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Abstract-This paper presents a genetic algorithmic approach to the shortest path (SP) routing problem. Variable-length chromosomes (strings) and their genes (parameters) have been used for encoding the problem. The crossover operation exchanges partial chromosomes (partial routes) at positionally independent crossing sites and the mutation operation maintains the genetic diversity of the population. The proposed algorithm can cure all the infeasible chromosomes with a simple repair function. Crossover and mutation together provide a search capability that results in improved quality of solution and enhanced rate of convergence. This paper also develops a population-sizing equation that facilitates a solution with desired quality. It is based on the gambler's ruin model; the equation has been further enhanced and generalized, however. The equation relates the size of the population, the quality of solution, the cardinality of the alphabet, and other parameters of the proposed algorithm. Computer simulations show that the proposed algorithm exhibits a much better quality of solution (route optimality) and a much higher rate of convergence than other algorithms. The results are relatively independent of problem types (network sizes and topologies) for almost all source-destination pairs. Furthermore, simulation studies emphasize the usefulness of the population-sizing equation. The equation scales to larger networks. It is felt that it can be used for determining an adequate population size (for a desired quality of solution) in the SP routing problem.Index Terms-Gambler's ruin model, genetic algorithms, population size, shortest path routing problem.

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Abstract. This paper describes a continuous estimation of distribution algorithm (EDA) to solve decomposable, real-valued optimization problems quickly, accurately, and reliably. This is the real-coded Bayesian optimization algorithm (rBOA). The objective is to bring the strength of (discrete) BOA to bear upon the area of real-valued optimization. That is, the rBOA must properly decompose a problem, efficiently fit each subproblem, and effectively exploit the results so that correct linkage learning even on nonlinearity and probabilistic building-block crossover (PBBC) are performed for real-valued multivariate variables. The idea is to perform a Bayesian factorization of a mixture of probability distributions, find maximal connected subgraphs (i.e. substructures) of the Bayesian factorization graph (i.e., the structure of a probabilistic model), independently fit each substructure by a mixture distribution estimated from clustering results in the corresponding partial-string space (i.e., subspace, subproblem), and draw the offspring by an independent subspacebased sampling. Experimental results show that the rBOA finds, with a sublinear scale-up behavior for decomposable problems, a solution that is superior in quality to that found by a mixed iterative density-estimation evolutionary algorithm (mIDEA) as the problem size grows. Moreover, the rBOA generally outperforms the mIDEA on well-known benchmarks for real-valued optimization.

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This paper provides empirical studies on MrBOA, which have been designed for strengthening diversity of nondominated solutions. The studies lead to modified sharing. A new selection scheme has been suggested for improving diversity performance. Empirical tests validate their effectiveness on uniformity and front-spread (i.e., diversity) of nondominated set. A diversity-preserving MrBOA (dp-MrBOA) has been designed by carefully combining all the promising components; i.e., modified sharing, dynamic crowding, and diversity-preserving selection. Experiments demonstrate that dp-MrBOA is able to significantly improve diversity performance (for the scaling problems), without weakening proximity of nondominated set.

This paper describes two elitism-based compact genetic algorithms (cGAs)-persistent elitist compact genetic algorithm (pe-cGA), and nonpersistent elitist compact genetic algorithm (ne-cGA). The aim is to design efficient compact-type GAs by treating them as estimation of distribution algorithms (EDAs) for solving difficult optimization problems without compromising on memory and computation costs. The idea is to deal with issues connected with lack of memory-inherent disadvantage of cGAs-by allowing a selection pressure that is high enough to offset the disruptive effect of uniform crossover. The point is to properly reconcile the cGA with elitism. The pe-cGA finds a near optimal solution (i.e., a winner) that is maintained as long as other solutions (i.e., competitors) generated from probability vectors are no better. It attempts to adaptively alter the selection pressure according to the degree of problem difficulty by employing only the pair-wise tournament selection strategy. Moreover, it incorporates the equivalent model of the (1 + 1) evolution strategy (ES) with self-adaptive mutation. The pe-cGA, apart from providing a high performance, also reveals the hidden connection between EDAs (e.g., cGA) and ESs (e.g., (1 + 1)-ES). On the other hand, the ne-cGA further improves the performance of the pe-cGA by avoiding strong elitism that may lead to premature convergence. The ne-cGA comes with all the benefits of the pe-cGA. In addition, it maintains genetic diversity as a bonus. This paper also proposes an analytic model for investigating convergence enhancement (i.e., speedup). Experimental results show that the proposed algorithms, ne-cGA in particular, generally exhibit a better quality of solution and a higher rate of convergence for most of the problems than do the existing cGA, sGA, and (1 + 1)-ES. The speedup model has been verified by experiments. The results also show that an adequate alleviation of elitism further improves the solution quality, as well as the convergence speed.

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