Abstract

In this article, we consider a fitness-level model of a non-elitist mutation-only evolutionary algorithm (EA) with tournament selection. The model provides upper and lower bounds for the expected proportion of the individuals with fitness above given thresholds. In the case of so-called monotone mutation, the obtained bounds imply that increasing the tournament size improves the EA performance. As corollaries, we obtain an exponentially vanishing tail bound for the Randomized Local Search on unimodal functions and polynomial upper bounds on the runtime of EAs on the 2-SAT problem and on a family of Set Cover problems proposed by E. Balas.

1  Introduction

Evolutionary algorithms are randomized heuristic algorithms employing a population of tentative solutions (individuals) and simulating an evolutionary type of search for optimal or near-optimal solutions by means of selection, crossover, and mutation operators. The evolutionary algorithms with crossover operator are usually called genetic algorithms (GAs). Evolutionary algorithms in general have a more flexible outline and include genetic programming, evolution strategies, estimation of distribution algorithms, and other evolution-inspired paradigms. Evolutionary algorithms are now frequently used in areas of operations research, engineering, and artificial intelligence.

Two major outlines of an evolutionary algorithm are the elitist evolutionary algorithm, that keeps a certain number of most promising individuals from the previous iteration, and the non-elitist evolutionary algorithm, that computes all individuals of a new population independently using the same randomized procedure. In this article, we focus on the non-elitist case.

One of the first theoretical results in the analysis of non-elitist GAs is the Schemata Theorem (Goldberg, 1989) which gives a lower bound on the expected number of individuals from some subsets of the search space (schemata) in the next generation, given the current population. A significant progress in understanding the dynamics of GAs with non-elitist outline was made in Vose (1995) by means of dynamical systems. However most of the findings in Vose (1995) apply to the infinite population case, and it is not clear how these results can be used to estimate the applicability of GAs to practical optimization problems. A theoretical possibility of constructing GAs that provably optimize an objective function with high probability in polynomial time was shown in Vitányi (2000) using rapidly mixing Markov chains. However, Vitányi (2000) provides only a very simple artificial example where this approach is applicable and further developments in this direction are not known to us.

One of the standard approaches to studying evolutionary algorithms in general is based on the fitness levels (Wegener, 2002). In this approach, the solution space is partitioned into disjoint subsets, called fitness-levels, according to values of the fitness function. In Lehre (2011), the fitness-level approach was first applied to upper-bound the runtime of non-elitist mutation-only evolutionary algorithms. Here and below, by the runtime we mean the expected number of fitness evaluations made until an optimum is found for the first time. Upper bounds of the runtime of non-elitist GAs, involving the crossover operators, were obtained later in Corus et al. (2014) and Eremeev (2017). The runtime bounds presented in Corus et al. (2014) and Lehre (2011) are based on the drift analysis. In Moraglio and Sudholt (2015), a runtime result is proposed for a class of convex search algorithms, including some non-elitist crossover-based GAs without mutation, on the so-called concave fitness landscapes.

In this article, we consider the non-elitist evolutionary algorithm, which uses a tournament selection and a mutation operator but no crossover. The -tournament selection randomly chooses individuals from the existing population and selects the best one of them (see, e.g., Thierens and Goldberg, 1994). The mutation operator is viewed as a randomized procedure, which computes one offspring with a probability distribution depending on the given parent individual. In this article, evolutionary algorithms with such outline are denoted as EA. We study the probability distribution of the EA population with regards to a set of fitness levels. The estimates of the EA behavior are based on a priori known parameters of a mutation operator. Using the proposed model we obtain upper and lower bounds on expected proportion of the individuals with fitness above certain thresholds. The lower bounds are formulated in terms of linear algebra and resemble the bound in the Schemata Theorem (Goldberg, 1989). Instead of schemata, here we consider the sets of genotypes with the fitness bounded from below. Besides that, the bounds obtained in this article may be applied recursively up to any given iteration.

The lower bounds on expected proportions of sufficiently fit individuals at iteration also imply the lower bounds on probabilities of finding a genotype with fitness above a specified threshold at any given iteration . Such results are closely related to the area of fixed budget computations, where one has a fixed budget of fitness evaluations that may be spent and the question is how good a solution one can expect to find with this budget (Jansen and Zarges, 2012).

