Anton Eremeev

Duc-Cuong Dang, Anton Eremeev, Per Kristian Lehre

This paper extends the runtime analysis of non-elitist evolutionary algorithms (EAs) with fitness-proportionate selection from the simple OneMax function to the linear functions. Not only does our analysis cover a larger class of fitness functions, it also holds for a wider range of mutation rates. We show that with overwhelmingly high probability, no linear function can be optimised in less than exponential time, assuming bitwise mutation rate $Θ(1/n)$ and population size $λ=n^k$ for any constant $k>2$. In contrast to this negative result, we also show that for any linear function with polynomially bounded weights, the EA achieves a polynomial expected runtime if the mutation rate is reduced to $Θ(1/n^2)$ and the population size is sufficiently large. Furthermore, the EA with mutation rate $χ/n=Θ(1/n)$ and modest population size $λ=Ω(\ln n)$ optimises the scaled fitness function $e^{(χ+\varepsilon)f(x)}$ for any linear function $f$ and any $\varepsilon>0$ in expected time $O(nλ\lnλ+n^2)$. These upper bounds also extend to some additively decomposed fitness functions, such as the Royal Road functions. We expect that the obtained results may be useful not only for the development of the theory of evolutionary algorithms, but also for biological applications, such as the directed evolution.

Anton Eremeev, Alexander Spirov

One of the main properties of biological systems is modularity, which manifests itself at all levels of their organization, starting with the level of molecular genetics, ending with the level of whole organisms and their communities. In a simplified form, these basic principles were transferred from the genetics of populations to the field of evolutionary computations, in order to solve applied optimization problems. Over almost half a century of development in this field of computer science, considerable practical experience has been gained and interesting theoretical results have been obtained. In this survey, the phenomena and patterns associated with modularity in genetics and evolutionary computations are compared. An analysis of similarities and differences in the results obtained in these areas is carried out from the modularity view point. The possibilities for knowledge transfer between the areas are discussed.

Anton Eremeev

The paper is devoted to upper bounds on the expected first hitting times of the sets of local or global optima for non-elitist genetic algorithms with very high selection pressure. The results of this paper extend the range of situations where the upper bounds on the expected runtime are known for genetic algorithms and apply, in particular, to the Canonical Genetic Algorithm. The obtained bounds do not require the probability of fitness-decreasing mutation to be bounded by a constant less than one.

Anton Eremeev

In this paper, we consider a fitness-level model of a non-elitist mutation-only evolutionary algorithm (EA) with tournament selection. The model provides upper and lower bounds for the expected proportion of the individuals with fitness above given thresholds. In the case of so-called monotone mutation, the obtained bounds imply that increasing the tournament size improves the EA performance. As corollaries, we obtain an exponentially vanishing tail bound for the Randomized Local Search on unimodal functions and polynomial upper bounds on the runtime of EAs on 2-SAT problem and on a family of Set Cover problems proposed by E. Balas.

Anton Eremeev

Sufficient conditions are found under which the iterated non-elitist genetic algorithm with tournament selection first visits a local optimum in polynomially bounded time on average. It is shown that these conditions are satisfied on a class of problems with guaranteed local optima (GLO) if appropriate parameters of the algorithm are chosen.

Benjamin Doerr, Anton Eremeev, Frank Neumann et al.

Recently, it has been proven that evolutionary algorithms produce good results for a wide range of combinatorial optimization problems. Some of the considered problems are tackled by evolutionary algorithms that use a representation which enables them to construct solutions in a dynamic programming fashion. We take a general approach and relate the construction of such algorithms to the development of algorithms using dynamic programming techniques. Thereby, we give general guidelines on how to develop evolutionary algorithms that have the additional ability of carrying out dynamic programming steps. Finally, we show that for a wide class of the so-called DP-benevolent problems (which are known to admit FPTAS) there exists a fully polynomial-time randomized approximation scheme based on an evolutionary algorithm.