S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19--44, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....or unconstrained optimization problems with one or several objective functions. The main advantage of this approach is that it requires a very low number of fitness function evaluations (between 2 and 10 of the number of evaluations required by the homomorphous maps of Koziel and Michalewicz [15], which is one of the best constraint handling techniques known to date) The technique has some problems to reach the global optima, but it produces very good approximations considering its low computation cost. The main drawback of the approach is that its implementation is considerably more ....

....our results are compared against a technique called stochastic ranking [22] which is representative of the state of the art in constrained evolutionary optimization. This approach has been found to be equally good or even better in some cases than the homomorphous maps of Koziel and Michalewicz [15]. 5.1 Examples The following parameters were adopted for IS PAES in all the experiments reported next: 8 , 5 # # . The maximum number of fitness function evaluations was set to 350,000, which is the number of ....

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Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19-- 44, 1999.

....such that the above inequality holds. The global optimum of this problem is located at x = 5; 5; 5) where f(x ) 1. 6 Comparison of Results To have a better idea of the degree of diculty of each of the test functions selected, we computed the metric , suggested by Koziel Michalewicz [16] to determine how dicult is to reach the feasible region of a certain constrained optimization problem. We determined =j F j = j S j experimentally, by generating one million random points in the search space S and determining whether or not they lied or not within the feasible region F . Table ....

....Table 2: Comparison of the results for the three test functions selected. which the maximum number of generations was set to 10000 (i.e. tness function evaluations) The summary of results is shown in Table 2. We compared our results with the homomorphous mappings of Koziel Michalewicz (KM) [16], which is the best constraint handling technique known to date. Koziel Michalewicz [16] performed 70 20000 = tness function evaluations to solve each of the examples previously described. For the rst example, the best solution known is the global optimum. It can be seen that for our ....

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Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19-44, 1999.

....of 10 encounters by individual (half the population size) the maximum depth of the octree is equal to the number of decision variables of the problem. These parameters were derived empirically after numerous experiments. Our results are compared to the homomorphous maps of Koziel Michalewicz [12] (one of the best current constraint handling techniques for evolutionary algorithms known to date) in Table 1. The results of Koziel and Michalewicz were obtained with 1,400,000 tness function evaluations, whereas our approach required only 50,020 tness function evaluations. As can be seen in ....

....support from CONACyT through a scholarship to pursue graduate studies at the Com Table 1: Comparison of the results for the test functions selected from [18] using 2 trees. Our approach is called CAEP (Cultural Algorithms with Evolutionary Programming) and Koziel Michalewicz s approach [12] is denoted by KM. BEST RESULT MEAN RESULT WORST RESULT TF OPTIMAL CAEP KM CAEP KM CAEP KM g01 15 15.0 14.7864 13.7574 14.7082 12.0 14.6154 g04 30665.539 30665.539 30664.5 30665.539 30655.3 30665.537 30645.9 g08 0.095825 0.095825 0.095825 0.095825 0.0891568 0.095825 0.0291438 g12 1.000 1.000 ....

Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19-44, 1999.

.... Such a scheme is based on an idea proposed by Hinterding and Michalewicz [22] The main advantage of this approach is that it requires a very low number of tness function evaluations (between 2 and 10 of the number of evaluations required by the homomorphous maps of Koziel and Michalewicz [27], which is one of the best constraint handling techniques known to date) The technique has some problems to reach the global optima, but it produces very good approximations considering its low computation cost. The main drawback of the approach is that its implementation is considerably more ....

....will nevertheless be considered as additional constraints by the new approach) The moment of inertia of each element can be di erent, thus the problem has 10 design variables. To get a measure of the diculty of solving each of these problems, a metric (as suggested by Koziel and Michalewicz [27]) was computed using the following expression: jF j=jSj (52) where jF j is the number of feasible solutions and jSj is the total number of solutions randomly generated. In this work, S = 1; 000; 000 random solutions. The di erent values of for each of the functions chosen are shown in ....

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Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19-44, 1999.

