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Subsections 5.3 and 5.4 provide two examples illustrating how Theorem may be used to import known results on Markov chains behavior. The example from Subsection 5.4 employs Theorem for finding a vector [MATH] so that Theorem may be applied to bound [MATH] |
from below. 3.2 Upper Bounds In this subsection, we obtain upper bounds on [MATH] using a reasoning similar to the proof of Proposition . Expression ( ) for all |
[MATH] yields: [EQUATION] which turns into equality in the case of level-based mutation. By the total probability formula we have: |
[EQUATION] [EQUATION] so [EQUATION] Under the expectation in the right-hand side we have a convex function on [MATH] . 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 [EQUATION] By means of iterative application of inequality ( 13 the components of the expected population vectors [MATH] |
may be bounded up to arbitrary [MATH] , starting from the initial vector [MATH] . The nonlinearity in the right-hand side of ( 13 ), however, creates an obstacle for obtaining an analytical result similar to the bounds of Theorems and |
Note that all of the estimates obtained up to this point are independent of the population size and valid for arbitrary [MATH] . In the Section we will see that the right-hand side of ( 13 ) reflects the asymptotic behavior of population under monotone mutation operator as [MATH] |
3.3 Comparison of EA to ( [MATH] [MATH] ) EA and ( [MATH] [MATH] ) EA This subsection shows how the probability of generating the optimal genotypes at a given iteration of the EA relates to analogous probabilities of ( [MATH] [MATH] ) EA and ( [MATH] [MATH] ) EA. The analysis here will be based on upper bound ( 13 ) an... |
Suppose, matrix gives the upper bounds for cumulative transition probabilities of the mutation operator [MATH] used in the EA. Consider the ( [MATH] [MATH] ) EA and the ( [MATH] [MATH] ) EA, based on a monotone mutation operator [MATH] for which is the matrix of cumulative transition probabilities and suppose that the ... |
[MATH] and [MATH] In what follows, for any [MATH] by [MATH] we denote the probability that current individual [MATH] on iteration [MATH] of the ( [MATH] [MATH] ) EA belongs to [MATH] . Analogously [MATH] denotes the probability [MATH] for the ( [MATH] [MATH] ) EA. |
The following proposition is based on upper bound ( 13 and the results from Borisovsky, 2001 Borisovsky and Eremeev, 2001 that allow to compare the performance of the EA, the ( [MATH] [MATH] ) EA and the ( [MATH] [MATH] ) EA. |
Proposition 5 Suppose that matrix is monotone. Then for any [MATH] holds [EQUATION] Proof. Let us compare the EA to the ( [MATH] [MATH] ) EA and to the [MATH] [MATH] ) EA using the mutation and initialization procedures as described above. Theorem (see the appendix) together with Proposition imply that |
[MATH] for all [MATH] . Furthermore, Theorem 5 from Borisovsky and Eremeev, 2001 (see the appendix) implies that [MATH] for all [MATH] . Using Proposition and monotonicity of , we conclude that both claimed inequalities hold. |
[MATH] EA with Monotone Mutation Operator First of all note that in the case of monotone mutation operator, two equal monotone matrices of lower and upper bounds [MATH] |
exist, so the bounds ( ) and ( 12 give equal results, and assuming [MATH] we get [EQUATION] This equality will be used several times in what follows. |
In general, the population vectors are random values whose distributions depend on [MATH] . To express this in the notation let us denote the proportion of genotypes from [MATH] in population [MATH] by [MATH] |
The following Lemma and Theorem based on this lemma indicate that in the case of monotone mutation, recursive application of the formula from right-hand side of upper bound ( 13 ) allows to compute the expected population vector of the infinite-population EA at any iteration [MATH] |
Lemma 1 Let the EA use a monotone mutation operator with cumulative transition probabilities matrix [MATH] , and let the genotypes of the initial population be identically distributed. Then |
(i) for all [MATH] and [MATH] holds [EQUATION] (ii) if the sequence of [MATH] -dimensional vectors [MATH] is defined as [EQUATION] |
[EQUATION] for [MATH] and [MATH] . Then [MATH] for all [MATH] at any iteration [MATH] The main step in the proof of Lemma (i) will consist in showing that for a supplementary random variable |
