Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and pro life techniques, where pro choice encompasses eliminating, decoding, and preserving, while prolife covers penalty based and repairing approaches. Overviews and comparisons published on evolutionary computation techniques for constraint handling so far mainly concern continuous domains, [29,30,32,34]. Constraint handling in continuous and discrete domains rely to a certain extent on the same ideas. There are, however, also differences, for instance in continuous domains constraints can be characterized as linear, non linear, etc. and in case of linear constraints special averaging ....

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....to be developed when the canonical representation is different from binary strings or real valued vectors. 2. Various constraints need to be taken into account by means of a suitable method (ranging from penalty functions to repair algorithms, constraint preserving operators, and decoders; see [169] for an overview) 3. Expert knowledge about the problem needs to be incorporated into the representation and the operators in order to guide the search process and increase its convergence velocity without running into the trap, however, to get confused and misled by expert beliefs and ....

Z. Michalewicz and M. Schoenauer, "Evolutionary algorithms for constrained parameter optimization problems," Evolutionary Computation, vol. 4, no. 1, pp. 1--32, 1996.

....Due to the complexity and unpredictability of nonlinear optimization, a general deterministic solution is impossible. This provides an opportunity for evolutionary algorithms. In recent years, several evolutionary algorithms have been proposed for nonlinear optimization problems. Michalewicz [6] provided an overview of these algorithms. In this paper, several evolutionary computation approaches were surveyed and a set of constrained numerical optimization test cases was provided. The key point in the constrained optimization process is to deal with the constraints. Many methods were ....

....one difference in the algorithm; instead of finding the gBest, each particle finds a neighborhood best (lBest) to update the new velocity. 3. EXPERIMENTAL DESIGN Twelve constrained numerical optimization problems were tested in the experiment. They were proposed by Michalewicz and Schoenauer [6]. These test cases include objective functions of various types with different types of constraints [5, 6] For detailed function information please refer to the references. Table 1: 12 constrained nonlinear optimization test cases (From Koziel, et al., 5] Func Dim. Type Relative size of ....

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Michalewicz, Z. and Schoenauer, M., "Evolutionary Algorithms for Constrained Parameter Optimization Problems," Evolutionary Computation, vol. 4, no. 1, pp. 1-32, 1996.

....and p is the number of equality constraints (in both cases, constraints could be linear or non linear) Only inequality constraints will be considered in this work. Although many constraint handling methods have been developed in the last few years for genetic algorithms (see for example [11]) most of them either require a large number of fitness function evaluations, complex encodings or mappings, or are limited to problems with certain (specific) characteristics. The aim of this work is to show that using concepts from multiobjective optimization [3] is possible to derive new ....

....tie break with accumulated constraint violation if (oldpop[candidate 1] sumviol oldpop[candidate 2] sumviol) end else if (flip(0. 5) pure probabilistic selection return(winner) 4 Comparison of Results To validate our approach, we have used the well known benchmark proposed in [11]. The specific test functions used are the following: i i n i n i i i ix x x 2 1 1 2 4 cos 2 cos (5) 0 75 . 0 1 = i i x x g ( 0 5 . 7 . 0 1 2 = n x x g where 20 = n , 10 0 i x ( n i , 1 L = 141 . 40792 293239 . 37 8356891 . 0 3578547 . 5 ....

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Michalewicz Z. and Schoenauer M. (1996) Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1--32.

....evolutionary algorithm (VEGA) in the mid eighties, there have been numerous authors who have worked along the same lines. Constraint handling with EA has usually been studied apart from multi objective optimization with, penalty methods, decoders and repair algorithms being the most commonly used [11]. These are, nevertheless, methods which are heavily dependent on the speci c problem of optimization. It is only in the last few years that greater interest has been awakened in the development of evolutionary techniques which enable optimization problems with constraints and multiple objectives ....

....operators proposed in the literature and with others, it was nally decided to use two cross types, uniform cross and arithmetical cross, and three types of mutation, uniform mutation, non uniform mutation and minimal mutation. The rst four have been studied and described in depth by other authors [11]. Minimal mutation causes a minimal change in the descendant as compared to the father, and it is especially appropriate in ne tuning real parameters. Hence it is the scheme for generating a new population in which the most innovative aspects of ENORA have their roots and which we describe ....

Michalewicz, Z., Schoenauer, M. (1996). Evolutionary Algorithms for constrained parameter optimization problems. Evolutionary Computation, vol. 4, no. 1, pp. 1-32.

