## Treating Constraints As Objectives For Single-Objective Evolutionary Optimization (1999)

Venue: | Engineering Optimization |

Citations: | 46 - 16 self |

### BibTeX

@ARTICLE{Coello99treatingconstraints,

author = {Carlos A. Coello Coello and Laboratorio Nacional},

title = {Treating Constraints As Objectives For Single-Objective Evolutionary Optimization},

journal = {Engineering Optimization},

year = {1999},

volume = {32},

pages = {275--308}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents a new approach to handle constraints using evolutionary algorithms. The new technique treats constraints as objectives, and uses a multiobjective optimization approach to solve the re-stated single-objective optimization problem. The new approach is compared against other numerical and evolutionary optimization techniques in several engineering optimization problems with different kinds of constraints. The results obtained show that the new approach can consistently outperform the other techniques using relatively small sub-populations, and without a significant sacrifice in terms of performance.

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Citation Context ...(x) g (x) 1 2 2 Sub-populations Old Sub-populations New m+1 3 1 2 1 2 3 m+1 genetic operators Apply g (x) m m Figure 1: Graphical representation of the approach introduced in this paper. Surry et al. =-=[25]-=- proposed the use of Pareto ranking [26] and VEGA [27] to handle constraints using this technique. In their approach, called COMOGA, the population was ranked based on constraint violations (counting ... |

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Citation Context ...that if i is feasible then Q i = 0. Ideally, the penalty should be kept as low as possible, just above the limit below which infeasible solutions are optimal (this is called, the minimum penalty rule =-=[13]-=-). However, although very simple, in practice it is quite difficult to implement this rule, because the exact location of the boundary between the feasible and infeasible regions is unknown in most pr... |

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Citation Context ...y the function with 3 inputs and 1 output shown in Table 3. To solve this problem, a bidimensional matrix representation of the circuit will be encoded in each chromosomic string as shown in Figure 4 =-=[29]-=-. In this matrix, each element is a gate (there are 5 types of gates: AND, NOT, OR, XOR and WIRE) that receives its 2 inputs from any gate at the previous column. More formally, we can say that the ci... |

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Citation Context ...45 (47) 27sx 5s45 (48) 6 4 2 360" 360" 360" 5 3 1 Figure 6: 10-bar plane truss used for Example No. 5. 4.5 Example 5 : Design of a 10-bar plane truss Consider the 10-bar plane truss sho=-=wn in Figure 6 [33]-=-. The problem is to find the cross-sectional area of each member of this truss, such that we minimize its weight, subject to stress and displacement constraints. The weight of the truss is given by: f... |

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Citation Context ...ch element can be different, thus the problem has 10 design variables. 5 COMPARISON OF RESULTS The genetic algorithm used for the experiments presented in this paper uses a fixed-point representation =-=[34, 35]-=-, according to which a chromosome is a string of the form hd 1 ; d 2 ; : : : ; dm i, where d 1 ; d 2 ; : : : ; dm are digits (numbers between zero and nine). Consider the examples shown in Figure 7, i... |

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Citation Context ...rent implementation was b = 5, as suggested by Michalewicz [8]. Binary tournament selection was used for all the examples presented next. 5.1 Example 1 This problem was solved before by Deb and Goyal =-=[37]-=- using GeneAS (Genetic Adaptive Search, which is a real-coded GA) and a traditional (binary) genetic algorithm (both approaches used a penalty function), and by Siddall [38] using ADRANS (Gall's adapt... |

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Citation Context ...ed are feasible). To solve this problem, it was necessary to add a module responsible for the analysis of the plane truss. This module uses the matrix factorization method included in Gere and Weaver =-=[43]-=- together with the stiffness method [43] to analyze the structure, and returns the values of the stress and displacement constraints, as well as the total weight of the structure. The solution shown f... |

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Citation Context ...problem for which no previous information is available. Another approach that does not seem to fit into any of the previous categories defined is tha mapping-based method proposed by Kim and Husbands =-=[23]-=-. This technique is a more theoretical approach that uses Riemann mappings to transform the feasible region into a shape that facilitates the search for the GA. The problem with this approach is that ... |

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Citation Context ...ct to the gates that are "next" to each other in the circuit, but without being necessarily connected. It is interesting to notice that if a row-order encoding is used, the problem becomes d=-=isruptive [30]-=-, making X Y Z F 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 Table 3: Truth table for the circuit of the third example. Input Output Figure 4: A gate in a two-dimensional template,... |

7 |
A Simple Genetic Algorithm for the Design of Reinforced Concrete
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(Show Context)
Citation Context ...ch element can be different, thus the problem has 10 design variables. 5 COMPARISON OF RESULTS The genetic algorithm used for the experiments presented in this paper uses a fixed-point representation =-=[34, 35]-=-, according to which a chromosome is a string of the form hd 1 ; d 2 ; : : : ; dm i, where d 1 ; d 2 ; : : : ; dm are digits (numbers between zero and nine). Consider the examples shown in Figure 7, i... |

6 |
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Citation Context ...ndom search with a penalty function). Their results were compared against those produced by the approach proposed in this paper, and are shown in Table 4. Notice that the solution reported by Siddall =-=[39]-=- is infeasible (it violates the first constraint). The solution shown for the technique proposed here is the best produced after 81 runs in which the crossover and mutation rates were iterated from 0:... |