## Running Time Analysis of a Multi-Objective Evolutionary Algorithm on a Simple Discrete Optimization Problem (2002)

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### BibTeX

@MISC{Laumanns02runningtime,

author = {Marco Laumanns and Lothar Thiele and Eckart Zitzler and Emo Welzl and Kalyanmoy Deb},

title = {Running Time Analysis of a Multi-Objective Evolutionary Algorithm on a Simple Discrete Optimization Problem},

year = {2002}

}

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### Abstract

For the first time, a running time analysis of a multi-objective evolutionary algorithm for a discrete optimization problem is given. To this end, a simple pseudo-Boolean problem (Lotz: leading ones - trailing zeroes) is defined and a population-based optimization algorithm (FEMO). We show, that the algorithm performs a black box optimization in #(n 2 log n) function evaluations where n is the number of binary decision variables. 1

### Citations

1233 |
Multi-objective optimization using evolutionary algorithms
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- 2001
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Citation Context ... probabilistic search heuristics that mimic principles of natural evolution. They are often used to solve optimization problems, in particular those with multiple objectives, for an overview see e.g. =-=[1]-=-. In multi-objective optimization, the aim is to find or to approximate the set of Pareto-optimal (or non-dominated)solutions. Existing theoretic work on convergence in evolutionary multi-objective op... |

576 |
Evolutionary Algorithms for Solving Multi-Objective Problems
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278 |
Non-Linear Multi-Objective Optimization
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Citation Context ...ase, the claimed bound of O(n 2 log n) holds. D. Comparing SEMO to a (1+1)-EA using Multistarts An alternative approach to find a set of Pareto-optimal solutions is the so-called scalarizing approach =-=[18]-=-. The different objective functions are aggregated to form a single-objective surrogate problem, on which a single-objective optimizer can be applied. This scalarization involves parameters to be set ... |

251 | Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy
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196 | Multiobjective Decision Making Theory and Methodology - Chankong, Haimes - 1983 |

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Citation Context ... algorithms many such results are contained in [8]. For the optimization of pseudoBoolean functions an extensive theory has been built up by Wegener et al., seese.g. [16], Droste, Jansen, and Wegener =-=[2, 3]-=-, or for a methodological overview [15]. Results on the running time of evolutionary algorithms in the multi-objective case are rare. Scharnow et al. [13] analyze a (1+1)-EA under multiple, nonconflic... |

152 |
Convergence Properties of Evolutionary Algorithms
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Citation Context ...ally the expected running time for a given class of problems and the success probability for a given optimization time. For single-objective evolutionary algorithms many such results are contained in =-=[8]-=-. For the optimization of pseudoBoolean functions an extensive theory has been built up by Wegener et al., seese.g. [16], Droste, Jansen, and Wegener [2, 3], or for a methodological overview [15]. Res... |

145 |
How genetic algorithms really work. I. Mutation and hillclimbing
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Citation Context ...orithms. In the case of a single objective and discrete search spaces, several results have been achieved regarding the optimization of pseudo-Boolean functions. Following first results by Mühlenbein =-=[19]-=- and Rudolph [20], a wide range of such problems was covered by Droste et al. [6], [7], who successfully applied and considerably extended analytical methods from the field of randomized algorithms. M... |

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- 2002
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Citation Context ...ccording to some probability distribution and mutated by flipping a randomly chosen bit. For Algorithm 1 we consider a uniform distribution for selecting the parent. An appropriate archiving strategy =-=[7]-=- is assumed to prevent the population from growing exponentially. For this study it suffices to ensure that a solution is only accepted if it has different objective values (line 7). 3.1 Running Time ... |

77 | Drift analysis and average time complexity of evolutionary algorithms
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Citation Context ...ed the distribution of the first hitting time of the optimum for the CountOnes problem [9] and for long-path problems [8]. He and Yao derived bounds for the expected running time using drift analysis =-=[13]-=- and exact expressions for the first hitting times of population-based EAs directly from the transition matrix of the associated Markov chains [14]. In the multiobjective case, only few theoretical re... |

