July 15, 2026
The paper develops the approach to the runtime analysis of evolutionary algorithms on the basis of limit theorems from probability theory. We consider the family of Jump\(_k\) benchmark functions, defined on the search space of binary strings of length \(n\), parametrized by the integer \(k\), which have multiple local optima at the Hamming distance \(k\) from a unique global optimum.
In this work, we consider the genetic algorithm \((1+(\lambda,\lambda))~GA\) from (Doerr, Doerr and Ebel, 2015) with tunable parameters of the mutation rate \(p\), crossover bias \(c\), and two intermediate population sizes \(\lambda_M\) and \(\lambda_C\), and study the time it escapes from the plateau in the case of Jump\(_k\) fitness function when \(np\) tends to infinity. The main result of this work is a tightened upper bound on the escape time from the work of Antipov, Doerr and Karavaev (2022). Besides that, the obtained bound applies to a wider range of algorithm parameters.
Genetic Algorithm, Optimization Time, Limit Theorems, Local Optima.
This work contains an extended version of the theoretical analysis presented in the report at the GECCO-26 conference (see [1]).
Given a pseudo-Boolean maximization problem \(f:\{0,1\}^n \to \mathbb{R}\), an evolutionary algorithm (EA) seeks for the maximum of \(f\), using a population of one or more individuals, corresponding to the points in the solution space \(\{0,1\}^n,\) and the fitness of the individuals is equal to the value of the objective function \(f(x)\). New test points in the EA are produced using the random operators of mutation and crossover. When using the latter, the EA is usually called a genetic algorithm. One of the most frequently used mutation operators is the standard mutation [2], where each bit of the given string \(\mathrm{x}\in \{0,1\}^n\), independently of the other bits, changes its value with a given probability \(p\). In this paper, we will assume that in the case of standard mutation, at each iteration of the EA, the number of bits \(\ell\) to be mutated is selected, using the binomial distribution \(\mathop{\mathrm{Bin}}(n, p)\) (next, we denote \(\ell \in {\rm Bin}(n,p)\)). Each new descendant is obtained from the parent individual by making changes in randomly selected \(\ell\) bits.
The authors of [3] developed the \((1 + (\lambda, \lambda))\) genetic algorithm (GA) for pseudo-Boolean optimization, where a crossover operator is able to eliminate some “unsuccessful” mutations. At each iteration of the \((1+(\lambda, \lambda))\) GA, \(\lambda\) offspring are generated from a single parent individual independently of each other, at the Hamming distance \(\ell\) from the parent, \(\ell \in {\rm Bin}(n,p)\). After that, the best individual in terms of fitness is selected from these \(\lambda\) solutions and the crossover operator is applied, combining it with the parent, given a crossover parameter \(c\): With probability \(c\) the crossover operator uses bits from the best child, and with probability \(1-c\), it uses bits from the parent solution. This way \(\lambda\) individuals are created, and the best of these \(\lambda\) individuals is accepted as a new parent, provided it is at least as fit as the previous parent. Otherwise the current individual remains unchanged. Theoretical analysis of optimization time showed that the algorithm \((1 + (\lambda, \lambda))\) GA for many values of configurable parameters turns out to be asymptotically faster on the fitness function than most classical evolutionary algorithms. Here and below, \(\text{\sc OneMax}(x):=\sum_{i=1}^n x_i\) for any \(\mathrm{x}\in\{0,1\}^n.\) In particular, as shown in [4] for the , the optimal choice of fixed mutation parameter \(p\) for the entire running time gives the optimization time \[{E[T]=\Theta\left(n \sqrt{{\log(n)\log\log\log(n)}/{\log\log n}}\right).}\]
In [5], the first mathematical runtime analysis for the \((1+(\lambda, \lambda))\) GA optimizing a multimodal optimization problem, namely the the classical Jump\(_k\) benchmark, was carried out. It was observed there that a combination of aggressive mutation with crossover as repair mechanism works even better on this benchmark. The \((1+(\lambda, \lambda))\) GA can optimize jump functions with gap size \(k \le n/4\) in expected time at most \(n^{(k+1)/2}e^{O(k)}k^{-k/2}.\) This is almost the square root of \(\Omega(n^k)\), the runtime many classic mutation-based algorithms have. For the problem of leaving the local optimum of , the authors of [5] perform parameters tuning that leads to a runtime \((n/k)^{k/2}e^{O(k)}\).