This article pays particular attention to a special case when mutation is monotone. Informally speaking, a mutation operator is monotone if fitter parents have a higher probability of producing fit offspring. One of the most well-known examples of monotone mutation is the bitwise mutation in the case of OneMax fitness function. As shown in Borisovsky and Eremeev (2008), in the case of monotone mutation, one of the most simple evolutionary algorithms, known as the (1+1) EA has the best-possible performance in terms of runtime and probability of finding the optimum.

In the case of monotone mutation, the lower bounds on expected proportions of the individuals turn into equalities for the trivial evolutionary algorithm (1,1) EA. This implies that the tournament selection at least has no negative effect on the EA performance in such a case. This observation is complemented by the asymptotic analysis of the EA with monotone mutation indicating that, given a sufficiently large population size and some technical conditions, increasing the tournament size , always improves the EA performance.

As corollaries of the general lower bounds on expected proportions of sufficiently fit individuals, we obtain polynomial upper bounds on the Randomized Local Search (RLS) runtime on unimodal functions and upper bounds on the runtime of EAs on 2-SAT problem and on a family of Set Cover problems proposed by Balas (1984). Unlike the upper bounds on the runtime of evolutionary algorithms with tournament selection from Corus et al. (2014), Eremeev (2017) and Lehre (2011), which require sufficiently large tournament size, the upper bounds on runtime obtained here hold for any tournament size.

The rest of the article is organized as follows. In Section 2, we give a formal description of the considered EA, introduce an approximating model of the EA population, and define some required parameters of the probability distribution of a mutation operator in terms of fitness levels. In Section 3, using the model from Section 2, we obtain lower and upper bounds on expected proportions of genotypes with fitness above some given thresholds. Section 4 is devoted to analysis of an important special case of monotone mutation operator, where the bounds obtained in the previous section become tight or asymptotically tight. In Section 5, we consider some illustrative examples of monotone mutation operators and demonstrate some applications of the general results from Section 3. In particular, in this section we obtain new lower bounds for probability to generate optimal genotypes at any given iteration for a class of unimodal functions, for 2-SAT problem and for a family of set cover problems proposed by E. Balas (in the latter two cases we also obtain upper bounds on the runtime of the EA). Besides that, in Section 5, we give an upper bound on expected proportion of optimal genotypes for OneMax fitness function. Section 6 contains concluding remarks.

This work extends the conference paper (Eremeev, 2000). The extension consists in comparison of the EA behavior to that of the (1,1) EA, the (1,) EA, and the (1+1) EA in Section 3 and in the new runtime bounds and tail bounds demonstrated in Section 5. The main results from the conference paper are refined and provided with more detailed proofs.

2  Description of Algorithms and Approximating Model

2.1  Notation and Algorithms

Let the optimization problem consist in maximization of an objective function on the set of feasible solutions , where is the search space of all binary strings of length .

The Evolutionary Algorithm EA. The EA searches for the optimal or suboptimal solutions using a population of individuals, where each individual (genotype) is a bitstring , and its components are called genes.

In each iteration, the EA constructs a new population on the basis of the previous one. The search process is guided by the values of a fitness function
formula
where is a penalty function.

The individuals of the population may be ordered according to the sequence in which they are generated; thus the population may be considered as a vector of genotypes , where is the size of population, which is constant during the run of the EA, and is the number of the current iteration. In this article, we consider a non-elitist algorithmic outline, where all individuals of a new population are generated independently from each other with identical probability distribution depending on the existing population only.

Each individual is generated through selection of a parent genotype by means of a selection operator, and modification of this genotype in mutation operator. During the mutation, a subset of genes in the genotype string is randomly altered. In general, the mutation operator may be viewed as a random variable with the probability distribution depending on .

The genotypes of the initial population are generated with some a priori chosen probability distribution. The stopping criterion may be, for example, an upper bound on the number of iterations . The result is the best solution generated during the run. The EA has the following scheme:

  1. Generate the initial population .

  2. For to do

    1. For to do

      1. Choose a parent genotype from by -tournament selection.

      2. Add to the population .

In theoretical studies, the evolutionary algorithms are usually treated without a stopping criterion (see, e.g., Neumann and Witt, 2010). Unless otherwise stated, in the EA we will also assume that

Note that in the special case of the EA with , we can assume that , since the tournament selection has no effect in this case.

(1,) EA and (1+1) EA. In the following sections, we will also need a description of two simple evolutionary algorithms, known as the (1,) EA and the (1+1) EA.

