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Unformatted text preview: 1.3 The Structure of Optimization 41 (1,3) (3,3) (0,2) (0,2) (0,2) (0,2) Genotype-Phenotype Mapping Objective Function(s) (0,0) (0,1) (0,2) (0,3) (1,0) (1,1) (1,2) (1,3) (2,0) (2,1) (2,2) (2,3) (3,0) (3,1) (3,2) (3,3) Problem Space X Search Space G 0100 1010 1100 1101 0000 1000 1001 0101 0110 1110 1111 0111 0001 0010 0011 1011 Objective Space R n Í Y Fitness Space R + Í V Fitness Assignment Process Genotype-Phenotype Mapping Objective Function(s) (0,0) (0,1) (0,3) (1,0) (1,1) (1,2) (1,3) (2,0) (2,1) (2,2) (2,3) (3,0) (3,1) (3,2) (3,3) Population (Phenotypes) Population (Genotypes) Objective Values Fitness Values Fitness Assignment Process 1111 1111 1110 1000 0100 0111 0111 0010 0010 0010 0010 1111 f i t n e s s f ( x ) 1 f ( x ) 1 Pop Í G X ´ Pop Í G X ´ Fitness and heuristic values (normally) have only a meaning in the context of a population or a set of solution candidates. Solution Space X Í S F(x Î X ) Î Y v(x) Î V The Involved Spaces The Involved Sets/Elements Figure 1.13: Spaces, Sets, and Elements involved in an optimization process. 42 1 Introduction Definition 1.19 (Solution Candidate). A solution candidate x is an element of the problem space X of a certain optimization problem. In the context of evolutionary algorithms, solution candidates are usually called pheno- types . In this book, we will use both terms synonymously. Somewhere inside the problem space, the solutions of the optimization problem will be located (if the problem can actually be solved, that is). Definition 1.20 (Solution Space). We call the union of all solutions of an optimization problem its solution space S . X ⋆ ⊆ S ⊆ X (1.22) This solution space contains (and can be equal to) the global optimal set X ⋆ . There may exist valid solutions x ∈ S which are not elements of the X ⋆ , especially in the context of constraint optimization (see Section 1.2.3 ). The Search Space Definition 1.21 (Search Space). The search space G of an optimization problem is the set of all elements g which can be processed by the search operations. As previously mentioned, the type of the solution candidates depends on the problem to be solved. Since there are many different applications for optimization, there are many different forms of problem spaces. It would be cumbersome to develop search operations time and again for each new problem space we encounter. Such an approach would not only be error-prone, it would also make it very hard to formulate general laws and to consolidate findings. Instead, we often reuse well-known search spaces for many different problems. Then, only a mapping between search and problem space has to be defined (see page 44 ). Although this is not always possible, it allows us to use more out-of-the-box software in many cases....
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