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Studies on Extremal Optimization and Its Applications in Solving Real World Optimization Problems

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Studies on Extremal Optimization and Its Applications in Solving Real World Optimization Problems Proceedings of the 2007 IEEE Symposium on Foundations of Computational Intelligence (FOCI 2007) 1 Studies on Extremal Optimization and Its Applications in Solving Real World Optimization Problems Yong-Zai Lu*, IEEE Fellow, Min-Rong Chen, Yu-Wang Chen Department of Automation Shanghai Jiaotong University Shanghai, China Abstract— Recently, a local-search heuristic algorithm called of surviving. Species in this model is located on the sites of a Extremal Optimization (EO) has been proposed and successfully lattice. Each species is assigned a fitness value randomly with applied in some NP-hard combinatorial optimization problems. uniform distribution. At each update step, the worst adapted This paper presents an investigation on the fundamentals of EO species is always forced to mutate. The change in the fitness of with its applications in discrete and numerical optimization problems. The EO was originally developed from the fundamental the worst adapted species will cause the alteration of the fitness of statistic physics. However, in this study we also explore the landscape of its neighbors. This means that the fitness values of mechanism of EO from all three aspects: statistical physics, the species around the worst one will also be changed randomly, biological evolution or co-evolution and ecosystem. Furthermore, even if they are well adapted. After a number of iterations, the we introduce our contributions to the applications of EO in solving system evolves to a highly correlated state known as SOC. In traveling salesman problem (TSP) and production scheduling, and that state, almost all species have fitness values above a certain multi-objective optimization problems with novel perspective in discrete and continuous search spaces, respectively. The threshold. In the SOC state, a little change of one species will simulation results demonstrate the competitive performance with result in co-evolutionary chain reactions called “avalanches” EO optimization solutions due to its extremal dynamics [8]. The probability distribution of the sizes “K” of these mechanism. avalanches is depicted by a power law PKK() , where τis a positive parameter. That is, the smaller avalanches are more Index Terms— Extremal optimization, Self-organized criticality, Multiobjective optimization, Traveling salesman problem, likely to occur than those big ones, but even the avalanches as Production scheduling. big as the whole system may occur with a small but non-negligible probability. Therefore, the large avalanches make any possible configuration (i.e., so-called “solution”in the I. INTRODUCTION evolutionary algorithms) accessible [9]. HE studies on NP-hard optimization problems have been a In contrast to genetic algorithms (GAs) which operate on an Tchallenge subject in optimization community. In addition to entire “gene-pool”of huge number of possible solutions, EO traditional operations research, the modern heuristics [1] have successively eliminates those worst components in the been attractive in fundamental research and real applications. sub-optimal solutions. Its large fluctuations provide significant The approaches to evolutionary algorithm (EA) [2], artificial hill-climbing ability, which enables EO to perform well life [3], simulated annealing (SA) [4] and Tabu search [5] et al. particularly at the phase transitions [9, 38, 39]. EO has been are developed from the natures of biological evolution, successfully applied to some NP-hard combinatorial statistical physics and artificial intelligence et al. In recent optimization problems such as graph bi-partitioning [9], TSP years, a novel general-purpose local search optimization [9], graph coloring [10], spin glasses [11], MAX-SAT [12] and approach, so-called “Extremal Optimization (EO)”has been dynamic combinatorial problems [42]. proposed by Boettcher and Percus [6, 32, 34, 35] based on the In this paper, we make a deep investigation on the fundamentals of statistical physics and self-organized criticality fundamental of EO from three points of view: statistical physics, (SOC) [8]. In contrast to SA which is inspired by equilibrium biological evolution and ecosystem. In this work, we also statistical physics, EO is based on Bak-Sneppen (BS) model [7] introduce our contributions to the applications of EO in discrete of biological evolution which simulates far-from equilibrium and numerical optimization problems. Finally, the advantages dynamics in statistical physics. The BS model is one of the and disadvantages of EO are discussed. models that show the nature of SOC [8, 33, 37, 40]. The SOC means that regardless of the initial state, the system always tunes II. EXTREMAL OPTIMIZATION itself to a critical point having a power-law behavior without Unlike GAs, which work with a population of candidate any tuning control parameter. In BS model, species has an solutions, EO evolves a