N.J.Radcliffe. Equivalence class analysis of genetic algorithms. In Foundations of Genetic Algorithms 3. Morgan Kaufmann, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... in conventional genetic search (for example, single parameter optimisation) and it has been strongly argued elsewhere the problem is highly detrimental to the effectiveness of the search process both in the specific context of neural networks [Radcliffe 1990] Belew 1990] and more generally [Radcliffe 1991a] Radcliffe 1991c] Radcliffe develops the notion of formae (in their original conception formae were introduced as equivalence classes induced by arbitrary equivalence relations over the search space) to deal with this problem. The counting arguement used to suggest that binary ....

N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems. 5(2):183- 205, 1991.

.... of solutions (but see [99] for an alternative view, and [100] for a transfer of schema theory to S expression representations used in genetic programming) Practical experience, as well as some theoretical hints regarding the binary encoding of continuous object variables [101] 102] 103] [104], 105] however, indicate that the binary representation has some disadvantages. The coding function might introduce an additional multimodality, thus making the combined objective function f = f ffi h (where f : M IR) more complex than the original problem f was. In fact, the ....

N. J. Radcliffe, "Equivalence class analysis of genetic algorithms, " Complex Systems, vol. 5, no. 2, pp. 183--206, 1991.

....There have been some debates on the cardinality of the genotype alphabet. Some have argued that the minimal cardinality, i.e. the binary representation, might not be the best [48] 114] Formal analysis of nonstandard representations and operators based on the concept of equivalent classes [115], 116] has given representations other than ary strings a more solid theoretical foundation. Real numbers have been proposed to represent connection weights directly, i.e. one real number per connection weight [27] 29] 30] 48] 63] 65] 74] 95] 96] 102] 110] 111] 117] ....

N. J. Radcliffe, "Equivalence class analysis of genetic algorithms, " Complex Syst., vol. 5, no. 2, pp. 183--205, 1991.

....sections to characterise the working of the genetic algorithm from a formal point of view. There are many formal versions based on other characterisations, e.g. overviews and set based in [Gol89, HB92, Hol92a] as well as algebra based, notably by Radcliffe and Surry [Rad91, Rad93, Rad94a, Rad94b] In a way similar to Hoffmeister and Bck in [HB92] a genetic algorithm GA can be modelled as a 5 tuple GA = INT (Z) # (4.44) using the symbols introduced in the preceding section. However, this model is static in nature, and does not take into account the more interesting dynamic ....

Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. In Foundations of Genetic Algorithms 3. Morgan Kaufmann, 1994.

....introduced in the preceding sections to characterise the working of the genetic algorithm from a formal point of view. There are many formal versions based on other characterisations, e.g. overviews and set based in [Gol89, HB92, Hol92a] as well as algebra based, notably by Radcliffe and Surry [Rad91, Rad93, Rad94a, Rad94b] In a way similar to Hoffmeister and Bck in [HB92] a genetic algorithm GA can be modelled as a 5 tuple GA = INT (Z) # (4.44) using the symbols introduced in the preceding section. However, this model is static in nature, and does not take into account the ....

Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, pages 183--205, Vol. 5, No. 2, 1991.

....Mills, Graham, 21, 22] Mitchell, Melanie, 35, 36, 37, 53, 54, 60, 61] Montana, David J. 68] Moriarty, David, 67] Muntean, Traian, 86, 87] Naillon, Martine, 24] Napliotis, Nicholas, 56] Ostermeier, Andreas, 69] Perkins, Sonya, 21] Petridis, V. 8, 18] Radcliffe, Nicholas J. [9, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81] Romaniuk, Steve G. 82] Santib a nez Koref, Ivan, 29, 30] Saravanan, N. 4] Schwefel, Hans Paul, 83] Silva, A. De, 22] Surry, Patrick, 9] Tackett, Walter Alden, 84] Talbi, El Ghazali, 85, 86, 87, 88, 89, 90] Tarroux, Philippe, 10] Thornton, Chris, 15] Voigt, Hans Michael, ....

....subject keywords of the papers given by the editor of this bibliography are shown next. The keywords neural networks , optimization , and evolution strategies have been omitted in this list because of their high occurrence rate. aerospace, 22] ALECSYS, 31, 34] algebra, 79] analyzing GA, [70, 71, 72, 74, 37] artificial life, 36] automata network, 10] bibliography evolutionary computing, 4] parallel GA, 7] Boltzmann tournament, 62] building blocks hypothesis, 35] CAD, 9] cell regulation proteins, 10] cellular automata, 60, 61] chaos, 60] classifiers, 31] comparison branch and ....

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Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205, 1991. (also as

....list contains references to all papers published as technical reports. The list is arranged in alphabetical order by the name of the institute. Aristotle University of Thessaloniki, 7] Bolt Beranek and Newman, 68] California Institute of Technology, 23] Edinburgh Parallel Computing Centre, [9, 70, 72, 73, 75, 78, 79] Florida Atlantic University, 4] International Computer Science Institute, 33, 34] International Computer Science Institute (ICSI) 31] Iowa State University, 55] Mitsubishi Electric Research Laboratories, 26] National University of Singapore, 82] Santa Fe Institute, 36, 60, 61] ....