The genotype of the current individual on iteration of the (1,) EA will be denoted by , and in the (1+1) EA it will be denoted by . The initial genotypes and are generated with some a priori chosen probability distribution. The only difference between the (1,) EA and the (1+1) EA consists in the method of construction of an individual for iteration using the current individual of iteration as a parent. In both algorithms the new individual is built with the help of a mutation operator, which we will denote by . In the case of the (1,) EA, the mutation operator is independently applied times to the parent genotype and out of offspring a single genotype with the highest fitness value is chosen as . (If there are several offspring with the highest fitness, the new individual is chosen arbitrarily among them.) In the (1+1) EA, the mutation operator is applied to once. If is such that then ; otherwise .

2.2  The Proposed Model

The EA may be considered as a Markov chain in a number of ways. For example, the states of the chain may correspond to different vectors of genotypes that constitute the population (see Rudolph, 1994). In this case, the number of states in the Markov chain is . Another model representing the GA as a Markov chain is proposed in Nix and Vose (1992), where all populations that differ only in the ordering of individuals are considered to be equivalent. Each state of this Markov chain may be represented by a vector of components, where the proportion of each genotype in the population is indicated by the corresponding coordinate and the total number of states is . In the framework of this model, Vose and collaborators have obtained a number of general results concerning the emergent behavior of GAs by linking these algorithms to the infinite-population GAs (Vose, 1995).

The major difficulties in application of the above-mentioned models to the analysis of GAs for combinatorial optimization problems are connected with the necessity to use the high-grained information about fitness value of each genotype. In the present article, we consider one of the ways to avoid these difficulties by means of grouping the genotypes into larger classes on the basis of their fitness.

Assume that and there are level lines of the fitness function fixed such that . The number of levels and the fitness values corresponding to them may be chosen arbitrarily, but they should be relevant to the given problem and the mutation operator to yield a meaningful model. Let us introduce the sequence of Lebesgue subsets of
formula
Obviously, . For the sake of convenience, we define . Also, we denote the level sets which give a partition of . Partitioning subsets are more frequently used in literature on level-based analysis, compared to the Lebesgue subsets . In this article, we will frequently state that a genotype has a sufficiently high fitness; therefore, the use of subsets will be more convenient in such cases. One of the partitions used in the literature, called the canonical partition, defines as the set of all fitness values on the search space .
Now suppose that for all and the a priori lower bounds and upper bounds on mutation transition probabilities from subset to are known; that is,
formula
Figure 1 illustrates the transitions considered in this expression.
Figure 1:

Transitions from to under mutation.

Figure 1:

Transitions from to under mutation.

Let denote the matrix with the elements where , and . The similar matrix of upper bounds is denoted by . Let the population on iteration be represented by the population vector
formula
where is the proportion of genotypes from in population . The population vector is a random vector, where for since .

Let be the probability that an individual, which is added after selection and mutation into , has a genotype from for , and According to the scheme of the EA this probability is identical for all genotypes of , i.e. .

Proposition 1:

for all .

Proof:

Consider the sequence of identically distributed random variables , where if the -th individual in the population belongs to , otherwise . By the definition, , consequently

Level-Based Mutation. If, for some mutation operator, there exist two equal matrices of lower and upper bounds and , that is, for all then the mutation operator will be called level-based. By this definition, in the case of level-based mutation, does not depend on a choice of genotype and the probabilities are well defined. In what follows, we call a cumulative transition probability. The symbol will denote the matrix of cumulative transition probabilities of a level-based mutation operator.

If the EA uses a level-based mutation operator, then the probability distribution of population is completely determined by the vector . In this case, the EA may be viewed as a Markov chain with states corresponding to the elements of
formula
which is the set of all possible vectors of population of size . Here and below, the symbol is used to denote a vector from the set of all possible population vectors .

The cardinality of set may be evaluated analogously to the number of states in the model of Nix and Vose (1992). Now levels replace individual elements of the search space, which gives a total of possible population vectors.

3  Bounds on Expected Proportions of Fit Individuals

In this section, our aim is to obtain lower and upper bounds on for arbitrary and if the distribution of the initial population is known.

