K. Deb, J. Horn, and D.E. Goldberg. Multimodal deceptive function. Complex Systems, 7:131--153, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to be solved provide misleading information. In this chapter, we will present an extension of the pcBHS model to cater for these two cases byintroducing redundancy and competition. 5.1 Introduction 5.1. 1 The challenge of deceptiveness Deception has been discussed rigorously in GA community[24,15,60,61,14,47, 25, 52] in the last decade. Briefl , deceptive problems contain deceptive attractors which mislead the algorithm to search for sub optima. Figure 5.1 shows a typical fully deceptive function for simple GAs on a maximization problem 1 . As shown in the figure, the global optimum and the suboptimum are ....

....and the suboptimum are located far apart with a big valley in between. The basin of attraction favoring for the suboptimum is much larger than the one favoring for the global one, making the problem deceptive. Intensive analysis and the definition of deception in the context of GA can be found in [15, 60]. Qualitatively, the degree of deception varies according to the comparative size of the global attractor and the sub optimal attractors. Full deception occurs when the size of the global attractor approaches zero (one global optimum only) when comparing with that of the sub optimal, while the ....

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K. Deb, J. Horn, and D.E. Goldberg. Multimodal deceptive function. Complex Systems, 7:131--153, 1993.

....to a common population. 1 Introduction There are various, different types of genetic algorithm (GA) difficulty in optimization problems which complicate the solution of those problems by canonical standard GAs. Examples for GA difficulty are deceptiveness (Das Whitley, 1991) multimodality (Deb, Horn, Goldberg, 1993), and noise (Kargupta Goldberg, 1994) A different type of GA difficulty is epistasis where random and highly ordered epistatic interactions are distinguished. In the first kind of epistasis a linkage between two genes has a random influence on the fitness like it is defined in the NK landscape ....

Deb, K., Horn, J., & Goldberg, D. E. (1993). Multimodal deceptive functions. Complex Systems, 7, 131--153.

.... Engineering, Tokushima University (Japan) 928] BYTE, 625] Chemical Physics Letters, 1076] Chemometrics and Intelligent Laboratory Systems, 460, 461, 462, 639, 643, 644] Chromatographia, 645] Chung Kuo Chi Hsueh Kung Ch eng Hsueh Pao, 1018] Clinical Chemistry, 456] Complex Systems, [381, 673, 736, 925] Computer Aided Design, 353] Computer Music Journal, 478] Computers and Geotechnics, 938] Computers Industrial Engineering, 197] Computers Mathematics with Applications, 60, 91, 260, 517] Computers Operations Research, 624, 945] Cryptologia, 689, 954, 956] Current Opinion in ....

....S. 198] Das, Rajarshi, 1065] Dasgupta, Dipankar, 199, 200, 201, 704, 705, 706, 707, 708, 709] Dastidar, D. Ghosh, 202] David, E. 963] Authors 17 Davidge, Robert, 203] Davidor, Yuval, 204, 205] Davis, Lawrence, 206, 207, 208, 209, 210] Davis, Thomas Elder, 973] Deb, Kalyanmoy, [217, 375, 376, 378, 381] Deboeck, Guido, 218, 219] deFigueiredo, Rui J. P. 944] Delaney, B. 239] Denham, M. J. 797] Deodhar, D. 538] Deugo, Dwight, 220, 221] Dhawan, Atam P. 222, 223, 657, 806] Dike, B. A. 834, 837] Dissanayake, M. W. M. G. 1072] Diver, D. A. 224] Dix, T. I. 815] Dobnikar, ....

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Kalyanmoy Deb, Jeffrey Horn, and David E. Goldberg. Multimodal deceptive functions. Complex Systems, 7(2):131--153, April 1993. ga:Goldberg93f.

....of problems singled out by such research are known as GA hard problems. In this paper, the GA hard problems are considered to be types of problem which the simple GA can t easily solve, and include non stationary problems, problems which have an enormous number of local optima, deceptive problems [Deb93], and so on. Whether they are GA hard or not, they may be optimized if there is some prior knowledge, but this situation never occurs as it is not necessary to use GAs for problems whose characteristics are already known. The reason for difficulties seems to be that, for the convergence of ....

Deb, K., Horn, J. and Goldberg, D. E., Multimodal Deceptive Functions, Complex Systems, vol. 7, no. 2, pp. 131-153, Complex Systems, (1993).