It is shown in [6] that choosing all parameter values (population size \(\lambda\), mutation rate \(p\), and crossover bias \(c\)) in each iteration randomly, using power-law distribution, is a robust approach. It was proved that this algorithm on the one hand can imitate simple hill-climbers on problems like , , or Minimum Spanning Tree. On the other hand, the algorithm is also efficient on functions, where the best static parameters are very different from those necessary to optimize simple problems.
In the present paper, we consider the \((1+(\lambda, \lambda))\) GA from [3] with tunable parameters of mutation rate \(p\), crossover bias \(c\), and two intermediate population sizes \(\lambda_M\) and \(\lambda_C\), and study the time to escape from the local optima in the case of Jump\(_k\) fitness function when \(np\) tends to infinity. The main result of this work is a tightened upper bound on the time to escape a local optimum \((n/k)^{k/2}e^{\Theta(k)}\) from [5]. Besides that, the obtained bound applies to a wider range of algorithm’s parameters.
For the sake of compactness, the number of ones in a bit string \(\mathrm{x}\) will be denoted as \(|\mathrm{x}|\), meaning the \(L^{1}\) norm of the vector \(\mathrm{x}\), or Hamming norm.
When introducing new notations in formulas, the symbol \(:=\) is used, and the colon is on the side of the quantity being defined. To describe the asymptotic behaviour of quantities associated with unbounded growth of the dimension \(n\), we will use the standard notations: \(g(n)=O(t(n))\) and \(g(n)=o(t(n))\), if the upper limit of the ratio \(t(n)/g(n)\) is bounded or converges to 0, respectively; \({g(n)\sim t(n)}\), if \(g(n)=t(n)(1+o(1))\). When using the latest asymptotic relations, we will usually omit the formal condition “for \(n\to\infty\)”. We say that a function \(t(n)\) belongs to the set \(\Theta(g(n))\) if there exist positive constants \(c_1\) and \(c_2\), as well as a non-negative integer \(n_0\) such that \(c_2 g(n) \le t(n) \le c_1 g(n)\) for all \(n\ge n_0\). Let us denote by \(\mathbb{N}_{0,n}\) the set of all integers from 0 to \(n\).
The notation \(\ell\in\mathcal{B}(n,p)\), where \(n\in \mathbb{N}\) and \(p\in(0,1)\), means that the random variable \(\ell\) has a binomial distribution with probability distribution \[\mathbf{P}(\ell=l)=\genfrac(){0pt}{0}{n}{l}p^{l}q^{n-l}=:P_{p}(n,l), \;l\in\mathbb{N}_{0,n}:=\{0,1,\ldots,n\},\] where \(q=1-p\).
The probability to have \(s\) successes in \(l\) random draws, without replacement, from a finite set of \(n\) objects, that contains exactly \(k\) objects with a required feature is described by the hypergeometric distribution. The probability mass function of a random variable that follows the hypergeometric distribution is given by \[P_{n,k}(l,s):={{k \choose s}{n-k \choose l-s}}\Big/{{n \choose l}}.\]
We consider the same version of the \((1+(\lambda,\lambda))\) GA as in [5], which is analogous to the one initially proposed in [7]. The algorithm has the following outline.
Algorithm 1. \((1+(\lambda,\lambda))\) GA with the offspring population sizes \(\lambda_M\) and \(\lambda_C\), mutation rate \(p\), and crossover bias \(c\), maximizing an objective function \(f:\{0,1\}^{n}\to \mathbb{R}\).