Let denote the probability that the genotype, chosen by the tournament selection from a population with vector , belongs to a subset . Note that if the current population is represented by the vector , then a genotype obtained by selection and mutation would belong to with a conditional probability
formula
1

3.1  Lower Bounds

Expression (1) and the definitions of bounds yield for all :
formula
2
which turns into an equality in the case of level-based mutation and .
Given a tournament size we obtain the following selection probabilities: , and, consequently, . This leads to the inequality:
formula
By the total probability formula,
formula
3
formula
4
where the last expression is obtained by regrouping the summation terms. Proposition 1 implies that . Consequently, since and , expression (4) gives a lower bound
formula
5

Note that Eq. (5) turns into an equality in the case of level-based mutation and . We would like to use Eq. (5) recursively times in order to estimate for any , given the initial vector . It will be shown in the sequel that such a recursion is possible under monotonicity assumptions defined below.

Monotone Matrices and Mutation Operators. In what follows, any -matrix with elements , will be called monotone iff for all from 1 to . Monotonicity of a matrix of bounds on transition probabilities means that the greater fitness level a parent solution has, the greater is its bound on transition probability to any subset . Note that for any mutation operator, the monotone upper and lower bounds exist. Formally, for any mutation operator a valid monotone matrix of lower bounds would be where is a zero matrix. A monotone matrix of upper bounds, valid for any mutation operator is , where is the matrix with all elements equal 1. These are extreme and impractical examples. In reality, a problem may be connected with the absence of bounds which are sharp enough to evaluate the mutation operator properly.

If given some set of levels there exist two matrices of lower and upper bounds such that and these matrices are monotone then operator is called monotone with regards to the set of levels. In this article, we will also call such operators monotone for short. Informally speaking, in the case of monotone mutation the fitter parents have a higher probability of producing fit offspring. Note that by the definition, any monotone mutation operator is level-based, since for all . The following proposition shows how the monotonicity property may be equivalently defined in terms of cumulative transition probabilities.

Proposition 2:
A mutation operator is monotone with regards to the set of levels iff for any such that for any genotypes holds
formula
Proof:
Indeed, suppose that and these matrices are monotone. Then for any genotypes and , holds
formula

Conversely, if for any level and any genotypes and , holds , then taking we note that is equal for all and one can assign The resulting matrices and are obviously monotone.

Proposition 2 implies that in the case of the canonical partition, that is, when is the set of all values of , operator is monotone w.r.t. iff for any genotypes and such that , for any holds
formula
The monotonicity of mutation operator with regards to a canonical partition is equivalent to the definition of monotone reproduction operator from Borisovsky and Eremeev (2001) in the case of single-parent, single-offspring reproduction. According to the terminology of Daley (1968), such random operators are also called stochastically monotone.
As a simple example of a monotone mutation operator we can consider a point mutation operator: with probability keep the given genotype unchanged; otherwise (with probability ) choose randomly from and change gene . As a fitness function we take the function , where . Let us assume and define the thresholds . All genotypes with the same fitness function value have equal probability to produce an offspring with any required fitness value; therefore, this is a case of level-based mutation. In such a case, identical matrices of lower and upper bounds and exist and they are both equal to the matrix of cumulative transition probabilities . The latter consists of the following elements: for all , since point mutation cannot reduce the fitness by more than one level; for because with probability any genotype is upgraded;
formula
because a genotype in can be obtained as an offspring of a genotype from in two ways: either the parent genotype has been upgraded (which happens with probability ) or it stays at level , which happens with probability ; finally because point mutation cannot increase the level number by more than 1. The elements of matrix obviously satisfy the monotonicity condition when . For the case of , we have which is non-negative if Therefore, with any the matrix is monotone in this example and the mutation operator is monotone as well.
Proposition 3:
If is monotone, then for any tournament size and holds
formula
6
besides that (6) is an equality if , operator is monotone and is its matrix of cumulative transition probabilities.
Proof:
Monotonicity of matrix implies that for all so the simple estimate may be applied to all terms of the sum in Eq. (5) and we get
formula
Regrouping the terms in the last bound we obtain the required Inequality (6).

Finally, note that lower bound Eq. (5) holds as an equality if the mutation operator is monotone and ; therefore, the last lower bound is an equality in the case of monotone and .

Lower Bounds from Linear Algebra. Let be a -matrix with elements let be the identity matrix of the same size, and denote . With these notations, Inequality (6) takes a short form Here and below, the inequality sign “” for some vectors and means the component-wise comparison, i.e. iff for all . The following theorem gives a component-wise lower bound on vector for any .

Theorem 1:
Suppose that is some matrix norm. If matrix is monotone and , then for all holds
formula
7
and Inequality (7) turns into an equation if the tournament size , the mutation operator used in the EA is monotone and is its matrix of cumulative transition probabilities.

