A.Homaifar, C.X.Qi and S.H.Lai, "Constrained Optimization Via Genetic Algorithms", Simulation 62(4), 1994, pp.252-254  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(denoted by ) x P ) has been tackled in the strategies we selected. In what follows we denote Homaiffar s, Joines Houck s, Schoenauer Xanthaki s, Powell Skolnick s and Kuri s methods as methods H, J, S, P and K, respectively. 2. 1 Method H This strategy was originally described in [2]. It defines l penalty levels depending on the magnitude of the violation of the constraints. To define such levels it demands to define intervals for each of the violations and a penalty value for every interval. M x x H R M x m i i j i 1 2 , 0 (3) M is the set al..l feasible ....

Homaiffar A., Qi C. & Lai S., "Constrained Optimization Via Genetic Algorithms". Simulation, 62:4, pp. 242-254, 1994.

....g12, and g13 were not included in their studies. TABLE III COMPARISON BETWEEN OUR (INDICATED BY RY) AND KOZIEL AND MICHALEWICZ S (INDICATED BY KM [14] ALGORITHMS; FOR PROBLEM g13, THE RESULT WAS TAKEN FROM [15, METHOD 4] THE TWO VALUES IN THE MEAN COLUMN FOR PROBLEM g13 REPRESENT MEDIANS [10] found a similar solution to ours for g04 using a genetic algorithm. Unfortunately, that solution violated two constraints. Another similar solution was found by Colville [3] using a mathematical programming technique. However, it is unclear how those two techniques [10] 3] would perform on a ....

....g13 REPRESENT MEDIANS [10] found a similar solution to ours for g04 using a genetic algorithm. Unfortunately, that solution violated two constraints. Another similar solution was found by Colville [3] using a mathematical programming technique. However, it is unclear how those two techniques [10], 3] would perform on a larger set of benchmark functions as we used here. For problem g05, which involves equality constraints, the algorithm given in [14] did not provide quality results. Hence, no results were given in their paper. Our algorithm has found consistently feasible solutions. ....

A. Homaifar, S. H.-Y. Lai, and X. Qi, "Constrained optimization via genetic algorithms," Simulation, vol. 62, no. 4, pp. 242--254, 1994.

....at all the use of a penalty function. 5 2.1 Static Penalties Under this category, we consider approaches in which the penalty factors do not depend on the current generation number in any way, and therefore, remain constant during the entire evolutionary process. Homaifar, Lai and Qi [78] proposed an approach in which the user de nes several levels of violation, and a penalty coecient is chosen for each in such a way that the penalty coecient increases as we reach higher levels of violation. This approach starts with a random population of individuals (feasible or infeasible) ....

....higher levels of violation. This approach starts with a random population of individuals (feasible or infeasible) An individual is evaluated using [104] R k;i max [0; g i ( x) 2 (8) where R k;i are the penalty coecients used, m is total the number of constraints (Homaifar et al. [78] transformed equality constraints into inequality constraints) f( x) is the unpenalized objective function, and k = 1; 2; l, where l is the number of levels of violation de ned by the user. The idea of this approach is to balance individual constraints separately by de ning a di erent ....

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A. Homaifar, S. H. Y. Lai, and X. Qi. Constrained Optimization via Genetic Algorithms. Simulation, 62(4):242-254, 1994.

.... if 1 j q jh j (X)j; if q 1 j m: However, these methods differ in many important details, how the penalty function is designed and applied to unfeasible solutions; we discuss them briefly in turn; for a full discussion, see [10] One of the methods was proposed by Homaifar, Lai, and Qi [6]. The method assumes that for every constraint we establish a family of intervals which determine appropriate penalty coefficient. For each constraint it creates several levels ( of violation, and for each level of violation and for each constraint, it creates a penalty coefficient R ij (i = 1; ....

.... other system produced variety of results (between 680.642 and 689.660, see [10] Of course, all resulting points X were feasible, which was not the case with other systems (e.g. Genocop II produced a value of 18.917 for the problem G5, the systems based on the methods of Homaifar, Lai, and Qi [6] and Powell and Skolnick [15] gave results of 2282.723 and 2101.367, respectively, for the problem G2) Clearly, Genocop III is a promising tool for constrained nonlinear optimization problems. However, there are many issues which require further attention and experiments. These include ....

Homaifar, A., Lai, S. H.-Y., Qi, X., Constrained Optimization via Genetic Algorithms, Simulation, Vol.62, No.4, 1994, pp.242--254.

