This work analyzes some concepts of genetic algorithms and explains why they may be applied with success to some problems in function optimization. In addition to other performance properties, it has been shown that genetic algorithms are able to overcome local minima in highly multimodal functions (e.g., Rastrigin, Schwefel). The performance of genetic algorithms is supported by an extensive theory, which is based on the assumption of additive gene effects. But the current work shows that the assumption of additive gene effects is not weak, and that the dependence on specific parameter settings is much stronger than often believed. Furthermore, the assumptions regarding the fitness function are so restricting that slight modifications of the standard test functions cause a failure of the optimization procedure even though the function's structure is preserved. The current experiments focus on a few widely-used scalable test functions. the results indicate that a standard g...

. The theory of evolutionary computation has been enhanced rapidly during the last decade. This survey is the attempt to summarize the results regarding the limit and finite time behavior of evolutionary algorithms with finite search spaces and discrete time scale. Results on evolutionary algorithms beyond finite space and discrete time are also presented but with reduced elaboration. Keywords: evolutionary algorithms, limit behavior, finite time behavior 1. Introduction The field of evolutionary computation is mainly engaged in the development of optimization algorithms which design is inspired by principles of natural evolution. In most cases, the optimization task is of the following type: Find an element x 2 X such that f(x ) f(x) for all x 2 X , where f : X ! IR is the objective function to be maximized and X the search set. In the terminology of evolutionary computation, an individual is represented by an element of the Cartesian product X \Theta A, where A is a possibly...

In the binary evolutionary optimization framework, two mutation operators are theoretically investigated. For both the standard mutation, in which all bits are flipped independently with the same probability, and the 1-bit-flip mutation, which flips exactly one bit per bitstring, the statistical distribution of the first hitting times of the target are thoroughly computed (expectation and variance) up to terms of order l (the size of the bitstrings) in two distinct situations: without any selection, or with the deterministic (1+1)-ES selection on the OneMax problem. In both cases, the 1-bit-flip mutation convergence time is smaller by a constant (in terms of l) multiplicative factor. These results extend to the case of multiple independent optimizers. Keywords Evolutionary algorithms, stochastic analysis, binary mutations, Markov chains, hitting times. 1 Introduction One known drawback of Evolutionary Algorithms as function optimizers is the amount of computational efforts they re...

. The role of the mutation rate in canonical genetic algorithms is investigated by comparing a constant setting, a deterministically varying, time-dependent mutation rate schedule, and a self-adaptation mechanism for individual mutation rates following the principle of self-adaptation as used in evolution strategies. The power of the self-adaptation mechanism is illustrated by a time-varying optimization problem, where mutation rates have to adapt continuously in order to follow the optimum. The strengths of the proposed deterministic schedule and the selfadaptation method are demonstrated by a comparison of their performance on difficult combinatorial optimization problems (multiple knapsack, maximum cut and maximum independent set in graphs). Both methods are shown to perform significantly better than the canonical genetic algorithm, and the deterministic schedule yields the best results of all control mechanisms compared. 1 Introduction Genetic Algorithms [11, 14] are the best kno...

It is shown by means of Markov chain analysis that unimodal binary long path problems can be solved by mutation and elitist selection in a polynomially bounded number of trials on average. 1 Unimodality of Binary Functions The notion of unimodal functions usually appears in the theory of optimization in IR 1 . Elster et al. (1977), pp. 228--230, provide a precise definition that is specialized to functions in IR 1 whereas the definition in Bronstein and Semendjajew (1988), p. 137, for functions in IR ` with ` 1 presupposes differentiability. Here, the following definition for functions over IB ` will be used: Definition 1 Let f be a real--valued function with domain IB ` where IB = f0; 1g. A point x 2 IB ` is called a local solution of f if f(x ) f(x) for all x 2 fy 2 IB ` : k y \Gamma x k 1 = 1g (1) where k x k 1 = P ` i=1 j x i j is the Hamming norm. If the inequality in (1) is strict, then x is termed a strictly local solution. The value f(x ) at a...

We investigate the effectiveness of stochastic hillclimbing as a baseline for evaluating the performance of genetic algorithms (GAs) as combinatorial function optimizers. In particular, we address four problems to which GAs have been applied in the literature: the maximum cut problem, Koza's 11-multiplexer problem, MDAP (the Multiprocessor Document Allocation Problem), and the jobshop problem. We demonstrate that simple stochastic hillclimbing methods are able to achieve results comparable or superior to those obtained by the GAs designed to address these four problems. We further illustrate, in the case of the jobshop problem, how insights obtained in the formulation of a stochastic hillclimbing algorithm can lead to improvements in the encoding used by a GA. Department of Computer Science, University of California at Berkeley. Supported by a NASA Graduate Fellowship. This paper was written while the author was a visiting researcher at the Ecole Normale Sup'erieure--rue d'Ulm, Group...

The most simple evolutionary algorithm, the so-called (1+1)EA accepts a child if its fitness is at least as large (in the case of maximization) as the fitness of its parent. The variant (1 + 1) # EA only accepts a child if its fitness is strictly larger than the fitness of its parent. Here two functions related to the class of long path functions are presented such that the (1 + 1)EA maximizes one of it in polynomial time and needs exponential time for the other while the (1+1) # EA has the opposite behavior. These results prove that small changes of an evolutionary algorithm may change its behavior significantly. Since the (1 + 1)EA and the (1 + 1) # EA di#er only on plateaus of constant fitness, the results also show how evolutionary algorithms behave on such plateaus. The (1 + 1)EA can pass a path of constant fitness and polynomial length in polynomial time. Finally, for these functions it is shown that local performance measures like the quality gain and the progress rate do not de...

For the first time, a running time analysis of a multi-objective evolutionary algorithm for a discrete optimization problem is given. To this end, a simple pseudo-Boolean problem (Lotz: leading ones - trailing zeroes) is defined and a population-based optimization algorithm (FEMO). We show, that the algorithm performs a black box optimization in #(n 2 log n) function evaluations where n is the number of binary decision variables. 1

The paper reports on the application of genetic algorithms, probabilistic search algorithms based on the model of organic evolution, to NP-complete combinatorial optimization problems. In particular, the subset sum, maximum cut, and minimum tardy task problems are considered. Except for the fitness function, no problem-specific changes of the genetic algorithm are required in order to achieve results of high quality even for the problem instances of size 100 used in the paper. For constrained problems, such as the subset sum and the minimum tardy task, the constraints are taken into account by incorporating a graded penalty term into the fitness function. Even for large instances of these highly multimodal optimization problems, an iterated application of the genetic algorithm is observed to find the global optimum within a number of runs. As the genetic algorithm samples only a tiny fraction of the search space, these results are quite encouraging.

Almost all analyses of time complexity of evolutionary algorithms (EAs) have been conducted for (1+1) EAs only. Theoretical results on the average computation time of population-based EAs are few. However, the vast majority of applications of EAs use a population size that is greater than one. The use of population has been regarded as one of the key features of EAs. It is important to understand in depth what the real utility of population is in terms of the time complexity of EAs, when EAs are applied to combinatorial optimization problems. This paper compares (1 + 1) EAs and (N + N) EAs theoretically by deriving their first hitting time on the same problems. It is shown that a population can have a drastic impact on an EA’s average computation time, changing an exponential time to a polynomial time (in the input size) in some cases. It is also shown that the first hitting probability can be improved by introducing a population. However, the results presented in this paper do not imply that population-based EAs will always be better than (1 + 1) EAs for all possible problems. I.