Convergence Rates of Evolutionary Algorithms for Quadratic Convex Functions with Rank-Deficient Hessian

The best achievable convergence rates of mutation-based evolutionary algorithms are known for various characteristic test problems. Most results are available for convex quadratic functions with Hessians of full rank. Here, we prove that linear convergence rates are achievable for convex quadratic functions even though the Hessians are rank-deficient. This result has immediate implications for recent convergence results for certain evolutionary algorithms for bi-objective optimization problems.

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

[2]  Kenneth A. De Jong,et al.  Artificial Evolution , 2021, Lecture Notes in Computer Science.

[3]  Nicola Beume,et al.  Convergence rates of SMS-EMOA on continuous bi-objective problem classes , 2011, FOGA '11.

[4]  Nicola Beume,et al.  Convergence Rates of (1+1) Evolutionary Multiobjective Optimization Algorithms , 2010, PPSN.

[5]  Ingo Rechenberg,et al.  Evolutionsstrategie : Optimierung technischer Systeme nach Prinzipien der biologischen Evolution , 1973 .

[6]  Robert Schaefer.  Parallel Problem Solving from Nature - PPSN XI, 11th International Conference, Kraków, Poland, September 11-15, 2010. Proceedings, Part II , 2010, PPSN.

[7]  Zbigniew Michalewicz,et al.  Evolutionary Computation 2 , 2000 .

[8]  Armin Scheel,et al.  Beitrag zur Theorie der Evolutionsstrategie , 1985 .

[9]  Hans-Paul Schwefel,et al.  Evolutionary Programming and Evolution Strategies: Similarities and Differences , 1993 .

[10]  Jens Jägersküpper,et al.  How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms , 2006, Theor. Comput. Sci..

[11]  Karim M. Abadir,et al.  Matrix Algebra: Notation , 2005 .

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

[13]  BeyerHans-Georg.  Toward a theory of evolution strategies , 1993 .

[14]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[15]  A I Oyman,et al.  Analysis of the (1, )-ES on the Parabolic Ridge , 2000, Evolutionary Computation.

[16]  Hans-Georg Beyer,et al.  The Theory of Evolution Strategies , 2001, Natural Computing Series.

[17]  Hans-Georg Beyer,et al.  Toward a Theory of Evolution Strategies: The (, )-Theory , 1994, Evolutionary Computation.

[18]  Zbigniew Michalewicz,et al.  Evolutionary Computation 1 , 2018 .

[19]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[20]  Thomas Bäck,et al.  Order Statistics for Convergence Velocity Analysis of Simplified Evolutionary Algorithms , 1994, FOGA.

[21]  Hans-Georg Beyer,et al.  Analysis of the (/, )-ES on the Parabolic Ridge , 2000, Evolutionary Computation.

[22]  Stephen R. Marsland,et al.  Convergence Properties of (μ + λ) Evolutionary Algorithms , 2011, AAAI.

[23]  G. Unter Rudolph.  Asymptotical Convergence Rates of Simple Evolutionary Algorithms under Factorizing Mutation Distributions , 2022 .

[24]  Hans-Georg Beyer,et al.  Toward a Theory of Evolution Strategies: Some Asymptotical Results from the (1,+ )-Theory , 1993, Evolutionary Computation.

[25]  Jens Jägersküpper,et al.  Algorithmic analysis of a basic evolutionary algorithm for continuous optimization , 2007, Theor. Comput. Sci..

[26]  W. Vent,et al.  Rechenberg, Ingo, Evolutionsstrategie — Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. 170 S. mit 36 Abb. Frommann‐Holzboog‐Verlag. Stuttgart 1973. Broschiert , 1975 .

[27]  Dipl. Ing. Karl Heinz Kellermayer.  NUMERISCHE OPTIMIERUNG VON COMPUTER-MODELLEN MITTELS DER EVOLUTIONSSTRATEGIE Hans-Paul Schwefel Birkhäuser, Basel and Stuttgart, 1977 370 pages Hardback SF/48 ISBN 3-7643-0876-1 , 1977 .