.... attraction of peaks and regions in the landscape to the GA s crossover operator is largely based on Holland s schema theorem (Holland, 1992) and schema average fitness calculations (Bethke, 1981) Goldberg (1987, 1989a, 1989b, 1989c) and later others (Whitley, 1991; Homaifar, Qi, Frost, 1991; Deb, Horn, Goldberg, 1993) defined and constructed deceptive landscapes, in which the GA should be attracted to suboptimal local optima and led away from the global optimum. Schema analysis has also been used to construct GA easy functions (Wilson, 1991) which have large basins of attraction for the global optimum 16 . ....

.... Gamma o Delta , at each order o min(k 1 ; k 2 ) where k i are the orders of bounded deception for two functions. Goldberg, Deb, and Horn found another way to construct and order partially deceptive functions without losing essential misleadingness (Goldberg, Deb, Horn, 1992; Deb, Horn, Goldberg, 1993). They constrain a bipolar deceptive function to be a function of folded unitation u fld (s) which is simply the number of ones minus the number of zeros: u fld (s) u(s) Gamma ( Gamma u(s) 2 u(s) Gamma . This leads to a symmetric function of unitation, f bip (u fld (s) Enforcing ....

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Deb, K., Horn, J., & Goldberg, D. E. (1993). Multimodal deceptive functions. Complex Systems, 7, 131--153.

....(Smith, Forrest, Perelson, 1992 ) have been gaining attention for maintaining multiple solutions. In this paper, we limit our study to fitness sharing, introduced by Goldberg and Richardson (1987) studied in detail in (Deb, 1989) and challenged by a massively multimodal problem in (Goldberg, Deb, Horn, 1992). Fitness sharing accomplishes niching by degrading the objective fitness (i.e. the unshared fitness) of an individual according to the presence of nearby (similar) individuals. Thus this type of niching requires a distance metric on the phenotype or genotype of the individuals. In this study, ....

....20 SigmaShare = 1.01 5 10 15 20 5 i j 10 15 20 SigmaShare = 1.30 Figure 4: Transition Matrices for Overlapping Niches. 3.2 OVERLAPPING NICHES It is often the case that niches overlap. In fitness sharing, we easily might estimate oe sh to be too large, sometimes on purpose (Goldberg, Deb, and Horn, 1992). In niched GAs and in life, while two species compete to fill a niche, the niches centered on them overlap. In this section, we examine the general case where oe sh 1. The niche count for ones, m 1 , is now i (N Gamma i) 1 Gamma 1=oe sh ) instead of just i, since we have to add in the ....

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Deb, K., Horn, J., & Goldberg, D. E. (1992). Multimodal deceptive functions. IlliGAL Report No. 92006 (April 1992), and submitted to Complex Systems (August 1992).

....Convergence to the true (or global) Pareto optimal front may not occur because of various features that may be present in a problem: 1. Multi modality, 2. Deception, 3. Isolated optimum, and 4. Collateral noise. All the above features are known to cause difficulty in single objective GAs (Deb et al. 1993) and, when present in a multi objective problem, may also cause difficulty for a multiobjective GA. In tackling a multi objective problem having multiple Pareto optimal fronts, a GA, like many other search and optimization methods, may converge to a local Pareto optimal front. Later, we create a ....

Deb, K., Horn, J. and Goldberg, D. E. (1993). Multi-Modal deceptive functions. Complex Systems, 7:131--153.

....front. The first task is a natural goal of any optimization algorithm. The second task is unique to multi criterion optimization 1 . Since no one solution in the 1 In multi modal optimization problems, often, the goal is also to find multiple global optimal solutions simultaneously [14, 22]. CLASSICAL METHODS v Pareto optimal set can be said to be better than the other, what an algorithm can do best is to find as many different Pareto optimal solutions as possible. We now review a couple of popular classical search and optimization methods briefly and discuss why there is a need ....

Deb, K., Horn, J., and Goldberg, D. E. (1993). Multi-Modal deceptive functions. Complex Systems, 7, 131--153.

....get stuck to a local Pareto optimal front. Later, we create a multimodal multi objective problem and show that a multi objective GA can get stuck at a local Pareto optimal front, if appropriate GA parameters are not used. Deception is a well known phenomenon in the studies of genetic algorithms (Deb and Goldberg, 1993; Goldberg 1989; Whitley, 1990) Deceptive functions cause GAs to get misled towards deceptive attractors. There is a difference between the difficulties caused by multi modality and by deception. For deception to take place, it is necessary to have at least two optima in the search space (a true ....

....almost the entire search space favors the deceptive (non global) optimum, whereas multimodality may cause difficulty to a GA, merely because of the sheer number of different optima where a GA can get stuck to. There even exists a study where both multi modality and deception coexist in a function (Deb, Horn, and Goldberg, 1993), thereby making these so called massively multi modal deceptive problems even harder to solve using GAs. We shall show how the concepts of single objective deceptive functions can be used to create multi objective deceptive problems, which are also difficult to solve using multi objective GAs. ....