1. \(\mathrm{x}\) \(\leftarrow\) random bit string of length \(n\);
2. while not terminated do
3. Mutation phase:
4. Choose \(\ell \in {\rm Bin}(n,p)\);
5. for \(i\in[1..\lambda_M]\) do
6. \(\mathrm{x}^{(i)}\) \(\leftarrow\) a copy of \(\mathrm{x}\);
7. Flip \(\ell\) bits in \(\mathrm{x}^{(i)}\) chosen uniformly at random;
8. end
9. \(\mathrm{x}' \leftarrow {\rm arg\, max}_{\mathrm{z}\in \{\mathrm{x}^{(1)},...,\mathrm{x}^{(\lambda_M)}\}} f(\mathrm{z})\);
10. Crossover phase:
11. for \(i\in[1..\lambda_C]\) do
12. Create \(\mathrm{y}^{(i)}\) by taking each bit from \(\mathrm{x}'\) with
13. probability \(c\) and from \(\mathrm{x}\) with probability \(1-c\);
14. end
15. \(\mathrm{y} \leftarrow {\rm arg\, max}_{\mathrm{z}\in \{\mathrm{y}^{(1)},...,\mathrm{y}^{(\lambda_C)}\}} f(\mathrm{z})\);
16. if \(f(\mathrm{y}) \geq f(\mathrm{x})\) then
17. \(\mathrm{x} \leftarrow \mathrm{y}\);
18. end
19. end
In what follows, we will refer to execution of Lines 3–18 of Algorithm 1 as one iteration of the algorithm. Note that in the one iteration may be considered in two perspectives: (i) treating \(\ell\) as a random variable, or (ii) considering a specific sample \(\ell=l\). Usage of notation \(\ell\) or \(l\) will clearly identify either of these two options below. We will refer to Lines 3–9 of Algorithm 1 as the mutation phase, and Lines 10–15 as the crossover phase.
The basic benchmark function \(\text{\sc OneMax}(\mathrm{x})=\sum\nolimits_{i=1}^{n} x_i,\) has been deeply studied in the literature on the theory of EAs. In terms of it is easy to define the function \(\text{\sc Jump}_k,\) where \(k\) is a parameter \(2\le k\le n\): \[\begin{align} &\text{\sc Jump}_k(\mathrm{x}) := \begin{cases} n + 1 & \text{if } \text{\sc OneMax}(\mathrm{x})=n\\ k + \text{\sc OneMax}(x) & \text{if } \text{\sc OneMax}(\mathrm{x})\leq n-k\\ n - \text{\sc OneMax}(\mathrm{x}) & \text{otherwise} \end{cases}. \end{align}\]
The global optimum of \(\text{\sc Jump}_k\) corresponds to the bit string of all ones, which we denote by \(\mathrm{x_{m}}\). The infinite family of functions \(\text{\sc Jump}_k\) serves as an example of a multimodal function, where all local optima are equally distant from the unique global optimum and a number of papers are devoted to the runtime analysis of evolutionary algorithms on these functions, see the references in [5].
Let an individual \(\mathrm{x}\) have the norm \(|\mathrm{x}|=n-k\), i.e. \(\mathrm{x}\) is in a local maximum of \(\text{\sc Jump}_k\).
In what follows, \(p_{M}^{(1)}(n,l,k,p)\) denotes probability that \(k\) zero bits and \(l-k\) one bits are flipped during a single mutatioin Line 7 of Algorithm 1 to the individual \(\mathrm{x}\). We denote this event by \(M^{(1)}(n,l,k,p)\). When the event \(M^{(1)}(n,l,k,p)\) occurs, the resulting individual \(\widetilde{\mathrm{x}}\) has the norm \(|\widetilde{\mathrm{x}} |=n-l+k\). Otherwise, \(\widetilde{\mathrm{x}}\) has the norm \(|\widetilde{\mathrm{x}}|<n-l+k\).
The explicit form of the probability of event \(M^{(1)}(n,l,k,p)\) is \[\label{main4}
p_{M}^{(1)}(n,l,k,p)=P_{n,k}(l,k)=\dfrac{(n-k)!l!}{(l-k)!n!},\;\;k\leq l,\tag{1}\] here \(P_{n,k}(l,s)\) is the hypergeometric distribution and we assume \(P_{n,k}(l,k)=0\) for
\(l<k\).
Let \(p_{M}^{(\lambda_{M})}(n,l,k,p)\) be the probability that, given \(\lambda_{M}\) independent mutations of \(\mathrm{x}\), the event \(M^{(1)}(n,l,k,p)\) occurs at least once throughout the mutation phase. We denote this event by \(M^{(\lambda_{M})}(n,l,k,p)\). The event \(M^{(\lambda_{M})}(n,l,k,p)\) leads to the choice of the individual \(\mathrm{x}'\) with the norm \(|\mathrm{x}'|=n-l+k\). If not all \(k\) positions with “0” components in \(\mathrm{x}\) have mutated, then \(\widetilde{\mathrm{x}}\) has a norm \(|\widetilde{\mathrm{x}}|<n-l+k\). According to this notation, the following relation holds: \[\label{Mlam}
p_{M}^{(\lambda_{M})}(n,l,k,p)=1-\Big(1-p_{M}^{(1)}(n,l,k,p)\Big)^{\lambda_{M}}.\tag{2}\]
Let \(p_{C}^{(1)}(n,l,k,p,c)=c^{k}(1-c)^{l-k}\) be the probability that in the crossover phase, within a single crossover of \(\mathrm{x}\) and \(\mathrm{x}'\) with \(|\mathrm{x}'|=n-l+k\) in Lines 12, 13, only the \(k\) zero-components in \(\mathrm{x}\) are replaced (i.e. these components are selected from \(\mathrm{x}'\)). We denote this event by \(C^{(1)}(n,l,k,p,c)\).
