By Maksimov V. I.

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**Extra resources for A Boundary Control Problem for a Nonlinear Parabolic Equation**

**Sample text**

For crossover basically goes the same. Selection operations33 choose the set of individuals which will take part in reproduction. They can either return a small group of best individuals or a wide spread of existing solution candidates. The same goes for archive pruning techniques which truncate the set of known good solutions if it becomes too large. While algorithms that favor exploitation have a fast convergence, they run a great risk of not ﬁnding the optimal solution and maybe get stuck at a local optimum.

It may however be an interesting fact to know that there exist proofs that some optimization algorithms (like simulated annealing and random optimiza- 12 1 Introduction tion) will always ﬁnd the global optimum (when granted a very long, if not inﬁnite, processing time). 2 on page 8 represents an example for the objective values of a function f : R2 → R. Such a function can be considered as a field 17 an assignment of a (quantity (the objective values) to every point of the (two-dimensional) space.

7 The Optimal Set 37 directly on the n = |F | objective values of the individuals and hence, can treat them as n-dimensional vectors. We view the n-dimensional space as a grid, creating d divisions in each dimension. The span of each dimension is deﬁned by the minimum and maximum objective values of the individuals in that dimension. The individuals with the minimum/maximum values are preserved always. Therefore, it is not possible to deﬁne maximum optimal set sizes k which are smaller then 2n.