Stadler, P. F. (1995) "Towards a Theory of Landscapes," Social Systems Research Institute Working Paper Number 9506. Madison: University of Wisconsin.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....relations since they are mapped one to one onto structures and thus are elements of an invertible map from sets of sequences onto structures and vice versa. An evaluation of structures resulting in a scalar quantity (fitness being an example) induces a landscape upon sequence space [49, 50, 58]. Fitness landscapes derived from folding RNA sequences into secondary structures represent the basis of the RNA model of molecular evolution [48, 52] which allows to treat phenotypes explicitly and which is accessible to computer simulation and mathematical modeling. Optimization processes on ....

P. F. Stadler. Towards a theory of landscapes. In R. Lop'ez-Pe~na, R. Capovilla, R. Garc'ia-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, pages 77--163, Berlin, New York, 1995. Springer Verlag.

....view sees evolutionary optimization as hill climbing on a tness landscape. These landscapes are built upon sequence space by assigning a numerical ( tness) value to every genotype. Recent developments focus on their mathematical characterization and statistical analysis (For reviews see [31, 32]) The primary active products derived from the genomes are biopolymers, RNA molecules and proteins. From biophysics we know that point mutations (single nucleotide exchanges in DNA) may have all kinds of e ects ranging from drastic changes in properties and functions to no change at all. ....

P. F. Stadler. Towards a theory of landscapes. In R. Lopez-Pe~na, R. Capovilla, R. Garca-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, pages 77-163, Berlin, New York, 1995. Springer Verlag. SFI preprint 95-03-030.

....routing schemes we study were originally based on graphs embedded in metric spaces. De nition 2.3 A metric graph, denoted [G; d] is a graph G = V; E) and a metric d : M M R such that V M . Fitness landscapes are a canonical framework for studying the eciency of local search procedures [5]. As we will see, they can be used to describe any metric graph. 3 De nition 2.4 [5] A (graph) tness landscape L, denoted (G; f) is a graph G = V; E) along with a function f : M R such that V M , and, for some x 2 V , f(x) 0. The following notion relates tness landscapes to metric ....

....nition 2.3 A metric graph, denoted [G; d] is a graph G = V; E) and a metric d : M M R such that V M . Fitness landscapes are a canonical framework for studying the eciency of local search procedures [5] As we will see, they can be used to describe any metric graph. 3 De nition 2. 4 [5] A (graph) tness landscape L, denoted (G; f) is a graph G = V; E) along with a function f : M R such that V M , and, for some x 2 V , f(x) 0. The following notion relates tness landscapes to metric graphs. De nition 2.5 Given a metric d : M M R and y 2 M , the tness function of ....

P. Stadler. Towards a theory of landscapes. In Complex Systems and Binary Networks, pages 77-163. Springer-Verlag, 1995.

....for each family of landscapes. It is important to note that each random walk was performed with respect the studied subspace. 5.1. Correlation Analysis The correlation structure of the landscape is investigated by measuring the autocorrelation functions of time series sampled on random walks [32, 23]. The autocorrelation function of time series n o p q r p s t is given by u v w x y f j o p p z m b f j o p m f j o p z m j o p (5) where f and j o p m are the expectation and variance, respectively. The measured autocorrelations are shown in Figure 7. The figure represents ....

P. F. Stadler. Towards theory of landscapes. In R. LopezPe #na, R. Capovilla, R. Garc ia-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, pages 77--163. Springer-Verlag, Berlin, 1995.

....values encountered in a HC walk that uses a steepest descent rule, choosing its best neighbor for its next step. It is unclear wether the assumption of isotropy of landscape still holds when comparing such different walks. Another point worth mentioning is that the correlation length is given in [3, 11] as a function of the size of the instance regardless of any other parameter. On the one hand, we acknowledge that being able to characterize the overall performance of a particular operator for any instance of a given size, is really interesting. On the other hand, there certainly exists ....

....We show that a city swap HC even if it improves the starting points is not able to give tours near the global optimum relatively to its own metric. We observe that 2opt move HC are able to come much closer to the global optimum. This last point is in accordance with Stadler s conjecture [11] about the existence of a massif central regrouping local optima in a area of restricted size. This property could be exploited, as an example by recombining local optima. On the opposite, local optima in the city swap landscape are separated by long distances. In future work we will be aiming ....

Peter F. Stadler. Towards a theory of landscapes. Technical Report SFI95 -03-030, Santa Fe Institute, 1995. 8

....spaces. Though this algorithm is completely useless for the purpose of optimization, the analysis of these walks is able to bring some interesting information about the structure of the space, the distribution of local optima and their quality. We consider this first work, its followers (such as [33, 30, 12]) and some other works dealing with the study of the correlation of the cost of neighborhing points [25, 11] as complementary to ours. In the following, we first present our study of the structure of the planar, euclidian TSP. From it, we derive a search algorithm and provide experimental ....