[MATH] the value of [MATH] is upper-bounded by an arbitrary small [MATH] . This step is made by splitting the range [MATH] of [MATH] into a “high-probability” area and a “low-probability” area in such a way that [MATH] is at most [MATH] in the “high-probability” area. Analogous technique is used e.g. in the proof of Le... |
Proof of Lemma From ( 14 ), we conclude that if statement (i) holds, then with [MATH] the convergence of [MATH] to [MATH] will imply that [MATH] . Thus, statement (ii) follows by induction on [MATH] |
Let us now prove statement (i). Given some [MATH] to prove ( 15 ) we recall the sequence of i.i.d. random variables [MATH] , where [MATH] , if the [MATH] -th individual of population [MATH] belongs to [MATH] , otherwise |
[MATH] . By the law of large numbers, for any [MATH] and [MATH] , we have [EQUATION] Note that [MATH] . Besides that, due to Proposition [MATH] (In the case of [MATH] this equality holds as well, since all individuals of the initial population are distributed identically.) Therefore, for any |
[MATH] the convergence [MATH] holds. Now by continuity of the function [MATH] , it follows that [EQUATION] Let us denote [MATH] . Then |
[EQUATION] [EQUATION] for arbitrary [MATH] , hence ( 15 ) holds. [MATH] Combining equality ( 14 ) with claim (i) of Lemma we obtain a recursive expression for [MATH] 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 [EQUATION] |
for all [MATH] For any [MATH] and [MATH] the term [MATH] of the sequence defined by ( 17 ) is nondecreasing in [MATH] and in [MATH] 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 below gives a rig... |
and [MATH] Theorem 4 Let [MATH] and [MATH] correspond to EAs with tournament sizes [MATH] and [MATH] , where [MATH] . Besides that, suppose that [MATH] is monotone with |
[MATH] for all [MATH] and the individuals of initial populations are identically distributed so that [MATH] for all [MATH] . Then for any [MATH] , given a sufficiently large [MATH] holds |
[EQUATION] Proof. Let the sequences [MATH] and [MATH] be defined as in Lemma , corresponding to tournament sizes [MATH] and [MATH] . By the above assumptions, |
[MATH] Now since [MATH] for all [MATH] we have [MATH] for any [MATH] . Thus, for all [MATH] holds [EQUATION] since [MATH] and [MATH] at least for one of the levels [MATH] according to the assumption that |
[MATH] . Due to the same reason, for all [MATH] from the last equality in ( 19 ) we get [MATH] Using the fact that [MATH] and re-arranging the terms as in the proof of Proposition we get |
[EQUATION] To sum up, for [MATH] we have [MATH] [MATH] and [MATH] Furthermore, if we assume that for all [MATH] holds [MATH] [MATH] and |
[MATH] then analogously to ( 19 ) we get [MATH] for all [MATH] . Besides that, just as in the case of [MATH] we get [MATH] and [MATH] So by induction we conclude that [MATH] for all |
[MATH] and all [MATH] Finally, by claim (ii) of Lemma , for any [MATH] and [MATH] given a sufficiently large [MATH] , holds [MATH] [MATH] |
Informally speaking, Theorem implies that in the case of monotone mutation operator an optimal selection mechanism consists in setting [MATH] which actually converts the EA into the ( [MATH] [MATH] ) EA. |
Applications and Illustrative Examples 5.1 Examples of Monotone Mutation Operators Let us consider two cases where the mutation is monotone and the matrices [MATH] have a similar form. |
First we consider the simple fitness function [MATH] Suppose that the EA uses the bitwise mutation operator, changing every gene with a given probability [MATH] , independently of the other genes. Let the subsets [MATH] be defined by the level lines [MATH] and [MATH] . The matrix [MATH] for this operator could be obtai... |
Let the representation of the problem admit a decomposition of the genotype string into [MATH] non-overlapping substrings (called blocks here) in such a way that the fitness function equals the number of blocks for which a certain property [MATH] holds. The functions of this type belong to the class of additively decom... |
[MATH] if [MATH] holds for the block [MATH] of genotype [MATH] and [MATH] otherwise (here [MATH] ). Suppose that during mutation, any block for which [MATH] did not hold, gets the property [MATH] with probability [MATH] , i.e. |
[EQUATION] On the other hand, assume that a block with the property [MATH] keeps this property during mutation with probability [MATH] , i.e. |
