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23.2 Global Optimization of Distributed Systems 401 -20 -10 0 10 20 30 -4 -3 -2 1 2 4 3 -5 -1 y j 1 (x) (x) º « y 2 (x) « Figure 23.2: ϕ ( x ), the evolved ψ 1 ( x ) ϕ ( x ), and ψ 2 ( x ). 3. A small population size decreases the diversity and furthers “incest” between similar solution candidates. Due to a lower rate of exploration, only a local minimum of the quality value will often be yielded. 4. Allowing functions of large depth and putting low pressure against bloat (see Section 4.10.3 on page 224 ) leads to uncontrolled function growth. The real laws ϕ that we want to approximate with symbolic regression do usually not consist of more than 40 expressions. This is valid for most physical, mathematical, or financial equations. Therefore, the evolution of large functions is counterproductive in those cases. Although we made some of these mistakes intentionally, there are many situations where it is hard to determine good parameter sets and restrictions for the evolution and they occur accidentally. 23.1.4 Limits of Symbolic Regression Often, we cannot obtain an optimal approximation of ϕ , especially if ϕ cannot be repre- sented by the basic expressions available to the regression process. One of these situations has already been discussed before: the case where ϕ has no closed arithmetical expression. Another possibility is that the regression method tries to generate a polynomial that ap- proximates the ϕ , but ϕ does contain different expressions like sin or e x or polynomials of an order higher than available. Yet another problem is that the values y i are often not results computed by ϕ directly but could, for example, be measurements taken from some physical entity and we want to use regression to determine the interrelations between this entity and some known parameters. Then, the measurements will be biased by noise and systematic measurement errors. In this situation, f ( ψ ,A ) will be greater than zero even after a successful regression. 23.2 Global Optimization of Distributed Systems 23.2.1 Introduction Optimization algorithms are methods for finding optimal configurations of different features of their solution candidates. Many aspects of distributed systems are configurable or depend
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402 23 Real-World Applications on parameter settings, such as the topology, security, and routing. Hence, there is a huge potential for using global optimization algorithms in order to improve them. And indeed, this potential is widely utilized. The study by Sinclair [1886] from 1999 re- ported that more than 120 papers had been published on work which employed Evolutionary Computation for optimizing network topologies and dimension, node placement, routing, and wavelength or frequency allocation. The comprehensive master’s thesis by Kampstra from 2005 [1087, 1088] builds on this aforementioned study and classifies over 400 papers. Ac- cording to Kampstra, communication networks was the field with the most researchers listed in EvoWeb, the European Network of Excellence in Evolutionary Computing, in 2005. The first workshop on this topic,
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