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Deb, K., Horn, J., and Goldberg, D. E. (1993). Multi-Modal deceptive functions. Complex Systems, 7, 131--153.

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Deb, K., Horn, J., & Goldberg, D. E. (1993). Multimodal deceptive functions. Complex Systems, 7, 131-153.

....immune system models (Smith, Forrest, Perelson, 1992 ) have been gaining attention as for maintaining multiple solutions. In this paper, we limit our study to fitness sharing, introduced by Goldberg and Richardson (1987) studied in detail in (Deb, 1989) and used extensively in (Goldberg, Deb, Horn, 1992). Fitness sharing accomplishes niching by degrading the objective fitness (i.e. the unshared fitness) of an individual according to the presence of nearby (similar) individuals. Thus this type of sharing requires a distance metric on the phenotype or genotype of the individuals. In this study, we ....

....Equation 2 predicts, namely at i = 15. Later, we will investigate this potential empirical confirmation of niched GA theory. 3.2 Overlapping niches It is often the case that niches overlap. In fitness sharing, we easily might estimate oe share to be too large, sometimes on purpose (Goldberg, Deb, and Horn, 1992). In niched GAs and in life, while two species compete to fill a niche, the niches centered on them overlap. In this section, we examine the general case where oe share 1. The calculation of transition probabilities for oe share 1 is somewhat more complex than for oe share 1, but is still ....

[Article contains additional citation context not shown here]

Deb, K., Horn, J., & Goldberg, D. E. (1992). Multimodal deceptive functions. IlliGAL Report No. 92006 (April 1992), and submitted to Complex Systems (August 1992).

....good individuals over from the customer population to the businessman population, thereby tapping into the power of selection and recombination to overcome a hard problem. 5. 1 A Worthy Challenger A family of difficult massively multimodal test functions was developed by Deb, Horn, and Goldberg (Deb, Horn, Goldberg, 1993) and one family member was used by Goldberg, Deb, and Horn (Goldberg, Deb, Horn, 1992) to test sharing augmented by a fitness scaling mechanism. Defining the unitation variable u as u(s) P 6 i=1 s i over the bit variables s i , a six bit bipolar deceptive subfunction is defined below and ....

.... i over the bit variables s i , a six bit bipolar deceptive subfunction is defined below and depicted in figure 4: f(s) 8 : 1 if u(s) 0; 6 0 if u(s) 1; 5 0:360384 if u(s) 2; 4 0:640576 if u(s) 3 (11) Further detail on the construction of the function is available elsewhere (Deb, Horn, Goldberg, 1993). It is deceptive in the sense that low order schemas lead to solutions maximally far from the global optima. In this paper as in Goldberg, Deb, and Horn (Goldberg, Deb, Horn, 1992) five copies of the functions are summed together over successive sets of six bits giving a 30 bit function. A ....

Deb, K., Horn, J., & Goldberg, D. E. (1993). Multimodal deceptive functions. Complex Systems, 7 (2), 131--153.

.... attraction of peaks and regions in the landscape to the GA s crossover operator is largely based on Holland s schema theorem (Holland, 1992) and schema average fitness calculations (Bethke, 1981) Goldberg (1987, 1989a, 1989b, 1989c) and later others (Whitley, 1991; Homaifar, Qi, and Frost, 1991; Deb, Horn, and Goldberg, 1993) defined and constructed deceptive landscapes, in which the GA should be attracted to suboptimal local optima and led away from the global optimum. Schema analysis has also been used to construct GA easy functions (Wilson, 1991) which have large basins of attraction for the global optimum 18 . ....

....partitions, i o j , at each order o min(k 1 ; k 2 ) where k i are the orders of bounded deception for two functions. Goldberg, Deb, and Horn found another way to construct and order partially deceptive functions without losing essential misleadingness (Goldberg, Deb, Horn, 1992; Deb, Horn, Goldberg, 1993). They constrained their bipolar deceptive function to be a function of folded unitation u fld (s) which is simply the number of ones minus the number of zeros: u fld (s) u(s) Gamma ( Gamma u(s) 2 u(s) Gamma . This leads to a symmetric function of unitation, f bip (u fld (s) By ....

[Article contains additional citation context not shown here]

Deb, K., Horn, J., & Goldberg, D. E. (1993). Multimodal deceptive functions. Complex Systems. 7 (1993), 131--153.

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Deb, K. et. al., (1993). Multimodal Deceptive Functions. Complex Systems 7:2, 131-153.

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