By Thomas Weise

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**Extra resources for Global Optimization Algorithms - Theory and Application**

**Sample text**

X ⋆ = extractOptimalSet(Xany , cF ) : X⋆ ⊆ X ∀x⋆ ∈ X ⋆ ⇒ ∃ i ∈ 0 . . 4 demonstrates how the extraction of an optimal set can be performed. 3 Pruning the Optimal Set As already mentioned, there may be very many if not inﬁnite many optimal solutions for a problem. On the other hand, the optimal set X ⋆ computed 36 1 Introduction by the optimization algorithms cannot grow inﬁnitely because we only have limited memory. Therefore we need to perform an action called pruning which reduces the size of the optimal set to a given limit [115, 116, 117].

Artiﬁcial ants are simulated little insects driven by a small program describing their behavior. They are placed on a map with food and obstacles. 4 on page 284 for the artificial ant problem and Chapter 4 on page 139 for genetic programming. 2 Optima, Gradient Descend, and Search Space maximum outside of the sub-domain global maximum local minimum 11 f local maximum X2 X1 global minimum inside X~ ~ outside X Fig. 4: Global and local optima of a two-dimensional function. used to ﬁnd a program x⋆ ∈ X allowing the ant to pile a maximum of food.

Therefore x1 ≻ x2 ⇔ cF (x1 , x2 ) < 0. 7 The Optimal Set Most multi-objective optimization algorithms return a set of optimal solutions X ⋆ instead of a single individual x⋆ . They keep internally track of the set of best solution candidates known. We use the prevalence-based deﬁnition for optimal sets as introduced on page 20 since it supersets all other deﬁnitions. 1 Updating the Optimal Set Whenever a new individual is created, X ⋆ may change. Possible, the new individual must be included in the optimal set or even prevails some of the solution candidates contained therein.