Let \(p_{C}^{(\lambda_{C})}(n,l,k,p,c)\) be the probability that throughout the crossover phase, conditioned that the event \(M^{(\lambda_{M})}(n,l,k,p)\) occurred (i.e., in the case of \(|\mathrm{x}|=n-k\) and \(|\mathrm{x}'|=n-l+k\)), the event \(C^{(1)}(n,l,k,p,c)\) will occur at least once. We denote this event by \(C^{(\lambda_{C})}(n,l,k,p,c)\). According to this notation, we have \[\begin{align} \label{Clam} p_{C}^{(\lambda_{C})}(n,l,k,p,c){=}1{-}\Big(1-p_{C}^{(1)}(n,l,k,p,c)\Big)^{\lambda_{C}}{=} 1-\big(1{-}c^{k}(1-c)^{l-k}\big)^{\lambda_{C}}. \end{align}\tag{3}\]
Let us denote by \(A_{\lambda_{M},\lambda_{C}}(n,l,k,p)= M^{(\lambda_{M})}(n,l,k,p)C^{(\lambda_{C})}(n,l,k,p,c)\) the event in which an iteration of the algorithm results in creating the optimum, i.e. we get \(\mathrm{y}=\mathrm{x_{m}}\).
By definition, \(p_{\lambda_{M},\lambda_{C}}(n,l,k,p):=\mathbf{P}(A_{\lambda_{M},\lambda_{C}}(n,l,k,p))\) is the probability of reaching the optimum as a result of an iteration, it is equal to \[p_{\lambda_{M},\lambda_{C}}(n,l,k,p,c)=p_{M}^{(\lambda_{M})}(n,l,k,p)p_{C}^{(\lambda_{M})}(n,l,k,p,c).\]
The transition from a non-random \(l\) to a random \(\ell\) is quite natural: we average it out using the total probability formula for a binomial distribution with probabilities \(P_{p}(n,l).\) Let us introduce the following notation for the resulting expectations: \[\begin{align} &p_{M}^{(\lambda_{M})}(n,k,p):=\mathbf{E}\big[ p_{M}^{(\lambda_{M})}(n,\ell,k,p)\big],\\ &p_{C}^{(\lambda_{C})}(n,k,p,c):=\mathbf{E}\big[ p_{C}^{(\lambda_{C})}(n,\ell,k,p,c)\big], \\ &p_{\lambda_{M},\lambda_{C}}(n,k,p,c):=\mathbf{E}[p_{\lambda_{M},\lambda_{C}}(n,\ell,k,p,c)]. \end{align}\]
The following proposition appears to be interesting by itself, as it provides simple expressions for \(p_{M}^{(\lambda_{M})}(n,k,p)\) and \(p_{C}^{(\lambda_{C})}(n,k,p,c)\), for the special case of \(\lambda_{M}=1\) and \(\lambda_{C}=1\).
Proposition 1. Suppose \(\ell\in \mathcal{B}(n,p)\), then \[\begin{align} \label{main000} &p_{M}^{(1)}(n,k,p)=\mathbf{E}[P_{n,k}(\ell,k)]=p^{k}, \\\label{main002} &p_{C}^{(1)}(n,k,p,c)=\mathbf{E}[c^{k}(1-c)^{\ell-k}]=c^{k} (1-c)^{-k} (1-pc)^{n}. \end{align}\] {#eq: sublabel=eq:main000,eq:main002}
Proof. The generating function of the binomial distribution is \[\phi_{\ell}(z):=\mathbf{E}[z^{\ell}]=(q+pz)^{n}, \quad|z|\leq1.\]
The factorial moment of order \(k\) is \[V^{(k)}: =\mathbf{E}\Big[\prod\nolimits_{i=0}^{k-1}(\ell-i)\Big]=\dfrac{n!}{(n-k)!}\mathbf{E}[P_{n,k}(\ell,k)].\]
By definition, \(V^{(k)}\) equals to the \(k\)-th derivative of \(\phi_{\ell}(z)\) at \(z=1\), which is \(p^{k}n!/(n-k)!\). This completes the proof of identity ?? .