single individual (i.e. chromosome) S. associated fitness value between 0 and 1 representing a time In EO, each decision variable in the current individual S is scale at which the species will mutate to a different species or considered “species”. There is only mutation operator in EO. become extinct. The species with higher fitness has more chance Through always performing mutation on the worst species and its neighbors successively, the individual can improve its components and evolve itself toward the optimal solution The corresponding author (email: [email protected]) 1-4244-0703-6/07/$20.00 ©2007 IEEE 162 Proceedings of the 2007 IEEE Symposium on Foundations of Computational Intelligence (FOCI 2007) 2 generation by generation. This requires that a suitable systems that have a critical point as an attractor. Their representation should be selected which permits each species to macroscopic behavior exhibits the spatial and temporal be assigned a quality measure (i.e. fitness). This differs from scale-invariance characteristics of the critical point of a phase holistic approaches such as evolutionary algorithms that assign transition [8]. It is interesting to note that in SOC, there is no equal-fitness to all species of a solution based on their collective need to tune control parameters to precise values. SOC is evaluation against an objective function. typically observed in slowly-driven non-equilibrium systems For a minimization problem with n decision variables, EO with extended degrees of freedom and a high level of proceeds as follows [9]: nonlinearity [8]. Inspired by SOC, EO drives the system far 1. Randomly generate an individual S. Set the optimal solution from equilibrium: aside from ranking, there exists no adjustable SS . best parameter, and new solutions are accepted indiscriminately. 2. For the “current”individual S Second, we can study the mechanism of EO from the (a) evaluate the fitness for each decision variable i xi, i (1, , n ) perspective of biological evolution. The EO heuristic was (b) find j satisfying j i for all i, i.e., x j creates the “worst motivated by the BS model which shows the emergence of SOC fitness”, in ecosystems. The fundamental idea behind this model is that (c) choose SNS' () such that x must change its state, N(S)is of co-evolutionary avalanches. It is well known that the j competitive and co-evolutionary activities are regarded as two the neighborhood of S, important factors that help the organisms evolve generation by (d) accept SS ' unconditionally, generation in the nature. Although co-evolution does not have (e) if the current cost function value is less than the minimum optimization as its exclusive goal, it serves as a powerful cost function value, i.e. CSCS()() , then set SS best best paradigm for EO [13]. EO follows the spirit of the BS model in 3. Repeat at Step 2 as long as desired. that it merely updates those components with worst fitness in the 4. Return S and CS() . best best current solution, replacing them by random values without ever To avoid getting stuck into a local optimum [9], a single explicitly improving them. At the same time, EO changes the parameter is introduced into EO and the new algorithm is fitness of all the species connected to the weakest one randomly. called -EO. In -EO, according to fitness , all x are Thus, the fitness of the worst species and its neighbors will i i always change together, which can be considered a ranked, i.e., a permutation of the variable labels i with co-evolutionary activity. This co-evolutionary activity gives (1) (2) (n ) . (1) rise to chain reactions or “avalanches”:large (non-equilibrium) is found. The worst variable x is of rank 1, j(1), and the fluctuations that rearrange major parts of the system, potentially j making any configuration accessible. Large fluctuations allow best variable is of rank n. Consider a scale-free probability the method to escape from local minima and explore the distribution over the ranks k, configuration space efficiently, while the extremal selection Pk k, 1 k n , (2) process enforces frequent returns to near-optimal solutions. for a fixed value of the parameter . At each update, select a Third, the mechanism of EO can be studied from the rank k according to . Then, modify Step 2 (c) so that the perspective of ecosystem. An ecosystem is defined as a Pk biological community of interacting organisms and their variable x with j() k changes its state. j surrounding environment. That is to say, the fitness of any species living in an ecosystem will be affected by the fitness of III. FUNDAMENTAL ASPECTS OF EXTREMAL OPTIMIZATION any other species in the same ecosystem, whereas the change in From Fig. 1, it can be seen that EO is a multi-disciplinary the fitness of any species will affect the fitness landscape (i.e. technique which is based on the fundamental and knowledge of environment) of the whole ecosystem. The interaction statistical physics, biological evolution and ecosystem. relationship between any two species in the ecosystem can be regarded as the inherent fundamental mechanism which drives all the species to co-evolving.
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