....Mills, Graham, 21, 22] Mitchell, Melanie, 35, 36, 37, 53, 54, 60, 61] Montana, David J. 68] Moriarty, David, 67] Muntean, Traian, 86, 87] Naillon, Martine, 24] Napliotis, Nicholas, 56] Ostermeier, Andreas, 69] Perkins, Sonya, 21] Petridis, V. 8, 18] Radcliffe, Nicholas J. [9, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81] Romaniuk, Steve G. 82] Santib a nez Koref, Ivan, 29, 30] Saravanan, N. 4] Schwefel, Hans Paul, 83] Silva, A. De, 22] Surry, Patrick, 9] Tackett, Walter Alden, 84] Talbi, El Ghazali, 85, 86, 87, 88, 89, 90] Tarroux, Philippe, 10] Thornton, Chris, 15] Voigt, Hans Michael, ....

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Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Technical Report TR-90-03, Edinburgh Parallel Computing Centre, 1990. (also as [74]; available via anonymous ftp cite ftp.epcc.ed.ac.uk directory /pub/tr/90 file tr9003.ps.Z) ga:Radcliffe90a.

..... Since length is variable, the chromosomes defined here are very different from those used with classic genetic algorithms. For this reason we needed to define new genetic operators to match their structure. The crossover operators need to have a property defined by Radcliff and termed respect ([10]) This prop 412 erty says that all children must inherit all the common genes of their parents. Figure 2 shows the way a crossover verifying this property for variable length chromosomes works : the common genes (here R1) are first copied in the children; then, a site is randomly chosen on ....

N. J. Radcliffe, "Equivalence class analysis of genetic algorithms ", in Complex Systems, vol. 5, pp. 183--205. 1991.

....do not necessarily capture relationships among meaningful properties that determine fitness [Altenberg, 1995] Generalized schemata can be defined by partitioning the space of structures with many other relations. Such attempts have been presented in the GA literature [Vose and Liepins, 1991; Radcliffe, 1991]. Relations analogous to the schema theorem will hold for other representations as well [Radcliffe, 1992] Indeed, schema theory explains the proliferation of substructures through selection but this fact is indicative more of when a schema hypothesis can be refuted. An argument for this remark is ....

....and complexity penalty ffl Alternative approaches for imposing parsimony pressure The rooted tree schema property Generalized schemata can be defined by partitioning the space of structures with other relations. Such attempts have been presented in the GA literature [Vose and Liepins, 1991; Radcliffe, 1991] Next we propose a simple structural property that defines a different type of partitioning of the space of programs. The space of programs will be partitioned based on the topmost structure of trees. We will call the relation induced by this property a rooted tree schema or tree schema. The ....

Nicholas J. Radcliffe, "Equivalence Class Analysis of Genetic Algorithms, " Complex Systems 5, (2):183--205, 1991.

....ES, and as possibly harmful for EP. The modern tendencies are more pragmatic, and put the discussions back to the representation issue: GAs users have turned to real number representations when dealing with real numbers, following experimental results by [JM91] and heuristic demonstrations by [Rad91]. ESs researchers have included recombination as a standard operator, as in [Sch81] and are designing specific operators for non real valued problems (e.g. in [BS95] And EP is being used on any representation, as for instance by [Ang93, FS95] But the whole EC community now takes for granted ....

N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--20, 1991.

....and Artificial Intelligence, 134] Artificial Intell. Eng. 234] Artificial Intelligence, 398, 424] Assem Autom, 350] Atmospheric Environment Part A General Topics, 488] Automotive Engineer, 616] Br. Telecommun. Eng. UK) 15] CC AI, 531] CIRP Ann. 46] Complex Systems, [677] Comput Control Eng. J. 333] Comput. Econ. Netherlands) 307] Comput. Geosci. UK) 41] Comput. Graph. UK) 337] Comput. Methods Appl. Mech. Eng. 188] Computer, 150] Computers Industrial Engineering, 324] Computers Operations Research, 264, 300, 345] Computing (UK) 710] ....

....344, 359, 364, 369, 370, 372, 373, 665, 666] Prager, Richard, 16] Probert, Penelope, 58, 331] Prosser, P. 185] Prugel Bennett, A. 51] Purchase, G. 427] Purvis, A. 645, 646] Pye, C. J. 266] Quick, R. J. 423] Quinn, L. 353, 668] Rabelo, L. C. 324] Radcliffe, Nicholas J. [18, 80, 93, 134, 148, 11, 437, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684] Rafiq, Y. 413] Rakowski, Tomasz, 295] Rana, A. S. 268] Rastogi, M. 318] Rattray, M. 51] Rayne, C. M. 79] Raywardsmith, V. J. 310] Rayward Smith, V. J. 329, 423] Rayward Smith, Vic J. 105, 54, 81, 233, 541, 542] Redmill, D. W. 286] Reeves, Colin R. 82, 135, 149, ....

