Z. Michalewicz & G. Nazhiyath, Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints, In D. B. Fogel (Ed.), Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647--651. IEEE Press (1995).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of the problem s objective function there is usually a number of constraints that must also be satisfied. GAs unfortunately do not incorporate a default mechanism for handling such constraints. This deficiency is alleviated by a number of methods proposed in the literature [5] 6] 7] [9], 11] 12] 13] The most important of these methods is the Penalty Assignment method [5] according to which penalty terms are added to the fitness function that depend on the constraint violation of the specific solution. In this way the invalid solutions are considered as valid but they are ....

Z. Michalewicz and G. Nazhiyath, "Genocop III: A Co-evolutionary Algorithm for Numerical Optimisation Problems with Nonlinear Constraints," in Proceedings of the 2nd IEEE International Conference on Evolutionary Computation, Vol. 2, Perth-Australia, 29 Nov. - 1 Dec. 1995, pp. 647-651.

....major problem is that it is di#cult to satisfy all the nonlinear constraints simultaneously, in the sense that enforcing the satisfaction of one constraint may cause other constraints to be violated. Furthermore, its results depend heavily on the order of constraints. A co evolutionary algorithm [146, 132] is based on a coevolutionary model 26 that utilizes two populations: one containing a set of constraints di#cult to be satisfied and the other composed of potential solutions to the problem. These two populations are competitively evolved in such a way that fitter constraints are violated by ....

Z. Michalewicz and G. Nazhiyath. GENOCOP III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. In Proc. of 2nd IEEE Int'l Conf. on Evolutionary Computation, pages 647--651, 1995.

....constraints. This ratio means that x is updated more often than #. In generating trial points in the x subspace, we have used a dynamically controlled neighborhood size in the SA part [162] based on the 1:1 ratio rule [56] whereas in the GA part, we have used the seven operators in Genocop III [116] and L d as our fitness function. In implementing CGA and CSAGA, we have used the default parameters of CSA [162] in the SA part and those of Genocop III [116] in the GA part. The generation of trial point # # in the # subspace is done by the following rule: j = # j r 1 # j where j = 1, ....

....size in the SA part [162] based on the 1:1 ratio rule [56] whereas in the GA part, we have used the seven operators in Genocop III [116] and L d as our fitness function. In implementing CGA and CSAGA, we have used the default parameters of CSA [162] in the SA part and those of Genocop III [116] in the GA part. The generation of trial point # # in the # subspace is done by the following rule: j = # j r 1 # j where j = 1, m. 4.2) Here, r 1 is randomly generated in [ 1 2, 1 2] if we choose to generate # probabilistically, and is randomly generated in [0, 1] if we choose to ....

[Article contains additional citation context not shown here]

Z. Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of IEEE International Conference on Evolutionary Computation, 2:647--651, 1995.

....requirement for each solver. We can see from the table that DONLP2 [150, 6] LANCELOT [52, 10, 102] LOQO [155, 14] MINOS [151, 15] KNITRO [44] SNOPT [71] FSQP [5] HQP OMUSES [2] and MOSEK [24, 13] are mainly used for solving continuous constrained NLPs with di#erentiable functions. Genocop [111, 9] and COBYLA2 [127] can solve constrained NLPs whose variables are continuous and whose functions are not di#erentiable or continuous. BARON [3, 137] BNB [1] MINLP BB [12] SBB [11] Mittlp [4] and AlphaEcp [170, 8] can solve all continuous, discrete and mixed integer constrained NLPs with ....

Z. Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of IEEE International Conference on Evolutionary Computation, 2:647--651, 1995.

....are used, a local minimum of an unconstrained penalty function is only a necessary but not a sufficient condition for the point to be a CLM dn of the original constrained NLP. Without a good method to select penalties, GA resorts to ad hoc tuning and has difficulty in achieving convergence [6]. Besides penalty methods, methods for handling nonlinear constraints directly have been studied. These include methods based on preserving feasibility with specialized genetic operators, methods searching along boundaries of feasible regions, methods based on decoders, repair of infeasible ....