.... representation building in the course of the search ( 21] 24] These approaches are related and have been coupled, like co evolution and penalty methods ( 13] and [30] or penalty and projection [16] Reviews on constraints handling in evolutionary optimization can be found in [17] and [18]. Among penalization strategies, one distinguishes static, dynamic and adaptive methods. Static penalties depend neither on the number of points sampled during the search nor on their performance ( 8] 13] Dynamic penalties ( 16] 10] are function of the number of points sampled while ....

....search which minimizes the penalized objective function, the best feasible point in terms of f p tends to x , therefore the curve drops to 0. The second test (from 2) has two variables and two constraints, that are reduced to one constraint through formulation. It is formulated as ([18]) min x1 ;x2 2[0:001;20] sin( 2 x1 ) sin(2 x2 ) 1 (x1 x2 ) such that g(x 1 ; x 2 ) max(g 1 (x 1 ; x 2 ) g 2 (x 1 ; x 2 ) 0 ; g 1 (x 1 ; x 2 ) x 1 x 2 1 ; g 2 (x 1 ; x 2 ) 1 x 1 (x 2 4) 28) Solutions of the primal and dual problems are : 1:228 ....

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Z. Michalewicz and M. Schoenauer, \Evolutionary algorithms for constrained parameter optimization", Evolutionary Computation, vol. 4, no. 1, pp. 1-32, 1997.

....ones, the resulting constraints have the same granularity and see http: www.wi.leidenuniv.nl home jvhemert csp ea therefore the same intrinsic difficulty. This rewriting of constraints, called constraint processing, is done in two steps: elimination of functional constraints (as in GENOCOP [16]) and decomposition of the CSP into primitive constraints. The choice of primitive constraints depends on the specification language. The primitive constraints chosen in the examples considered in [13] the N Queens Problem and the Five Houses Puzzle, are linear inequalities of the form: v i ....

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....constraint satisfaction problems but may not work well for constrained NLPs, because it does not consider the objective seriously. Besides, it calculates fitness based on historical records, making it easy to get stuck in local minima. All these methods have at least one of the following problems [129, 133, 130]: a) di#culty in finding feasible regions or in maintaining feasibility for nonlinear constraints, b) requiring specific domain knowledge or problem dependent genetic operators, and (c) tendency to get stuck at local minima. A series of software packages, GENOCOP I, II, and III [76] utilize ....

....(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, ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....thesis a framework [159] for solving constrained NLPs that unifies simulated annealing (SA) genetic algorithms (GA) and greedy searches in looking for saddle points. The framework allows us to show that many leading algorithms, such as DLM [169] CSA [162] and GA search of penalty formulations [117, 114] are similar algorithms that di#er only in some components of the framework. Based on the first order necessary and su#cient conditions in Theorem 1.1, Figure 1.1 depicts a general stochastic optimization procedure to look for SP dn . The procedure maintains a list of candidate points to be ....

.... dynamic penalty formulations in [94, 97, 114, 125, 115, 77, 32, 138, 122, 137, 42] Besides requiring domainspecific knowledge, most of these heuristics have di#culties either in finding feasible regions or in maintaining feasibility for nonlinear constraints, and get stuck easily in local minima [117, 114]. Some typical constraint handling techniques are explained next. Note that these techniques are all heuristic in nature. In general, methods based on dynamic penalty formulations can at best, but have no guarantee to, find CLM dn . Consider a simple problem with only one constraint function h 1 ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....of a solution. In this thesis I restrict the discussion to finding one solution. With this terminology, solving a CSP means finding one feasible element of the search space while solving a COP means finding a feasible and optimal element. Solving COPs by GAs is extensively treated in [21, 22] and [25], the present investigation concerns solving CSPs by GAs. In this thesis binary constraint satisfaction problems over finite domains are considered, this means that constraints act between pairs of variables. This is not restrictive however since any CSP can be reduced to a binary CSP by using a ....

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....and pro life techniques, where pro choice encompasses eliminating, decoding, and preserving, while pro life covers penalty based and repairing approaches. Overviews and comparisons published on evolutionary computation techniques for constraint handling so far mainly concern continuous domains, [29, 30, 31, 34]. Constraint handling in continuous and discrete domains rely to a certain extent on the same ideas. There are, however, also di erences, for instance in continuous domains constraints can be characterized as linear, non linear, etc. and in case of linear constraints special averaging ....