....cases proposed since then [3, 5, 19] Only recently several approaches were reported to handle general nonlinear programming problems. However, their description is supported by experimental evidence based on different test cases; some of them provide results for test cases with very few variables [11, 16, 7]; some of them did not use functions which have a closed form [14] Also, they differ in details (representation, selection method, operators and their frequencies, etc. so it is quite hard to make any comparisons. This paper surveys these methods (next Section) and provides five test cases ....

....function is designed and applied In the rest of the paper we assume minimization problems. to unfeasible solutions. In the following subsections we discuss them in turn; the methods are sorted in decreasing order of parameters they require. This method was proposed by Homaifar, Lai, and Qi [7]. The method assumes that for every constraint we establish a family of intervals which determine appropriate penalty coefficient. It works as follows: ffl for each constraint, create several ( levels of violation, ffl for each level of violation and for each constraint, create a penalty ....

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Homaifar, A., Lai, S. H.-Y., Qi, X., Constrained Optimization via Genetic Algorithms, Simulation, Vol.62, No.4, 1994, pp.242--254.

....de Rouen, INSA CNRS UMR 6138, Rouen, France. E mail: Rodolphe.Leriche insa rouen.fr . F. Guyon is with the Bio statistics Bio mathematics lab. Paris 7 univ. France. E mail: guyon urbb.jussieu.fr . Four types of methods for handling constraints exist: penalization of infeasible solutions([8], 11] 23] 25] 29] 2] 6] projection of infeasible solutions onto the feasible domain ( 20] 27] 28] co evolution of populations which together solve the constrained optimization problem ( 22] and constraints representation building in the course of the search ( 21] 24] These ....

....[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 adaptive penalties ( 2] 6] 23] 11] vary with points evaluations. Mixed approaches exist, e.g. in [29] Duality and related concepts such as Lagrange multipliers have yielded some of the most ecient ....

A. Homaifar, S. H.-Y. Lai, and X. Qi, \Constrained optimization via genetic algorithms", Simulation, vol. 62, no. 4, pp. 242 - 254, 1994.

....constraint in the following way: f j ( x) maxf0; g j ( x)g; if 1 j q jh j ( x)j; if q 1 j m: However, these methods di er in many important details, how the penalty function is designed and applied to infeasible solutions. For example, a method of static penalties was proposed [14]; it assumes that for every constraint we establish a family of intervals which determine appropriate penalty coecient. The method of dynamic penalties was examined [15] where individuals are evaluated (at the iteration t) by the following formula: eval( x) f( x) C t) where C, and ....

Homaifar, A., S. H.-Y. Lai, and X. Qi (1994). Constrained optimization via genetic algorithms. Simulation 62 (4), 242-254. xxii EVOLUTIONARY ALGORITHMS FOR ENGINEERING APPLICATIONS

....reaching every point from every other point in a search space. Therefore, these methods assure the reachability of optimal solutions, albeit asymptotic convergence. Some variants of penalty formulations have also been used in GAs to handle constraints. The method of multi level static penalties [101] divides constraint violations into levels, each of which has its own penalty values. This method is very problem dependent and cannot be generalized to other optimization problems. Generation based dynamic penalties [112] annealing penalties [130] and adaptive penalties [27, 89, 151] can be ....

A. Homaifar, S. H. Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62:242--254, 1994.

....to the final goal of finding CLM dn or CGM dn when penalties are not large enough. Approximations to the process that sacrifice global optimality of solutions have been developed [102, 111] Various constraint handling techniques have been developed based on 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 ....

....for the search to converge to a CGM dn of the original constrained problem. Otherwise, the search may end up finding only CLM dn or even infeasible solutions. Some variants of penalty formulations have been used in GAs to handle constraint. These include methods with multi level static penalties [94], generation based dynamic penalties [97] annealing penalties [114] and adaptive penalties [42, 84, 125] Although they di#er in their ways of modifying the penalties, all of them adjust penalties at the end of each generation, instead of when the unconstrained problem at previous penalty levels ....

A. Homaifar, S. H. Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62:242--254, 1994.

....to the final goal of finding CLM dn or CGM dn when penalties are not large enough. Approximations to the process that sacrifice the global optimality of solutions have been developed [117, 129] Various constraint handling techniques have been developed based on dynamic penalty formulations in [99, 113, 133, 148, 134, 76, 8, 170, 145, 169]. Besides requiring domainspecific knowledge, most of these heuristics have di#culties in finding feasible regions or in maintaining feasibility for nonlinear constraints and get stuck easily in local minima [135, 133] Some typical constraint handling techniques are explained next. Note that ....

