The (1+1) evolution strategy (ES), a simple, mutation-based evolutionary algorithm for continuous optimization problems, is analyzed. In particular, we consider the most common type of mutations, namely Gaussian mutations, and the 1 5 -rule for mutation adaptation, and we are interested in how the runtime/number of function evaluations to obtain a predefined reduction of the approximation error depends on the dimension of the search space.The most discussed function in the area of ES is the so-called SPHERE-function given by SPHERE: R n → R with x → x Ix (where I ∈ R n×n is the identity matrix), which also has already been the subject of a runtime analysis. This analysis is extended to arbitrary positive definite quadratic forms that induce ellipsoidal fitness landscapes which are "close to being spherically symmetric", showing that the order of the runtime does not change compared to SPHERE. Furthermore, certain positive definite quadratic forms f : R n → R with x → x Qx, where Q ∈ R n×n , inducing ellipsoidal fitness landscapes that are "far away from being spherically symmetric" are exemplarily investigated, namely f (x) = · x 2 1 + · · · + x 2 n/2 + x 2 n/2+1 + · · · + x 2 n with = poly(n) such that 1/ → 0 as n → ∞. It is proved that the optimization very quickly stabilizes and that, subsequently, the runtime to halve the approximation error is ( · n) compared to (n) for SPHERE.
In practical optimization, applying evolutionary algorithms has nearly become a matter of course. Their theoretical analysis, however, is far behind practice. So far, theorems on the runtime are limited to discrete search spaces; results for continuous search spaces are limited to convergence theory or even rely on validation by experiments, which is unsatisfactory from a theoretical point of view.The simplest, or most basic, evolutionary algorithms use a population consisting of only one individual and use random mutations as the only search operator. Here the so-called (1+1) evolution strategy for minimization in R n is investigated when it uses isotropically distributed mutation vectors. In particular, so-called Gaussian mutations are analyzed when the so-called 1/5-rule is used for their adaptation.Obviously, a reasonable analysis must respect the function to be minimized, and furthermore, the runtime must be measured with respect to the approximation error. A first algorithmic analysis of how the runtime depends on n, the dimension of the search space, is presented. This analysis covers all unimodal functions that are monotone with respect to the distance from the optimum. It turns out that, in the scenario considered, Gaussian mutations in combination with the 1/5-rule indeed ensure asymptotically optimal runtime; namely, Θ(n) steps/function evaluations are necessary and sufficient to halve the approximation error.
Numerical simulation is already an important cornerstone for aircraft design, although the application of highly accurate methods is mainly limited to the design point. To meet future technical, economic and social challenges in aviation, it is essential to simulate a real aircraft at an early stage, including all multidisciplinary interactions covering the entire flight envelope, and to have the ability to provide data with guaranteed accuracy required for development and certification. However, despite the considerable progress made there are still significant obstacles to be overcome in the development of numerical methods, physical modeling, and the integration of different aircraft disciplines for multidisciplinary analysis and optimization of realistic aircraft configurations. At DLR, these challenges are being addressed in the framework of the multidisciplinary project Digital-X (4/ 2012-12/2015). This paper provides an overview of the project objectives and presents first results on enhanced disciplinary methods in aerodynamics and structural analysis, the development of efficient reduced order methods for load analysis, the development of a multidisciplinary optimization process based on a multi-level/variable-fidelity approach, as well as the development and application of multidisciplinary methods for the analysis of maneuver loads.
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