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Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205, 1991. (also as [672]; available via anonymous ftp cite ftp.epcc.ed.ac.uk directory /pub/tr/90 file tr9003.ps.Z ) ga:Radcliffe91c.

....Wales, 618] total 8 thesis in 8 schools 4.4 Report series The following list contains references to all papers published as technical reports. The list is arranged in alphabetical order by the name of the institute. Coventry University, 222] EPRI, 160] Edinburgh Parallel Computing Centre, [18, 672, 675, 676, 678, 681, 682, 707, 708] IBM, 698] Loughborough University of Technology, 667] Patents 11 Plymouth Engineering Design Centre, 502, 503] The University of Sheffield, 515] University College London, 601] University of Cambridge, 472] University of Durham, 639] University of East Anglia, 138, 539] ....

....344, 359, 364, 369, 370, 372, 373, 665, 666] Prager, Richard, 16] Probert, Penelope, 58, 331] Prosser, P. 185] Prugel Bennett, A. 51] Purchase, G. 427] Purvis, A. 645, 646] Pye, C. J. 266] Quick, R. J. 423] Quinn, L. 353, 668] Rabelo, L. C. 324] Radcliffe, Nicholas J. [18, 80, 93, 134, 148, 11, 437, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684] Rafiq, Y. 413] Rakowski, Tomasz, 295] Rana, A. S. 268] Rastogi, M. 318] Rattray, M. 51] Rayne, C. M. 79] Raywardsmith, V. J. 310] Rayward Smith, V. J. 329, 423] Rayward Smith, Vic J. 105, 54, 81, 233, 541, 542] Redmill, D. W. 286] Reeves, Colin R. 82, 135, 149, ....

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Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Technical Report TR-90-03, Edinburgh Parallel Computing Centre, 1990. (also as [677]; available via anonymous ftp cite ftp.epcc.ed.ac.uk directory /pub/tr/90 file tr9003.ps.Z ) ga:Radcliffe90a.

....344, 359, 364, 369, 370, 372, 373, 665, 666] Prager, Richard, 16] Probert, Penelope, 58, 331] Prosser, P. 185] Prugel Bennett, A. 51] Purchase, G. 427] Purvis, A. 645, 646] Pye, C. J. 266] Quick, R. J. 423] Quinn, L. 353, 668] Rabelo, L. C. 324] Radcliffe, Nicholas J. [18, 80, 93, 134, 148, 11, 437, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684] Rafiq, Y. 413] Rakowski, Tomasz, 295] Rana, A. S. 268] Rastogi, M. 318] Rattray, M. 51] Rayne, C. M. 79] Raywardsmith, V. J. 310] Rayward Smith, V. J. 329, 423] Rayward Smith, Vic J. 105, 54, 81, 233, 541, 542] Redmill, D. W. 286] Reeves, Colin R. 82, 135, 149, ....

....keywords neural networks , optimization , and evolution strategies have been omitted in this list because of their high occurrence rate. A , 421] acoustics, 705] adaptation, 555] directed, 26] adaptive filters, 174] adaptive plan, 645, 646] AI, 525] algebra, 682] analysing GA, [672, 674, 675, 677, 502, 640, 80, 82, 134, 270, 297, 394, 404, 424, 463] analysing GA crossover, 503, 44] deception, 87, 276] hardness, 276] long path problems, 468] reproduction, 259] Royal Road functions, 423] selection, 216] statistical mechanics formulation, 51] analysis of GA, 271] animation, 605] ant colony search, 385] antennas, 190] ....

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Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. In ? [731], page ? y(ssq) ga94aRadcliffe.

....[32] total 12 books 4.2 Journal articles The following list contains the references to every journal article included in this bibliography. The list is arranged in alphabetical order by the name of the journal. Archives of Control Sciences, 245] Artificial Intelligence, 311] Complex Systems, [358] DIMACS, 155] Evolutionary Computation, 66, 67, 73, 78, 89, 92, 367] Fuzzy Systems Artificial Intelligence Reports and Letters, 135] IEE Proceedings C: Generation, Transmission and Distribution, 109] IEE Proceedings, Vision, Image, Signal Processing, 47] IEEE Transactions on Power ....

....[270] Phon Amnuaisuk, Somnuk, 95] Postaire, Jack G erard, 279] Potter, Mitchell A. 82] Preux, Philippe, 167] Probert, Penelope, 124, 188] Proenca, Luis Miguel, 21, 24, 26] Punch, William F. 55, 84] Punch, III, William F. 49, 79] Pyylampi, Tero, 101, 294] Radcliffe, Nicholas J. [111, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365] Ranito, J. V. 21] Rasanen, Petri, 121] Raymer, Michael L. 55, 84] Reeves, Colin R. 126, 138, 215] Romaniuk, Steve G. 366] Rooij, A. J. F. Van, 64] Rosca, Justinian, 281] Rost, Ursula, 272] Rowe, Jon, 169, 229] Ryan, Conor, 287] Ryynanen, Matti, 127, 230, 240, 291] Saarinen, ....

