on using the


  In the year 2000 our coleague Matej Leps has published his paper where he focused on optimization of reinforced concrete beam which he has studied in a particular case of a certain loading state. In our opinion this task belongs to the most difficult ones, because the fitness functions appears to be very uncomfortable: it has many local extremes of different values and different locations and moreover it is non-continuos within the whole domain. To resolve this problem Matej uses method based upon a binary genetic algoritms recombined with augmented simulated annealing (AUSA) which decreases probability of premature convergence. By the next step he has improve this technology in such a way: at first he generates ten starting populations and then chooses ten best chromosomes from each of them; these are then going to be used as an initial population of final computation.

  We have tried to solve the same problem using differential evolution, a method developed by Rainer Storn and Kenneth Price (see homepage of differential evolution).

  As the domain is discrete in this case, it is necessary to round all coordinates of solution vectors (chromosomes) and together check them if they didn't fall out from a domain. We have used modified source code published by R. Storn with strategy marked as "rand_to_best/1/exp", values of its parameters were: F=0.85 and CR=1.0. This method itself doesn't contain any technologies that could help it to get out from the local extremes. All the same it deports surprisingly good results, as shown in this table.

Method SGA+AUSA Differential evolution
Best Result
Worst Result
579 CZK
660 CZK
574 CZK
621 CZK
Average 603 CZK 584 CZK

  In both cases maximum number of possible fitness calls was restricted to 200 000. Convergence process is shown on this picture:

    Appears that algoritms was caught in local extreme several times but everytime has achieved to get out of it in itself without help of other technologies.

    Source code of the fitness function was made by Matej and is too complicated to be presented here. Instead, I would like to show sections be each of 18 variables through a point of the found best extreme that we made after.


    Matej Leps & Michal Sejnoha: New approach to optimization of reinforced concrete beams, CIVIL-COMP, 2000
    Rainer Storn: On the usage of differential evolution for function  optimization, NAPHIS, 1996


    Ondra Hrstka email:<ondra@klobouk.fsv.cvut.cz> hmpg:http://klobouk.fsv.cvut.cz/~ondra
    Anicka Kucerovaemail: <anicka@klobouk.fsv.cvut.cz> hmpg:http://cml.fsv.cvut.cz/~anicka
    Matej Leps email:<matej.leps@fsv.cvut.cz>