@phdthesis{Krejca2019,
  author    = {Krejca, Martin Stefan},
  title     = {Theoretical analyses of univariate estimation-of-distribution algorithms},
  doi       = {10.25932/publishup-43487},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-434870},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xii, 243},
  year      = {2019},
  abstract  = {Optimization is a core part of technological advancement and is usually heavily aided by computers. However, since many optimization problems are hard, it is unrealistic to expect an optimal solution within reasonable time. Hence, heuristics are employed, that is, computer programs that try to produce solutions of high quality quickly. One special class are estimation-of-distribution algorithms (EDAs), which are characterized by maintaining a probabilistic model over the problem domain, which they evolve over time. In an iterative fashion, an EDA uses its model in order to generate a set of solutions, which it then uses to refine the model such that the probability of producing good solutions is increased. In this thesis, we theoretically analyze the class of univariate EDAs over the Boolean domain, that is, over the space of all length-n bit strings. In this setting, the probabilistic model of a univariate EDA consists of an n-dimensional probability vector where each component denotes the probability to sample a 1 for that position in order to generate a bit string. My contribution follows two main directions: first, we analyze general inherent properties of univariate EDAs. Second, we determine the expected run times of specific EDAs on benchmark functions from theory. In the first part, we characterize when EDAs are unbiased with respect to the problem encoding. We then consider a setting where all solutions look equally good to an EDA, and we show that the probabilistic model of an EDA quickly evolves into an incorrect model if it is always updated such that it does not change in expectation. In the second part, we first show that the algorithms cGA and MMAS-fp are able to efficiently optimize a noisy version of the classical benchmark function OneMax. We perturb the function by adding Gaussian noise with a variance of σ², and we prove that the algorithms are able to generate the true optimum in a time polynomial in σ² and the problem size n. For the MMAS-fp, we generalize this result to linear functions. Further, we prove a run time of Ω(n log(n)) for the algorithm UMDA on (unnoisy) OneMax. Last, we introduce a new algorithm that is able to optimize the benchmark functions OneMax and LeadingOnes both in O(n log(n)), which is a novelty for heuristics in the domain we consider.},
  language  = {en}
}