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The space spanned by all possible fitness values V is normally a subset of the positive real numbers V ⊆ R+ . 3 on page 27. 3 The Structure of Optimization 47 The origin of the term fitness has been borrowed biology36 [1924, 1633] by the evolutionary algorithms community. When the first applications of genetic algorithms were developed, the focus was mainly on single-objective optimization. Back then, they called this single function fitness function and thus, set objective value ≡ fitness value. This point of view is obsolete in principle, yet you will find many contemporary publications that use this notion.
26, we will use the notation Op(x) for the application of any of the operations searchOp ∈ Op to the genotype x. With Opk (x) we denote k successive applications of (possibly different) search operators. If the parameter x is left away, i. , just Opk is written, this chain has to start with a search operation with zero arity. 27 (Completeness). A set Op of search operations searchOp is complete if and only if every point g1 in the search space G can be reached from every other point g2 ∈ G by applying only operations searchOp ∈ Op.
From the concept of Pareto optimization to the Method of Inequalities, the need to compare elements of the problem space in terms of their quality as solution for a given problem winds like a read thread through this matter. Even the weighted sum approach and the External Decision Maker do nothing else than mapping multi-dimensional vectors to the real numbers in order to make them comparable. If we compare two solution candidates x1 und x2 , either x1 is better than x2 , vice versa, or both are of equal quality.