A New Method for Lower Bounds on the Running Time of Evolutionary Algorithms

@article{Sudholt2013ANM,
  title={A New Method for Lower Bounds on the Running Time of Evolutionary Algorithms},
  author={Dirk Sudholt},
  journal={IEEE Transactions on Evolutionary Computation},
  year={2013},
  volume={17},
  pages={418-435}
}
  • Dirk Sudholt
  • Published 7 September 2011
  • Computer Science, Mathematics
  • IEEE Transactions on Evolutionary Computation
In this paper a new method for proving lower bounds on the expected running time of evolutionary algorithms (EAs) is presented. It is based on fitness-level partitions and an additional condition on transition probabilities between fitness levels. The method is versatile, intuitive, elegant, and very powerful. It yields exact or near-exact lower bounds for LO, OneMax, long k-paths, and all functions with a unique optimum. Most lower bounds are very general; they hold for all EAs that only use… Expand
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References

SHOWING 1-10 OF 61 REFERENCES
General Lower Bounds for the Running Time of Evolutionary Algorithms
We present a new method for proving lower bounds in evolutionary computation based on fitness-level arguments and an additional condition on transition probabilities between fitness levels. TheExpand
Runtime Analysis of the ( + 1) EA on Simple Pseudo-Boolean Functions
  • C. Witt
  • Computer Science, Medicine
  • Evolutionary Computation
  • 2006
TLDR
A newproof technique is developed that bounds the runtime of the ( + 1) EA and investigates the stochastic process for creating family trees of individuals; the depth of these trees is bounded and the progress of the population towards the optimum is captured. Expand
Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation
TLDR
The present paper picks up Hajek's line of thought to prove a drift theorem that is very easy to use in evolutionary computation and shows how previous analyses involving the complicated theorem can be redone in a much simpler and clearer way. Expand
Optimizing Linear Functions with Randomized Search Heuristics - The Robustness of Mutation
  • C. Witt
  • Mathematics, Computer Science
  • STACS
  • 2012
TLDR
The standard mutation probability p = 1/n is optimal for all linear functions, and the (1+1) EA is found to be an optimal mutation-based algorithm that turns out to be surprisingly robust since large neighborhood explored by the mutation operator does not disrupt the search. Expand
Runtime Analysis of the ( μ +1) EA on Simple Pseudo-Boolean Functions
Although Evolutionary Algorithms (EAs) have been successfully applied to optimization in discrete search spaces, theoretical developments remain weak, in particular for population-based EAs. ThisExpand
The use of tail inequalities on the probable computational time of randomized search heuristics
TLDR
A new inequality that is based on the general form of the Chernoff inequality and previous methods such as ''fitness-based partitions'' and ''potential functions'', which have been used to analyze the expected running time of RSHs are demonstrated. Expand
A rigorous analysis of the compact genetic algorithm for linear functions
TLDR
First rigorous runtime analyses of a simple EDA, the compact genetic algorithm (cGA), for linear pseudo-Boolean functions on n variables are presented, proving a general lower bound for all functions and a general upper bound forall linear functions. Expand
Time complexity of evolutionary algorithms for combinatorial optimization: A decade of results
TLDR
This paper presents a survey of the results obtained in the last decade along computational time complexity analyzes of evolutionary algorithms, and the most common mathematical techniques are introduced. Expand
Adaptive population models for offspring populations and parallel evolutionary algorithms
We present two adaptive schemes for dynamically choosing the number of parallel instances in parallel evolutionary algorithms. This includes the choice of the offspring population size in a (1+λ) EAExpand
Tight Bounds on the Optimization Time of the (1+1) EA on Linear Functions
TLDR
The standard mutation probability $p=1/n$ is optimal for all linear functions, and the (1+1) EA is found to be an optimal mutation-based algorithm. Expand
...
1
2
3
4
5
...