It does not require any special properties of the objective function and is low-sensitive to noise. The numerical procedure possesses a simple structure and can be applied very flexibly in dependence on the problem by adjusting several effective control parameters. This concept permits an implementation of arbitrary nonlinear algorithms  as mapping models, e.g. for structural analyses, into a fuzzy analysis with α-level optimization.

The primary algorithm of the modified evolution strategy is formulated for continuous coordinates and constant constraints, which corresponds to a fuzzy analysis with non-interactive fuzzy input variables. An extension to more general conditions, in particular, for dealing with discrete optimization problems is straightforward.

The solution technique targets at a minimum computational cost. This increases approximately linearly with the number of dimensions of the problem. The modified evolution strategy may be characterized as a generally applicable, numerically efficient and robust optimization technique.

numerical procedure

The optimization problem is described by the decision/design variables x_i, the objective function z = f(x₁, ..., x_n), and some constraints specifying the permissible domain for the x_i. With respect to a fuzzy analysis with α-level optimization, the decision variables are defined by the input variables x_i ∈ , i = 1, ..., n, the objective function is described by the mapping model M, and the constraints are given by (x₁, ..., x_n) ∈ , see Fig. 1.






The numerical procedure is formulated as a (1+1) evolution strategy that is modified by incorporating some elements from a gradient descend method and from a Monte Carlo method.

The starting point representing the first parent point x^[q] = x^[0] is specified with the aid of uniform distribution over the permissible domain . For generating offspring points  x^[q+1] a maximum and a minimum distance

from the parent point x^[q] are specified for each coordinate x_i, which forms a local search domain. A definition of max_d_i  and min_d_i by compressing  transfers its proportions to the local search domain.

A first offspring point x^[q+1] of x^[q] is generated within this local search domain (between the green and red lines in Fig. 1) by means of a uniform distribution. Its objective function value z^[q+1] = f(x₁^[q+1], ..., x_n^[q+1]) is checked for an improvement compared with z^[q] = f(x₁^[q], ..., x_n^[q]) of the parent point. If an improvement is obtained, the search proceeds along the randomly selected direction until no further improvement in the objective function value occurs. If the boundary of the permissible domain is crossed over, the non-permissible points are drawn back on the permissible boundary to comply with the constraints (x₁, ..., x_n) ∈ .

If the randomly determined offspring point x^[q+1] does not lead to an improvement, the next offspring point x^[q+2] is positioned at the same distance as x^[q+1] from the parent point x^[q] but in the opposite direction

If the offspring point x^[q+2] also shows no further improvement in the value of z^[q+2], the next point x^[q+3] is again determined randomly starting from x^[q] as in the computation of x^[q+1], and so on.

This procedure is continued until no further improvement is achieved within a given number of tests. Then, the distance bounds are reduced by a factor to refine the search. After some refinements and further unsuccessful steps the procedure is terminated. The last parent point is taken as the optimum point x_opt.

To increase the numerical efficiency, existing points x^[q] with known objective function values z^[q] from previous optimizations are reused. In a defined close neighborhood of new points this available information is exploited instead of evaluating the objective function.

A further efficiency feature is caused by the determination procedure for offspring points. This guides the search, by higher probability, towards the corners of the permissible domain , where the x_opt are located in the case of monotonic mappings. Also, the direction of the largest extent of  is preferred compared with other directions.

Finally, several control parameters are defined to adjust the optimization procedure to the problem in each particular case and thus to optimize the numerical search performance. These control parameters specify, for example, the size of the local search domain, the neighborhood for reusing existing points, the refinement factor for the local search domain, and termination criteria.