Since the generating function of the binomial distribution is \(\mathbf{E}[x^{l}]=(q+px)^{n}\), so \(\mathbf{E}[(1-c)^{l}]=(q+p(1-c))^{n}\), which proves identity ?? . ◻
Our goal is to obtain upper bounds on the expected number of fitness evaluations and the expected number of iterations till the global optimum is reached, if the algorithm is started in a local optimum, i.e. the initial solution is such that \(|x|=n-k\). Let us denote these random variables by \(T_{F}\) and \(T_{I}\), respectively.
The overall proof scheme is quite natural and similar to the proof of Lemmas 6 and 7 and Theorem 8 from [5]. We present our approach in this section.
Suppose that on some set \(B=B(n)\subseteq \mathbb{N}_{n}:=\{ 1,2,\ldots,n\}\) of some positive probability \(\mathbf{P}(B)>C_{B}>0\) the following bound is valid \[\label{M0l} p_{\lambda_{M}}(n,l,k,p)>C_{M}>0,\;l\in B.\tag{4}\]
Here the sequences \(C_{B}=C_{B}(n)>0\) and \(C_{M}=C_{M}(n)>0\) may depend on \(n\). If the condition 4 is satisfied and the following estimate holds \[\sum\nolimits_{l\in B}p_{C}^{(\lambda_{C})}(n,l,k,p,c)P_{p}(n,l)\geq C_{C,B}>0,\] for binomial distribution \(P_{p}(n,l)\) and some sequence \(C_{C,B}{=}C_{C,B}(n)>0\) then the inequality 5 below describes the probability of reaching the optimum as a result of one iteration: \[\label{1lam0l1} p_{\lambda_{M},\lambda_{C}}(n,k,p,c) \geq C_{M} \sum_{l\in B}p_{C}^{(\lambda_{C})}(n,l,k,p,c)P_{p}(n,l)\geq C_{M}C_{C,B}>0.\tag{5}\] The expected number of iterations \(\mathbf{E}[T_{I}]\) and the expected number of fitness evaluations \(\mathbf{E}[T_{F}]\) till reaching the optimum may be expressed as \[\label{TI0} \mathbf{E}[T_{I}]=p_{\lambda_{M},\lambda_{C}}^{-1}(n,k,p,c)\leq \frac{1}{C_{M}C_{C,B}},\;\mathbf{E}[T_{F}]\leq \frac{\lambda_{M}+\lambda_{C}}{C_{M}C_{C,B}}.\tag{6}\] Here we have used the fact that \(T_{I}\) has a geometric distribution with a probability of success \(p_{\lambda_{M},\lambda_{C}}(n,k,p,c)\).
If \(p_{\lambda_{C}}(n,l,k,p,c)>C_{C}>0\), \(l\in B\), then analogously to 5 , one can obtain the lower bound \[\label{1lam0l01} p_{\lambda_{M},\lambda_{C}}(n,k,p,c) \geq C_{C} \sum\nolimits_{l\in B}p_{M}^{(\lambda_{M})}(n,l,k,p)P_{p}(n,l) \geq C_{C}C_{M,B}>0\tag{7}\] for some sequence \(C_{M,B}=C_{M,B}(n)>0.\) In particular, if \(p_{C}^{(\lambda_{C})}(n,l,k,p,c)\geq C_{C}\) and \(p_{M}^{(\lambda_{M})}(n,l,k,p)\geq C_{M}\) for \(l\in B\), then \[\label{TI11} p_{\lambda_{M},\lambda_{C}}(n,k,p,c)\geq C_{C}C_{M}C_{B}, \;\;\mathbf{E}[T_{I}]\leq C_{M}^{-1}C_{C}^{-1}C_{B}^{-1}.\tag{8}\]
Let us discuss the choice of the set \(B\). In [5] (in the proof of Lemma 6) it is chosen in the form \(B=[np,2np]\), where \(np\geq2k\). We propose to choose it in the form \(B\subseteq[\max\{k,np - N\sqrt{npq}\},np + N\sqrt{npq}]\), where \(N\in(0,\infty)\). In this paper, we will analyze only the case \(k\leq np - N \sqrt{npq}\) and in the final asymptotic steps we set \(B=\Delta_{n}(N):=[np - N\sqrt{npq},np + N\sqrt{npq}]\). Dropping the \(k\leq np - N \sqrt{npq}\) condition is easy, but it will lead to cumbersome calculations.