[EQUATION] Let [MATH] and the subsets [MATH] correspond to the level lines [MATH] again. In this case the element [MATH] of cumulative transition probabilities matrix [MATH] equals the probability to obtain a genotype containing [MATH] or more blocks with property [MATH] after mutation of a genotype which contained [MA... |
[MATH] denote the probability that during mutation [MATH] blocks without property [MATH] would produce [MATH] blocks with this property and let [MATH] denote the probability that after mutation of a set of [MATH] blocks with property [MATH] , there will be at least [MATH] |
blocks with property [MATH] among them. (If [MATH] then [MATH] With these notations, [EQUATION] Clearly, [MATH] and [MATH] Thus, |
[EQUATION] It is shown in Eremeev, 2000 Borisovsky and Eremeev, 2008 that if [MATH] then matrix [MATH] defined by ( 20 ) is monotone. |
Now matrix [MATH] for the bitwise mutation on OneMax function is obtained assuming that [MATH] and [MATH] This operator is monotone in view of the above mentioned result, if [MATH] , since in this case [MATH] . 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 [MATH] be a graph with a set of vertices [MATH] and the edge set |
[MATH] where [MATH] . A subset [MATH] is called a vertex cover of [MATH] if every edge has at least one endpoint in [MATH] . The vertex cover problem is to find a vertex cover [MATH] of minimal cardinality. |
Suppose that the VCP is handled by the EA with the following representation: each gene [MATH] corresponds to an edge [MATH] of [MATH] , assigning one of its endpoints which has to be included in the cover [MATH] . To be specific, we can assume that [MATH] means that [MATH] and [MATH] means that [MATH] . The vertices, n... |
Note that most publications on evolutionary algorithms for VCP use the vertex-based representation with [MATH] genes, where [MATH] implies inclusion of vertex [MATH] into [MATH] |
(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 [MATH] the graph consisting of [MATH] 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 a... |
In the case of [MATH] , the fitness [MATH] coincides with the number of optimally covered triangles in [MATH] (i.e. triangles where only two different vertices are chosen), since covering non-optimally all triangles gives [MATH] and each optimally covered triangle decreases the size of the cover by one. Let the genes r... |
Computational Experiments. Below we present some experimental results in comparison with the theoretical estimates obtained in Section . To this end we consider an application of the EA to the VCP on graphs [MATH] . The average proportion of optimal genotypes in the population for different population sizes is presente... |
[MATH] and [MATH] (these parameters are chosen to ensure clear visibility on plots). The statistics is accumulated in 1000 independent runs of the algorithm where for each [MATH] only one individual [MATH] was checked for optimality. Thus for each [MATH] we have a series of 1000 Bernoully trials with a success probabil... |
The experimental results are shown in dashed lines. The solid lines correspond to the lower and upper bounds given by the expressions ( ) and ( 13 ). The plot shows that upper bound ( 13 ) gives a good approximation to the value of [MATH] even if the population size is not large. The lower bound ( ) coincides with the ... |
Another series of experiments was carried out to compare the behavior of EAs with different tournament sizes. Figure presents the experimental results for 1000 runs of the EA with [MATH] [MATH] and |
[MATH] solving the VCP on [MATH] . 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 . The 95%-confidence intervals are found as described above. |
5.2 Lower Bound for Randomized Local Search on Unimodal Functions. First of all let us describe a Randomized Local Search algorithm (RLS) which will be implicitly studied in this subsection. At each iteration of RLS the current genotype [MATH] is stored. In the beginning of RLS execution, [MATH] is initialized with som... |
by flipping exactly one randomly chosen bit in [MATH] . If [MATH] then [MATH] is replaced by the new genotype [MATH] . The process continues until some termination condition is met. |
Below we will illustrate the usage of Theorem on the class of [MATH] Unimodal functions. In this class, each function has exactly [MATH] distinctive fitness values |
[MATH] , 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 [MATH] and that level [MATH] consists of optimal solutions. |