The proof of this theorem is similar to the well-known inductive proof of the formula for a sum of terms in a geometric series Note that the recursion is similar to the recursive formula assuming . However, in our case, matrices and vectors replace numbers, we have to deal with inequalities rather than equalities and the initial element may be nonzero unlike .

Proof of Theorem 4:

Let us consider a sequence of -dimensional vectors where , . We will show that for any , using induction on . Indeed, for the inequality holds by the definition of . Now note that the right-hand side of Inequality (6) will not increase if the components of are substituted with their lower bounds. Therefore, assuming we already have for some and substituting for we make an inductive step .

By properties of the linear operators (see, e.g., Kolmogorov and Fomin 1999, Chapter III, § 29), due to the assumption that we conclude that matrix exists.

Now, using the induction on for any we will obtain the identity
formula
which leads to Inequality (7). Indeed, for the base case of by the definition of we have the required equality. For the inductive step, we use the following relationship
formula

In conditions of Theorem 4, the right-hand side of Inequality (7) approaches when tends to infinity; thus, the limit of this bound does not depend on distribution of the initial population.

In many evolutionary algorithms, an arbitrary given genotype may be produced with a nonzero probability as a result of mutation of any given genotype . Suppose that the probability of such a mutation is lower bounded by some for all . Then one can obviously choose some monotone matrix of lower bounds that satisfies for all . Thus, for all . In this case, one can consider the matrix norm . Due to the monotonicity of we have , so , and the conditions of Theorem 4 are satisfied. A trivial example of a matrix that satisfies the above description would be a matrix where all elements are equal to .

The framework of fixed budget computations, proposed in Jansen and Zarges (2012) for the RLS and the (1+1) EA, naturally extends to the EAs. In this framework, one has a fixed budget of fitness evaluations that may be spent by the EA and it is required to estimate the expectation where is the greatest fitness value found during iterations . Theorem 4 implies lower bounds on probabilities to generate a genotype with fitness above the specified thresholds at any given iteration : where is the right-hand side of Inequality (7). These bounds may be used in the fixed budget framework: Note that , so using the Abel transform we get a fixed budget estimate
formula

Application of Theorem 4 may be complicated due to difficulties in finding the vector and in estimation the effect of multiplication by matrix Some known results from linear algebra can help to solve these tasks, as the example in Subsection 5.2 shows. However, sometimes it is possible to obtain a lower bound for via analysis of the (1,1) EA algorithm, choosing an appropriate mutation operator for it. This approach is discussed below.

Lower Bounds from Associated Markov Chain. Suppose that a partition defined by contains no empty subsets and let denote a -matrix, with components
formula
Note that is a stochastic matrix so it may be viewed as a transition matrix of a Markov chain, associated to the set of lower bounds . This chain is a model of the (1,1) EA, which is a special case of the (1,) EA with (see Subsection 2.1). Suppose that the (1,1) EA uses an artificial monotone mutation operator where the cumulative transition probabilities are defined by the bounds , corresponding to the EA mutation operator . Namely, given a parent genotype , for any we have , where is such that . Operator may be simulated, for example, by the following two-stage procedure. At the first stage, a random index of the offspring level is chosen with the probability distribution where is the level of parent . At the second stage, the offspring genotype is drawn uniformly at random from . (Simulation of the second stage may be computationally expensive for some fitness functions but the complexity issues are not considered now.) The initial search point of the (1,1) EA is generated at random with probability distribution defined by the probabilities . Denoting , by properties of Markov chains we get . The following theorem is based on a comparison of to the distribution of the Markov chain .
Theorem 2:
Suppose all level subsets are non-empty and matrix is monotone. Then for any holds
formula
8
where is a triangular -matrix with components if and otherwise. Besides that Inequality (8) turns into an equation if , the EA mutation operator is monotone and is its matrix of cumulative transition probabilities.
Proof:
The (1,1) EA described above is identical to an EA' with , and mutation operator . Let us denote the population vector of EA' by . Obviously,
formula
9
Proposition 3 implies that in the original EA with population size and tournament size , the expectation is lower bounded by the expectation since Inequality (6) holds as an equality for the whole sequence of and the right-hand side of Inequality (6) is non-decreasing on . Equality together with Eq. (9) imply the required bound Inequality (8).

Note that Inequalities (7) and (8) in Theorems 4 and 5 turn into equalities if these theorems are applied to the EA with and monotone mutation operator defined above. Therefore, both theorems guarantee equal lower bounds on , given equal matrices .