P. Stadler, Towards A Theory Of Landscapes, in R. Lopes-Pena, R. Capovilla, R. Garcia-Pelayo, H. Waelbroeck, F. Zertuche (eds), Complex Systems and Binary Networks, Springer-Verlag, 1995

....2 X : Usually, the distance measure d is de ned by the neighbourhood structure: y 2 N(x) d(x; y) 1: A locally optimal vector x 2 X is then one such that g(x ) g(y) 8 y 2 N(x ) We shall denote the set of such optima as X . For a fuller discussion on landscapes, see [3] or [4]; for some illustrations of the way landscapes are formed, and how they depend on neighbourhood structure, see [5] Many things make an instance dicult, but one of them is the number of optima induced in the landscape = jX j. Besides this an estimate of might be useful in de ning ....

Stadler, P.F.: Towards a Theory of Landscapes. In: Lopez-Pe~na, R., Capovilla, R., Garc  ia-Pelayo, R., Waelbroeck, H. and Zertuche, F. (eds.): Complex Systems and Binary Networks. Springer-Verlag, Berlin (1995) 77-163

....problems, while they are outperformed on other problems. It is conjectured that this depends upon the agent ability to exploit the tness landscape characteristics. Among such properties are: ruggedness, number of local optima, distribution of local optima and topology of the basins of attraction [42, 51, 52, 36, 31, 32]. The choice of tness function and operator de nes such characteristics and it is reasonable that the combination of searches on di erent tness landscapes (derived from the same original problem) can smooth the problem hardness (or, at least, e ectively face it) Cooperative search [30, 3, 12, ....

P.F. Stadler. Towards a theory of landscapes. In R. Lopez-Pe~na, R. Capovilla, R. GarcaPelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, volume 461 of Lecture Notes in Physics, pages 77-163, Berlin, New York, 1995. Springer Verlag. SFI preprint 95-03-030.

....the landscape and extract useful properties (number of local minima, size of basins of attraction of global minima, auto correlation, etc. and observe the resulting dynamics. Most of the knowledge and results available about optimization on tness landscapes can be directly applied [Kau93, Kau95, Sta95, KM95] The following questions can be answered: Given the strategies, does the protocol produce a dynamics such that the agents reach always often never a global minimum Dually, given the protocol, can the strategies lead the agents to reach their goal In case of heterogeneous ....

Peter F. Stadler. Towards a theory of landscapes. In R. Lopez-Pe~na, R. Capovilla, R. Garca-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, volume 461 of Lecture Notes in Physics, pages 77-163, Berlin, New York, 1995. Springer Verlag. SFI preprint 95-03-030.

....fitness is such a feature that will present a problem. Jones and Forrest [Jones and Forrest, 1995] used Fitness Distance Correlation, which is the degree in which the (Hamming ) distance to the global optimum is correlated to the fitness, to predict GA performance. Jones [Jones, 1995] and Stadler [Stadler, 1995] generalised the landscape concept to include arbitrary operator distance functions, such as the crossover landscape. A lot of research has also been done into the role of the mutation and recom bination operators, parameter settings, and convergence. Spears [Spears, 1998] investigated the ....

P. Stadler. Towards a theory of landscapes. In R. L'opez Pe na, editor, Complex Systems and Binary Networks. Santa Fe Institute Preprint, 1995. ftp://ftp. itc. univie. ac. at/pub/PREPRINTS/sta-95-01. ps. gz.

....operators and studying their performance. Difficulties in iGP occur because the fitness landscapes are anisotropic in sense of being on non regular underlying graphs. The iGP components may be improved through measurements of: the fitness distance correlation [4] the autocorrelation function [13], and some information characteristic [15] of their fitness landscapes. The performance investigation is another crucial issue in iGP, which is still under discussion and on which there is no consensus. We propose to observe the behavior of evolutionary search with: 1) diversity measures, such as ....

....on the correspondence between the fitness and the distance to some known global optima. This is evaluated with the fitness distance correlation [4] The local landscape characteristics depend on the relation between the fitness and the operator. This is estimated by the autocorrelation function [13]. Experiments for inducing regular expressions [14] were performed with three operators: contxt pvscr ving mutation (CPM) hicr ar chical var iabl length mutation (HVLM) 10] and replacement mutation (RM) 12] HVLM inserts a sub tree before a chosen vertex, deletes a node so that its largest ....

