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However, if too many ants perform task j, sj also decreases by the factor nact In this case the probability increases that ants will switch tasks, and start to specialize in a new task
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This section shows how AAs can be used to optimize functions de ned over continuous spaces, and to solve MOPs as well as dynamically changing problems
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Continuous Ant Colony Optimization
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Ant colony optimization algorithms were originally developed to solve discrete optimization problems, where the values assigned to variables of the solution are constrained by a xed nite set of discrete values In order to apply ant colony algorithms to solve continuous optimization problems, the main problem is to determine a way to map the continuous space problem to a graph search problem A simple solution to this problem is to encode oating-point variables using binary string representations [523] If a oating-point variable is encoded using an n bit-string, the graph representation G = (V, E), contains 2n nodes two nodes per bit (one for each possible value, ie 0 and 1) A link exists between each pair of nodes Based on this representation, the discrete ACO algorithms can be used to solve the problem It is, however, the case that binary representation of oating-point values loses precision This section discusses an ACO algorithm for optimizing continuous spaces without discretizing the solution variables The rst ACO algorithm for continuous function optimization was developed by Bilchev and Parmee [67, 68] This approach focused on local optimization Wodrich and Bilchev [915] extended the local search algorithm to a global search algorithm, which was further improved by Jayaraman et al [414], Mathur et al [564] and Rajesh et al [701] This global search algorithm, referred to as continuous ACO (CACO), is described next The CACO algorithm performs a bi-level search, with a local search component to exploit good regions of the search space, and a global search component to explore bad regions With reference to Algorithm 178, the search is performed by nk ants, of which nl ants perform local searches and ng ants perform global searches The rst step of the algorithm is to create nr regions Each region represents a point in the continuous search space If xi denotes the i-th region, then xij U (xmin,j , xmax,j ) for each dimension j = 1, , nx and each region i = 1, , nr ; xmin,j and xmax,j are respectively the minimum and maximum values of the domain in the j-th dimension After initialization of the nr regions, the tness of each region is evaluated, where the tness function is simply the continuous function being optimized Let f (xi ) denote the tness of region xi The pheromone, i , for each region is initialized to one The global search identi es the ng weakest regions, and uses these regions to nd ng
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174 Advanced Topics Algorithm 178 Continuous Ant Colony Optimization Algorithm Create nr regions; i (0) = 1, i = 1, , nr ; repeat Evaluate tness, f (xi ), of each region; Sort regions in descending order of tness; Send 90% of ng global ants for crossover and mutation; Send 10% of ng global ants for trail di usion; Update pheromone and age of ng weak regions; Send nl ants to probabilistically chosen good regions; for each local ant do if region with improved fitness is found then Move ant to better region; Update pheromone; end else Increase age of region; Choose new random direction; end Evaporate all pheromone; end until stopping condition is true; Return region xi with best tness as solution;
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new regions to explore Most of the global ants perform crossover to produce new regions, with a few ants being involved in trail di usion For the crossover operation, for each variable xij of the o spring, choose a random weak region xi and let xij = xij , with probability Pc (referred to as the crossover probability) After the crossover step, the o spring xi is mutated by adding Gaussian noise to each variable xij : xij xij + N (0, 2 ) where the mutation step size, , is reduced at each time step: max (1 r (1 t/nt ) )
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where r U (0, 1), max is the maximum step size, t is the current time step, nt is the maximum number of iterations, and 1 controls the degree of nonlinearity The nonlinear reduction in mutation steps limits exploration in later iterations to allow for better exploitation Trail di usion implements a form of arithmetic crossover Two weak regions are randomly selected as parents For parents xi and xl , the o spring x is calculated as xj = 2 xij + (1 2 )xlj where 2 U (0, 1) (1770)
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