By the de MoivreLaplace Theorem (see e.g. Theorem 5.3.1 from [8]) for \(N=N(n)=o\big((npq)^{1/6}\big)\) (possibly dependent on \(n\)) we have \[\label{1} \mathbf{P}(\ell\in\Delta_{n}(N))\sim 2\Phi(N)-1=: c_{\Phi}(N),\tag{9}\] where \(\Phi(N)=(2\pi)^{-0.5}\int_{-\infty}^{N}e^{-u^{2}/2}\mathsf{d}u\) is the cumulative distribution function of the standard normal random value. In the case of \(N\to\infty\) we get \[\mathbf{P}(\ell\in\Delta_{n}(N))\to1, \;\;\mathbf{P}(\ell\notin\Delta_{n}(N))\to0.\] Note that \(2\Phi(2)-1= 0.99730020\) for \(N=2\) and \(2\Phi(5)-1=0.99999942\) for \(N=5\), and then converges very quickly to 1. But increasing of \(N\) worsens the estimates for \(p_{C}^{(\lambda_{C})}(n,l,k,p,c)\geq C_{C}\) and \(p_{M}^{(\lambda_{M})}(n,l,k,p)\geq C_{M}\) for \(l\in B\).
The length of the interval \(\Delta_{n}(N)\) is Infinitesimal to \([np,2np]\), this is the reason why the upper bounds on the runtime obtained by means of \(B=[np,2np]\) are by more than a factor of \(2^{k(1+o(1))}\) higher than ours (for more details, see the comments below the proof of Lemma 5).
Let’s introduce the notations \[\epsilon_{\mu}(x):= \sum_{i\geq \mu+1}\frac{x^{i}}{i(i-1)},\quad \mu\in \mathbb{N},\quad \text{where}\;\;\sum_{i=2}^{1}\frac{x^{i}}{i(i-1)}:=0,\] and consider the asymptotic behavior of \(p_{M}^{(1)}(n,l,k,p)\) as \(n\to\infty\). First, we present some technical results.
Lemma 1. Let \(x\in[0,1)\), \(m,\mu\in\mathbb{N}\) and \(\epsilon_{\mu}(x):=\sum_{i\geq \mu+1}x^{i}(i(i-1))^{-1}\), then the following representations are valid \[\begin{align} \label{main1} &&(1-x)^{m(1-x)}=\exp\Big\{m\Big(-x+\sum_{i\geq 2}x^{i}(i(i-1))^{-1}\Big)\Big\} \\ \label{main2} &=:&\exp\bigg\{m\bigg(-x+\sum_{i=2}^{\mu}x^{i}(i(i-1))^{-1}+\epsilon_{\mu}(x)\bigg)\bigg\},\\ \label{main02} &&where\;0\leq\epsilon_{\mu}(x)= \dfrac{\theta x^{\mu+1}}{(1-x)\mu(\mu+1)},\;\theta\in(0,1). \end{align}\] {#eq: sublabel=eq:main1,eq:main2,eq:main02}
Proof. The Taylor series expansion of the logarithm for \(|x|<1\) is \[\label{ln1} \ln(1-x)=-\sum_{i\geq 1}x^{i}/i.\tag{10}\] Noting that the identity \((1-x)^{y}=e^{y\ln(1-x)}\), in view of the representation \[-(1-x)\ln(1-x)=(1-x)\sum_{i\geq 1}x^{i}/i=x-\sum_{i\geq 2}x^{i}(i(i-1))^{-1},\] which is valid for absolutely convergent series if \(x\in[0,1)\), after substitution \(y=m(1-x)\), entails the statement ?? . In the expression ?? , \(\epsilon_{\mu}(x)= \sum_{i> \mu}x^{i}(i(i-1))^{-1}\), therefore after using the formula for the sum of a geometric progression \(\sum_{i>\mu}x^{i}=x^{\mu+1}/(1-x)\) we obtain the bound ?? . ◻
Let’s describe some relevant properties of the functions \(\epsilon_{1}(x)\)
Lemma 2. The functions \(\epsilon_{1}(x)\) and \(\epsilon_{1}(x)/x\), \(x\in[0,1)\), are monotonically increasing, \(\epsilon_{1}(x),\epsilon_{1}(x)/x\in[0,1)\) and the following relations hold: \[\label{main5} \epsilon_{1}(x)=(1-x)\ln(1-x)+x=\sum_{i=2}^{\infty}\dfrac{x^{i}}{i(i-1)},\qquad{(1)}\] and \(\epsilon_{1}'(x){=}-\ln(1-x)>0\).