A. Homaifar, S. H. Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62:242--254, 1994.

....ranges from two to ten and number of clusters ranges from two to nine. Note that we are encoding the centres of the clusters, which will be #oating point numbers, in the chromosomes. One way in which this could have been implemented is by performing real representation with a binary encoding [24]. However, in order to keep the mapping between the actual cluster centres and the encoded centres straight forward, for convenience we have implemented real coded GAs over here [3] In this context one may note the observations in Ref. 25] after they experimentally compared binary and #oating ....

A. Homaifar, S.H.Y. Lai, X. Qi, Constrained Optimization via genetic algorithms, Simulation 62 (1994) 242}254.

....of finding CLM dn or CGM dn since penalties in those subproblems are not large enough. Approximations to the process of sacrificing global optimality of solutions have been developed [98, 107] A variety of constraint handling techniques have been developed based on dynamicpenalty 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 ....

....order for a search to converge to a CGM dn of the original constrained problem. Otherwise, the search may end up with only CLM dn or even infeasible solutions. Some variants of penalty formulations have been applied in GAs to handle constraints, including methods with multilevel static penalties [91], generation based dynamic penalties [96] annealing penalties [109] and adaptive penalties [34, 82, 125] However, those methods still have di#culties in choosing appropriate penalties for di#erent kinds of constrained NLPs. Besides SA and GA, some other stochastic global optimization ....

A. Homaifar, S. H. Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62:242--254, 1994.

....they are considered as a good alternativefor solving CO problems. Promising results have been reported during the past few years and several variants of Genetic Algorithms (GA) 6] Evolutionary Programming [3] and Evolution Strategies (ES) 20] have been proposed to cope with the CO problem [7], 8] 12] 22] The most common approach for solving CO problems is the use of a penalty function. The constrained problem is transformed to an unconstrained one, by penalizing the constraints and building a single objective function, which in turn is minimized using an unconstrained ....

....the algorithm s current iteration number# and H(x)isapenalty factor, defined as H(x) m (q i (x) q i (x) fl(q i (x) # (4) where q i (x) maxf0#g i (x)g, i = 1#: #m. The function q i (x) is a relative violated function of the constraints# (q i (x) is a multi stage assignment function [7]# fl(q i (x) is the power of the penalty function# and g i (x) are the constraints described in Eq. 2) The functions h( andfl( are problem dependent. In our experiments, a non stationary multi stage assignment penalty function was used. Details concerning the penalty function used ....

Homaifar, A., Lai, A.H.--Y., Qi, X.: Constrained Optimization via Genetic Algorithms. Simulation 2(4) (1994) 242--254

....the EC approaches have used penalty and barrier function concepts to handle constraints. Several penalty functions based genetic algorithms appear in the literature, namely static, dynamic, annealing, adaptive, death penalties and superiority of feasible points. The method of static penalties [8] uses a family of intervals for every constraint that determines the appropriate penalty coecient. Joines and Houck [9] proposed dynamic penalties that vary with the generations. The method of annealing penalties, called Genocop II (for Genetic algorithms for Numerical Optimization of Constrained ....

A. Homaifar, S.H.-Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62(4):242-254, 1994. see also paper by Fogel the following year (same journal).

....of finding CLM dn or CGM dn since penalties in those subproblems are not large enough. Approximations to the process of sacrificing global optimality of solutions have been developed [98, 107] A variety of constraint handling techniques have been developed based on dynamicpenalty 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 ....

....order for a search to converge to a CGM dn of the original constrained problem. Otherwise, the search may end up with only CLM dn or even infeasible solutions. Some variants of penalty formulations have been applied in GAs to handle constraints, including methods with multilevel static penalties [91], generation based dynamic penalties [96] annealing penalties [109] and adaptive penalties [34, 82, 125] However, those methods still have di#culties in choosing appropriate penalties for di#erent kinds of constrained NLPs. 20 Besides SA and GA, some other stochastic global optimization ....

A. Homaifar, S. H. Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62:242--254, 1994.

....to the final goal of finding CLM dn or CGM dn when penalties are not large enough. Approximations to the process that sacrifice global optimality of solutions have been developed [102, 111] 20 Various constraint handling techniques have been developed based on dynamic penalty formulations in [94, 97, 114, 125, 115, 77, 32, 138, 122, 137, 42]. Besides requiring domainspecific knowledge, most of these heuristics have difficulties 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 ....

....for the search to converge to a CGM dn of the original constrained problem. Otherwise, the search may end up finding only CLM dn or even infeasible solutions. Some variants of penalty formulations have been used in GAs to handle constraint. These include methods with multi level static penalties [94], generation based dynamic penalties [97] annealing penalties [114] and adaptive penalties [42, 84, 125] Although they differ in their ways of modifying the penalties, all of them adjust penalties at the end of each generation, instead of when the unconstrained problem at previous penalty ....

