Learning and exploiting problem structure is one of the key challenges in optimization. This is especially important for black-box optimization (BBO) where prior structural knowledge of a problem is not available. Existing model-based Evolutionary Algorithms (EAs) are very efficient at learning structure in both the discrete, and in the continuous domain. In this article, discrete and continuous model-building mechanisms are integrated for the Mixed-Integer (MI) domain, comprising discrete and continuous variables. We revisit a recently introduced model-based evolutionary algorithm for the MI domain, the Genetic Algorithm for Model-Based mixed-Integer opTimization (GAMBIT). We extend GAMBIT with a parameterless scheme that allows for practical use of the algorithm without the need to explicitly specify any parameters. We furthermore contrast GAMBIT with other model-based alternatives. The ultimate goal of processing mixed dependences explicitly in GAMBIT is also addressed by introducing a new mechanism for the explicit exploitation of mixed dependences. We find that processing mixed dependences with this novel mechanism allows for more efficient optimization. We further contrast the parameterless GAMBIT with Mixed-Integer Evolution Strategies (MIES) and other state-of-the-art MI optimization algorithms from the General Algebraic Modeling System (GAMS) commercial algorithm suite on problems with and without constraints, and show that GAMBIT is capable of solving problems where variable dependences prevent many algorithms from successfully optimizing them.

We describe a parameter-free estimation-of-distribution algorithm (EDA) called the adapted maximum-likelihood Gaussian model iterated density-estimation evolutionary algorithm (AMaLGaM-ID A, or AMaLGaM for short) for numerical optimization. AMaLGaM is benchmarked within the 2009 black box optimization benchmarking (BBOB) framework and compared to a variant with incremental model building (iAMaLGaM). We study the implications of factorizing the covariance matrix in the Gaussian distribution, to use only a few or no covariances. Further, AMaLGaM and iAMaLGaM are also evaluated on the noisy BBOB problems and we assess how well multiple evaluations per solution can average out noise. Experimental evidence suggests that parameter-free AMaLGaM can solve a wide range of problems efficiently with perceived polynomial scalability, including multimodal problems, obtaining the best or near-best results among all algorithms tested in 2009 on functions such as the step-ellipsoid and Katsuuras, but failing to locate the optimum within the time limit on skew Rastrigin-Bueche separable and Lunacek bi-Rastrigin in higher dimensions. AMaLGaM is found to be more robust to noise than iAMaLGaM due to the larger required population size. Using few or no covariances hinders the EDA from dealing with rotations of the search space. Finally, the use of noise averaging is found to be less efficient than the direct application of the EDA unless the noise is uniformly distributed. AMaLGaM was among the best performing algorithms submitted to the BBOB workshop in 2009.

Markov chain Monte Carlo (MCMC) algorithms are sampling methods for intractable distributions. In this paper, we propose and investigate algorithms that improve the sampling process from multi-dimensional real-coded spaces. We present MCMC algorithms that run a population of samples and apply recombination operators in order to exchange useful information and preserve commonalities in highly probable individual states. We call this class of algorithms Evolutionary MCMCs (EMCMCs). We introduce and analyze various recombination operators which generate new samples by use of linear transformations, for instance, by translation or rotation. These recombination methods discover specific structures in the search space and adapt the population samples to the proposal distribution. We investigate how to integrate recombination in the MCMC framework to sample from a desired distribution. The recombination operators generate individuals with a computational effort that scales linearly in the number of dimensions and the number of parents. We present results from experiments conducted on a mixture of multivariate normal distributions. These results show that the recombinative EMCMCs outperform the standard MCMCs for target distributions that have a nontrivial structural relationship between the dimensions.

In this paper, we study two recent theoretical models—a population-sizing model and a convergence model—and examine their assumptions to gain insights into the conditions under which selecto-recombinative GAs work well. We use these insights to formulate several design rules to develop competent GAs for practical problems. To test the usefulness of the design rules, we consider as a case study the map-labeling problem, an NP-hard problem from cartography. We compare the predictions of the theoretical models with the actual performance of the GA for the map-labeling problem. Experiments show that the predictions match the observed scale-up behavior of the GA, thereby strengthening our claim that the design rules can guide the design of competent selecto-recombinative GAs for realistic problems.

Scalable evolutionary computation has. become an intensively studied research topic in recent years. The issue of scalability is predominant in any field of algorithmic design, but it became particularly relevant for the design of competent genetic algorithms once the scalability problems of simple genetic algorithms were understood. Here we present some of the work that has aided in getting a clear insight in the scalability problems of simple genetic algorithms. Particularly, we discuss the important issue of building block mixing. We show how the need for mixing places a boundary in the GA parameter space that, together with the boundary from the schema theorem, delimits the region where the GA converges reliably to the optimum in problems of bounded difficulty. This region shrinks rapidly with increasing problem size unless the building blocks are tightly linked in the problem coding structure. In addition, we look at how straightforward extensions of the simple genetic algorithm—namely elitism, niching, and restricted mating are not significantly improving the scalability problems.