evolutionary algorithms and its convergence properties

In this paper, a general framework of quantum-inspired multiobjective evolutionary algorithms is proposed based on the basic principles of quantum computing and general schemes of multi-objective evolutionary algorithms. One of the sufficient convergence conditions to Pareto optimal set is presented and it is proved under partially order set theory. Moreover, two algorithms are given as examples meeting this convergence condition, in which two improved Q-gates are used. Their convergence properties are discussed. Additionally, one counterexample is also given

Diversity as Width

It is argued that if the population of options is a finite poset, diversity comparisons may be conveniently based on widths i.e. on the respective maximum numbers of pairwise incomparable options included in the relevant subposets. The width-ranking and the undominated width-ranking are introduced and characterized