If \(x,y\in[0,1)\) and \(x>y\), then the following inequalities hold: \[\label{main5D} 0.5(x-y)\leq D(x,y)\leq (1/6+(3(1-x))^{-1})(x-y)\leq 0.5\dfrac{x-y}{1-x},\qquad{(2)}\] where \(D(x,y):=\epsilon_{1}(x)/x-\epsilon_{1}(y)/y\).
Proof. The relations ?? are written out by definition, where the series converges absolutely on the interval \(x\in[0,1)\), and the function \((1-x)\ln(1-x)\) at zero is defined by continuity, i.e., equal to zero. Therefore \(\epsilon_{1}(0)=0\) and \(\epsilon_{1}(1)=1\).
By definition of \(D(x,y)\) and by the formula for the sum of the geometric progression, we obtain \[D(x,y)=\sum_{i=2}^{\infty}\dfrac{x^{i-1}-y^{i-1}}{i(i-1)}=(x-y)\sum_{i=2}^{\infty}\dfrac{\sum\nolimits_{j=0}^{i-2}x^{j}y^{i-2-j}}{i(i-1)}.\]
Replacing all variables in the last representation with \(y\) or \(x\), we obtain the inequalities \[\begin{align} &0.5(x-y)\leq(x-y){\displaystyle\sum\limits_{i=2}^{\infty}}\dfrac{y^{i-2}}{i}\leq D(x,y)\leq (x-y){\displaystyle\sum\limits_{i=2}^{\infty}}\dfrac{x^{i-2}}{i}\\ &\leq \left (0.5+\dfrac{x}{3(1-x)}\right )(x-y)\leq 0.5\dfrac{x-y}{1-x}, \end{align}\] equivalent to ?? , which is the last statement of the lemma. ◻
Recall the Stirling formula (see e.g. [8] p. 117, Section 2.2.) \[s!=\sqrt{2\pi s}s^{s}e^{-s}R(s), \;\;s\in\mathbb{N},\] for the function \(R(s)\) that satisfies the inequalities \(1<e^{\frac{1}{12s+1}}<R(s)<e^{\frac{1}{12s}}<1.08690405\). Note that \(R(s)=1+o(1)\) if \(s\to \infty\).
Lemma 3. Suppose \(k\leq (1-\varepsilon)n\) and \(l\geq(1+\varepsilon)k\) for some \(\varepsilon\in(0,1)\), then the following representation is valid: \[\label{main03} p_{M}^{(1)}(n,l,k,p)=\sqrt{\dfrac{1-k/n}{1-k/l}}e^{-kD(k/l,k/n)}(l/n)^{k} R_{n,l,k},\qquad{(3)}\] where \(R_{n,l,k}=R(n-k)R(l)R^{-1}(l-k)R^{-1}(n)\).
Proof. Using the Stirling formula, and taking into account the expression 1 , we obtain \[\begin{align} \nonumber &&p_{M}^{(1)}(n,l,k,p)=\dfrac{(n-k)!l!}{(l-k)!n!}=\sqrt{\dfrac{(n- k)l}{(l-k)n}}\dfrac{(n-k)^{n-k}l^{l}}{(l-k)^{l-k}n^{n}}R_{n,l,k}\\ \label{bpM} &=& \sqrt{\dfrac{1-k/n}{1-k/l}}\dfrac{(1-k/n)^{n-k}}{(1-k/l)^{l-k}}(l/n)^{k} R_{n,l,k}. \end{align}\tag{11}\] By Lemma 1 \[\label{bpM1} \dfrac{(1-k/n)^{n-k}}{(1-k/l)^{l-k}}=\dfrac{\exp\{- n(k/n{-}\epsilon_{1}(k/n))\}}{\exp\{-l(k/l{-} \epsilon_{1}(k/l))\}} = \exp\{n\epsilon_{1}(k/n){-}l\epsilon_{1}(k/l)\}.\tag{12}\] Assertion ?? of this lemma follows from Lemma 2 and equations 11 and 12 . ◻
We will analyze the properties of the second bound from formulas 6 .