Subsections 5.3 and 5.4 provide two examples illustrating how Theorem 5 may be used to import known results on Markov chains behavior. The example from Subsection 5.4 employs Theorem 5 for finding a vector so that Theorem 4 may be applied to bound from below.

3.2  Upper Bounds

In this subsection, we obtain upper bounds on using a reasoning similar to the proof of Proposition 3. Expression (1) for all yields:
formula
10
which turns into equality in the case of level-based mutation. By the total probability formula we have:
formula
11
so
formula
12

Under the expectation in the right-hand side we have a convex function on . Therefore, in the case of monotone matrix , using Jensen's inequality (see, e.g., Rudin 1987, Chapter 3) we obtain the following proposition.

Proposition 4:
If is monotone then
formula
13

By means of iterative application of Inequality (13) the components of the expected population vectors may be bounded up to arbitrary , starting from the initial vector . The nonlinearity in the right-hand side of Inequality (13), however, creates an obstacle for obtaining an analytical result similar to the bounds of Theorems 4 and 5.

Note that all of the estimates obtained up to this point are independent of the population size and valid for arbitrary . In the Section 4 we will see that the right-hand side of Inequality (13) reflects the asymptotic behavior of population under monotone mutation operator as .

3.3  Comparison of EA to (1,) EA and (1+1) EA

This subsection shows how the probability of generating the optimal genotypes at a given iteration of the EA relates to analogous probabilities of (1,) EA and (1+1) EA. The analysis here will be based on upper bound Inequality (13) and on some previously known results provided in the appendix.

Suppose, matrix gives the upper bounds for cumulative transition probabilities of the mutation operator used in the EA. Consider the (1,) EA and the (1+1) EA, based on a monotone mutation operator for which is the matrix of cumulative transition probabilities and suppose that the initial solutions and have the same distribution over the fitness levels as the best incumbent solution in the EA population . Formally: for any and In what follows, for any by we denote the probability that current individual on iteration of the (1,) EA belongs to . Analogously denotes the probability for the (1+1) EA.

The following proposition is based on upper bound Inequality (13) and the results from Borisovsky (2001) and Borisovsky and Eremeev (2001) that allow us to compare the performance of the EA, the (1,) EA, and the (1+1) EA.

Proposition 5:
Suppose that matrix is monotone. Then for any holds
formula
Proof:

Let us compare the EA to the (1,) EA and to the (1+1) EA using the mutation and initialization procedures as described above. Theorem 16 (see the appendix) together with Proposition 1 imply that for all . Furthermore, Theorem 15 from Borisovsky and Eremeev (2001) (see the appendix) implies that for all . Using Proposition 6 and monotonicity of , we conclude that both claimed inequalities hold.

4  EA with Monotone Mutation Operator

First of all, note that in the case of the monotone mutation operator, two equal monotone matrices of lower and upper bounds exist, so the bounds Eq. (5) and Eq. (12) give equal results, and assuming we get
formula
14
This equality will be used several times in what follows.

In general, the population vectors are random values whose distributions depend on . To express this in the notation, let us denote the proportion of genotypes from in population by .

The following Lemma 8 and Theorem 9 based on this lemma indicate that in the case of monotone mutation, recursive application of the formula from right-hand side of upper bound Eq. (13) allows to compute the expected population vector of the infinite-population EA at any iteration .

Lemma 1:

Let the EA use a monotone mutation operator with cumulative transition probabilities matrix , and let the genotypes of the initial population be identically distributed. Then

  • for all and holds
    formula
  • if the sequence of -dimensional vectors is defined as
    formula
    formula

for and . Then for all at any iteration .

The main step in the proof of Lemma 8 (i) will consist in showing that for a supplementary random variable the value of is upper-bounded by an arbitrary small . This step is made by splitting the range of into a “high-probability” area and a “low-probability” area in such a way that is at most in the “high-probability” area. An analogous technique is used, for example, in the proof of Lebesgue Theorem (see, e.g., Kolmogorov and Fomin, 1999, Chapter VII, Section 44).

Proof of Lemma 8:

From Eq. (14), we conclude that if statement (i) holds, then with the convergence of to will imply that . Thus, statement (ii) follows by induction on .