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Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205, 1991. (also as [354]; ftp.epcc.ed.ac.uk: /pub/tr/90/ tr9003.ps.Z) Key: ga:Radcliffe91c.

....by the name of the institute. Akademie der Wissenschaft der DDR, 306] Argonne National Laboratory, 136] Aristotle University of Thessaloniki, 107] Bolt Beranek and Newman, 350] California Institute of Technology, 299] Carnegie Mellon University, 11, 44] Edinburgh Parallel Computing Centre, [111, 354, 356, 357, 359, 362, 363] Florida Atlantic University, 4] International Computer Science Institute (ICSI) 310, 312, 313] Iowa State University, 337] Leiden University, 194] Michigan State University, 48] Mitsubishi Electric Corp. 303] Mitsubishi Electric Research Laboratories, 302] National University of ....

....[270] Phon Amnuaisuk, Somnuk, 95] Postaire, Jack G erard, 279] Potter, Mitchell A. 82] Preux, Philippe, 167] Probert, Penelope, 124, 188] Proenca, Luis Miguel, 21, 24, 26] Punch, William F. 55, 84] Punch, III, William F. 49, 79] Pyylampi, Tero, 101, 294] Radcliffe, Nicholas J. [111, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365] Ranito, J. V. 21] Rasanen, Petri, 121] Raymer, Michael L. 55, 84] Reeves, Colin R. 126, 138, 215] Romaniuk, Steve G. 366] Rooij, A. J. F. Van, 64] Rosca, Justinian, 281] Rost, Ursula, 272] Rowe, Jon, 169, 229] Ryan, Conor, 287] Ryynanen, Matti, 127, 230, 240, 291] Saarinen, ....

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Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Technical Report TR-90-03, Edinburgh Parallel Computing Centre, 1990. (also as [358]; ftp.epcc.ed.ac.uk: /pub/tr/90/ tr9003.ps.Z) Key: ga:Radcliffe90a. Bibliography 61

....9 schools 4.4 Report series The following list contains references to all papers published as technical reports. The list is arranged in alphabetical order by the name of the institute. Army Strategic Defense Command, 106] Carnegie Mellon University, 176] Edinburgh Parallel Computing Centre, [243] General Motors Research Laboratories, 53] IBM, 256] Indian Institute of Technology, 245] Institute of Psychology CNR, 211] Los Alamos National Laboratory, 108] NASA Ames Research Center, 55] Naval Ocean Systems Center, 105] Navy Research Laboratory, 49] Oregon Graduate Center, ....

....Carsten, 230] Petry, Frederick E. 16, 17, 44] Pettey, Chrisila Cheri Baxter, 231] Pichler, E. E. 232] Pitney, Gilbert, 233] Poon, P. W. 234] Porto, Vincent W. 98, 104, 235] Powell, David J. 236, 237] Preis, K. 205, 302] Quinlan, J. R. 238] Radcliffe, Nicholas J. [242, 243, 244] Raedt, L. De, 72] Rajeev, S. 245] Ralston, Patricia A. S. 293] Ramsey, Connie Loggia, 132] Rao, Vasant B. 258] Rasmussen, Steen, 246] Ray, Thomas S. 247] Reynolds, Robert G. 248] Rice, James P. 187, 188] Rich, S. S. 249] Richards, Dana S. 50] Richards, G. G. 241] ....

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Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Technical Report TR-90-03, Edinburgh Parallel Computing Centre, 1990. (also as [?]; available via anonymous ftp cite ftp.epcc.ed.ac.uk directory /pub/tr/90 file tr9003.ps.Z ) ga:Radcliffe90a.

....theorem relies on the static building block hypothesis which does not take the dynamic of the process into account. Though extensions of the schema theorem have been performed either for non binary encoded chromosomes (e.g. see [1] or for other structures than schemata (predicates [39] formae [30, 31], interval for real coded individuals [13] Other works on the behavior of GAs focus on giving a proof of convergence of the algorithm, either by enhancing the schema theorem [43] 41, 26, 40] or by modeling the behavior of the GA with Markov s chain [11] 29] 8] A very interesting review of ....

Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--205, 1991.

....the sequence of departments, bay by bay, read from top to bottom, left to right. The second chromosome contains an encoding of the number of bays, and where in the sequence the breaks between bays occur. The encoding is shown in Fig. 1. For breeding, we used a variant of uniform crossover [15]. Each location in the offspring s sequence is occupied by the department in the corresponding location from one or the other parent with equal probability, so that all common locations in the parents are carried over to the child. Conflicts are then resolved to ensure that each department occurs ....

Radcliffe, N. J., "Equivalence Class Analysis of Genetic Algorithms, Complex Systems 5, 183-205 (1991).

