Standard Steady State Genetic Algorithms Can Hillclimb Faster than Mutation-only Evolutionary Algorithms

Explaining to what extent the real power of genetic algorithms lies in the ability of crossover to recombine individuals into higher quality solutions is an important problem in evolutionary computation. In this paper we show how the interplay between mutation and crossover can make genetic algorithms hillclimb faster than their mutation-only counterparts. We devise a Markov Chain framework that allows to rigorously prove an upper bound on the runtime of standard steady state genetic algorithms to hillclimb the ONEMAX function. The bound establishes that the steady-state genetic algorithms are 25% faster than all standard bit mutation-only evolutionary algorithms with static mutation rate up to lower order terms for moderate population sizes. The analysis also suggests that larger populations may be faster than populations of size 2. We present a lower bound for a greedy (2+1) GA that matches the upper bound for populations larger than 2, rigorously proving that 2 individuals cannot outperform larger population sizes under greedy selection and greedy crossover up to lower order terms. In complementary experiments the best population size is greater than 2 and the greedy genetic algorithms are faster than standard ones, further suggesting that the derived lower bound also holds for the standard steady state (2+1) GA

When hypermutations and ageing enable artificial immune systems to outperform evolutionary algorithms

We present a time complexity analysis of the Opt-IA artificial immune system (AIS). We first highlight the power and limitations of its distinguishing operators (i.e., hypermutations with mutation potential and ageing) by analysing them in isolation. Recent work has shown that ageing combined with local mutations can help escape local optima on a dynamic optimisation benchmark function. We generalise this result by rigorously proving that, compared to evolutionary algorithms (EAs), ageing leads to impressive speed-ups on the standard Image 1 benchmark function both when using local and global mutations. Unless the stop at first constructive mutation (FCM) mechanism is applied, we show that hypermutations require exponential expected runtime to optimise any function with a polynomial number of optima. If instead FCM is used, the expected runtime is at most a linear factor larger than the upper bound achieved for any random local search algorithm using the artificial fitness levels method. Nevertheless, we prove that algorithms using hypermutations can be considerably faster than EAs at escaping local optima. An analysis of the complete Opt-IA reveals that it is efficient on the previously considered functions and highlights problems where the use of the full algorithm is crucial. We complete the picture by presenting a class of functions for which Opt-IA fails with overwhelming probability while standard EAs are efficient

Artificial immune systems can find arbitrarily good approximations for the NP-hard number partitioning problem

Typical artificial immune system (AIS) operators such as hypermutations with mutation potential and ageing allow to efficiently overcome local optima from which evolutionary algorithms (EAs) struggle to escape. Such behaviour has been shown for artificial example functions constructed especially to show difficulties that EAs may encounter during the optimisation process. However, no evidence is available indicating that these two operators have similar behaviour also in more realistic problems. In this paper we perform an analysis for the standard NP-hard Partition problem from combinatorial optimisation and rigorously show that hypermutations and ageing allow AISs to efficiently escape from local optima where standard EAs require exponential time. As a result we prove that while EAs and random local search (RLS) may get trapped on 4/3 approximations, AISs find arbitrarily good approximate solutions of ratio (1+) within n(−(2/)−1)(1 − )−2e322/ + 2n322/ + 2n3 function evaluations in expectation. This expectation is polynomial in the problem size and exponential only in 1/

Self-adaptation of Mutation Rates in Non-elitist Populations

The runtime of evolutionary algorithms (EAs) depends critically on their parameter settings, which are often problem-specific. Automated schemes for parameter tuning have been developed to alleviate the high costs of manual parameter tuning. Experimental results indicate that self-adaptation, where parameter settings are encoded in the genomes of individuals, can be effective in continuous optimisation. However, results in discrete optimisation have been less conclusive. Furthermore, a rigorous runtime analysis that explains how self-adaptation can lead to asymptotic speedups has been missing. This paper provides the first such analysis for discrete, population-based EAs. We apply level-based analysis to show how a self-adaptive EA is capable of fine-tuning its mutation rate, leading to exponential speedups over EAs using fixed mutation rates.Comment: To appear in the Proceedings of the 14th International Conference on Parallel Problem Solving from Nature (PPSN