....P is used. 4 Experimental Results We present in this section our experimental results in evaluating CSA ID , CGA ID and CSAGA ID on discrete constrained NLPs. In implementing CSA ID , CGA ID and CSAGA ID , we have used the defaualt parameters of CSA [9] in the SA part and those of Genocop III [6] in the GA part. In addition, for iterative deepening to work, we have set the following parameters: ae = 2, K = 3, N 0 = 10 Delta n v , and Nmax = 1:0 Theta 10 8 n v , where n v is the number of variables, and N 0 and Nmax are, respectively, initial and maximum number of probes. Based on the ....

Z. Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of IEEE International Conference on Evolutionary Computation, 2:647--651, 1995.

....requirement for each solver. We can see from the table that DONLP2 [150, 6] LANCELOT [52, 10, 102] LOQO [155, 14] MINOS [151, 15] KNITRO [44] SNOPT [71] FSQP [5] HQP OMUSES [2] and MOSEK [24, 13] are mainly used for solving continuous constrained NLPs with di#erentiable functions. Genocop [111, 9] and COBYLA2 [127] can solve constrained NLPs whose variables are continuous and whose functions are not di#erentiable or continuous. BARON [3, 137] BNB [1] MINLP BB [12] SBB [11] Mittlp [4] and AlphaEcp [170, 8] can solve all continuous, discrete and mixed integer constrained NLPs with ....

Z. Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of IEEE International Conference on Evolutionary Computation, 2:647--651, 1995.

....of constraints. This ratio means that x is updated more often than . In generating trial points in the x subspace, we have used a dynamically controlled neighborhood size in the SA part [162] based on the 1:1 ratio rule [56] whereas in the GA part, we have used the seven operators in Genocop III [116] and L d as our fitness function. In implementing CGA and CSAGA, we have used the default parameters of CSA [162] in the SA part and those of Genocop III [116] in the GA part. The generation of trial point 0 in the subspace is done by the following rule: 0 j = j r 1 OE j where j = 1; ....

....size in the SA part [162] based on the 1:1 ratio rule [56] whereas in the GA part, we have used the seven operators in Genocop III [116] and L d as our fitness function. In implementing CGA and CSAGA, we have used the default parameters of CSA [162] in the SA part and those of Genocop III [116] in the GA part. The generation of trial point 0 in the subspace is done by the following rule: 0 j = j r 1 OE j where j = 1; Delta Delta Delta ; m: 4.2) Here, r 1 is randomly generated in [ Gamma1=2; 1=2] if we choose to generate probabilistically, and is randomly generated in [0; ....

[Article contains additional citation context not shown here]

Z. Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of IEEE International Conference on Evolutionary Computation, 2:647--651, 1995.

....are used, a local minimum of an unconstrained penalty function is only a necessary but not a sufficient condition for the point to be a CLM dn of the original constrained NLP. Without a good method to select penalties, GA resorts to ad hoc tuning and has difficulty in achieving convergence [6]. Besides penalty methods, methods for handling nonlinear constraints directly have been studied. These include methods based on preserving feasibility with specialized genetic operators, methods searching along boundaries of feasible regions, methods based on decoders, repair of infeasible ....

....P is used. 4 Experimental Results We present in this section our experimental results in evaluating CSA ID , CGA ID and CSAGA ID on discrete constrained NLPs. In implementing CSA ID , CGA ID and CSAGA ID , we have used the defaualt parameters of CSA [9] in the SA part and those of Genocop III [6] in the GA part. In addition, for iterative deepening to work, we have set the following parameters: ae = 2, K = 3, N 0 = 10 Delta n v , and Nmax = 1:0 Theta 10 8 n v , where n v is the number of variables, and N 0 and Nmax are, respectively, initial and maximum number of probes. Based on the ....

Z. Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of IEEE International Conference on Evolutionary Computation, 2:647--651, 1995.

....of constraints. This ratio means that x is updated more often than . In generating trial points in the x subspace, we have used a dynamically controlled neighborhood size in the SA part [13] based on the 1:1 ratio rule [3] whereas in the GA part, we have used the seven operators in Genocop III [10] and L d as our fitness function. In implementing CSA ID , CGA ID and CSAGA ID , we have used the default parameters of CSA [13] in the SA part and those of Genocop III [10] in the GA part. The generation of trial point 0 in the subspace is done by the following rule: 0 j = j r 1 OE j ....

