Probabilistic analysis of evolution strategies using isotropic mutations

This dissertation deals with optimization in high-dimensional Euclidean space. Namely, a particular type of direct-search methods known as Evolution Strategies (ESs) are investigated. Evolution Strategies mimic natural evolution, in particular mutation, in order to "evolve" an approximate solution. As this dissertation focuses on theoretical investigation of ESs in the way randomized approximation algorithms are analyzed in theoretical computer science (rather than by means of convergence theory or dynamical-system theory), very basic and simple ESs are considered. Namely, the only search operator that is applied are so-called isotropic mutations. That is, a new candidate solution is obtained by adding a random vector to the current candidate solution the distribution of which is spherically symmetric. General lower bounds on the number of steps/isotropic mutations which are necessary to reduce the approximation error in the search space are proved, where the focus is on how the number of optimization steps depends on (and scales with) the dimensionality of the search space. These lower bounds hold independently of the function to be optimized and for large classes of ESs. Moreover, for several concrete optimization scenarios where certain ESs optimize a unimodal function, upper bounds on the number of optimization steps are proved

Analyses of evolutionary algorithms

Evolutionary algorithms (EAs) are a highly successful tool commonly used in practice to solve algorithmic problems. This remarkable practical value, however, is not backed up by a deep theoretical understanding. Such an understanding would facilitate the application of EAs to further problems. Runtime analyses of EAs are one way to expand the theoretical knowledge in this field. This thesis presents runtime analyses for three prominent problems in combinatorial optimization. Additionally, it provides probability theoretical tools that will simplify future runtime analyses of EAs. The first problem considered is the Single Source Shortest Path problem. The task is to find in a weighted graph for a given source vertex shortest paths to all other vertices. Developing a new analysis method we can give tight bounds on the runtime of a previously designed and analyzed EA for this problem. The second problem is the All-Pairs Shortest Path problem. Given a weighted graph, one has to find a shortest path for every pair of vertices in the graph. For this problem we show that adding a crossover operator to a natural EA using only mutation provably decreases the runtime. This is the first time that the usefulness of a crossover operator was shown for a combinatorial problem. The third problem considered is the Sorting problem. For this problem, we design a new representation based on trees. We show that the EA nat- urally arising from this representation has a better runtime than previously analyzed EAs.Evolutionäre Algorithmen (EAs) werden in der Praxis sehr erfolgreich eingesetzt. Bisher werden die theoretischen Grundlagen von EAs jedoch nicht zufriedenstellend verstanden. Laufzeitanalysen für einfache EAs sollen dieses Verständnis erweitern. Diese Dissertation enthält Laufzeitanalysen für EAs für drei wohlbekannte kombinatorische Probleme. Zusätzlich werden wahrscheinlichkeitstheoretische Hilfsmittel zur Analyse von EAs eingeführt. Zuerst behandeln wir das Single Source Shortest Path Problem. Die Aufgabe besteht darin, in einem gewichteten Graphen einen kürzesten Weg von einem Startknoten zu jedem anderen Knoten zu finden. Durch die Entwick- lung einer neuen Analysemethode konnten wir scharfe Schranken für die Laufzeit eines bereits zuvor präsentierten und analysierten EAs angeben. Als nächstes betrachten wir das All-Pairs Shortest Path Problem. Hierbei will man für jedes Paar von Knoten in einem gewichteten Graphen einen kürzesten Weg berechnen. Für dieses Problem zeigen wir, dass das Hinzufügen eines Crossover Operators die Laufzeit gegenüber einem natürlichen EA, der nur Mutationen nutzt, verbessert. Dies ist das erste Mal, dass für ein kombinatorisches Problem bewiesen wurde, dass ein Crossover Operator die Laufzeit reduziert. Für das Sortierproblem entwickeln wir eine neue, auf Bäumen beruhende Repräsentation und zeigen, dass der natürlich daraus entstehende EA eine bessere Laufzeit hat als vorherige EAs

Towards a Theory of Randomized Search Heuristics

There is a well-developed theory about the algorithmic complexity of optimization problems. Complexity theory provides negative results which typically are based on assumptions like NP#=P or NP#=RP