## Scalable Test Problems for Evolutionary Multi-Objective Optimization (2001)

Venue: | Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH |

Citations: | 94 - 17 self |

### BibTeX

@TECHREPORT{Deb01scalabletest,

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

title = {Scalable Test Problems for Evolutionary Multi-Objective Optimization},

institution = {Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

After adequately demonstrating the ability to solve di#erent two-objective optimization problems, multi-objective evolutionary algorithms (MOEAs) must now show their e#cacy in handling problems having more than two objectives. In this paper, we have suggested three di#erent approaches for systematically designing test problems for this purpose. The simplicity of construction, scalability to any number of decision variables and objectives, knowledge of exact shape and location of the resulting Pareto-optimal front, and introduction of controlled di#culties in both converging to the true Pareto-optimal front and maintaining a widely distributed set of solutions are the main features of the suggested test problems. Because of the above features, they should be found useful in various research activities on MOEAs, such as testing the performance of a new MOEA, comparing di#erent MOEAs, and better understanding of the working principles of MOEAs.

### Citations

1273 |
Multi-Objective Optimization using Evolutionary Algorithms
- Deb
- 2001
(Show Context)
Citation Context ...6) test problem VNT has a discrete set of Pareto-optimal fronts, but was designed 1 for three objectives only. Similar simplicity prevails in the existing constrained test problems (Veldhuizen, 1999; =-=Deb, 2001-=-). However, in 1999, the first author introduced a systematic procedure of designing test problems which are simple to construct and are scalable to the number of decision variables (Deb, 1999). In th... |

471 | A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II
- Deb, Agrawal, et al.
- 2002
(Show Context)
Citation Context ...isting MOEAs cannot be applied to problems having more than two objectives. Developers of the state-of-the-art MOEAs (such as PAES (Knowles and Corne, 1999), SPEA (Zitzler and Thiele, 1999), NSGA-II (=-=Deb et al., 2000-=-) and others) have all considered the scalability aspect while developing their algorithms. The domination principle, non-dominated sorting algorithms, elite-preserving and other EA operators can all ... |

469 | SPEA2: Improving the strength pareto evolutionary algorithm
- Zitzler, Laumanns, et al.
- 2001
(Show Context)
Citation Context ...ont, more computationally e#ective distribution metrics may be needed. Although a widely distributed set of solutions can be found, as using NSGA-II or SPEA2 (a modified version of SPEA suggested in (=-=Zitzler, Laumanns, and Thiele, 2001-=-)) shown in the next section, the obtained distribution can be far from being a uniformly distributed set of 100 points on the Pareto-optimal surface. The niching method are usually designed to attain... |

447 | Comparison of multiobjective evolutionary algorithms: Empirical results - Zitzler, Deb, et al. - 2000 |

412 | An overview of evolutionary algorithms in multiobjective optimization - Fonseca, Fleming - 1995 |

339 | Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications
- Zitzler
- 1999
(Show Context)
Citation Context ...esigned with respect to an arbitrary number of objectives and have been used already on 2, 3, and 4-objective knapsack problems (Zitzler and Thiele, 1999). The mutual non-covered hyper-volume metric (=-=Zitzler, 1999-=-), which is similar to the set coverage metric, is applicable in higher-objective problems as well. Nevertheless, the convergence to coverage issue may not become clear from one such metric. Furthermo... |

290 | Nonlinear Multiobjective Optimization - Miettinen - 1999 |

169 | Multi-objective genetic algorithms: problem difficulties and construction of test problems
- Deb
- 1999
(Show Context)
Citation Context ..., 1999; Deb, 2001). However, in 1999, the first author introduced a systematic procedure of designing test problems which are simple to construct and are scalable to the number of decision variables (=-=Deb, 1999-=-). In these problems, the exact shape and location of the Pareto-optimal solutions are also known. The basic construction used two functionals g and h # with non-overlapping sets of decision variables... |

122 |
The Pareto archived evolution strategy: A new baseline algorithm for multiobjective optimization
- Knowles, Corne
- 1999
(Show Context)
Citation Context ...re more than two objectives are used. This is not to say that the existing MOEAs cannot be applied to problems having more than two objectives. Developers of the state-of-the-art MOEAs (such as PAES (=-=Knowles and Corne, 1999-=-), SPEA (Zitzler and Thiele, 1999), NSGA-II (Deb et al., 2000) and others) have all considered the scalability aspect while developing their algorithms. The domination principle, non-dominated sorting... |