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P. Stadler, 'Towards a Theory of Landscapes', In: Complex Systems and Binary Networks, R.Lopz-Pefia et al.(eds.), Springer-Verlag, Berlin, 77-163, 1995.

....kinds of problems, while they are outperformed on other problems. It is conjectured that this depends upon the agent ability to exploit the FL characteristics. Among such properties are: ruggedness, number of local optima, distribution of local optima and topology of the basins of attraction [36, 44, 45, 30, 25, 26]. The choice of tness function and operator de nes such characteristics and it is reasonable that the combination of searches on di erent FLs (derived from the same model) can overcome the problem hardness (or, at least, e ectively face it) Cooperative search [24, 2, 9, 51, 43] consists of a ....

P.F. Stadler. Towards a theory of landscapes. In R. Lopez-Pe~na, R. Capovilla, R. Garca-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, volume 461 of Lecture Notes in Physics, pages 77-163, Berlin, New York, 1995. Springer Verlag. SFI preprint 95-03-030.

....of TSP and QAP instances which partly explain the observed performance differences and motivates important aspects of the algorithmic design of MMAS. 3. 1 Analysis of Fitness Landscapes Central to the search space analysis of combinatorial optimization problems is the notion of fitness landscape [42,53]. Intuitively, the fitness landscape can be 7 imagined as a mountainous region with hills, craters, and valleys. A local search algorithm can be pictured as a wanderer that performs a biased walk in this landscape. In a minimization problem such as the TSP or the QAP, the goal is to find the ....

P. F. Stadler. Towards a theory of landscapes. Technical Report SFI--95--03--030, Santa Fe Institute, 1995.

....relations since they are mapped one to one onto structures and thus are elements of an invertible map from sets of sequences onto structures and vice versa. An evaluation of structures resulting in a scalar quantity (fitness being an example) induces a landscape upon sequence space [49, 50, 58]. Fitness landscapes derived from folding RNA sequences into secondary structures represent the basis of the RNA model of molecular evolution [48, 52] which allows to treat phenotypes explicitly and which is accessible to computer simulation and mathematical modeling. Optimization processes on ....

P. F. Stadler. Towards a theory of landscapes. In R. Lop'ez-Pe~na, R. Capovilla, R. Garc'ia-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, pages 77--163, Berlin, New York, 1995. Springer Verlag.

....for each family of landscapes. It is important to note that each random walk was performed with respect the studied subspace. 5.1. Correlation Analysis The correlation structure of the landscape is investigated by measuring the autocorrelation functions of time series sampled on random walks [32, 23]. The autocorrelation function of time series ff t g n t=0 is given by ae(s) E[f t f t s ] Gamma E[f t ]E[f t s ] V [f t ] 5) where E[f t ] and V [f t ] are the expectation and variance, respectively. The measured autocorrelations are shown in Figure 7. The figure represents the average ....

P. F. Stadler. Towards theory of landscapes. In R. LopezPe # na, R. Capovilla, R. Garc ia-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, pages 77--163. Springer-Verlag, Berlin, 1995.

....walk: do a random walk, and compute the correlation between the fitness at time t and time t i. This can be used to give an estimate of how rugged (how many false optima are in the landscape) a 67 landscape is. This approach has been taken by Manderick, et al. 37] Hordijk [28] and Stadler [48]. Manderick, et al. however, use random mutations which means they use the landscape of a hill climber, but apply this analysis to GAs. This is problematic. Crossover can do random walks if two parents produce two children rather than a single child, so random walk analysis is not limited to ....

....bounds on a particular landscape, or prove some expected level of performance. In On Searching ff ary Hypercubes [10] certain characteristics of hypercubes are explored both theoretically and experimentally. For example, we show that the ff ary hypercube has a maximum of n=ff peaks. Stadler [48] relates points of graph theory to landscapes. Chapters 2 5 and Appendix A of this thesis are theoretical analyses of landscapes (Appendix A explores Gray codes, but this can be related to Hamiltonian paths and cycles in multary based graphs) 8.3 Discriminating Functions, and Interesting ....

Stadler, Peter F. (1995). "Towards a Theory of Landscapes." Technical Report SFI-TR-95-03-030, The Santa Fe Institute, Santa Fe, New Mexico.

....the influence of crossover, mutation and selection on the flow of the population s distribution in the landscape. 5.2 Multimodality: Number of Local Optima One important characteristic of a landscape is its number of local optima, as they are obstacles for local search algorithms. Some authors [39,50] even conjecture that there is a close relation between ruggedness (nearest neighbor correlation, see section 6 for a detailed discussion) and the number of local optima: fewer local optima probably result in a larger correlation and easier optimization. In the extreme and intractable case of a ....

