An analytic expression of relative approximation error for a class of evolutionary algorithms

An important question in evolutionary computation is how good solutions evolutionary algorithms can produce. This paper aims to provide an analytic analysis of solution quality in terms of the relative approximation error, which is defined by the error between 1 and the approximation ratio of the solution found by an evolutionary algorithm. Since evolutionary algorithms are iterative methods, the relative approximation error is a function of generations. With the help of matrix analysis, it is possible to obtain an exact expression of such a function. In this paper, an analytic expression for calculating the relative approximation error is presented for a class of evolutionary algorithms, that is, (1+1) strictly elitist evolution algorithms. Furthermore, analytic expressions of the fitness value and the average convergence rate in each generation are also derived for this class of evolutionary algorithms. The approach is promising, and it can be extended to non-elitist or population-based algorithms too.

[1]  G. Unter Rudolph.  Local Convergence Rates of Simple Evolutionary Algorithms with Cauchy Mutations , 1998 .

[2]  Xin Yao,et al.  On the approximation ability of evolutionary optimization with application to minimum set cover , 2010, Artif. Intell..

[3]  Günter Rudolph,et al.  Convergence Rates of Evolutionary Algorithms for Quadratic Convex Functions with Rank-Deficient Hessian , 2013, ICANNGA.

[4]  G. Unter Rudolph.  Convergence Rates of Evolutionary Algorithms for a Class of Convex Objective Functions , 1997 .

[5]  Joe Suzuki,et al.  A Markov chain analysis on simple genetic algorithms , 1995, IEEE Trans. Syst. Man Cybern..

[6]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[7]  Carsten Witt,et al.  UNIVERSITY OF DORTMUND REIHE COMPUTATIONAL INTELLIGENCE COLLABORATIVE RESEARCH CENTER 531 Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Worst-Case and Average-Case Approximations by Simple Randomized Search Heuristics , 2004 .

[8]  Yuren Zhou,et al.  Performance Analysis of Evolutionary Algorithms for the Minimum Label Spanning Tree Problem , 2014, IEEE Transactions on Evolutionary Computation.

[9]  Carsten Witt,et al.  Approximating Covering Problems by Randomized Search Heuristics Using Multi-Objective Models , 2007, Evolutionary Computation.

[10]  C. P. Huang.  An efficient algorithm for computing powers of triangular matrices , 1978, ACM '78.

[11]  Thomas Jansen,et al.  Fixed budget computations: a different perspective on run time analysis , 2012, GECCO '12.

[12]  Franz Rothlauf,et al.  On the importance of the second largest eigenvalue on the convergence rate of genetic algorithms , 2001 .

[13]  Thomas Jansen,et al.  Performance analysis of randomised search heuristics operating with a fixed budget , 2014, Theor. Comput. Sci..

[14]  Lishan Kang,et al.  On the Convergence Rates of Genetic Algorithms , 1999, Theor. Comput. Sci..

[15]  Lishan Kang,et al.  Convergence rates for a class of evolutionary algorithms with elitist strategy , 2001 .

[16]  Yiu-Ming Cheung,et al.  On Convergence Rate of a Class of Genetic Algorithms , 2006, 2006 World Automation Congress.

[17]  XI FachbereichInformatik.  Finite Markov Chain Results in Evolutionary Computation: a Tour D'horizon , 1998 .

[18]  Walter Shur.  A simple closed form for triangular matrix powers , 2011 .

[19]  Xin Yao,et al.  Towards an analytic framework for analysing the computation time of evolutionary algorithms , 2003, Artif. Intell..

[20]  Joe Suzuki,et al.  A Further Result on the Markov Chain Model of Genetic Algorithms and Its Application to a Simulated Annealing-like Strategy , 1998, FOGA.

[21]  Frank Neumann,et al.  Approximating Covering Problems by Randomized Search Heuristics Using Multi-Objective Models , 2010, Evolutionary Computation.

[22]  X. Yao,et al.  An analysis of evolutionary algorithms for finding approximation solutions to hard optimisation problems , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[23]  Jun He,et al.  Average Convergence Rate of Evolutionary Algorithms , 2015, IEEE Transactions on Evolutionary Computation.

[24]  Xin Yao,et al.  On the Easiest and Hardest Fitness Functions , 2012, IEEE Transactions on Evolutionary Computation.

[25]  Pietro Simone Oliveto,et al.  Analysis of the $(1+1)$-EA for Finding Approximate Solutions to Vertex Cover Problems , 2009, IEEE Transactions on Evolutionary Computation.