We present an empirical study of a range of evolutionary algorithms applied to various noisy combinatorial optimisation problems. There are three sets of experiments. The first looks at several toy problems, such as O NE M AX and other linear problems. We find that UMDA and the Paired-Crossover Evolutionary Algorithm (PCEA) are the only ones able to cope robustly with noise, within a reasonable fixed time budget. In the second stage, UMDA and PCEA are then tested on more complex noisy problems: S UBSET S UM , K NAPSACK and S ET C OVER . Both perform well under increasing levels of noise, with UMDA being the better of the two. In the third stage, we consider two noisy multi-objective problems (C OUNTING O NES C OUNTING Z EROS and a multi-objective formulation of S ET C OVER ). We compare several adaptations of UMDA for multi-objective problems with the Simple Evolutionary Multi-objective Optimiser (SEMO) and NSGA-II. We conclude that UMDA, and its variants, can be highly effective on a variety of noisy combinatorial optimisation, outperforming many other evolutionary algorithms.

This article presents an exploratory landscape analysis of three NP-hard combinatorial optimisation problems: the number partitioning problem, the binary knapsack problem, and the quadratic binary knapsack problem. In the article, we examine empirically a number of fitness landscape properties of randomly generated instances of these problems. We believe that the studied properties give insight into the structure of the problem landscape and can be representative of the problem difficulty, in particular with respect to local search algorithms. Our work focuses on studying how these properties vary with different values of problem parameters. We also compare these properties across various landscapes that were induced by different penalty functions and different neighbourhood operators. Unlike existing studies of these problems, we study instances generated at random from various distributions. We found a general trend where some of the landscape features in all of the three problems were found to vary between the different distributions. We captured this variation by a single, easy to calculate parameter and we showed that it has a potentially useful application in guiding the choice of the neighbourhood operator of some local search heuristics.

A genetic algorithm is invariant with respect to a set of representations if it runs the same no matter which of the representations is used. We formalize this concept mathematically, showing that the representations generate a group that acts upon the search space. Invariant genetic operators are those that commute with this group action. We then consider the problem of characterizing crossover and mutation operators that have such invariance properties. In the case where the corresponding group action acts transitively on the search space, we provide a complete characterization, including high-level representation-independent algorithms implementing these operators.

In a previous paper (Rowe et al., 2002), aspects of the theory of genetic algorithms were generalised to the case where the search space, Ω, had an arbitrary group action defined on it. Conditions under which genetic operators respect certain subsets of Ω were identified, leading to a generalisation of the term schema . In this paper, search space groups with more detailed structure are examined. We define the class of structural crossover operators that respect certain schemata in these groups, which leads to a generalised schema theorem. Recent results concerning the Fourier (or Walsh) transform are generalised. In particular, it is shown that the matrix group representing Ω can be simultaneously diagonalised if and only if Ω is Abelian. Some results concerning structural crossover and mutation are given for this case.

It is supposed that the finite search space Ω has certain symmetries that can be described in terms of a group of permutations acting upon it. If crossover and mutation respect these symmetries, then these operators can be described in terms of a mixing matrix and a group of permutation matrices. Conditions under which certain subsets of Ω are invariant under crossover are investigated, leading to a generalization of the term schema . Finally, it is sometimes possible for the group acting on Ω to induce a group structure on Ω itself.

We define an abstract normed vector space where the genetic operators are elements. This is used to define the disturbance of the generational operator g as the distance between the crossover and mutation operator (combined) and the identity. This quantity appears in a bound on the variance of fixed-point populations, and in a bound on the force ‖ v – g ( v )‖ that applies to the optimal population v . When analyze for the case of fixed-length binary strings, a connection is shown between these measures and the size of the search space. Guides for parameter settings are given, if population convergence is required as the string length tends to infinity.