Keywords: genetic algorithm, simulated annealing, differential evolution.

This contribution focuses on the comparison of several stochastic optimization algorithms used by authors in their previous works for solution of some problems arising in civil engineering. Since each of these algorithms was designed for one specific task only, we expect that comparison of the performance of individual methods should give valuable hints about the robustness and the applicability of a particular approach for the solution of real-world problems.

For the sake of completeness, brief characteristics of examined functions and algorithms follow.

Test functions to optimize

Chebychev trial polynomial problem. Very well known optimization problem that has been used as a benchmark many times. Its solution was one of the first successful applications of Differential Evolution method [5].

"Type 0" function. This function was proposed in reference [1] to study the behaviour of selected optimization methods for high-dimensional problems on real domains. It is constructed in such a way that it has a single extreme, is continuous and monotone in all directions with respect to the distance from the extreme. Tests for dimensions ranging from 1 to 200 are performed in this study.

Reinforced concrete beam layout. This problem was studied in reference [2,4]. The goal is to find the least expensive structure which yet satisfies all strength and serviceability requirements. Moreover, due to technological reasons the optimized parameters were represented as integer/binary variables. Therefore, this function should test the ability of an algorithm to optimize the discrete function with a moderate number of unknowns and constraints.

Formulation of a periodic unit cell. The motivation of this problem comes from the study of fiber-reinforced composite materials with complicated microstructures. Our task is to replace such a material system by a more simple object - the Periodic Unit Cell, consisting of a few reinforcements only, which resembles the original structure as closely as possible. Since the resulting objective function is multi-modal and noisy, it should verify the ability of a tested algorithm to escape from local minima.

Applied methods

Differential evolution (DE). This method differs from classical genetic algorithms in many points of view. It uses a single recombination operator that can be presented as a cross-over. No mutation operators are introduced and the selection method consists of replacing worse performing solution vectors with better ones only. It is reported to perform very well for a large number of optimization problems[5].

Simplified Atavistic Differential Evolution (SADE). This method was proposed in reference[1] to eliminate the strong dependence of the Differential Evolution on a problem dimension. It enriches DE by some features taken from classical genetic algorithms, like mutation operators and tournament selection method.

Integer Augmented Simulated Annealing (IASA). This method is a modification of the standard version of the Augmented Simulated Annealing algorithm[3] with integer coding and operators. Moreover, to improve its local searching abilities, two operators were adopted from a real-coded approach. The performance of this method for optimization of structural design with discrete variables was very satisfactory.

Real-valued Augmented Simulated Annealing (RASA). This algorithm incorporates real-valued operators into the framework of the Augmented Simulated Annealing. This algorithm showed to be well-suited for the problem of periodic unit cell formulation[4].

1

O. Hrstka and A. Kucerová, "Search for optimization method on multidimensional real domains", Contributions to Mechanics of Materials and Structures, CTU Reports, CTU Publishing House, 4, 87-104, 2000.

2

M. Leps and M. Sejnoha: "New Approach to Optimization of Reinforced Concrete Beams", Computational Concrete Structures Technology, Civil-Comp Press, Edinburgh, 143-151, 2000. doi:10.4203/ccp.69.6.3

3

S.W. Mahfoud and D. E. Goldberg, "A genetic algorithm for parallel simulated annealing.", In Parallel Problem Solving from Nature, 2, 301-310, 1992.

4

K. Matous, M. Leps, J. Zeman and M. Sejnoha, "Applying genetic algorithms to selected topics commonly encountered in engineering practice", Computer Methods in Applied Mechanics and Engineering, 190, 13-14, 1629-1650, 2000. doi:10.1016/S0045-7825(00)00192-4

5

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

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