Mathematical functions for testing optimization algorithms

"ANDRE"

• This set of twenty mathematical functions is taken from [1].
• Functions have one to twenty variables.
• Some of them are smooth or not.
• Most of them are multi-modal.
• Download:
  • zip archiv of matlab codes: andre-matlab-v0.0.2.zip (25.5kB, version 0.0.2 released 19 Feb 2009)
  • C++ classes for particular functions as offsprings of class function.h:
    andre.cpp(15kB, version 0.0.1 released 23 Jan 2009)
    andre.h(2kB, version 0.0.1 released 23 Jan 2009)
    function.h(1kB, version 0.0.1 released 23 Jan 2009)
    
    Note: These functions in cpp are multiplied by (-1) in order to be then maximized.
    
  • analytical expressions and some other figures: andre.pdf (839kB, version 0.0.2 released 16 Feb 2009)

Comparison of some algorithms - test with known optimum

• Algorithms were started 100times and stopped when the optimum was found with the given precission or the objective function was evaluated 2,000,000times (in case of GRBFN, the function calls limit was set to 1000 because of insufficient memory).
• Test was performed on andre-matlab-v0.0.2.
• Number of runs, were optimum was found with given precision - so-called succes rate (SR) and average number of fitness calls (ANFC) over succesful runs are given in following Table:

• Other publications:
  • Comparison of Standard Binary Genetic Algorithm (SBGA), Extended Binary Genetic Algorithm (EBGA), Differential Evolution (DE) and Siplified Atavistic Differential Evolution (SADE) and SADE+CERAF is presented in [2].
  • Comparison of SADE algorithm, GRADE algorithm, GRADE algorithm extended by CERAF strategy and GRBFN strategy is presented in Chapter 4 of [3].
  • Comparison in modified version of DE and evolutionary strategy is presented in Master thesis of Josef Nosek [4] in czech language.
• News:
  • 19 Feb 2009: Version 0.0.2 of matlab implementation is released. In previous version, there were errors in definition of PShubert2, Shekel2 and 3, Hartman2config, F5n and F15n. Those errors are corrected. Function Shekel1, Hartman1 and Hartman2 are rewrited for simplification. All config files are modified in order to be easily implemented to matlab version of GRADE algorithm. For the same purpose, *eval functions were added.
  • 16 Feb 2009: There was some an error in PShubert1,2 expression in andre.pdf, v0.0.1 and it was corrected in v0.0.2. The same error is in publications [2] and [3]. C++ code is correct, the error was in matlab codes only in PShubert2 and it will be corrected soon.