Analysis of the (1 + 1) EA on subclasses of linear functions under uniform and linear constraints

Linear functions have gained great attention in the run time analysis of evolutionary computation methods. The corresponding investigations have provided many effective tools for analyzing more complex problems. So far, the runtime analysis of evolutionary algorithms has mainly focused on unconstrained problems, but problems occurring in applications frequently involve constraints. Therefore, there is a strong need to extend the current analyses and used methods for analyzing unconstrained problems to a setting involving constraints. In this paper, we consider the behavior of the classical Evolutionary Algorithm on linear functions under linear constraint. We show tight bounds in the case where the constraint is given by the OneMax function and the objective function is given by either the OneMax or the BinVal function. For the general case we present upper and lower bounds.Tobias Friedrich, Timo Kötzing, J.A. Gregor Lagodzinski, Frank Neumann, Martin Schirnec

Running Time Analysis of the (1+1)-EA for Robust Linear Optimization

Evolutionary algorithms (EAs) have found many successful real-world applications, where the optimization problems are often subject to a wide range of uncertainties. To understand the practical behaviors of EAs theoretically, there are a series of efforts devoted to analyzing the running time of EAs for optimization under uncertainties. Existing studies mainly focus on noisy and dynamic optimization, while another common type of uncertain optimization, i.e., robust optimization, has been rarely touched. In this paper, we analyze the expected running time of the (1+1)-EA solving robust linear optimization problems (i.e., linear problems under robust scenarios) with a cardinality constraint $k$. Two common robust scenarios, i.e., deletion-robust and worst-case, are considered. Particularly, we derive tight ranges of the robust parameter $d$ or budget $k$ allowing the (1+1)-EA to find an optimal solution in polynomial running time, which disclose the potential of EAs for robust optimization.Comment: 17 pages, 1 tabl

Analysis of Evolutionary Algorithms in Dynamic and Stochastic Environments

Many real-world optimization problems occur in environments that change dynamically or involve stochastic components. Evolutionary algorithms and other bio-inspired algorithms have been widely applied to dynamic and stochastic problems. This survey gives an overview of major theoretical developments in the area of runtime analysis for these problems. We review recent theoretical studies of evolutionary algorithms and ant colony optimization for problems where the objective functions or the constraints change over time. Furthermore, we consider stochastic problems under various noise models and point out some directions for future research.Comment: This book chapter is to appear in the book "Theory of Randomized Search Heuristics in Discrete Search Spaces", which is edited by Benjamin Doerr and Frank Neumann and is scheduled to be published by Springer in 201

Analysis of the (1+1) EA on subclasses of linear functions under uniform and linear constraints

Linear functions have gained a lot of attention in the area of run time analysis of evolutionary computation methods and the corresponding analyses have provided many effective tools for analyzing more complex problems. In this paper, we consider the behavior of the classical (1+1) Evolutionary Algorithm for linear functions under linear constraint. We show tight bounds in the case where both the objective and the constraint function is given by the OneMax function and present upper bounds as well as lower bounds for the general case. We also consider the LeadingOnes fitness function.Tobias Friedrich, Timo Kötzing, Gregor Lagodzinski, Frank Neumann, Martin Schirnec