First paragraph: In combinatorial optimization, the goal is to find an optimal solution, according to some objective function, from a discrete search space. These problems arise widely in industry and academia and, unfortunately, many of them are NP-hard and no polynomial time algorithm can guarantee their solution to a certified optimality unless. Therefore, in the last decades researchers have investigated the use of stochastic search algorithms to find near optimal solutions to these problems. In particular, great research efforts have been devoted to the development and application of metaheuristic algorithms to solve combinatorial optimization problems.

Unconventional computing devices operating on nonlinear chemical media offer an interesting alternative to standard, semiconductor-based computers. In this work we study in-silico a chemical medium composed of communicating droplets that functions as a database classifier. The droplet network can be "programmed" by an externally provided illumination pattern. The complex relationship between the illumination pattern and the droplet behavior makes manual programming hard. We introduce an evolutionary algorithm that automatically finds the optimal illumination pattern for a given classification problem. Notably, our approach does not require us to prespecify the signals that represent the output classes of the classification problem, which is achieved by using a fitness function that measures the mutual information between chemical oscillation patterns and desired output classes. We illustrate the feasibility of our approach in computer simulations by evolving droplet classifiers for three machine learning datasets. We demonstrate that the same medium composed of 25 droplets located on a square lattice can be successfully used for different classification tasks by applying different illumination patterns as its externally supplied program.

Recently, ant colony optimization (ACO) algorithms have proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses have focused on combinatorial problems such as path finding. We rigorously analyze an ACO algorithm optimizing linear pseudo-Boolean functions under additive posterior noise. We study noise distributions whose tails decay exponentially fast, including the classical case of additive Gaussian noise. Without noise, the classical [Formula: see text] EA outperforms any ACO algorithm, with smaller [Formula: see text] being better; however, in the case of large noise, the [Formula: see text] EA fails, even for high values of [Formula: see text] (which are known to help against small noise). In this article, we show that ACO is able to deal with arbitrarily large noise in a graceful manner; that is, as long as the evaporation factor [Formula: see text] is small enough, dependent on the variance [Formula: see text] of the noise and the dimension n of the search space, optimization will be successful. We also briefly consider the case of prior noise and prove that ACO can also efficiently optimize linear functions under this noise model.

Complex combinatorial problems are most often optimised with heuristic solvers, which usually deliver acceptable results without any indication of the quality obtained. Recently, predictive diagnostic optimisation was proposed as a means of characterising the fitness landscape while optimising a combinatorial problem. The scalars produced by predictive diagnostic optimisation appear to describe the difficulty of the problem with relative reliability. In this study, we record more scalars that may be helpful in determining problem difficulty during the optimisation process and analyse these in combination with other well-known landscape descriptors by using exploratory factor analysis on four landscapes that arise from different search operators, applied to a varied set of quadratic assignment problem instances. Factors are designed to capture properties by combining the collinear variances of several variables. The extracted factors can be interpreted as the features of landscapes detected by the variables, but disappoint in their weak correlations with the result quality achieved by the optimiser, which we regard as the most reliable indicator of difficulty available. It appears that only the prediction error of predictive diagnostic optimisation has a strong correlation with the quality of the results produced, followed by a medium correlation of the fitness distance correlation of the local optima.

Geometric crossover is a formal class of crossovers that includes many well-known recombination operators across representations. In previous work, it was shown that all evolutionary algorithms with geometric crossover (but no mutation) do the same form of convex search regardless of the underlying representation, the specific selection mechanism, offspring distribution, search space, and problem at hand. Furthermore, it was suggested that the generalised convex search could perform well on generalised forms of concave and approximately concave fitness landscapes regardless of the underlying space and representation. In this article, we deepen this line of enquiry and study the runtime of generalised convex search on concave fitness landscapes. This is a first step toward linking a geometric theory of representations and runtime analysis in the attempt to (1) set the basis for a more general, unified approach for the runtime analysis of evolutionary algorithms across representations, and (2) identify the essential matching features of evolutionary search behaviour and landscape topography that cause polynomial performance. We present a general runtime result that can be systematically instantiated to specific search spaces and representations and present its specifications to three search spaces. As a corollary, we obtain that the convex search algorithm optimises LeadingOnes in [Formula: see text] fitness evaluations, which is faster than all unbiased unary black box algorithms.