....(no search strategy can given access only to such a representation) but rather that the Schema Theorem still holds in this case. The careless interpretation that this suggests that a genetic algorithm will out perform a random enumeration is incorrect in these circumstances. Radcliffe [16] also showed that the Fundamental Theorem can be applied to arbitrary subsets of the search space. An extreme example of Radcliffe s observation was given by Liepins and Vose [13] who showed how to convert a fully deceptive GA problem to a fully easy problem by changing the representation. 4.2.4: ....

Radcliffe, N. J., "Equivalence Class Analysis of Genetic Algorithms," Complex Systems, 5, 2, 1991, pp. 183-205.

....solutions of related fitness is mean forma variance. By generating random formae of a particular size and measuring the fitness variance within them, we can estimate the mean variance for formae of a given size. This was shown to be a good qualitative indicator of relative algorithmic performance (Radcliffe Surry, 1994b) 120 For Dedekind representation, all formae are convex simplices R m in bounded by hyper planes perpendicular to the coordinate axes, while for the Isodedekind representation formae are convex simplices bounded by arbitrary hyper planes. In order to investigate analytically the forma variance ....

N. J. Radcliffe, 1991a. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205.

....There have been some debates on the cardinality of the genotype alphabet. Some have argued that the minimal cardinality, i.e. the binary representation, might not be the best [48, 114] Formal analysis of nonstandard representations and operators based on the concept of equivalent classes [115, 116] has given representations other than k ary strings a more solid theoretical foundation. Real numbers have been proposed to represent connection weights directly, i.e. one real number per (a) b) r r r r Node2 Node1 r 6 6 6 Gamma Gamma Gamma Gamma Gamma Gamma Delta Delta Delta ....

N. J. Radcliffe, "Equivalence class analysis of genetic algorithms," Complex Systems, vol. 5, no. 2, pp. 183--205, 1991.

....that determines the direction of movement [89] The two basic components of this approach are: a) an initialization procedure that can generate feasible points, and (b) genetic operators that explore the feasible region. Additionally, the genetic operators must satisfy the following conditions [111, 83]: 1) crossover should be able to generate all points between the parents, 2) small mutations must result in small changes in the fitness function. In the work done by Schoenauer and Michalewicz [120] several examples are presented and special genetic operators are designed for each using ....

Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--220, 1991.

.... and or adequation of parameters for a given problem has made this experimental comparative method very popular since the early days of GAs (see e.g. Schaffer et al. 1989) Nevertheless, some studies have tried to propose heuristical approaches to the a priori estimation of problem difficulty (Radcliffe 1991; Grefenstette 1995; Kargupta 1995) Among these, the Fitness Distance Correlation (Jones Forrest 1995) seems suited to comparing initialization procedures: it allows to compare the behavior of a GA on different problems, on the basis of a random sample of individuals built using the ....

Radcliffe, N. J. 1991. Equivalence class analysis of genetic algorithms. Complex Systems 5:183--20.

....randomly flips the bits of the parent according to a fixed user supplied probability. In the replacement phase, all P offspring replace all parents. Due to that generational replacement, the best fitness in the population can decrease: the original GA strategy is not elitist. In more recent works [84, 112], the genotype space can be almost any space, as long as some crossover and mutation operators are provided. Moreover, proportional selection has been gradually replaced by ranking selection (the selection is performed on the rank of the individuals rather than on their actual fitness) or ....

....real numbers and bitstrings, respectively. However, recent tendencies (as discussed briefly in section 2.5) indicate that this is not a major difference. More important is the adequation of the operators to the chosen representation and the objective function (i.e. the fitness landscape) [84, 112]. ffl Bottom up versus top down, and the usefulness of crossover: According to the Schema Theorem [72, 62] see also the complete survey in this volume [135] GA main strength comes from the crossover operator: better and better solutions are built by exchanging building blocks from partially ....

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N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5pp.183--20, 1991.

....and operators are illustrated for instance in the domain of combinatorial Address for correspondence: LMS and CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France M. Sebag, M. Schoenauer and M. Peyral Revisiting the Memory of Evolution 2 optimization [36] or shape design [44] see [35] for general recommendations about representation operators) As noted by Janikow [26] this transition is quite similar to what happened in the field of artificial intelligence (see [41] for a survey) at first, people were fascinated by the generality of the principles at hand and they aimed at ....

....In summary, the more a bit was recently mutated, the more it is mutated. The set S of bits that are candidates to mutation does not depend on the individual, but rather converges to a fixed set, making mimetic mutation unable to explore the whole search space (violating the ergodicity requirement [35, 40]) If M stands for the Leader (Entrepreneur strategy) the population converges toward M . Incidentally, mimetic mutation here resembles the BSC operator of Syswerda [50] but BSC actually uses the average of the current population to evolve the individuals, instead of the memory L. If M stands ....

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N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183-- 20, 1991.

