List of Symbols V solution space for the global optimization problem d(·, ·) distance function in the space V dH (·, ·) Hamming distance function · 2 Euclidean norm in the space V · arbitrary norm in the space V N dimension of the global optimization problem (N =dim(V )) D admissible set of solutions to the global optimization problem Φ : D→R+ objective function of a maximization problem Φ˜ : D→R+ objective function of a minimization problem x∗ global maximizer (or minimizer) to the global optimization problem x+ local maximizer (or minimizer) to the global optimization problem meas(·) Lesbegue measure function Loc set of local optimization methods loc(·) local optimization method Rloc attractor of the local minimizer x+ with respect to the strictly x+ decreasing local optimization method loc(·) B + x+ basin of attraction of the local minimizer x P random sample, population Pr(·) probability measure M(A) space of probabilistic measures over the set A b(·) function that selects the best fitted individual in a population Dr ⊂D grid of encoded points (set of phenotypes) r =#Dr the cardinality of phenotype set U genetic universum (set of genotypes) code : U →Dr encoding bijective function dcode : D −→ U partial function of inverse encoding (decoding partial function) Ω binary genetic universum in Chapters 3 - 5, space of elementary events in Section 6.1.5 l binary code length Z2 group {0, 1} with the addition modulo 2 204 List of Symbols ⊕ the addition operator in Z2, coordinate-by-coordinate addition of binary vectors from Z2×,...,×Z2 codea(·) affine binary encoding codeG(·) Gray binary encoding f : U → R+ fitness function Scale(·) nonlinear function that modifies fitness µ number of individuals in the parental population self (·) probability distribution of selection Elite subset of individuals that pass to the next epoch with the probability 1 1 vector of l ones ˆi the inverse of the binary vector i (ˆi = 1 ⊕ i) pm rate of binary mutation pc rate of binary crossover [·] evaluation function for boolean expressions ⊗ coordinate-by-coordinate multiplication of binary vectors type type of binary crossover mutx(·) probability distribution for the binary mutation of an individual x crossx,y(·) probability distribution for the binary crossover of individuals x and y N (e, C) realization (result of sampling) of the N-dimensional Gauss random variable with the mean vector e and the covariance matrix C N (e, σ) realization (result of sampling) of the one dimensional Gauss random variable with the mean value e and the standard deviation σ U[0, 1] realization (result of sampling) of the random variable of the uniform probability distribution over the real interval [0, 1] λ number of offspring individuals in a single genetic epoch κ individual life time parameter of the evolutionary algorithm in section 5.3.2, raster resolution in section 6.1.10 S space of schemata H schemata from the space S ∆(H) length of the schemata H ℵ(H) degree of the schemata H E(·) expected value operator E space of states of genetic algorithms eqp equivalence relation among vectors of genotypes Sµ group of permutations of the µ-element set Λr−1 unit r − 1 simplex in Rr r−1 Xµ ⊂ Λ finite set of states of the genetic algorithm with finite population (µ<+∞) #A cardinality of the set or multiset A n =#E cardinality of the space of states πt genetic algorithm state probability distribution in the epoch t τ Markov transition function of genetic algorithm states Q probability transition matrix F (·) selection operator for the simple genetic algorithm List of Symbols 205 M(·) mixing operator for the simple genetic algorithm G(·) genetic (heuristics) operator for the simple genetic algorithm M mixing matrix K⊂Λr−1 set of fixed points of the genetic operator G dim(V ) dimension of the vector space V diam(A) diameter of the subset A of a metric space N set of natural numbers Z set of integers Z+ set of nonnegative integers Z+ = N ∪{0} R set of rational numbers R+ set of nonnegative rational numbers top(V ) topology on V (the family of open sets in V ) A the topological closure of the set A in the proper topology (A)− the complement of the set A, i.e. (A)− = V \ A if it is contained in the space V χA the characteristic function of the set A. 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