....in the SA part [13] based on the 1:1 ratio rule [3] whereas in the GA part, we have used the seven operators in Genocop III [10] and L d as our fitness function. In implementing CSA ID , CGA ID and CSAGA ID , we have used the default parameters of CSA [13] in the SA part and those of Genocop III [10] in the GA part. The generation of trial point 0 in the subspace is done by the following rule: 0 j = j r 1 OE j where j = 1; Delta Delta Delta ; m: 1.14) Here, r 1 is randomly generated in [ Gamma1=2; 1=2] if we choose to generate probabilistically, and is randomly generated in ....

Z. Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of IEEE International Conference on Evolutionary Computation, 2:647--651, 1995.

....using a conventional optimization technique its functionality would had been compromised by the quite diverse nature of the objective functions. If a non linear component would be present in the constraint set, the overall system could easily be adapted to this new situation using the proposal in [14]. In this way, the technique broached in this paper exhibits a flexibility otherwise achieved by traditional optimization methods. ACKNOWLEDGEMENTS The authors express their gratitude to Eng. Jos Luis Amaral (Portucel Viana) for the used plant data and information and to Dr. C. M. Fonseca for ....

Z. Michalewicz and G. Nazhiyath. Genocop III: A Co-evolutionary Algorithm for Numerical Optimization Problems with Nonlinear Constraints. In D. B. Fogel, editor, Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647-651, IEEE Press, 1995.

....infeasible solution in Lamarckian evolution, or the un repaired infeasible solution can be used for the reproduction process. There is no general guideline how to repair infeasible solutions. Most of the repair heuristics are problem dependent. The repairing process used in GENOCOP III system, [7], which is similar to the retraction operation from Complex Method, is an example of a problem independent repairing method that can be applied to various problems. Constraint handling with a Special Representation Some elaborate representation methods like [8] to make all the possible ....

Z. Michalewicz and G. Nazhiyath. Genocop III: A coevolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of the second IEEE ICEC, Perth, Australia, 1995

.... F(x) Examples of penalty formulations include static penalties, dynamic penalties, annealing penalties, and adaptive penalties [5] In general, these problemdependent methods may require extensive tuning and lack a strong mathematical foundation, making them hard to guarantee convergence [4]. In addition to penalty methods, other methods have been studied in GA for handling constraints. These include methods based on preserving feasibility with specialized genetic operators, methods searching along boundaries of feasible regions, methods based on decoders, repair of infeasible ....

....d (x; t) as the fitness function. Lines 6 9 search the x subspace by selecting from P(t) individuals to reproduce using genetic operators and by inserting the individuals generated into P(t) according to their fitness values. In our experiments, we have used the seven operators in Genocop III [4]. Line 10 updates according to the vector of maximum violations H(h; P(t) where the maximum violation of a constraint is evaluated over all the individuals in P(t) That is, H i (h; P(t) max x2P(t) H(h i (x) i = 1; 2; Delta Delta Delta ; m; 10) where h i (x) is the i th ....

Z. Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. Proceedings of IEEE International Conference on Evolutionary Computation, 2:647--651, 1995.

....proposed by Joines and Houck [68] the quality of the solution found was very sensitive to changes in the values of the parameters. Even when a certain set of values for these parameters (C = 0:5, fi = 1 or 2) were found by the authors of this method to be a reasonable choice, Michalewicz [84, 88] reported that these values produced premature convergence most of the time in other examples. Also, it was found that the technique normally either converged to an infeasible solution or to a feasible one that was far away from the global optimum [84, 23] Apparently, this technique provides very ....

....in which the feasible search space is convex and constitutes a reasonably large portion of the whole search space. This approach has the drawback of not exploiting any information from the infeasible points that might be generated by the evolutionary algorithm to guide the search. Michalewicz [84, 88, 89] has shown that the use of death penalty is inferior to the use of penalties that are defined in terms of the distance to the feasible region. Kuri s approach does not use information about the amount of constraint violation, but only about the number of constraints that were violated. Although ....

