CONTEXTO HUMANO.

In this introductory exposition, we shall only consider binary genetic al­ gorithms which use the underlying alphabet However, in applications where real parameters are optimized in a compact domain of

it may be advantageous to consider a larger, discrete alphabet representing a finite, equidistant set of real numbers2. Such an approach is advocated and used, e.g., in work by Markus, Renner, & Vanza (Márkus et al., 1997, p. 48), Kondoh & Schmitt (Schmitt and Kondoh, 2000), and Savchenko& Schmitt (Savchenko and Schmitt, 2001). See also work by Nomura & Shimo­ hara (Nomura and Shimohara, 2001). One task in future work is certainly to generalize the approach taken here and in (Márkus et al., 1997; Savchenko and Schmitt, 2001; Schmitt and Kondoh, 2000; Schmitt, 2001; Schmitt, 2002) to the case of a continuous alphabet.

We shall consider (the genome of) creatures or candidate solutions in the model world to which the genetic algorithm is applied as strings of length over the alphabet where usually Let denote the set of creatures.

The set of populations to which the genetic algorithm is applied, is the set of of creatures, We shall assume that is even, and if not explicitly stated otherwise. Set Then every population is (canonically identified with) a string of length L over Let

If is a population, then we define

and If then we shall write

if A spot in the genome is, by definition, the position of one of the letters in a word over representing a creature or population. For we define the Hamming distance as the number of spots in the genome where and differ.

The vector space underlying our model for genetic algorithms is the free complex vector space3 over Thus, becomes the basis of which is con­ sistent with the notation introduced in section 1.1. Every population can be identified canonically with an integer in i.e., the letters com­

prising are used as digits to define the integer in binary representation. This

induces a natural order on Now, we identify with

2_{If a regular programming language such as Fortran or C is employed for the Implementation of the genetic}

algorithm, then only a finite set of real numbers is used. In many cases, the search space can be restricted further by a rough analysis of the given optimization problem to a finite interval

3_{(Schmitt, 2001, Sec. 2.6) discusses the identification of with the L-fold tensor product of the free}

vector space over . This can be used for analysis of mutation as in (Schmitt, 2001, Prop. 3.3, Prop. 3.6) and various crossover operators as in (Schmitt, 2002, Sec. 2.4–5).

and by linear extension, this defines an isomorphism Let

By suppressing notation for the particular isomorphism just de­ fined, we can consistently denote the of where X stands for a linear operator acting on and the corresponding matrix acting on as

Let be the free vector space over all populations which are uniform, i.e., which consist of copies of a single creature. Consequently, shall denote the set of uniform populations. In addition, shall denote the orthog­

onal projection onto The following simple result whose proof is listed in section 5 allows for compact notation in some of the subsequent results and proofs.

1.4.1. Proposition. Let be a linear map such that

for every Then X satisfies and

2.

THE GENETIC OPERATORS

The genetic operators can be categorized into two groups: the mixing opera­ tors mutation and crossover which are used concurrently in a genetic algorithm, and the selection operators such as proportional fitness selection, tournament selection or simulated annealing type selection which provide alternatives for implementation of a genetic algorithm. However, crossover has some common features with selection such as leaving uniform populations invariant.

Mutation-crossover has been investigated by many researchers. The most common and simple framework investigated is multiple-spot mutation com­ bined with single-cutpoint crossover in the multi-set model for populations over a binary alphabet. Earlier references include the work of Davis & Principe (Davis and Principe, 1991; Davis and Principe, 1993), Vose & Liepins (Vose and Liepins, 1991), and Nix & Vose (Nix and Vose, 1992). Other work which has significance to the present work are the papers by Suzuki (Suzuki, 1997; Suzuki, 1998) (compare (Schmitt, 2001, Sec. 8.3)). In (Vose and Wright, 1998a; Vose and Wright, 1998b), Vose & Wright discuss the mutation-crossover matrix via the Walsh-transform. Note that the Vose-Liepins version of mutation- crossover is different than the mutation-crossover operation discussed here. Section 3.5 discusses how to embed the Vose-Liepins version of mutation- crossover into the model presented here.