106 | A variant of evolution strategies for vector optimization - Kursawe - 1991 |

98 |
and Lamout, “Multiobjective evolutionary algorithms : analyzing the state of the art
- David, Veldhuizen, et al.
- 2000
(Show Context)
Citation Context ...es. Viennet's (1996) test problem VNT has a discrete set of Pareto-optimal fronts, but was designed 1 for three objectives only. Similar simplicity prevails in the existing constrained test problems (=-=Veldhuizen, 1999-=-; Deb, 2001). However, in 1999, the first author introduced a systematic procedure of designing test problems which are simple to construct and are scalable to the number of decision variables (Deb, 1... |

72 |
Some experiments in machine learning using vector evaluated genetic algorithms, Unpublished doctoral dissertation
- Schaffer
- 1984
(Show Context)
Citation Context ...fferent objective functions are simply used as different translations of a single objective function. For example, the problem SCH1 uses the following two single-objective functions for minimization (=-=Schaffer, 1984-=-): f1(x) =x 2 , f2(x) =(x − 2) 2 . 3sSince the optimum x ∗(1) for f1 is not the optimum for f2 and vice versa, the Pareto-optimal set consists of more than one solution, including the individual minim... |

52 | Fault Tolerant Design Using Single and Multicriteria Genetic Algorithm Optimization - Schott - 1995 |

45 | Multi-objective optimization by genetic algorithms: A review - Chen, Tamaki, et al. - 1996 |

26 | Constrained test problems for multi-objective evolutionary optimization - Deb, Pratap, et al. - 2001 |

20 |
GA-based decision support system for multi-criteria optimization
- Tanaka
- 1995
(Show Context)
Citation Context ...search space. This can be easily achieved by using non-linear functionals for f i with the decision variables. Interestingly, there exist twovariable and three-variable constrained test problems TNK (=-=Tanaka, 1995-=-) and Tamaki (1996) in the literature using the above concept. In this problem, only two objectives (with f i = x i ) and two constraints were used. The use of f i = x i made a uniform density of solu... |

16 | On the Convergence and Diversity-Preservation Properties of Multi-Objective Evolutionary Algorithms
- Laumanns, Thiele, et al.
- 2001
(Show Context)
Citation Context ...recognized that this feature of problems can cause MOEAs di#culty in finding the true Pareto-optimal solutions. However, in handling such problems, MOEAs with the newly-suggested #-dominance concept (=-=Laumanns et al., 2001-=-) intro24 f 1 f 2 f 3 A B Pareto-optimal line Figure 32: The shaded region is non-dominated with Pareto-optimal solutions A and B. duced by the authors may be found useful. However, it is worth highli... |

15 | Mutation control and convergence in evolutionary multi-objective optimization
- Laumanns, Rudolph, et al.
- 2001
(Show Context)
Citation Context ...[-1/ # n, 1/ # n] for all i. Veldhuizen (1999) lists a number of such test problems. It is interesting to note that such a construction procedure can be extended to higher-objective problems as well (=-=Laumanns, Rudolph, and Schwefel, 2001-=-). In a systematic procedure, each optimum may be assumed to lie on each of M (usuallysn) coordinate directions. However, the Pareto-optimal set resulting from such a construction depends on the chose... |

15 | Hybridization of a multiobjective genetic algorithm, a neural network and a classical optimizer for a complex design problem - Poloni, Giurgevich, et al. - 2000 |

13 |
Failure of Pareto-based MOEAs: Does non-dominated really mean near to optimal
- Ikeda, Kita, et al.
- 2001
(Show Context)
Citation Context ... note that this di#culty can also occur in problems having an M-dimensional Pareto-optimal front, as long as the Pareto-optimal surface is weakly non-dominated with adjoining surfaces. Another study (=-=Kokolo, Kita, and Kobayashi, 2001-=-) has also recognized that this feature of problems can cause MOEAs di#culty in finding the true Pareto-optimal solutions. However, in handling such problems, MOEAs with the newly-suggested #-dominanc... |

10 | Multicriteria Optimization Using A Genetic Algorithm for Determining A Pareto Set - Viennet, Marc, et al. - 1996 |

8 | Some experiments in machine learning using vector evaluated genetic algorithms - Schaer - 1984 |