As a mutation operator in the EA we will use a routine denoted by [MATH] : given a genotype [MATH] , this routine first changes one randomly chosen gene and if this modification improves the genotype fitness, then [MATH] outputs the modified genotype, otherwise [MATH] outputs the genotype [MATH] unchanged. Note that in... |
Mutation operator [MATH] never decreases the genotype fitness and improves any non-optimal genotype with probability at least [MATH] , so we have [MATH] for all |
[MATH] and [MATH] for [MATH] . The chances for improvements by more that one fitness level are not foreseeable, so we put [MATH] for all [MATH] . Note that this matrix |
is monotone. Now [MATH] and the matrix [MATH] consists of the following elements: [EQUATION] In order to apply Theorem we also need to choose an appropriate matrix norm and evaluate this norm for matrix [MATH] In this particular application we will use [MATH] which is the matrix norm induced by the Euclidean vector nor... |
[MATH] where [MATH] is the maximal eigenvalue of matrix [MATH] Here and below [MATH] denotes the transpose of matrix [MATH] It is easy to check that matrix [MATH] is composed of zero elements everywhere except for [MATH] diagonal elements, [MATH] |
superdiagonal and [MATH] subdiagonal elements. In particular, it has identical elements [MATH] on the diagonal and all superdiagonal and subdiagonal elements are equal to [MATH] This matrix [MATH] belongs to the class of tridiagonal Toeplitz matrices and its maximal eigenvalue is |
[EQUATION] (see Theorem in the appendix). Therefore [EQUATION] So [MATH] and since matrix is monotone we can apply Theorem Let us denote [MATH] . The vector [MATH] satisfies the equation [MATH] and since [MATH] , the right-hand side in inequality ( ) of Theorem tends to [MATH] as [MATH] |
In order to obtain an explicit lower bound on [MATH] for any given [MATH] , we will evaluate the speed of convergence of the right-hand side in inequality ( ) to . Note that by properties of matrix norms we have |
[EQUATION] Thus for any distribution of initial population Theorem gives a lower bound [EQUATION] where the last inequality holds because each component of vector |
[MATH] is upper-bounded by [MATH] which is at most [MATH] by inequality ( 21 ). Finally, independently of population size [MATH] and tournament size [MATH] we get a lower bound for the proportion of optimal genotypes in the EA population: |
[EQUATION] The Taylor expansion for [MATH] gives [EQUATION] Now since [MATH] and [MATH] we obtain [EQUATION] In the case of RLS, i.e. when [MATH] , this gives the following tail bound |
Corollary 1 The probability that the maximum of a fitness function from [MATH] Unimodal is first reached after more than [MATH] iterations of RLS is at most [MATH] |
A positive feature of this tail bound is that it approaches to 0 exponentially fast in [MATH] . A weakness of Corollary is that its bound is grater than 1 (and therefore useless) when [MATH] The obtained tail bound may be improved for some relatively small [MATH] using the expected RLS runtime bound and Markov inequali... |
5.3 Lower Bounds and Runtime Analysis for 2-SAT Problem The Satisfiability problem (SAT) in general is known to be NP-complete Garey and Johnson, 1979 , but it is polynomially solvable in the special case denoted by 2-SAT: given a Boolean formula with CNF where each clause contains at most two literals, find out whethe... |
Let [MATH] be the number of logical variables and let [MATH] be the number of clauses in the CNF. A natural encoding of solutions is a binary string [MATH] where [MATH] if the [MATH] -th logical variable has the value ”true” and otherwise [MATH] |
We consider an EA with the tournament size [MATH] and the following mutation operator [MATH] : Draw randomly a clause which is not satisfied, choose one variable among the variables of the clause at random, and modify this variable. Otherwise keep the solution unchanged. This method of random perturbation was proposed ... |
Schöning, 1999 A fitness function does not influence the EA execution when [MATH] but it will be useful for our theoretical analysis. Let us assume that [MATH] equals the Hamming distance to a satisfying assignment [MATH] . Here and below, we assume that at least one satisfying assignment [MATH] exists. |
For any non-satisfying truth assignment the improvement probability is 1/2, so we can apply the following monotone bounds: [MATH] for all [MATH] |