Level-Based Analysis of the Population-Based Incremental Learning Algorithm

The Population-Based Incremental Learning (PBIL) algorithm uses a convex combination of the current model and the empirical model to construct the next model, which is then sampled to generate offspring. The Univariate Marginal Distribution Algorithm (UMDA) is a special case of the PBIL, where the current model is ignored. Dang and Lehre (GECCO 2015) showed that UMDA can optimise LeadingOnes efficiently. The question still remained open if the PBIL performs equally well. Here, by applying the level-based theorem in addition to Dvoretzky--Kiefer--Wolfowitz inequality, we show that the PBIL optimises function LeadingOnes in expected time $\mathcal{O}(n\lambda \log \lambda + n^2)$ for a population size $\lambda = \Omega(\log n)$, which matches the bound of the UMDA. Finally, we show that the result carries over to BinVal, giving the fist runtime result for the PBIL on the BinVal problem.Comment: To appea

On inversely proportional hypermutations with mutation potential

Artificial Immune Systems (AIS) employing hypermutations with linear static mutation potential have recently been shown to be very effective at escaping local optima of combinatorial optimisation problems at the expense of being slower during the exploitation phase compared to standard evolutionary algorithms. In this paper we prove that considerable speed-ups in the exploitation phase may be achieved with dynamic inversely proportional mutation potentials (IPM) and argue that the potential should decrease inversely to the distance to the optimum rather than to the difference in fitness. Afterwards we define a simple (1+1)~Opt-IA, that uses IPM hypermutations and ageing, for realistic applications where optimal solutions are unknown. The aim of the AIS is to approximate the ideal behaviour of the inversely proportional hypermutations better and better as the search space is explored. We prove that such desired behaviour, and related speed-ups, occur for a well-studied bimodal benchmark function called \textsc{TwoMax}. Furthermore, we prove that the (1+1)~Opt-IA with IPM efficiently optimises a third bimodal function, \textsc{Cliff}, by escaping its local optima while Opt-IA with static potential cannot, thus requires exponential expected runtime in the distance between the cliff and the optimum

OMAE2002-28586 MODERN APPROACHES TO CRACK ARREST

ABSTRACT In modern day structures fracture initiation cannot be excluded in an absolute sense, defects may exist that when subjected to the operating stress may form brittle propagating cracks. In such cases an assessment of the crack arrest capability of the steel may be required to ensure structural integrity. This paper aims to give a review of existing crack arrest assessment procedures discussing their respective advantages and limitations. In addition a number of modern approaches currently being developed will be introduced and their capabilities compared to existing procedures

On steady-state evolutionary algorithms and selective pressure: Why inverse rank-based allocation of reproductive trials is best

We analyse the impact of the selective pressure for the global optimisation capabilities of steady-state evolutionary algorithms (EAs). For the standard bimodal benchmark function TwoMax, we rigorously prove that using uniform parent selection leads to exponential runtimes with high probability to locate both optima for the standard (+1) EA and (+1) RLS with any polynomial population sizes. However, we prove that selecting the worst individual as parent leads to efficient global optimisation with overwhelming probability for reasonable population sizes. Since always selecting the worst individual may have detrimental effects for escaping from local optima, we consider the performance of stochastic parent selection operators with low selective pressure for a function class called TruncatedTwoMax, where one slope is shorter than the other. An experimental analysis shows that the EAs equipped with inverse tournament selection, where the loser is selected for reproduction and small tournament sizes, globally optimise TwoMax efficiently and effectively escape from local optima of TruncatedTwoMax with high probability. Thus, they identify both optima efficiently while uniform (or stronger) selection fails in theory and in practice. We then show the power of inverse selection on function classes from the literature where populations are essential by providing rigorous proofs or experimental evidence that it outperforms uniform selection equipped with or without a restart strategy. We conclude the article by confirming our theoretical insights with an empirical analysis of the different selective pressures on standard benchmarks of the classical MaxSat and multidimensional knapsack problems