....the same number of decodings d. Additionally, it is reasonable to request that (4) the 16 transformation T is computationally fast and (5) it has locality feature in the sense that small changes in the decoded solution result in small changes in the solution itself [34] Koziel and Michalewicz [91, 92] have recently proposed a homomorphous mapping between an n dimensional cube and a feasible search space (either convex or non convex) The main idea of this approach is to transform the original problem into another (topologically equivalent) function that is easier to optimize by the EA. Kim ....

....that is easier to optimize by the EA. Kim and Husbands [86, 87] had an earlier proposal of a similar approach that used Riemann mappings to transform the feasible region into a shape that facilitated the search for the EA. Despite the several advantages of Koziel and Michalewicz s approach [92], it also has some disadvantages [92] It uses an extra parameter v which has to be found empirically, performing a set of runs. Requires extra computational e ort because of the binary search required to nd the intersection of a line with the boundary of the feasible region (which is the ....

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Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19-44, 1999.

.... region [33, 13] There are also studies about using multiobjective concepts to handle constraints in EAs [22] These approaches find or approximate the optimal solution with less fitness function evaluations than other competitive approaches like the Homomorphous Maps of Koziel and Michalewicz [21]. Two of the most recent techniques to handle constraints in EAs found in the literature, the Stochastic Ranking by Runarsson Yao [28] and the Adaptive Segregational Contraint Handling Evolutionary Algorithm (ASCHEA) by Hamida Schoenauer [18, 19] are both based on an ES. The quality and ....

....will nevertheless be considered as additional constraints by the new approach) The moment of inertia of each element can be different, thus the problem has 10 design variables. To get a measure of the difficulty of solving each of these problems, a metric (as suggested by Koziel and Michalewicz [21]) was computed using the following expression: 40) where is the number of feasible solutions and is the total number of solutions randomly generated. In this work, 0( random solutions. The different values of for each of the functions chosen are ....

Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19--44, 1999.

....selected from [14] Best Result Mean Result Worst Result TF optimal CAEP KM CAEP KM CAEP KM G08 0.095825 0.095825 0.095825 0.09525552 0.0891568 0.0901302 0.0291438 G11 0.750 0.7402695 0.75 0.79299844 0.75 0.8380483 0. 75 Our results are compared to the homomorphous maps of Koziel Michalewicz [10] in Table 1. The results of Koziel and Michalewicz were obtained with 1,400,000 fitness function evaluations, whereas our approach required only 97,540 fitness function evaluations. Table 1 indicates that our approach produces very good results with respect to the homomorphous maps at a fraction ....

Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19-44, 1999.

....leads to scalability and aggregation problems. Moreover, the algorithm did not include any niching or diversification mechanism to ensure a uniform spread of points along the Pareto frontier that is required for multiobjective problems. There have also been attempts by Koziel and Michalewicz[21] to handle single objective constrained optimization problems through the use of homomorphous mapping. It is clear from the above discussion that a generic evolutionary algorithm for constrained single and multiobjective optimization should avoid aggregation and scaling of objectives and ....

Koziel, S. and Michalewicz, Z.: Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization, Evolutionary Computation, 7(1)(1999), 19-44.

....(3.12) in CSA, where # is a constant smaller than 1. In our experiments, we have used four cooling rates: # = 0.1, # = 0.5, # = 0.8, and # = 0.95. 4. 6 Selected Test Benchmarks To evaluate various strategies used in CSA, we have chosen 12 di#cult benchmark problems: G1, G2 and G5 from G1 G10 [133, 119], and 2.1, 2.7.5, 5.2, 5.4, 6.2, 6.4, 7.2, 7.3 and 7.4 from a collection of optimization benchmarks [68] The former were originally developed for testing and tuning various constraint handling techniques in evolutionary algorithms (EAs) while the latter were derived from practical applications, ....

....with respect to the whole search space varies from 0 to almost 100 , and the topologies of feasible regions are quite di#erent. Table 4.1 shows the statistics of these problems. Due to a lack of discrete and mixed integer benchmarks, we derive them from the two sets of continuous benchmarks [133, 119, 68] as follows. In generating a constrained MINLP, we assume that variables with odd indices are continuous and those with even indices are discrete. In discretizing continuous variable x i in range [l i , u i ] where l i and u i are lower and 86 upper bounds of x i , respectively, we force x i to ....