60 | Rigorous hitting times for binary mutations
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Citation Context ...cal methods from the field of randomized algorithms. Modeling the EA as a Markov process, Garnier et al. calculated the distribution of the first hitting time of the optimum for the CountOnes problem =-=[9]-=- and for long-path problems [8]. He and Yao derived bounds for the expected running time using drift analysis [13] and exact expressions for the first hitting times of population-based EAs directly fr... |

58 |
A rigorous complexity analysis of the (1+1) evolutionary algorithm for separable functions with Boolean inputs
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- 1998
(Show Context)
Citation Context ... algorithms many such results are contained in [8]. For the optimization of pseudoBoolean functions an extensive theory has been built up by Wegener et al., seese.g. [16], Droste, Jansen, and Wegener =-=[2, 3]-=-, or for a methodological overview [15]. Results on the running time of evolutionary algorithms in the multi-objective case are rare. Scharnow et al. [13] analyze a (1+1)-EA under multiple, nonconflic... |

57 | On a multi–objective evolutionary algorithm and its convergence to the Pareto set
- Rudolph
- 1998
(Show Context)
Citation Context ... approximate the set of Pareto-optimal (or non-dominated)solutions. Existing theoretic work on convergence in evolutionary multi-objective optimization has so far mainly dealt with the limit behavior =-=[9, 10, 12, 11, 4, 5, 14]-=-. Under appropriate conditions for the variation and the selection operators, global convergence to the Pareto set can be guaranteed in the limit. In addition to that, we are often interested in a qua... |

49 |
On the convergence of multiobjective evolutionary algorithms
- Hanne
- 1999
(Show Context)
Citation Context ... approximate the set of Pareto-optimal (or non-dominated)solutions. Existing theoretic work on convergence in evolutionary multi-objective optimization has so far mainly dealt with the limit behavior =-=[9, 10, 12, 11, 4, 5, 14]-=-. Under appropriate conditions for the variation and the selection operators, global convergence to the Pareto set can be guaranteed in the limit. In addition to that, we are often interested in a qua... |

39 |
Fitness landscapes based on sorting and shortest path problems
- Scharnow, Tinnefeld, et al.
- 2002
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Citation Context ... seese.g. [16], Droste, Jansen, and Wegener [2, 3], or for a methodological overview [15]. Results on the running time of evolutionary algorithms in the multi-objective case are rare. Scharnow et al. =-=[13]-=- analyze a (1+1)-EA under multiple, nonconflicting objectives. The purpose of this paper is to present a first analysis of different population-based multi-objective evolutionary algorithms (MOEAs)on ... |

39 | From an Individual to a Population: An Analysis of the First Hitting Time of Population-Based Evolutionary Algorithms
- He, Yao
- 2002
(Show Context)
Citation Context ...s for the expected running time using drift analysis [13] and exact expressions for the first hitting times of population-based EAs directly from the transition matrix of the associated Markov chains =-=[14]-=-. In the multiobjective case, only few theoretical results are available. Most of the corresponding studies were concerned with the limit behavior, i.e., the question whether the search algorithm conv... |

37 | Convergence Properties of Some Multi-Objective Evolutionary Algorithms
- Rudolph, Agapie
- 2000
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Citation Context |

35 | Theoretical aspects of evolutionary algorithms
- Wegener
- 2001
(Show Context)
Citation Context ... For single-objective evolutionary algorithms many such results are contained in [8]. For the optimization of pseudoBoolean functions an extensive theory has been built up by Wegener et al., seese.g. =-=[16]-=-, Droste, Jansen, and Wegener [2, 3], or for a methodological overview [15]. Results on the running time of evolutionary algorithms in the multi-objective case are rare. Scharnow et al. [13] analyze a... |

34 |
On a bicriterion formulation of the problems of integrated system identification and system optimization
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Citation Context ...n. In this study, we want to address this question by a theoretical analysis of the different approaches. As a representative for the single-objective multistart class, we use the ɛ-constraint method =-=[10]-=-. The ɛ-constraint method works by choosing one objective function as the only objective and the remaining objective functions as constraints. By a systematic variation of the constraint bounds, diffe... |