[Article contains additional citation context not shown here]

Zbigniew Michalewicz and G. Nazhiyath. Genocop III: A co-evolutionary algorithm for numerical optimization with nonlinear constraints. In David B. Fogel, editor, Proceedings of the Second IEEE International Conference on Evolutionary Computation, pages 647--651, Piscataway, New Jersey, 1995. IEEE Press.

....a number of ideas for using (in)feasible niche individuals which enable to explore new feasible areas and to make the population quickly to evolve towards a feasible global minimum is proposed. The performance of the EVOSLINOC can be successfully evaluated on many benchmark optimization problems [11, 10]. 1 Introduction 1.1 Optimization problem The general numerical scalar optimization problems with linear and nonlinear constraints can be formally stated as: min x f(x) where x = x 1 ; xn ) T 2 F S. The search space S is usually defined as a rectangle of the n dimensional space ....

....of the population for Test Case 1 after less than 10 generations; the population moves then into the feasible global minimum and concentrates on it in less than 400 generations. It is interesting to note that the ES is more robust and stable by finding the global feasible minimum than other tools [11, 2, 7]. 5.2 Test Case 2 The optimization problem (taken from [6, 11] is minimize f(x) Gammax 1 Gamma x 2 ; subject to nonlinear constraints: 2x 4 1 Gamma 8x 3 1 8x 2 1 2 Gamma x 2 0 4x 4 1 Gamma 32x 3 1 88x 2 1 Gamma 96x 1 36 Gamma x 2 and bounds: 0 x 1 3 and 0 x 2 4. ....

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Z. Michalewicz and C. Nazhiyath. Genocop III: A co--evolutionary algorithm for numerical optimization problems with nonlinear constraints. In D. B. Fogel (Ed.), Proceedings of the 2 th IEEE International Conference on Evolutionary Computation, pages 647--651, 1995.

....rule dictates that 5 of the infeasible individuals have their genetic representation updated to reflect the repaired feasible solution. This partial updating was shown on a set of combinatorial problems to be better than either no updating or always updating. However, Michalewicz and Nazhiyath [31] determined that a higher percentage 20 25 of update did better when using repair functions to solve continuous constrained nonlinear programming problems. Previous research has concentrated either on the comparison of pure Lamarckian and pure Baldwinian search, or the effectiveness of partial ....

Z. Michalewicz and G. Nazhiyath. Genocop iii: A co-evolutionary algorithm for numerical optimization problems with non-linear constraints. In The Second IEEE Conference on Evolutionary Computation, pages 647--651, Perth, Australia, December 1995.

....original domain, see Figure 1.c. They will be stacked together to make one domain, Figure 1.e, where the evolutionary search can work, and an inverse mapping to the original domains, Figure 1.a, will be used for tness calculation of each individual. 1 The repairing process used in GENOCOP III, [11], is an example of a problem independent repairing method, which is similar to the one from complex method, 14] g = f 1 f 1 f 1 1 g = f 1 Change of Coordinate Stacking a. Original e. Mapped Circles Domains Circles Polar Coordinate Domain b. Unit c. Adjusted d. Circles in Figure 1: Handling ....

Zbigniew Michalewicz and Girish Nazhiyath. Genocop III: A Co-evolutionary Algorithm for Numerical Optimization Problems with Nonlinear Constraints. In Proceedings of the second IEEE ICEC, Perth, Australia, 1995. http://www.coe.uncc.edu/ ~gnazhiya/gchome.html.

....for numerical optimization problems only special cases allowed the use of either specialized operators which preserve feasibility of solutions or repair algorithms, which attempt to convert an infeasible solution into feasible one. For example, a possible use of a repair algorithm was described in [6], but in that approach it was necessary to maintain two separate populations with feasible and infeasible solutions: a set of reference feasible points was used to repair infeasible points. Consequently, most evolutionary techniques for numerical optimization problems with constraints are based on ....

Michalewicz, Z. and G. Nazhiyath (1995). Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. In D. B. Fogel (Ed.), Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647--651. IEEE Press.

....constraint violations. However, normalization of constraints f j ( x) is suggested. On a number of test problems and on an engineering design problem, this approach is better able to nd constrained optimum solutions than Powell and Skolnick s approach. The third method (Genocop III) proposed in [26] is based on the idea of repairing infeasible individuals. Genocop III incorporates the original Genocop system, but also extends it by maintaining two separate populations, where a development in one population in uences evaluations of individuals in the other population. The rst population P s ....