....de recherche comprend des sous espaces performants (building blocks) et si les solutions optimales appartiennent a l intersection de ces sous espaces. Le th eor eme des sch emas de Holland [31] indique que les AG sont bien adapt es pour explorer un espace de recherche ainsi structur e. Radcliffe [51] s int eresse aux propri et es et au comportement des AGs dans des espaces de repr esentation quelconques ; introduisant une g en eralisation des sch emas de Holland, il propose six principes liant la repr esentation choisie et les op erateurs d evolution agissant sur cette repr esentation, dont ....

....le croisement d efini ci dessus est adapt e au contexte binaire. L approche des strat egies d evolution (ES) 60] s est focalis ee sur la prise en compte d un espace de recherche r eel( Omega = IR N ) et a propos e des op erateurs de croisement sp ecifiquement adapt es a ce contexte (voir aussi [42, 51]) que nous d ecrirons en 3.3. De fa con g en erale, il est en effet souhaitable de repenser les op erateurs de croisement et de mutation compte tenu de l espace de repr esentation adopt e (cf 3.2.3) 2.4.3 La mutation Principe La mutation consiste a reproduire un individu de la population ....

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N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--20, 1991.

....operator (Modified PMX [7] used in the initial study may have been unsuitable as it did not respect the correct type of building block. This idea was advanced by Goldberg as the Principle of Meaningful Building Blocks [3] Modified PMX is a Positionbased crossover operator it respects [8] building blocks such as 1 4 5. Crossover operators have been devised that respect edge and precedence building blocks. In fact, recent work has proposed such a taxonomy of sequencing operators [6] An argument can be made for assuming that precedences are sensible building blocks for this ....

N. J. Radcliffe. Equivalence Class analysis of genetic Algorithms. Complex Systems, 5(2):183--205, 1991.

.... principles have been stated: ffl the variance of the fitness values of all individuals sharing some genetic materials (the schemata in GA terminology) should decrease when the amount of common material increases [43] ffl crossing over two individual should respect their common genetic material [42]; ffl mutation should be ergodic [42] i.e. a finite number of mutations should be able to join any two points of the search space; this point is also crucial for all theoretical convergence results based on Markov chain analysis [11] ffl mutation should respect the principle of strong ....

.... variance of the fitness values of all individuals sharing some genetic materials (the schemata in GA terminology) should decrease when the amount of common material increases [43] ffl crossing over two individual should respect their common genetic material [42] ffl mutation should be ergodic [42], i.e. a finite number of mutations should be able to join any two points of the search space; this point is also crucial for all theoretical convergence results based on Markov chain analysis [11] ffl mutation should respect the principle of strong causality [44] i.e. small changes of the ....

N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--20, 1991.

.... experiments [Constantinescu 1994] Some criterions investigated in the literature will guide systematic experiments: ffl The fitness variance theory of Radcliffe [Radcliffe and Surry 1994] studies the variance of the fitness as a function of the order of an extension of schemas called formae [Radcliffe 1991], and, simply put, shows that the complexity and difficulties of evolution increases with the average variance of the fitness. But if the formae and their order (or their precision) are well defined on any binary representation, including the bit array representation of section 3, it is not ....

Radcliffe, N. J. (1991). Equivalence class analysis of genetic algorithms. Complex Systems, 5:183-- 20.

....Since the seminal work of Holland [27] and the comprehensive study of Goldberg [21] Genetic Algorithms (GAs) have gradually been recognized as powerful stochastic optimization algorithms. More recently, the initial framework of fixed length bitstrings has been widened to other search spaces [42, 39, 7, 6], emphasizing the need for problem specific modifications of the basic algorithms. The field of Evolutionary Computation covers all alternate evolutionary algorithms [49, 19, 4] The main interest of stochastic methods in Engineering Sciences is to break the limits of standard deterministic ....

....optimization [38, 5] provided genetic operators are defined on the search space. However, the design of good genetic operators is still a matter of experience, and a posteriori numerical experimentation remains the only possible validation, though some general guidelines have been stated [42]. Section 4.4 will give examples of specific operators, justified by the performance of the resulting algorithm. On the other hand, it is well known that the main drawback of GAs is their slowness. In particular, when both a GA and some deterministic method (e.g. a gradient method) can be used on ....

N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--20, 1991.

.... k using either the standard digit or the Gray coding [3] However, beside the loss of information induced by the discretization step, Genetic Algorithms using binary coding and the corresponding standard binary operators violate some basic principles related to the so called schemata analysis [20]. Most people working in the area of parameter optimization using Genetic Algorithms now rely on real encoded GAs, where both the genotype space and the phenotype space are subsets of IR n . This implies the definition of specific real valued crossover and mutation operators. The most popular ....

N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--20, 1991.

....the case of graph partitioning. However, in this paper we want to concentrate on formulating a systematic framework for investigating the GA performance. GAs are clearly rather more complex than normal neighbourhood search methods, and in any 1 The term respect was first used by Radcliffe [5, 6]. Informally, respect implies that where parents share a particular characteristic, their offspring should always inherit this characteristic. In Radcliffe s work the shared characteristic may be a more general construct (or forma) but in many practical cases it is simply a particular element of ....