In contrast to popular belief, one must observe that mutation and not cross­ over is the main thriving force for mixing in a genetic algorithm. Mutation assures weak ergodicity, and as an immediate consequence strong ergodicity of the Markov chain describing the mathematical model for the genetic al­

gorithm (see section 3.2 and Theorem 3.3.2). In the generic situation of a blind search with a fitness function of largely unknown behavior, it is muta­ tion and not crossover that drives the algorithm. In particular, mutation cre­ ates the noise that destroys uniform populations containing suboptimal solu­ tions which is something crossover cannot do. Note that Banzhaf, Francone & Nordin (Banzhaf et al., 1996) report experimental results that favor larger mu­ tation rates, i.e., strong mixing by mutation. There are quite natural situations where crossover asymptotically plays no role in the probabilistic outcome of a genetic algorithm ( cf. (Schmitt, 2001, Thm. 8.3.3, Thm. 8.5.2–3)). Crossover assists mutation in accelerating the mixing process towards the uniquely deter­ mined fix-point of the fully positive, symmetric mutation-crossover operator

( cf. Proposition 2.2.3.4). This statement is made very precise in (Schmitt et al., 1998, Prop. 10) and (Schmitt, 2001, Thm. 6.1). However, there are “royal road cases” where by the design of the fitness landscape the acceleration by crossover is significant (cf. work by Jansen & Wegener (Jansen and Wegener, 2001)). In regard to the family of selection operators, we shall restrict us here to the case of proportional fitness selection which is discussed in detail in section 2.3. With respect to tournament selection, we refer the reader to work by Goldberg (Goldberg, 1990), (Goldberg, ) as well as Goldberg & Deb (Goldberg and Deb, 1991), and to the monographs by Mitchell (Mitchell, 1996, p. 170) and Michalewicz (Michalewicz, 1994, p. 59). In regard to sim­ ulated annealing type selection, we refer the reader to the introductory paper by Aarts & Van Laarhoven (Aarts and van Laarhoven, 1989), work by Lozano, Larrañaga, Graña & Albizuri (Lozano et al., 1999) and Mahfoud & Goldberg (Mahfoud and Goldberg, 1992; Mahfoud and Goldberg, 1995), (Schmitt et al., 1998, p. 124: Rem. on Simulated Annealing) and (Schmitt, 2001, Sec. 6.2).

2.1

MULTIPLE-SPOT MUTATION

Mutation models random change in the genetic information of creatures, and is inspired by random change of genetic information in living organisms, e.g., through the effects of radiation or chemical mismatch. Multiple-spot mu­ tation has been studied theoretically by many authors as discussed in the introductory paragraphs to section 2. See also (Schmitt et al., 1998, Sec. 2.1, p. 110 ff., “multiple-bit mutation”), (Schmitt, 2001, Sec. 3.3), (Schmitt, 2002, Sec. 2.2). In this section, we shall repeat some of the analysis in (Schmitt et al., 1998). However, our discussion here will be limited to the absolute minimum. Multiple-spot mutation is the most commonly used procedure for mutation in implementations of genetic algorithms.

2.1.1. Definition (multiple-spot mutation Let

in the current population. The decision for change is made positively with probability

the letter at spot

(STEP 2) If the decision has been made positively in step 1, then (STEP 1) Decide probabilistically whether or not to change the letter at spot

is altered, i.e., the bit at spot is flipped.

Let also denote the stochastic matrix associated with multiple-spot mu­ tation. acts on in the sense of line (12) and describes transition proba­ bilities for entire populations.

If then mutation is the identity operation. If then the cur­ rent population is bit-wise complemented. In what follows, we shall usually exclude these two trivial cases, if we discuss mutation.

2.1.2. Proposition. Let denote multiple-spot mutation as in Def­

inition 2.1.1 with mutation rate Suppose that Then we have:

then

1 The coefficients of are given as follows: In particular, is a fully positive and symmetric. 2 If

3 If then In addition,

4 If is sufficiently small, then is an invertible matrix. 5 is the uniquely determined invariant probability distribution of PROOF: In order to pass from to one has to make the decision to change one of bits times, and one has to retain bits at spots. Independently from the order of such steps, the combined probability for the

required procedure is given by One has

This shows statement (1). If

then This combined with statement

(1) shows statement (2). By statement (1), one has

Observing that is stochastic then yields statement (3). The determinant is a continuous (polynomial) function in the

coefficients of Statement (3) shows that as

This implies that for sufficiently small Consequently,

is invertible (Lang, 1970, p. 108: Thm. 8) for such This shows statement (4). Since is symmetric, it follows that is an invariant vector of

Proposition 2.1.2.4 is contained in (Schmitt et al., 1998, Prop. 3.4) or the slightly stronger (Schmitt, 2001, Prop. 3.6.3) which show that is invertible4

for

2.2

SINGLE-CUTPOINT REGULAR