Tobias Storch

Real royal road functions for constant population size

Evolutionary and genetic algorithms (EAs and GAs) are quite successful randomized function optimizers. This success is mainly based on the interaction of different operators like selection, mutation, and crossover. Since this interaction is still not well understood, one is interested in the analysis of the single operators. Jansen and Wegener [Proceedings of GECCO2001, 2001, pp. 375-382] have described so-called real royal road functions where simple steady-state GAs have a polynomial expected optimization time while the success probability of mutation-based EAs is exponentially small even after an exponential number of steps. This success of the GA is based on the crossover operator and a population whose size is moderately increasing with the dimension of the search space. Here new real royal road functions are presented where crossover leads to a small optimization time, although the GA works with the smallest possible population size--namely 2.

Faster Evolutionary Algorithms by Superior Graph Representation

We present a new representation for individuals in problems that have cyclic permutations as solutions. To demonstrate its usefulness, we analyze a simple randomized local search and a (1+1) evolutionary algorithm for the Eulerian cycle problem utilizing this representation. Both have an expected runtime of Theta(m2 log(m)), where m denotes the number of edges of the input graph. This clearly beats previous solutions, which all have an expected optimization time of Theta(m3 or worse (PPSN 06, CEC 04). We are optimistic that our representation also allows superior solutions for other cyclic permutation problems. For NP-complete ones like the TSP, however, other means than theoretical run-time analyses are necessary

When the Plus Strategy Outperforms the Comma Strategyand When Not

Occasionally there have been long debates on whether to use elitist selection or not. In the present paper the simple (1, lambda) EA and {1 + lambda) EA operating on {0, l}n are compared by means of a rigorous runtime analysis. It turns out that only values for lambda that are logarithmic in n are interesting. An illustrative function is presented for which newly developed proof methods show that the (1, lambda) EA - where lambda is logarithmic in n - outperforms the (1 + lambda) EA for any lambda. For smaller offspring populations the (1, lambda) EA is inefficient on every function with a unique optimum, whereas for larger lambda the two randomized search heuristics behave almost equivalently.

How randomized search heuristics find maximum cliques in planar graphs

Surprisingly, general search heuristics often solve combinatorial problems quite sufficiently, although they do not outperform specialized algorithms. Here, the behavior of simple randomized optimizers on the maximum clique problem on planar graphs is investigated rigorously. The focus is on the worst-, average-, and semi-average-case behaviors. In semi-random planar graph models an adversary is allowed to modify moderately a random planar graph, where a graph is chosen uniformly at random among all planar graphs. With regard to the heuristics particular interest is given to the influences of the following four popular strategies to overcome local optima: local- vs. global-search, single- vs. multi-start, small vs. large population, and elitism vs. non-elitism selection. Finally, the black-box complexities of the planar graph models are analyzed.

On the Choice of the Population Size

Evolutionary Algorithms (EAs) are population-based randomized optimizers often solving problems quite successfully. Here, the focus is on the possible effects of changing the parent population size. Therefore, new functions are presented where for a simple mutation-based EA even a decrease of the population size by one leads from an efficient optimization to an enormous running time with an overwhelming probability. This is proven rigorously for all feasible population sizes. In order to obtain these results, new methods for the analysis of the EA are developed.

How comma selection helps with the escape from local optima

We investigate (1,λ) ESs using isotropic mutations for optimization in ℝn by means of a theoretical runtime analysis. In particular, a constant offspring-population size λ will be of interest. n nWe start off by considering an adaptation-less (1,2) ES minimizing a linear function. Subsequently, a piecewise linear function with a jump/cliff is considered, where a (1+λ) ES gets trapped, i.e., (at least) an exponential (in n) number of steps are necessary to escape the local-optimum region. The (1,2) ES, however, manages to overcome the cliff in an almost unnoticeable number of steps. n nFinally, we outline (because of the page limit) how the reasoning and the calculations can be extended to the scenario where a (1,λ) ES using Gaussian mutations minimizes Cliff, a bimodal, spherically symmetric function already considered in the literature, which is merely Sphere with a jump in the function value at a certain distance from the minimum. For λ a constant large enough, the (1,λ) ES manages to conquer the global-optimum region – in contrast to (1+λ) ESs which get trapped.

Finding large cliques in sparse semi-random graphs by simple randomized search heuristics

Surprisingly, general heuristics often solve some instances of hard combinatorial problems quite sufficiently, although they do not outperform specialized algorithms. Here, the behavior of simple randomized optimizers on the maximum clique problem is investigated. We focus on semi-random models for sparse graphs, in which an adversary is even allowed to insert a limited number of edges (and not only to remove them). In the course of these investigations the approximation behavior on general graphs and the optimization behavior for sparse graphs and further semi-random graph models are also considered. With regard to the optimizers, particular interest is given to the influences of the population size and the search operator.

On the impact of objective function transformations on evolutionary and black-box algorithms

Different objective functions characterize different problems. However, certain fitness transformations can lead to easier problems although they are still a model of the considered problem. In this article, the class of not worsening transformations for a simple population-based evolutionary algorithm (EA) is described completely. That is the class of functions that transfers easy problems in easy ones and difficult problems in difficult ones. Surprisingly, this class

Design und Analyse randomisierter Suchheuristiken

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