N.J.Radcliffe (1991) Equivalence class analysis of genetic algorithms. Complex Systems, 5, 183-205.

....Holland in that it uses bit strings to encode floating point numbers. The second software package, described below, was a hybridized GA which directly exploits the native floating point representation of the worktations which it was run on. The hybridization towards real numbers is described in [Radcliffe 91] and [Michalewicz 92] The results obtained with both programs were consistent, and only the experiments with the floating point package are detailed in this document. The genetic algorithm progresses in discrete time steps called generations. At every generation the fitness values of all the ....

N. J. Radcliffe, Equivalence Class Analysis of Genetic Algorithms, in Complex Systems 5, pp 183-205, 1991.

....makes it more difficult to choose among them when facing a specific instance of a problem. Some promising directions are given in the literature. The fitness variance theory of Radcliffe [22] studies the variance of the fitness as a function of the order of an extension of schemas called formae [21], and, simply put, shows that the complexity and difficulties of evolution increases with the average variance of the fitness as a function of the formae order. But if the formae and their order (or their precision) are well defined on any binary representation, including the bit array ....

N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--20, 1991.

....sont des m ethodes d optimisation stochastiques maintenant bien connues, depuis le travail s eminal de J. Holland [21] consolid e par le livre de D. Goldberg [15] Plus r ecemment, le strict cadre initial des chaines de bits de longueur fixe a et e etendu a des espaces de recherches quelconques [33, 31, 6, 5], montrant clairement que les algorithmes de bases devaient etre modifi es pour tenir compte des particularit es du probl eme a traiter. Le terme d algorithmes evolutionnaires ( Evolutionary Algorithmes) recouvre ainsi maintenant toutes ces variantes [38, 13, 4] De telles m ethodes peuvent ....

.... etant g en eralement non bijectif, non isom etrique, Le choix de l espace g enotypique r esulte alors d un compromis entre la simplicit e du codage et la possibilit e de disposer dans l espace ph enotypique d op erateurs g en etiques performants, respectant certaines heuristiques [33]. Les AGs traditionnels insistent sur l usage des chaines de bits comme g enotypes, la puissance des op erateurs g en etiques devant, dans de cas, compenser toute eventuelle perte d information due au codage. Les autres ecoles en pr esence insistent sur la n ecessit e d avoir deux espaces aussi ....

N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5:183--20, 1991.

....of this is the choice of network topology. Secondly, the failure modes of the genetic algorithm seen in neural network applications are common to a broader class of problems, and their study can yield more general insights. This paper is a study in the application of forma analysis (Radcliffe [22, 23]) to this and related problems. It begins in section 2 witha brief review and a discussion of the difficulties with previous genetic approaches to problems in neural networks. This is followed, in section 3, by a short review of schema and forma analysis and a discussion of the permutation ....

.... genetic search (for example, simple parameter optimisation) and it has been strongly argued elsewhere the problem is highly detrimental to the effectiveness of the search process both in the specific context of neural networks (Radcliffe [20, 21] Belew et al. 1] and more generally (Radcliffe [22, 23]) 3 Schema and Forma Analysis 3.1 Formulation and Principles In order to understand the motivations for the ideas put forward in this paper it is necessary briefly to review some of the theory of genetic algorithms. Holland s ground breaking formulation and analysis of genetic algorithms ....

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Nicholas J. Radcliffe. Equivalence Class Analysis of Genetic Algorithms. Complex Systems, 5(2):183--205, 1991.

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:183--205, 1991.

....of solutions over the space C of chromosomes. These considerations and others (including Goldberg s principles of minimal alphabets and meaningful building blocks, 4] led to the proposal of six design principles for constructing useful representations, formae and genetic operators (Radcliffe [10]) In the following, the number of formae induced by an equivalence relation will be referred to as the precision of both the relation and the formae it induces. 3 The set of equivalence relations which induce the formae (equivalence classes) under consideration will be written Psi and the set ....

....random edges in such a way as to complete a legal tour. The lack of separability simply ensures that R 3 does not properly assort the formae. 10 Locality Formae All of the formae discussed thus far have been fairly similar to traditional schemata. We now introduce locality formae, Radcliffe [10]) which are rather different in character. Locality formae relate chromosomes on the basis of their closeness to each other. Suppose our function is defined over a real interval [a; b) We then define formae which divide the interval up into strips of arbitrary width. Thus, a forma is a half open ....

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N. J. Radcliffe, Equivalence Class Analysis of Genetic Algorithms, to appear in Complex Systems.

....in Foundations of Genetic Algorithms III , Ed: L.D. Whitley, M.D. Vose, Morgan Kaufmann (San Mateo) pp51 72, 1994. Fitness Variance of Formae and Performance Prediction Nicholas J. Radcliffe njr epcc.ed.ac.uk Edinburgh Parallel Computing Centre University of Edinburgh King s Buildings EH9 3JZ Scotland Patrick D. Surry pds epcc.ed.ac.uk Edinburgh Parallel Computing Centre University of Edinburgh King s Buildings EH9 3JZ Scotland Abstract Representation is widely ....