A. Homaifar, S. H. Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62:242--254, 1994.

....focus on the first evolutionary approach, and we will consider the second one as future works. Problems as in (4) for a given value ff p of the parameter ff, are nonlinear constrained optimization problems. The most usual technique in EA based constrained optimization is the penalty method [4], in which a constrained problem is transformed into a unconstrained one by associating a cost or penalty with all the constraint violations. The success of this approach depends on the way in which the penalties are dealt with. Other approaches such as decoders or repair algorithms [6] also ....

Homaifar, A., Qi, C.X., Lai, S.H. (1994). Constrained optimization via genetic algorithms. Simulation, vol. 62, no. 4, pp. 242254.

....function approach involves a number of penalty parameters which must be set right in any problem to obtain feasible solutions. This dependency of GA s performance on penalty parameters has led researchers to devise sophisticated penalty function approaches such as multi level penalty functions [3], dynamic penalty functions [4] and penalty functions involving temperaturebased evolution of penalty parameters with repair operators [5] All these approaches require extensive experimentation for setting up appropriate parameters needed to define the penalty function. Michalewicz [6] ....

....of R j (or R) to find what value would steer the search towards the feasible region. This requires extensive experimentation to find any reasonable solution. This problem is so severe that some researchers have used different values of R j (or R) depending on the level of constraint violation [3], and some have used sophisticated temperature based evolution of penalty parameters through generations [5] involving a few parameters describing the rate of evolution. 2) The inclusion of the penalty term distorts the objective function [1] For small values of R j (or R) the distortion is ....

[Article contains additional citation context not shown here]

A. Homaifar, S. H.-V. Lai, X. Qi, Constrained optimization via genetic algorithms. Simulation 62/4 (1994) 242--254.

....constraint in the following way: f j ( x) maxf0; g j ( x)g; if 1 j q jh j ( x)j; if q 1 j m: However, these methods di er in many important details, how the penalty function is designed and applied to infeasible solutions. For example, a method of static penalties was proposed [14]; it assumes that for every constraint we establish a family of intervals which determine appropriate penalty coecient. The method of dynamic penalties was examined [15] where individuals are evaluated (at the iteration t) by the following formula: eval( x) f( x) C t) P m j=1 f j ....

Homaifar, A., S. H.-Y. Lai, and X. Qi (1994). Constrained optimization via genetic algorithms. Simulation 62 (4), 242-254.

....in the 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 ....

Homaifar, A., Qi, C., and Lai, S. Constrained optimization via genetic algorithms. Simulation 62, 4 (Apr. 1994), 242--253.

....a solution better than the best o ered in [14] For problem g04, 30664:5 was reported as being by far the best value reported by any evolutionary system for this test case [14] Our algorithm has now improved this record substantially by nding the optimum consistently. 6 Homaifar et al. [10] found a similar solution to ours for g04 using a genetic algorithm. Unfortunately that solution violated two constraints. Another similar solution was found by Colville [3] using 5 The minus sign was added to the average result because we transformed the maximization problem into the ....

....7372:613 7559:192 5:3E 02 8835:655 642 g11 0:750 0:750 0:750 0:750 8:0E 05 0:750 57 g12 1:000000 1:000000 1:000000 1:000000 0:0E 00 1:000000 82 g13 0:053950 0:053957 0:057006 0:067543 3:1E 02 0:216915 349 a mathematical programming technique. However, it is unclear how those two techniques [10], 3] would perform on a larger set of benchmark functions as we used here. For problem g05 which involves equality constraints, the algorithm given in [14] did not provide quality results. Hence no results were given in their paper. Our algorithm has found consistently feasible solutions. Some ....

A. Homaifar, S.H.-Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62(4):242-254, 1994.

....rejection takes place in the selection step) it can be viewed as a penalty method with in nite penalty [1] This method might have great diculties to nd even a rst feasible point. The static penalty methods use user de ned values for the penalty coef cients. Some improvement was brought in [4] by increasing these values with the level of violation but the main diculty results from the lack of any hint about the value these penalty coecient should have. 2.2 Dynamic methods In these methods, the penalty coecients are modi ed along evolution according to a user de ned schedule ....

A. Homaifar, S. H.-Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62(4):242-254, 1994.

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A.Homaifar, C.X.Qi and S.H.Lai, "Constrained Optimization Via Genetic Algorithms", Simulation 62(4), 1994, pp.252-254

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A. Homaifar, S.H.-Y. Lai, and X. Qi. Constrained optimization via genetic algorithms. Simulation, 62(4):242--254, 1994.

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Homaifar, A., S. H.-Y. Lai, and X. Qi (1994). Constrained optimization via genetic algorithms. Simulation 62(4), pp. 242--254.

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