P. Stadler. Towards a theory of landscapes. In R. Lopz-Pena, R. Capovilla, R. GarciaPelayo, H. Waelbroeck, and F. Zertouche, editors, Complex systems and binary networks, pages 77--173. Springer Verlag, 1995.

....of TSP and QAP instances which partly explain the observed performance differences and motivates important aspects of the algorithmic design of MMAS. 3. 1 Analysis of Fitness Landscapes Central to the search space analysis of combinatorial optimization problems is the notion of fitness landscape [42,53]. Intuitively, the fitness landscape can be 7 imagined as a mountainous region with hills, craters, and valleys. A local search algorithm can be pictured as a wanderer that performs a biased walk in this landscape. In a minimization problem such as the TSP or the QAP, the goal is to find the ....

P. F. Stadler. Towards a theory of landscapes. Technical Report SFI--95--03--030, Santa Fe Institute, USA, 1995.

....correlation matrix. Until now, mainly two di#erent correlation functions have been used to describe fitness landscapes; these are often referred to as auto correlation functions. Based on these functions, however, rough approximations of the behavior of the evolutionary process were possible (Stadler 1995, Manderick 1997) One type of these functions is based on the auto correlation of fitness values in time series that result from random walks on a fitness landscape. The dynamics of such random walks are only determined by mutation and fitness values or selective pressures are not considered. ....

....initial conditions for the random walk. The second type of auto correlation function is based on considering the correlation of fitness values between pairs of genotypes at varying mutational distances. It can be shown that these two types of auto correlation functions carry equivalent information (Stadler 1995). The correlation measure presented in this study at the same time reduces and enhances the statistical information provided: the focus of this new measure is on single mutational steps (reduced information) but the initial fitness of considered individuals is maintained in the measure (enhanced ....

Stadler P.F. (1995). Towards a Theory of Landscapes, In: "Complex Systems and Binary Networks" (Proc. of the Guanajuato Lectures 1995) Lopez-Pena R., Capovilla R., Garc ia-Pelayo R., Waelbroeck H. and Zertuche F. (eds.), Springer.

....(see [48, pp. 304 317] and became important in theories of molecular evolution and the origin of life [15, 14, 20, 22, 35, 38, 47, 50, 49] and in evolutionary computer science [36] Similar developments exist in physics, where free energy landscapes are considered, and in search theory [53]. The main challenge to landscape theory is to determine which features of the fitness landscape determine the evolvability of the systems on the landscape. A landscape results from the combination of three elements. A set of types V , which in genetics can be genotypes or gametes, a set of ....

....i.e. there are exactly two nodal domains. Following a suggestion by Kauffman [37] we term this type of landscape Fujijama landscapes because they consist of a single mountain Gamma . For Hamming graphs a much stronger result holds: Fujijama landscapes have no non global minima or maxima [53]. ffl The geometry of the nodal domains of a landscape f is of course a very important characteristic of the landscape. Courant s nodal domain theorem 3 , which has been proved recently for graphs [10] states that, if all eigenvalues of Gamma Delta are labeled in ascending order, 0 = 0 1 ....

P. F. Stadler. Towards a theory of landscapes. In R. Lop'ez-Pe~na, R. Capovilla, R. Garc'ia-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, pages 77--163, Berlin, New York, 1995. Springer Verlag.

No context found.

Stadler, P. F. (1995) "Towards a Theory of Landscapes," Social Systems Research Institute Working Paper Number 9506. Madison: University of Wisconsin.

No context found.

P. F. Stadler. Towards a theory of landscapes. In R. Lopez-Pena, R. Capovilla, R. Garca-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, pages 77--163. Springer Verlag, 1995.

No context found.

P. F. Stadler, "Towards a Theory of Landscapes," in Complex Systems and Binary Networks, (R. Lopez-Pena, R. Capovilla, R. Garca-Pelayo, H. Waelbroeck, and F. Zertuche, eds.), (Berlin, New York), pp. 77--163, Springer Verlag, 1995.

No context found.

Stadler, P.F. (1995) "Towards a Theory of Landscapes." Social Systems Research Institute Working P1aper Number 9506. Madison: University of Wisconsin.

No context found.

P. F. Stadler. Towards a theory of landscapes. In R. Lopez-Pena, R. Capovilla, R. Garca-Pelayo, H. Waelbroeck, and F. Zertuche, editors, Complex Systems and Binary Networks, pages 77--163, Berlin, New York, 1995. Springer Verlag.

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