[MATH] for [MATH] [EQUATION] [MATH] [MATH] . These lower bounds define the Markov chain transition probabilities with [MATH] and [MATH] according to Subsection 3.1 . It turns out that this matrix is the same as the transition matrix of the symmetric Gambler’s Ruin random walk with one reflecting barrier (state 0) and o... |
[MATH] for [MATH] [MATH] , all other elements [MATH] are equal to zero. The result from Papadimitriou, 1991 implies that, regardless of the initial state, there exists a constant [MATH] , such that after [MATH] transitions the absorbing probability of this random walk is at least 1/2. This means that [MATH] and the [MA... |
Corollary 2 If the EA for 2-SAT has the tournament size [MATH] and the mutation operator [MATH] then the probability to generate a satisfying assignment in population [MATH] is at least [MATH] for some constant [MATH] |
It makes sense to apply Theorem only in the case of [MATH] in this example, since for [MATH] the tournament selection is impossible without computing the Hamming distance to a satisfying assignment which is unknown. |
If the EA with [MATH] and mutation [MATH] is restarted every [MATH] iterations and [MATH] , then the overall runtime of this iterated EA is [MATH] by Corollary and Markov inequality. Note that Corollary holds for any distribution of the initial population, so the runtime bound [MATH] applies to the EA without restarts ... |
5.4 Lower Bounds and Runtime Analysis for Balas Set Cover Problems In general the set cover problem (SCP) is formulated as follows. Given: a ground set [MATH] and a set of covering subsets |
[MATH] , with indices [MATH] . A subset of indices [MATH] is called a cover if [MATH] The goal is to find a cover of minimum cardinality. In what follows, for any [MATH] we denote by [MATH] the set of numbers of the subsets that cover an element [MATH] , i. e. [MATH] . Note that an instance of SCP may be defined by a f... |
Suppose the binary representation of the SCP solutions is used, i.e. genes [MATH] are the indicators of the elements from [MATH] , so that [MATH] . If |
[MATH] is a cover then we assign its fitness [MATH] otherwise [MATH] , where [MATH] is a decreasing function of the number of non-covered elements from [MATH] |
Consider a family [MATH] of set cover problems introduced by Balas, 1984 . Here it is assumed that [MATH] and that all [MATH] -element subsets of [MATH] |
are given as subsets [MATH] . Thus any collection of less than [MATH] elements from [MATH] belongs to [MATH] for some [MATH] and does not cover the element [MATH] . At the same time any subset [MATH] of size [MATH] covers all elements of [MATH] and therefore it is an optimal cover. Larger subsets are non-optimal covers... |
Since any [MATH] -element subset of [MATH] is an optimal cover, family [MATH] is solvable trivially. Nevertheless this family is known to be hard for general-purpose integer programming algorithms Balas, 1984 Saiko, 1989 . In particular, it was shown in Saiko, 1989 that problems from this class are hard to solve using ... |
Note that any [MATH] -element subset [MATH] for [MATH] leaves [MATH] elements of the ground set uncovered, regardless of the choice of elements in [MATH] . So in the case of tournament selection, equivalently to studying the EA on family [MATH] we may study the EA where the fitness is given by a function of unitation, ... |
[EQUATION] where function [MATH] is decreasing, function [MATH] is increasing and [MATH] Consider the point mutation operator with tunable parameter [MATH] |
defined in Subsection 3.1 . Let [MATH] and let the thresholds [MATH] be equal to fitness of genotypes that contain [MATH] genes ”1” accordingly. Note that [MATH] is a cover iff [MATH] |
We have the following lower bounds: [MATH] for all [MATH] [MATH] for [MATH] [EQUATION] [MATH] [MATH] . These lower bounds [MATH] coincide with the corresponding cumulative transition probabilities except for level [MATH] , where we pessimistically assume [MATH] (in fact we could safely put [MATH] but [MATH] is chosen t... |
In case we are interested in runtime bounds for the EA, rather than expected values of vector [MATH] , we can assume [MATH] . All other non-zero lower bounds [MATH] defined above could be relaxed by putting [MATH] . In this case we would have the associated Markov chain with a transition matrix [MATH] the same as in Su... |
Corollary 3 Suppose that the EA with a tournament size [MATH] uses the point mutation operator with parameter [MATH] . Then given |
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