The theory of evolutionary computation (EC) has experienced rapid and productive growth in recent years. New proof techniques and novel theoretical frameworks have allowed advances in our understanding of the processes and structures inherent in evolutionary optimization. As a result, the frontiers of our knowledge have been expanded further than ever before. Some recent trends in this field, which are covered in this issue, include developments in the understanding of the behavior of evolutionary algorithms (EAs) in dynamic environments rather than just static settings, a theoretical appreciation of the advantages arising from the parallelization of evolutionary algorithms through a greater comprehension of the underlying dynamics, and an understanding of algorithm behavior on broad function classes, including -hard problems.The primary goal of this special issue is to provide extended and polished versions of diverse examples of the best theoretical work presented at conferences in 2014, and to serve as a forum for researchers to advance the theoretical understanding of evolutionary computation methods. The papers included in this special issue span a plurality of topics and offer the reader a cross section of recent outstanding work in EC theory.In dynamic optimization the objective function changes over time, and optimization algorithms face the additional challenge of tracking these changes to be successful. The article “Analysis of Randomised Search Heuristics for Dynamic Optimisation,” by Thomas Jansen and Christine Zarges, presents a novel analytical framework for the analysis of randomized search heuristics on dynamic problems inspired by the fixed-budget computations perspective. The authors introduce a new interesting class of bi-stable dynamic functions where the optimum oscillates between two complementary strings, and apply the framework to analyze and compare the performance of evolutionary algorithms and artificial immune systems on the novel class of functions.Over three decades ago, László Lovász observed that in discrete optimization, submodularity is the counterpart to convexity. However, in contrast to the focus on convex functions in continuous evolutionary optimization, so far submodular functions have received comparatively little attention from EC theoreticians studying discrete functions. The article “Maximizing Submodular Functions under Matroid Constraints by Evolutionary Algorithms,” by Tobias Friedrich and Frank Neumann, addresses this gap by analyzing the performance of evolutionary algorithms on different classes of submodular functions. The maximization of submodular functions is -hard in general, and the authors present several approximation results for monotone submodular and nonmonotone symmetric submodular functions under different kinds of matroid constraints.The idea behind parallel evolutionary algorithms is to evolve multiple subpopulations in parallel and allow interprocess communication at given time intervals. During these migration phases, fractions of each subpopulation can be shared among the subpopulations. There is very little understanding of how the migration frequency affects algorithmic performance, so setting the migration interval parameter appropriately may be difficult. In the article “Design and Analysis of Schemes for Adapting Migration Intervals in Parallel Evolutionary Algorithms,” Andrea Mambrini and Dirk Sudholt propose two schemes to automatically adapt the migration interval of parallel EAs during execution and provide a rigorous analytical framework that yields upper bounds on the expected runtime and expected communication effort of the parallel EAs with different migration topologies for various function classes.In the field of genetic programming, a long-standing open problem is how to address the issue of bloat, the emergence during evolution of solution elements that do not contribute significantly or at all to program fitness or semantics but increase program complexity. The article “On the Performance of Different Genetic Programming Approaches for the SORTING Problem,” by Markus Wagner, Frank Neumann, and Tommaso Urli, tackles the issue of bloat control in the context of sorting. As a basis for their study, they consider program trees and use some measure of sortedness of an in-order traversal to evaluate their fitness. The authors investigate single- and multiobjective variants of genetic programming algorithms with and without bloat control mechanisms, give rigorous upper bounds on their running times, and complement the study with experiments.The topic of constraint handling has recently gained traction in the continuous domain. The article “Markov Chain Analysis of Cumulative Step-Size Adaptation on a Linear Constrained Problem,” by Alexandre Chotard, Anne Auger, and Nikolaus Hansen, presents a rigorous analysis of a (1,)-Evolution Strategy using resampling on a linear function with a linear constraint. The authors prove the previously assumed property that a Markov chain, describing the behavior of the algorithm, exhibits stability in cases with constant step-size and with cumulative step-size adaptation with cumulation parameter equal to 1. This property characterizes the divergence of the algorithm with constant step-size and the geometric divergence or convergence with step-size adaptation, implying fast convergence of Monte Carlo simulations of the divergence rate.In their seminal 2006 paper, Droste, Jansen, and Wegener introduced the concept of black box complexity in order to establish a complexity theory for general-purpose randomized search heuristics. Generally speaking, the black box complexity of a problem is a lower bound on the number of function evaluations needed by any black box algorithm to solve it. Recently, black box models have been refined and developed extensively, allowing the hardness of objective function classes to be more precisely understood. The article “Unbiased Black Box Complexities of Jump Functions,” by Benjamin Doerr, Carola Doerr, and Timo Kötzing, analyzes the unbiased black box complexity of a Jump function class where in each function a local optimum is k bits in distance from the global optimum. The authors provide polynomial upper bounds on the black box complexity of Jump for different sizes of the gap. In particular, they show that an unbiased polynomial-time black box algorithm exists even when almost all of the search space is a plateau of constant fitness.The guest editors would like to thank the authors for their contributions, the referees for their careful reviewing and constructive comments, and the editor-in-chief, Hans-Georg Beyer, for his support in preparing this special issue.