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Journal of Evolutionary Computation, 4(1):1-32, 1996.

....Repairing infeasible candidates requires a repair procedure that modi es a given chromosome such that it will not violate constraints. This technique is thus problem dependent but if a good repair procedure can be developed then it works well in practice, see for instance Section 4. 5 in [33] for a comparative case study. The preserving approach amounts to designing and applying problem speci c operators that do preserve the feasibility of parent chromosomes. Using such operators the search becomes quasi free, because the o spring remains in the feasible search space, if the parents ....

....problem formulation. Elements of the new search space S serve as inputs for a decoding procedure that creates feasible solutions, and it is assumed that a free (modulo preserving operators) search can be performed in S by an EA. For a nice illustration we refer again to Section 4. 5 in [33]. In case of indirect constraint handling the optimization objectives replacing the constraints are traditionally viewed as penalties for constraint violation, hence to be minimized. In general penalties are given for violated constraints although some (problem speci c) EA allocate penalties for ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1-32, 1996.

....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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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....formulations in [91, 96, 109, 125, 110, 73, 27, 142, 122, 141, 34] Most of these techniques require domain specific knowledge. The main di#culties of these heuristics are in finding feasible regions, maintaining feasibility for nonlinear constraints, or getting stuck easily in local minima [112, 109]. 18 In general, methods based on penalty formulations have no guarantee to find CLM dn . Consider a problem with only one constraint h 1 (x) and an objective f(x) If a penaltybased algorithm starts from x # and 1 (x # ) min x#N dn (x # )# x # h 1 (x) 0 and f(x # ) min x#N dn ....

....problems when penalties are not chosen properly. That is, the success of SA in constrained optimization depends heavily on the proper choice of penalties. Moreover, SA requires a very slow cooling schedule in order to converge to an optimal solution with high probabilities. Genetic algorithm (GA) [112, 108, 77, 62, 109, 136, 118, 123, 86, 58, 35] is a stochastic global optimization algorithm . It maintains a population of candidate points in each generation. In each generation, it uses some genetic operators, such as crossover and mutation, to generate new candidate points. All the old and new candidate points are then ranked according to ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996. 121

....are discussed in this section. 2.1 Penalty Formulations This approach first transforms (1) into an unconstrained optimization problem or a sequence of unconstrained problems, and then solves it by using existing unconstrained minimization methods. Many heuristics developed to handle constraints [7] are normally problem dependent, have di#culties in finding feasible regions or in maintaining feasibility, and get stuck easily in local minima. Static penalty formulations [3, 6] transform (1) into an unconstrained problem, min x L # (x, #) f(x) i (x) #m j max where # ....

....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; ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....2.1) We have also extended simulated annealing (SA) 9] and greedy search [10] to look for discreteneighborhood saddle points SP dn (Section 2. 2) At the same time, new problem dependent constraint handling heuristics have been developed in the GA community to handle nonlinear constraints [7] (Section 2.3) Up to now, there is no clear understanding on how to unify these algorithms into one that can be applied to find CGM dn for a wide range of problems. Based on our previous work, our goal in this paper is to develop an effective framework that unifies SA, GA, and greedy search for ....

.... b b b b b b b b b b b b 3000 4000 5000 6000 7000 PR (NgP ) b b b b b b b b b a) PR(NgP ) approaches one asymptotically b) Existence of absolute minimum NoptP in PR (NgP ) Figure 1: An example showing the application of CSAGA with P = 3 to solve a discretized version of G1 [7] (NoptP 2000) CGM dn is hit in any of the N probes: PR (N) 1 Gamma (1 Gamma p j ) where N 0: 2) Reachability can be maintained by reporting the best solution found by the algorithm when it stops. As an example, Figure 1a plots PR (N g P ) when CSAGA (see Section 3.2) was run under ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....metric may also be difficult to be determined depending on the problem considered. Anyway, the MPF method was successfully used in constrained optimization of real world problems, e.g. SS96] For a survey on constrained optimization using evolutionary algorithms, see Michalewicz and Schoenauer [MS96]. This paper uses a penalty function approach proposed for GAs by the second author [Deb98] This method uses two metrics. One is directly based on violated constraints obtained by simple addition and the other on the worst fitness value of feasible individuals entering the selection operator in ....

