**MR.BEAM PROJECT**

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.

Rainer Storn: On the usage of differential evolution for function optimization, NAPHIS, 1996

**Authors:**

**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>