Nicholas J. Radcliffe, 1991. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205.

.... which group together the chromosomal representatives of solutions which share properties that might be expected to influence their fitness, not only do not form schemata, but are not even contained in any schema except the most general one ( Delta Delta Delta , where is the don t care symbol) (Radcliffe, 1990, 1991a) For example, coding the integer range 0 to 15 on four bits in the conventional manner, the representatives of 7 and 8 (0111 2 and 1000 2 respectively) share membership of no schema except . Similarly, multiples of three are grouped together by no (traditional binary) schema except . 3. ....

....binary) schema except . 3. Not Only Schemata Obey the Schema Theorem Finally, the counting argument which gives rise to the notion of intrinsic parallelism suggests that binary representations maximise the degree of intrinsic parallelism only if attention is restricted to conventional schemata (Radcliffe, 1990). This restriction is inappropriate, as has been argued by Antonisse (1989) Radcliffe (1991a) and Vose (1991) and is detailed below in section 3. 2.3. Representations and Operators Vose Liepins (1991) have pointed out that the difference between the simplest problems for genetic search and ....

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Radcliffe, 1991a. Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms.

....A Language and Parallel Framework for Evolutionary Computing Patrick D. Surry and Nicholas J. Radcliffe Edinburgh Parallel Computing Centre, King s Buildings University of Edinburgh, Scotland, EH9 3JZ Abstract. The Reproductive Plan Language RPL2 is an extensible, interpreted language for writing and using evolutionary computing programs. It supports arbitrary genetic representations, all ....

Nicholas J. Radcliffe, 1991. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205.

....than their higher variance counterparts. This suggests strongly that developing further measurements of representation quality, together with further ways of characterising representation independentoperators (such as the notions of respect, assortment and transmission, developed by Radcliffe, 1991, 1994) and a theory linking those together, is an urgent task. 7 Outlook We have provided in this paper a formal demonstration of various fundamental limitations on search algorithms, with particular reference to evolutionary algorithms. These results establish clearly the central role of ....

....research community. On this basis, we conclude by suggesting some of the key open questions that we see. Given a class of problems about which we have partial knowledge, how can that knowledge be formalised in a way useful to constructing representations algorithms Ideas from forma analysis (Radcliffe, 1991, 1994) offer some directions here, but there is clearly vastly more to be done. Can the quality of a representation be measured in any useful way What kinds of predictive models of performance (if any) can be built given a welldefined problem domain, representation and algorithm ....

N. J. Radcliffe, 1991. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205.

.... of these perceived limitations, various extensions and generalisations have been proposed (Goldberg Lingle, 1985; Antonisse, 1989; Radcliffe, 1990; Vose, 1991) This paper is concerned only with the particular generalisation called forma analysis, which has been developed in a series of papers (Radcliffe, 1990, 1991a, 1991b, 1993, 1992a, 1992b) though the relationship between forma analysis and other generalisations is touched on in the discussion at the end of the paper (section 7) The purpose of the present paper is to define forma analysis rather more rigorously than has before been attempted and to ....

....contradicting orthogonality (equation 36) Since it is obvious that the identity equivalence relation cannot be constructed by intersection of other equivalence relations, this concludes the proof. Following the analogy with linear algebra, a basis can now be defined in the obvious way. 2 In Radcliffe (1991b) orthogonality was defined to be what is here defined as orthogonality to order two (pair wise orthogonality) This definition is unsatisfactory because pair wise orthogonality does not imply full orthogonality in the sense of this exposition. This was not realised at the time the earlier ....

[Article contains additional citation context not shown here]

Radcliffe, 1991a. Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205, 1991.

....or reviewing them all. In addition to the countless empirical studies which pertain to the debate, there have been a few attempts to re examine and generalise schema analysis and the notion of implicit parallelism, notably those by Goldberg Lingle (1985) Wright (1990) Antonisse (1989) Radcliffe (1990, 1991), Vose (1991) Vose Liepins (1991) and Eshelman Schaffer (1992) In particular, Antonisse argued for a re interpretation of schemata under which the counting argument actually suggests that higher cardinality representations will exhibit greater implicit parallelism than their lower ....

Radcliffe, 1991. Nicholas J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183--205, 1991.

No context found.

N.J.Radcliffe. Equivalence class analysis of genetic algorithms. In Foundations of Genetic Algorithms 3. Morgan Kaufmann, 1994.

No context found.

N.J.Radcliffe. Equivalence class analysis of genetic algorithms. In Foundations of Genetic Algorithms 3. Morgan Kaufmann, 1994.

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N. J. Radcliffe. Equivalence Class Analysis of Genetic Algorithms. Complex Systems, 5:183--205, 1991.

No context found.

N. J. Radcliffe. Equivalence class analysis of genetic algorithms. Complex Systems 5, (2):183--205, 1991.

No context found.

Radcliffe, N. Equivalence class analysis of genetic algorithms. Complex Systems, 5(2):183-205, 1991.

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