Z. Michalewicz and M. Schoenauer. Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1 32, 1996.

.... In the community of evolutionary computing, various evolutionary algorithms have been proposed for complex constraint problems, including non differentiable or discontinuous problems, and have been evaluated on various test cases such as problems G1 G11 proposed by Michalewicz and Schoenauer [Michalewicz and Schoenauer, 1996] . Although these test cases include objective functions of various types with various numbers of variables and different numbers and types of constraints, they failed to provide meaningful conclusion on what characteristics make a constraint problem difficult for an evolutionary technique, or in ....

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....is also based on dynamic penalties and was described by Michalewicz and Attia [10, 11] Adaptive transformation attempt to use the information from the search to adjust the control parameters. This is usually done by examining the tness of feasible and infeasible members in the current population [12]. The death penalty method just rejects infeasible individuals. In this method the initial population must be feasible. The method of superiority of feasible points was developed by Powell and Skolnick [13] and is based on a classical penalty approach with one notable exception. Each individual is ....

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1-32, 1996.

....formulations in [91, 96, 109, 125, 110, 73, 27, 142, 122, 141, 34] Most of these techniques require domain specific knowledge. The main di#culties of these heuristics are in finding feasible regions, maintaining feasibility for nonlinear constraints, or getting stuck easily in local minima [112, 109]. 18 In general, methods based on penalty formulations have no guarantee to find CLM dn . Consider a problem with only one constraint h 1 (x) and an objective f(x) If a penaltybased algorithm starts from x # and h 1 (x # ) min x#N dn (x # )# x # h 1 (x) 0 and f(x # ) min x#N ....

....problems when penalties are not chosen properly. That is, the success of SA in constrained optimization depends heavily on the proper choice of penalties. Moreover, SA requires a very slow cooling schedule in order to converge to an optimal solution with high probabilities. Genetic algorithm (GA) [112, 108, 77, 62, 109, 136, 118, 123, 86, 58, 35] is a stochastic global optimization algorithm . It maintains a population of candidate points in each generation. In each generation, it uses some genetic operators, such as crossover and mutation, to generate new candidate points. All the old and new candidate points are then ranked according to ....

[Article contains additional citation context not shown here]

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996. 121

....feasible solutions. It must be understood that penalty functions severely complicate the decision space of an application and will require an iterative approach to problem formulation to ensure that both the fitness standard deviation is sufficiently small and feasible solutions are being sought. Michalewicz Schoenauer (1996) and Hilton Culver (2000) provide a review and guidance for constraint handling methods for evolutionary computation applications. Proper constraint handling methods can reduce the overall computational complexity of solving an application by both reducing the required population size and ....

Michalewicz, Z. & Schoenauer, M. (1996) Evolutionary Algorithms for Constrained Parameter Optimization Problems, Evolutionary Computation, 4(1): 1-32.

....a Pentinum III 500 MHz computer with Solaris 7. #L d represents the number of L d (x; function evaluations. The best T ID (f ) for each problem is boxed. 77 5. 1 The coefficients of determination R 2 on linear fits of Q and log 2 (T opt (1; Q) The benchmarks evaluated are G1, G2 [117], Rastrigin, Floudas and Pardalos Problem 5.2 and 5.4 [65] 84 5.2 Summary of existing constrained NLP benchmarks. 92 5.3 Summary of existing constrained NLP solves. ....

....the existence of a minimum in N ff PR (N ff ) when CSA was applied to solve (1.7) Each cooling schedule is run 200 times. 70 xiv 4. 4 An example showing the existence of a minimum in Ng PR (Ng ) when CSAGA with a population size P = 3 was applied to solve G1 [117]. The experiments were run 200 times at each N g . 71 4.5 CSA ID : CSA with iterative deepening, called with desired solution quality Q. 72 4.6 CGA ID : CGA with iterative deepening, called with desired solution quality Q. 73 5.1 A ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....2.1) We have also extended simulated annealing (SA) 9] and greedy search [10] to look for discreteneighborhood saddle points SP dn (Section 2. 2) At the same time, new problem dependent constraint handling heuristics have been developed in the GA community to handle nonlinear constraints [7] (Section 2.3) Up to now, there is no clear understanding on how to unify these algorithms into one that can be applied to find CGM dn for a wide range of problems. Based on our previous work, our goal in this paper is to develop an effective framework that unifies SA, GA, and greedy search for ....