Michalewicz, Z. and G. Nazhiyath (1995). Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. In D. B. Fogel (Ed.), Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647-651. IEEE Press.

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Z. Michalewicz and G. Nazhiyath. Genocop III: A Co-evolutionary Algorithm for Numerical Optimization Problems with Nonlinear Constraints. In [109], pp.647--651.

....set 45 parameters Clearly, the results are parameter dependent. It is quite likely that for a given problem there exist one optimal set of parameters for which the system returns feasible near optimum solution, however, it might be quite hard to find it. A limited set of experiments reported by Michalewicz (1995a) indicates that the method can provide good results if violation levels and penalty coefficients R ij are tuned to the problem. For example, Homaifar et al. 1994) experimented with the following problem (Himmelblau, 1992) Minimize a function of 5 variables: G4( x) 5:3578547x 2 3 ....

....but the sum of the violated constraints was quite small (10 Gamma4 ) On the other hand, the method seems to provide too strong penalties: often the factor (C Theta t) ff grows too fast to be useful. The system has little chances to escape from local optima: in most experiments reported by Michalewicz (1995a) the best individual was found in early generations. The method gave very good results for test cases where the objective functions were quadratic. 3.2.3 Method of annealing penalties This method, called Genocop II, is also based on dynamic penalties and was described by Michalewicz and Attia ....

[Article contains additional citation context not shown here]

Michalewicz, Z. and G. Nazhiyath (1995). Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. In D. B. Fogel (Ed.), Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647--651.

....during this phase. The number of genes to be mutated in a chromosome is controlled by , the mean for a Poisson distributed random number. Hence if we have n genes in a chromosomes, each with a 1=n probability of mutation, we would use = 1. We compare the results with those of Genocop [8] [12] and show it produces comparable or better results. For these experiments a population size of 100 was used, 50 of the new individuals were produced by crossover and 50 by mutation. It should be noted that most of the parameters of the EA were set to values which were thought to be reasonable , ....

Michalewicz, Z. and G. Nazhiyath (1995). Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints. In D. B. Fogel (Ed.), Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647--651. IEEE Press.

....some problems the process of repairing infeasible individuals might be as complex as solving the original problem. This is the case for the nonlinear transportation problem [33] most scheduling and timetable problems, and many others. On the other hand, the recently completed Genocop III system [38] for constrained numerical optimization (nonlinear constraints) is based on repair algorithms. Genocop III incorporates the original Genocop system [33] which handles linear constraints only) but also extends it by maintaining two separate populations, where a development in one population ....

....by specialized operators (as in Genocop) The second population, P r , consists of fully feasible reference points. These reference points, being feasible, are evaluated directly by the objective function, whereas search points are repaired for evaluation. The first results are very promising [38]. 2.5 Replacement of individuals by their repaired versions The question of replacing repaired individuals is related to so called Lamarckian evolution, which assumes that an individual improves during its lifetime and that the resulting improvements are coded back into the chromosome. Recently ....

Michalewicz, Z. and Nazhiyath, G., Genocop III: A Co-evolutionary Algorithm for Numerical Optimization Problems with Nonlinear Constraints, Proceedings of the Second IEEE ICEC, Perth, Australia, 29 November -- 1 December 1995.

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Z. Michalewicz & G. Nazhiyath, Genocop III: A co-evolutionary algorithm for numerical optimization problems with nonlinear constraints, In D. B. Fogel (Ed.), Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647--651. IEEE Press (1995).

No context found.

Michalewicz, Z. and G. Nazhiyath (1995). Genocop III: A coevolutionary algorithm for numerical optimization problems with nonlinear constraints. In D. B. Fogel (Eds.), Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647--651. IEEE Press.

No context found.

Z. Michalewicz , G. Nazhiyath, "Genocop III: A Co-Evolutionary Algorithm for Numerical Optimization Problems with Nonlinear Constraints", Proceedings of the 2nd IEEE International Conference on Evolutionary Computation, 1995, pp 647-651.

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