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S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19--44, 1999.

....in the Search Framework The purpose of this section is to evaluate the 24 possible combinations of strategies in the general framework in Figure 1.1, and determine the best combination for generating probes and for organizing candidates. Table 4. 2 shows the evaluation results on problem G2 [117, 105] using each of the 24 combinations of strategies in Figure 1.1 when SA, GA, or combined SA and GA was used to generate probes in the x subspace. We show the average time of 10 runs for each combination of strategies in order to reach two solution quality levels (1 or 10 worse than CGM dn ) We ....

....inserting candidates in both the x and # subspaces by annealing rules leads to good and stable performance. For this reason, we use this combination of strategies in our following experiments. Next, we show experimental results of evaluating our proposed algorithms on ten constrained NLPs G1 G10 [117, 105]. These problems have objective functions of various types (linear, quadratic, cubic, polynomial, and nonlinear) and constraints of linear inequalities, nonlinear equalities, and nonlinear inequalities. The number of variables is up to 20, and that of constraints, including simple bounds, is up to ....

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S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19--44, 1999.

....and dynamic weights on convergence time and solution quality from 20 randomly generated starting points for the discretized version of Problem 2.6 in [57] Weight w is the initial weight in the dynamic case. 90 xi NLPs derived from continuous constrained NLPs G1 G10 [135, 121]. All timing results in seconds were collected on a Pentinum III 500 MHz computer with Solaris 7. For all problems except G2, CSA was able to find the optimal solutions in the times reported. For G2, CSA has a 97 success ratio. stands for no solution found for the solution quality ....

....DLM runs. SR stands for success ratio of finding solutions with specified quality within 100 feasible DLM runs. 109 4. 3 Performance comparison of DLM General and CSA in solving continuous constrained NLPs: G1 G10 [135, 121]. All timing results in seconds were collected on a Pentinum III 500 MHz computer with Solaris 7. For all problems except G2, CSA was able to find the optimal solutions in the times reported. stands for no solution found for the solution quality specified within 100 feasible DLM runs. SR ....

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S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19--44, 1999.

....time to arrive at the global solution. 5 Experimental Results on Constrained Problems In this section, we apply CSA to solve general discrete, continuous, and mixed nonlinear constrained problems. Due to a lack of discrete mixed benchmarks, we derive them from some existing continuous benchmarks [7, 5] as follows. In generating a mixed problem, we assume that variables with odd indices are continuous and those with even indices are discrete. In discretizing continuous variable x i in the range [a i , b i ] if b i i 1, we force the variable to take values from the set A i = b i , s; ....

....in order to improve its chance of satisfaction. Note that w i is adjusted using T as a reference because constraint violations are expected to decrease when T decreases. 5. 2 Evaluation Results In this section, we show the results of applying CSA on 10 constrained optimization problems G1 G10 [7, 5] with objective functions of various types (linear, quadratic, cubic, polynomial, and nonlinear) and constraints of linear inequalities, nonlinear equalities, and nonlinear inequalities. The number of variables is up to 20, and that of constraints, including simple bounds, is up to 42. These ....

S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19--44, 1999.

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S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19--44, 1999.

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Koziel, S., Michalewicz, Z.: Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation 7 (1999) 19--44

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Koziel, S., Michalewicz, Z.: Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation 7 (1999) 19--44

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S. Koziel/ & Z. Michalewicz, Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization, Evolutionary Computation, vol. 7 (1), pp. 19-44 (1999).

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S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19--44, 1999.

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S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19--44, 1999.

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S. Koziel and Z. Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary

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S. Koziel and Z. Michalewicz. Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7(1):19 -- 44, 1999.

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Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19-- 44, 1999.

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Koziel,S. and Michalewicz,Z. Evolutionary Algorithms, Homomorphours Mappings, and Constrained Parameter Optimization, Evolutionary computation 7(1),1999,pp.19-44

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Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19-44, 1999.

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Koziel S, Michalewicz Z. Evolutionary algorithms, homomorphous mappings , and constrained parameter optimization. Evolutionary Computation, 1999, 7(1): 19-44

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Slawomir Koziel and Zbigniew Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1):19--44, 1999.

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