.... 6000 7000 0 1000 2000 3000 4000 5000 NgP PR (NgP ) N g P b b b b b b b b b b a) PR(NgP ) approaches one asymptotically b) Existence of absolute minimum NoptP in NgP PR (NgP ) Figure 1: An example showing the application of CSAGA with P = 3 to solve a discretized version of G1 [7] (NoptP 2000) CGM dn is hit in any of the N probes: PR (N) 1 Gamma N Y j=1 (1 Gamma p j ) where N 0: 2) Reachability can be maintained by reporting the best solution found by the algorithm when it stops. As an example, Figure 1a plots PR (N g P ) when CSAGA (see Section 3.2) was ....

[Article contains additional citation context not shown here]

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....2.1) We have also extended simulated annealing (SA) 13] and greedy search [14] to look for discrete neighborhood saddle points SP dn (Section 2. 2) At the same time, new problem dependent constraint handling heuristics have been developed in the GA community to handle nonlinear constraints [11] (Section 2.3) Up to now, there is no clear understanding on how to unify these algorithms into one that can be applied to find CGM dn for a wide range of problems. 1 For two vectors v and w of the same number of elements, v w means that each element of v is not less than the corresponding ....

....6000 7000 0 1000 2000 3000 4000 5000 NgP PR (NgP ) N g P b b b b b b b b b b a) PR(NgP ) approaches one asymptotically b) Existence of absolute minimum NoptP in NgP PR (NgP ) Figure 1.1. An example showing the application of CSAGA with P = 3 to solve a discretized version of G1 [11] (NoptP 2000) Based on our previous work, our goal in this chapter is to develop an effective framework that unifies SA, GA, and greedy search for finding CGM dn . In particular, we propose constrained genetic algorithm (CGA) and combined constrained SA and GA (CSAGA) that look for SP dn . We ....

[Article contains additional citation context not shown here]

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....are dealt with. Other approaches such as decoders or repair algorithms [6] also suffer from the disadvantage of being tailored to the specific problem and are not sufficiently general to handle a variety of problems. An overview of EA for constrained parameter optimization problems can be found in [7]. In this section we describe a problem independent evolutionary technique to solve the problem. Obviously, the main interest is in solving prob procedure EA begin initialize population evaluate population while (not termination condition) do begin generate new population evaluate ....

....evaluation of the first is smaller than the evaluation of the second. Many variation operators have been proposed during the last years . In the context in which we are here concerned, and after a long experimentation process, we finally decided to use uniform crossover and nonuniform mutation [7]. 3 Numerical Example We consider for the sake of illustration the following nonlinear fuzzy problem: Min 2x 2 1 2x 1 x 2 x 2 2 Gamma 10x 1 Gamma 10x 2 s:t: x 2 1 x 2 2 . 5 2x 1 x 2 . 5 with violations d 1 = 4, d 2 = 3, and x 1 ; x 2 2 [0; 5] Table 1 shows simulation ....

Michalewicz, Z., Schoenauer, M. (1996). Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, vol. 4, no. 1, pp. 1-32.

....require extensive experimentation for setting up appropriate parameters needed to define the penalty function. Michalewicz [6] describes the difficulties in each method and compares the performance of these algorithms on a number of test problems. In a similar study, Michalewicz and Schoenauer [7] concluded that the static penalty function method (without any sophistication) is a more robust approach than the sophisticated methods. This is because one such sophisticated method may work well on some problems but may not work so well in another problem. 2 In this paper, we develop a ....

....limit. In a later section, we shall revisit this welded beam design problem and show how the proposed constrained handling method finds solutions very close to the true optimum reliably and without the need of using any penalty parameter. Michalewicz [6] and later Michalewicz and Schoenauer [7] have discussed different constraint handling methods used in GAs. They have classified most of the evolutionary constraint handling methods into five categories: 1) methods based on preserving feasibility of solutions, 2) methods based on penalty functions, 3) methods making distinction ....

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Z. Michalewicz and M. Schoenauer, Evolutionary algorithms for constrained parameter optimization problems, Evolutionary Computation 4/1 (1996) 1--32.

....previous section. 3. 1 Constraint Manipulation Techniques In order to manipulate the restriction set there are several methods which can be grouped in three major categories: i) methods which preserve the feasibility of solutions [10] ii) methods based in penalty functions [16] 22] 6] 8][12][11] 17] and (iii) methods based in the search of feasible solutions[21] 15] Among these the method proposed in [10] is the only one with significant results when applied to high order problems. Studies conducted in [18] showed that the other two categories are perfectly suitable only when ....