Let us now prove statement (i). Given some to prove Eq. (15) we recall the sequence of i.i.d. random variables , where , if the -th individual of population belongs to , otherwise . By the law of large numbers, for any and , we have
formula
Note that . Besides that, due to Proposition 1, (In the case of this equality holds as well, since all individuals of the initial population are distributed identically.) Therefore, for any the convergence holds. Now by continuity of the function , it follows that
formula
Let us denote . Then
formula
For arbitrary we can split the integration domain into two subsets and and bound by 1 in the first case and by in the second case. Then in view of the definition of we get
formula
The first term tends to 0 as . The second term is at most for any . Hence Eq. (15) holds.

Combining Equality (14) with claim (i) of Lemma 8 we obtain a recursive expression for in the infinite-population EA, which is formulated as

Theorem 3:
If the mutation operator is monotone and individuals of the initial population are distributed identically, then
formula
18
for all .

For any and the term of the sequence defined by Eq. (17) is nondecreasing in and in as well. With this in mind, we can expect that the components of population vector of the infinite-population EA will typically increase with the tournament size. Theorem 10 below gives a rigorous proof of this fact under some technical conditions on distributions of and .

Theorem 4:
Let and correspond to EAs with tournament sizes and , where . Besides that, suppose that is monotone with for all and the individuals of initial populations are identically distributed so that for all . Then for any , given a sufficiently large holds
formula
Proof:

Let the sequences and be defined as in Lemma 8, corresponding to tournament sizes and . By the above assumptions,

Now since for all we have for any . Thus, for all holds
formula
19
since and at least for one of the levels according to the assumption that . Due to the same reason, for all from the last equality in Eq. (19) we get Using the fact that and rearranging the terms as in the proof of Proposition 3 we get
formula
To sum up, for we have and .

Furthermore, if we assume that for all holds and then analogously to Eq. (19) we get for all . Besides that, just as in the case of we get and So by induction we conclude that for all and all .

Finally, by claim (ii) of Lemma 8, for any and , given a sufficiently large , holds .

Informally speaking, Theorem 10 implies that in the case of the monotone mutation operator an optimal selection mechanism consists in setting which actually converts the EA into the (1,) EA.

5  Applications and Illustrative Examples

5.1  Examples of Monotone Mutation Operators

Let us consider two cases where the mutation is monotone and the matrices have a similar form.

First, we consider the simple fitness function . Suppose that the EA uses the bitwise mutation operator, changing every gene with a given probability , independently of the other genes. Let the subsets be defined by the level lines and . The matrix for this operator could be obtained using the result from Bäck (1992), but here we shall consider this example as a special case of a more general setting.

Let the representation of the problem admit a decomposition of the genotype string into nonoverlapping substrings (called blocks here) in such a way that the fitness function equals the number of blocks for which a certain property holds. The functions of this type belong to the class of additively decomposed functions, where the elementary functions are Boolean and substrings are non-overlapping (see, e.g., Mühlenbein et al., 1999). Let if holds for the block of genotype , and otherwise (here ).

Suppose that during mutation, any block for which did not hold, gets the property with probability , that is,
formula
On the other hand, assume that a block with the property keeps this property during mutation with probability , that is,
formula
Let and the subsets correspond to the level lines again. In this case, the element of cumulative transition probabilities matrix equals the probability to obtain a genotype containing or more blocks with property after mutation of a genotype which contained blocks with this property. Let denote the probability that during mutation blocks without property would produce blocks with this property and let denote the probability that after mutation of a set of blocks with property , there will be at least blocks with property among them. (If then ) With these notations,
formula
Clearly, and Thus,
formula
20
It is shown in Eremeev (2000) and Borisovsky and Eremeev (2008) that if , then matrix defined by Expression (20) is monotone.

Now matrix for the bitwise mutation on OneMax function is obtained assuming that and . This operator is monotone in view of the above mentioned result, if , since in this case . The monotonicity of bitwise mutation on OneMax is used in works of Doerr et al. (2010) and Witt (2013).

Expression (20) may be also used for finding the cumulative transition matrices of some other optimization problems with a regular structure. As an example, below we consider the vertex cover problem (VCP) on graphs of a special structure.

In general, the vertex cover problem is formulated as follows. Let be a graph with a set of vertices and the edge set where . A subset is called a vertex cover of if every edge has at least one endpoint in . The vertex cover problem is to find a vertex cover of minimal cardinality.