32 | Evolutionary search for minimal elements in partially ordered finite sets
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- 1998
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31 | Methods for the analysis of evolutionary algorithms on pseudoBoolean functions,” in Evolutionary Optimisation
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Citation Context ...ed in [8]. For the optimization of pseudoBoolean functions an extensive theory has been built up by Wegener et al., seese.g. [16], Droste, Jansen, and Wegener [2, 3], or for a methodological overview =-=[15]-=-. Results on the running time of evolutionary algorithms in the multi-objective case are rare. Scharnow et al. [13] analyze a (1+1)-EA under multiple, nonconflicting objectives. The purpose of this pa... |

25 | How to analyse evolutionary algorithms
- Beyer, Schwefel, et al.
- 2002
(Show Context)
Citation Context ..., but also more precise statements about the performance of EA variants and for better understanding the dynamics of EAs. A recent overview of the theoretical analysis of EAs is given by Beyer et al. =-=[2]-=-. A major part of this theory is the running time analysis, which addresses the question of how long a certain algorithm takes to find the optimal solution for a specific problem or a class of problem... |

20 | Evolutionary search under partially ordered sets,” Dept
- Rudolph
- 1999
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Citation Context |

18 | Archiving with guaranteed convergence and diversity in multi-objective optimization
- Laumanns, Thiele, et al.
- 2002
(Show Context)
Citation Context ...onding studies were concerned with the limit behavior, i.e., the question whether the search algorithm converges if the number of iterations goes to infinity [21], [22], [23], [24], [11], [12], [26], =-=[16]-=-. Only recently, Scharnow et al. [25] provided a running time analysis of a (1+1)-EA on the shortest path problem and showed that a multiobjective formulation of the problem can reduce the time to fin... |

15 | On the Convergence and Diversity-Preservation Properties of MultiObjective Evolutionary Algorithms. TIK-Report No. 108. Institut für Technische Informatik und Kommunikationsnetze - Laumanns - 2001 |

14 | Statistical distribution of the convergence time of evolutionary algorithms for long-path problems
- Garnier, Kallel
- 2000
(Show Context)
Citation Context ...andomized algorithms. Modeling the EA as a Markov process, Garnier et al. calculated the distribution of the first hitting time of the optimum for the CountOnes problem [9] and for long-path problems =-=[8]-=-. He and Yao derived bounds for the expected running time using drift analysis [13] and exact expressions for the first hitting times of population-based EAs directly from the transition matrix of the... |

8 |
Global multiobjective optimization with evolutionary algorithms: Selection mechanisms and mutation control
- Hanne
- 2001
(Show Context)
Citation Context |

2 |
A random process for searching a graph (comment),” Pers
- Alon
- 2002
(Show Context)
Citation Context ...nd to mutation probabilities. Instead of analyzing GEMO directly, we first define and analyze a more general randomized graph search algorithm, which is similar to a process described and analyzed in =-=[1]-=-. The purpose of this approach is two-fold. First, it gives a more intuitive view of how the different population-based algorithms behave on the Pareto front. Second, it provides a general tool, like ... |

2 |
How to analyze evolutionary algorithms,” Theor
- Beyer, Schwefel, et al.
- 2002
(Show Context)
Citation Context ..., but also more precise statements about the performance of EA variants and for better understanding the dynamics of EAs. A recent overview of the theoretical analysis of EAs is given by Beyer et al. =-=[2]-=-. A major part of this theory is the running time analysis, which addresses the question of how long a certain algorithm takes to find the optimal solution for a specific problem or a class of problem... |

1 |
the analysis of the (1+1) evolutionary algorithm,” Theor
- “On
- 2002
(Show Context)
Citation Context ...have been achieved regarding the optimization of pseudo-Boolean functions. Following first results by Mühlenbein [19] and Rudolph [20], a wide range of such problems was covered by Droste et al. [6], =-=[7]-=-, who successfully applied and considerably extended analytical methods from the field of randomized algorithms. Modeling the EA as a Markov process, Garnier et al. calculated the distribution of the ... |

1 |
multiobjective optimization with evolutionary algorithms: Selection mechanisms and mutation control
- “Global
(Show Context)
Citation Context ...s are available. Most of the corresponding studies were concerned with the limit behavior, i.e., the question whether the search algorithm converges if the number of iterations goes to infinity [11], =-=[12]-=-, [16], [21]–[24], [26]. Only recently, Scharnow et al. [25] provided a running time analysis of a (1+1)-EA on the shortest path problem and showed that a multiobjective formulation of the problem can... |