Michalewicz, Z., and Schoenauer, M. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation Journal (1996). To appear in.

....produce an strong dependency betwen the problem and the EA. Homaifar et al. 1994) proposed multi stage optimization techniques based on penalty functions, which reduce the above mentioned drawbacks. An overview of EA based techniques for constrained parameter optimization problems is in Michalewicz and Schoenauer (1996). Mathematically, a general constrained multiobjective optimization problem can be written as follows: Minimize f i (x) i = 1; p subject to g j (x) 0 j = 1; m (1) where x = x 1 ; xn ) 2 IR n is an n dimensional vector, with l k x k u k (k = 1; n) i.e. ....

....Stochastic sampled with replacement (Goldberg, 1989) is used to produce copies of individuals for the next generation according to their selection probabilities. 2. 4 Variation operators We consider four variation operators (two crossovers and two mutations) which have been described and used by Michalewicz and Schoenauer (1996) for numerical optimization problems. 2 point crossover is directly designed from the well known bitstring crossover operators, except that the information exchanged between the two parents are real valued coordinates. With the arithmetical crossover, given two individuals X s = x s 1 ; ....

[Article contains additional citation context not shown here]

Michalewicz, Z., Schoenauer, M. (1996). Evolutionary Algorithms for constrained parameter optimization problems. Evolutionary Computation, vol. 4, no. 1, pp. 1-32.

....simulated annealing (SA) 7] and greedy search [9] to look for saddle points in discrete neighborhoods of x (SP dn ) Section 2. 2) At 1 the same time, new problem dependent constrainthandling heuristics have been developed in the genetic algorithm (GA) community to handle nonlinear constraints [5] (Section 2.3) Up to now, there is no clear understanding on whether these algorithms can be unified and which strategy is the most effective in finding CGM dn . Based on our previous work, we develop in this paper a framework that unifies SA, greedy search, and GA in looking for SP dn . We ....

.... that transform (3) into an unconstrained function F(x) consisting of a sum of the objective and the constraints weighted by penalties, and use GA to minimize 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 ....

[Article contains additional citation context not shown here]

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....of any sort (linear, non linear, equality or inequality) into the fitness function as to guide the search properly. The approach most commonly used to incorporate constraints is the penalty function, and there have been many successful applications of this approach reported in the literature [1, 15, 35, 31, 12]. However, penalty functions have some well known limitations [41] from which the most remarkable is the difficulty to define good penalty factors. These penalty factors are normally generated by trial and error, although their definition may severely affect the results produced by the GA [41] ....

Zbigniew Michalewicz and Marc Schoenauer. Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1--32, 1996.

....Design and Automation Journal, 887] Engineering with Computers, 621] Eur. J. Oper. Res. 558] Eur. J. Oper. Res. Netherlands) 75] European Journal of Operational Research, 35, 53, 79, 196, 210, 275, 285, 327, 365, 386, 408, 516, 599, 628, 740, 788] Evolutionary Computation, [109, 532, 596, 650, 744] Evolutionary Theory, 241] EvoNews, 123, 140, 173, 235, 266, 293, 379, 389, 400, 500, 506, 716, 839, 881] Exp. Fluids (Germany) 717] Expert Syst. Appl. UK) 590] Expert Syst. Appl. UK) 634] Expert Systems Applications (UK) 801] Flugwiss. Weltraumforsch. 605] Folding and Design, ....

....[209] Melis, Marcello, 293] Melis, M. 347] Melius, C. 706] Melssen, W. J. 47] Menozzi, Roberto, 294] Merelo, J. J. 295] Meservy, R. D. 167] Metzger, G. J. 296] Meyer, Jean Arcady, 643] Meza, J. C. 297] Michalewicz, M. 894] Michalewicz, Zbignew, 894] Michalewicz, Zbigniew, [298, 744, 844, 878] Michielssen, Eric, 292, 598, 895, 905] Mielonen, Matti, 299] Mierendorff, H. 300] Miescke, K. J. 780] Migowsky, S. 531] Miikkulainen, Risto, 308] Miller, J. F. 503, 771] Miller, John H. 198] Miller, S. T. 430] Ming, Lei, 301] Mitchell, Melanie, 302] Mitchell, R. J. 376] ....

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Zbigniew Michalewicz and Marc Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):?, Spring 1996. y[?] ga96bMichalewicz.