Suppose that the VCP is handled by the EA with the following representation: each gene corresponds to an edge of , assigning one of its endpoints which has to be included in the cover . To be specific, we can assume that means that and means that . The vertices, not assigned by one of the chosen endpoints, do not belong to . On one hand, this edge-based representation is degenerate in the sense that one vertex cover may be encoded by different genotypes . On the other hand, any genotype defines a feasible cover . A natural way to choose the fitness function in the case of this representation is to assume .

Note that most publications on evolutionary algorithms for VCP use the vertex-based representation with genes, where implies inclusion of vertex into (see, e.g., Neumann and Witt, 2010, § 12.1). In contrast to the edge-based representation, the vertex-based representation is not degenerate but some genotypes in this representation may define infeasible solutions.

Following Saiko (1989) we denote by the graph consisting of disconnected triangle subgraphs. Each triangle is covered optimally by two vertices and the redundant cover consists of three vertices. In spite of simplicity of this problem, it is proven in Saiko (1989) that some well-known algorithms of branch and bound type require exponential in number of iterations if applied to the VCP on graph .

In the case of , the fitness coincides with the number of optimally covered triangles in (i.e., triangles where only two different vertices are chosen), since covering nonoptimally all triangles gives and each optimally covered triangle decreases the size of the cover by one. Let the genes representing the same triangle constitute a single block, and let the property imply that a triangle is optimally covered. Then by looking at the two possible ways to produce a gene triplet that redundantly covers a triangle, (i) given a redundant triangle and (ii) given an optimally covered triangle, we conclude that (i) and (ii) . Using Expression (20) we obtain the cumulative transition matrix for this mutation operator. It is easy to verify that in this case the inequality holds for any mutation probability , and therefore the operator is always monotone.

Computational Experiments. Below we present some experimental results in comparison with the theoretical estimates obtained in Section 3. To this end we consider an application of the EA to the VCP on graphs . The average proportion of optimal genotypes in the population for different population sizes is presented in Figure 2. Here , , , and (these parameters are chosen to ensure clear visibility on plots). The statistics is accumulated in 1000 independent runs of the algorithm where for each only one individual was checked for optimality. Thus for each we have a series of 1000 Bernoulli trials with a success probability which is estimated from the experimental data. The 95%-confidence intervals for success probability in Bernoulli trials are computed using the Normal approximation as described in Cramer (1946, Chapter 34).
Figure 2:

Average proportion of optimal VCP solutions and the theoretical lower and upper bounds as functions of the iteration number. Here , , and 10.

Figure 2:

Average proportion of optimal VCP solutions and the theoretical lower and upper bounds as functions of the iteration number. Here , , and 10.

The experimental results are shown in dashed lines. The solid lines correspond to the lower and upper bounds given by Expressions (7) and (13). The plot shows that upper bound Expression (13) gives a good approximation to the value of even if the population size is not large. The lower bound Expression (7) coincides with the experimental results when , up to a minor sampling error.

Another series of experiments was carried out to compare the behavior of EAs with different tournament sizes. Figure 3 presents the experimental results for 1000 runs of the EA with , and solving the VCP on . This plot demonstrates the increase in the average proportion of the optimal genotypes as a function of the tournament size, which is consistent with Theorem 10. The 95%-confidence intervals are found as described above.
Figure 3:

Average proportion of optimal solutions to VCP and the theoretical upper bound, as functions of the iteration number. Here , , and 10.

Figure 3:

Average proportion of optimal solutions to VCP and the theoretical upper bound, as functions of the iteration number. Here , , and 10.

5.2  Lower Bound for Randomized Local Search on Unimodal Functions.

First of all, let us describe the RLS algorithm which will be implicitly studied in this subsection. At each iteration of RLS the current genotype is stored. In the beginning of RLS execution, is initialized with some probability distribution (e.g., uniformly over ). An iteration of RLS consists in building an offspring of by flipping exactly one randomly chosen bit in . If , then is replaced by the new genotype . The process continues until some termination condition is met.

Below we will illustrate the usage of Theorem 4 on the class of -Unimodal functions. In this class, each function has exactly distinctive fitness values , and each solution in the search space is either optimal or its fitness may be improved by flipping a single bit. Naturally we assume that and that level consists of optimal solutions.

As a mutation operator in the EA we will use a routine denoted by : given a genotype , this routine first changes one randomly chosen gene and if this modification improves the genotype fitness, then outputs the modified genotype, otherwise outputs the genotype unchanged. Note that in the case of the EA with mutation becomes a version of RLS. The lower bounds from Section 3 are tight for