....is difficult to achieve in practice. If any of the unconstrained problems is not solved optimally, then the process is not guaranteed to find a CGM. In addition to penalty formulations, many heuristics have been developed to handle constraints. These include constraint handling techniques in GA [16], such as annealing penalties, adaptive penalties, preserving feasibility with specialized genetic operators, searching along the boundary of feasible regions, death penalty methods, behavioral memory with a linear order of constraints, repair of infeasible solutions, co evolutionary methods, and ....

....Journal on Artificial Intelligence Tools, June 2000 8 4 Experimental Results on Constrained NLPs In this section, we first evaluate various strategies to improve the performance of CSA in solving a set of continuous constrained NLPs. Then we compare it with evolutionary algorithms (EAs) [16, 14], a sequential quadratic programming (SQP) 5, 4, 21] package DONLP2 [22] as well as interval methods [8] We also apply CSA to solve derived discrete and mixed integer constrained NLPs that cannot be solved efficiently by existing methods. 4.1 Nonlinear Constrained Benchmarks We chose two sets ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....ff when CSA is applied to solve (6) The dotted line shows the trace taken in a run of CSAAT GammaI D . Table 2. The averages and standard deviations of coefficient of determination R 2 on linear fits of f 0 and log2 (N ff ) for fixed PR(N ff ) Benchmark Mean(R 2 ) Std. Dev. R 2 ) G1 [10] 0.9389 0.0384 G2 [10] 0.9532 0.0091 Rastrigin 0.9474 0.0397 Problem 5.2 [5] 0.9461 0.0342 fixed PR (N ff ) and a monotonically non decreasing relationship between PR (N ff ) and N ff at fixed f 0 . These observations lead to the following exponential model: N ff = ke Gammaaf 0 for fixed ....

....to solve (6) The dotted line shows the trace taken in a run of CSAAT GammaI D . Table 2. The averages and standard deviations of coefficient of determination R 2 on linear fits of f 0 and log2 (N ff ) for fixed PR(N ff ) Benchmark Mean(R 2 ) Std. Dev. R 2 ) G1 [10] 0.9389 0. 0384 G2 [10] 0.9532 0.0091 Rastrigin 0.9474 0.0397 Problem 5.2 [5] 0.9461 0.0342 fixed PR (N ff ) and a monotonically non decreasing relationship between PR (N ff ) and N ff at fixed f 0 . These observations lead to the following exponential model: N ff = ke Gammaaf 0 for fixed PR (N ff ) and ....

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

....Michalewicz, 1992) are global optimization methods that aim at complex objective functions and constraints. Most research into applications of EA to nonlinear programming problems has been concerned with complex objective functions but not with constraints, and only recently several approaches (Michalewicz and Schoenauer, 1996) have extended evolutionary techniques by some constraint handling methods. For particular constrained optimization problems, specialized EA have been developed by incorporating problem specific knowledge into the EA, e.g. the Transportation Problem (Jim enez and Cadenas, 1995; Jim enez and ....

....Other approaches such as decoders or repair algorithms (Michalewicz, 1992) also suffer from the disadvantage of being tailored to the specific problem and are not sufficiently general to handle a variety of problems. An overview of EA for constrained parameter optimization problems can be found in (Michalewicz and Schoenauer, 1996). With this background, we are interested in problem independent evolutionary techniques to solve general constrained optimization problems such as linear programming problems, nonlinear programming ones, integer programming ones, boolean programming ones, and mixed programming ones. Obviously, ....

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Michalewicz, Z., Schoenauer, M. (1996). Evolutionary Algorithms for constrained parameter optimization problems. Evolutionary Computation, vol. 4, no. 1, pp. 1-32.

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Z. Michalewicz and M. Schoenauer. Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1-32, 1996.

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4:1--32, 1996. 49

....by the FEM under loading i, and D lim its prescribed limit. Introducing the positive penalty parameters i , the tness function to minimize is W eight X i (D max D lim ) 1) However, adjusting i is not an easy task, and many speci c methods exist in Evolutionary Computation [40]. The adaptive penalty method used here updates the penalty parameter based upon global statistics of feasibility in the population. Its main goal is to explore the neighborhood of the boundary of the feasible region by trying to keep in the population individuals that are on both sides of that ....

Z. Michalewicz and M. Schoenauer. Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1-32, 1996.

....of variable length stuctures for parsimony pressure [144] it may provide a useful mechanism for increasing the performance of an evolutionary algorithm. When dealing with constrained optimization problems, penalty functions are often used. A common technique is the method of static penalties [92], which requires xed user supplied penalty parameters. The main reason for its wide spread use is that it is the simplest technique to implement: It requires only the straightforward modi cation of the evaluation function eval as follows: eval( x) f( x) W penalty( x) where f is the ....

....EAs applied to constrained problems and EAs operating on variable length representation in light of parsimony, for instance in GP. In both cases the definition of the evaluation function contains a penalty term. For constrained problems this term is to suppress constraint violations [91] 89] [92], in case of GP it represents a bias against growing tree size and depth [101] 122] 123] 144] Obviously, the amount of penalty can be di erent for di erent individuals, but if the penalty term itself is not varied along the evolution then we do not see these cases as examples of controlling ....

Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4:1-32, 1996.

....constraints where f , g i and h j are real valued functions on the search space S. The satisfaction of the set of constraints (g i , h j ) de nes the feasible region F . Though many speci c methods to handle such constrained problems within Evolutionary Algorithms have been proposed (see e.g. [10] for a survey) this paper will concentrate on the penalty function method. The idea of penalty methods is to penalize infeasible solutions by adding to the objective function f a positive quantity (when the goal is to minimize f) in order to decrease the quality of such infeasible individuals: ....

....a priori discusses their advantages. Section 4 is devoted to a detailed study of the behavior of the algorithm on two selected test cases. Finally, an extensive experimental study is presented in section 5, giving some results obtained by this technique on some reference test problems taken from [10, 6]. 2 Evolutionary penalty methods First note that the general penalty approach to handle constraints is not speci c to Evolutionary Computation, as any optimization method can be applied to the penalized objective function. For instance, the static methods reviewed here are very general. The ....

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Z. Michalewicz and M. Schoenauer. Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1-32, 1996.

....18.1, 19.9 and 21.6 for the 10 20, 20 40 and 40 80 meshes. 4 Adaptive penalty The problem at hand is a constrained problem (see section 2. 4) Constrained optimization has been recently a very active field in Evolutionary Computation, and many specific methods have been designed (see e.g. [30] for a survey) Moreover, it is clear from mechanical considerations that the solution lies on the boundary of the feasible domain at least for the continuous problem. Furthermore, specific methods exist to explore the boundary of the feasible domain when the constraint is know to be active at ....

Z. Michalewicz and M. Schoenauer. Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1--32, 1996.

....[5, 11] And, to a less extend, some works addressed the initialization issue [23, 12] But it is amazing to see that the design of the tness function is generally omitted from the discussions. One exception is given by the eld of constraint handling, for which a range of penalty methods (see [14] for a survey) can be seen as addressing the issue, though somehow indirectly: the tness function is modi ed in order to take into account the constraints, but the main objective function is itself hardly discussed. Of course, only real world problems make the issue of the objective function ....

....like the one of Figure 4: the idea would be to forbid velocity distribution that look too chaotic but roughness is not easy to measure. A rst could be to forbid too high discontinuities between adjacent Vorono cells. But the implementation of such constraint is not straightforward (see [14]) should one penalize high jumps in velocities with xed or dynamic penalties or repair bad individuals by modifying their genotype, Moreover, whatever the constraints on the genotype, there is absolutely no guarantee at all that other types of weirdos will not emerge, meeting these ....

Z. Michalewicz and M. Schoenauer. Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1-32, 1996.

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Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996.

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Michalewicz, Z., Schoenauer, M.: Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation 4 (1996) 1--32

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Z. Michalewicz and M. Schoenauer. Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation, 4(1):1--32, 1996.

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Michalewicz, Z., Schoenauer, M.: Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation 4 (1996) 1--32

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Z. Michalewicz and M. Schoenauer, "Evolutionary algorithms for constrained parameter optimization problems," Evol. Comput., vol. 4, no. 1, pp. 1--32, 1996.

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Michalewicz, Z. and Schoenauer, M. (1996), Evolutionary algorithms for constrained parameter optimization problems, Evolutionary Computation 4(1), 1--32.

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Michalewicz, Z. and Schoenauer, M. (1996). Evolutionary algorithms for constrained parameter optimization problems, Evolutionat?/ Computation, 4(1): pp. 1-32.

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Michalewicz, Z. and Schoenauer, M. (1996). Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1), 1--32.

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Michalewicz, Z. and Schoenauer, M. (1996). Evolutionary Algorithms for constrained parameter optimization problems. Evolutionary Computation, Vol.4, No1, pp. 1--32.

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