For the sake of brevity, let us temporarily denote \(\lambda_{M}\) and \(\lambda_{C}\) by \(L_{1}\) and \(L_{2}\), \(p_{M}^{(1)}(n,l,k,p)\) and \(p_{C}^{(1)}(n,l,k,p,c)\) by \(p_{1}=p_{1}(n)\) and \(p_{2}=p_{2}(n)\), and \(p_{M}^{(\lambda_{M})}(n,l,k,p)\) and \(p_{C}^{(\lambda_{C})}(n,l,k,p,c)\) by \(p_{1}^{(L_{1})}\) and \(p_{2}^{(L_{2})}\). Then the relations 2 and 3 imply \[\label{e1} 1-e^{-L_{i}p_{i}}\leq p_{i}^{(L_{i})}=1-(1-p_{i})^{L_{i}}=1-e^{L_{i}\ln(1-p_{i})},\;\;i=1,2,\tag{13}\] and \[\label{0e1} p_{i}^{(L_{i})}=1-e^{-L_{i}p_{i}(1+o(1))},\;\;i=1,2,\tag{14}\] in the case \(0<p_{i}=p_{i}(n)\to 0\), for \(n\to\infty\). Note that by 14 \[\label{111} p_{i}^{(L_{i})}=1-e^{L_{i}\ln(1-p_{i})}=L_{i}p_{i}(1+o(1))\tag{15}\] as \(L_{i}p_{i}\to0\).
Let’s denote the main part of \(\mathbf{E}[T_{F}]\) in the case \(p_{i}=o(1)\), \(i=1,2\), \[\varphi_{L_{1},L_{2}}(n):= \dfrac{L_{1}+L_{2}}{(1-e^{-L_{1}p_{1}})(1-e^{-L_{2}p_{2}})}.\]
Lemma 4. Let conditions 14 be satisfied, where the sequences \(p_{i}=p_{i}(n)>0\), \(i=1,2\), are fixed, and the representations \(L_{i}=(\widetilde{c}_{i} +o(1))p_{i}^{-1}\), \(\widetilde{c}_{i}\in\mathbb{R}^{+}\), are valid, then the function \[\widetilde{\varphi}_{L_{1},L_{2}}(n)=\dfrac{L_{1}+L_{2}}{\big(1{-}e^{-\widetilde{c}_{1}}\big)\big(1{-}e^{-\widetilde{c}_{2}}\big)}=\varphi_{L_{1},L_{2}}(n)(1+o(1))\] is increasing of \(\widetilde{c}_{i}\).
If \(L_{1}(n)=L_{2}(n)=L(n)\), \(L(n)p_{2}(n)\to 0\), and \(\widetilde{c}_{1}>0\), then \[\label{mam1} \varphi_{L,L}(n)=\dfrac{2L(n)}{L(n)p_{2}(n)(1-e^{-\widetilde{c}_{1}})}(1+o(1)).\qquad{(4)}\]
Proof. We give an asymptotically equivalent form for the function \(\varphi_{L_{1},L_{2}}(n)\) for \(\widetilde{c}_{i}>0\) \[\dfrac{\widetilde{c}_{1}/p_{1}^{(L_{1})}+\widetilde{c}_{2}/p_{2}^{(L_{2})}}{(1-e^{-\widetilde{c}_{1}}) (1-e^{-\widetilde{c}_{2}})}.\]
The function \(x/(1-e^{-x})\) is monotonically increasing, so according to estimates 6 , the average number of fitness evaluations decreases as \(\widetilde{c}_{i}>0\) tends to zero. Relation ?? clearly follows from the conditions of the lemma. ◻
Let’s describe the exact asymptotic behavior of \(p_{M}^{(\lambda_{M})}(n,l,k,p)\) for \(n\to\infty\), \(\lambda_{M}\to\infty\), and a number of additional constraints. The imposed conditions can be relaxed, but this will lead to cumbersome calculations and formulations of results.
Let’s introduce the notation \(\mathcal{P}_{k,p}(c_{1})\) for the set of sequences \(\{k\}=\{k(n)\}_{n\in\mathbb{N}}\) and \(\{p\}=\{p(n)\in(0,p_{0})\subseteq(0,1)\}_{n\in\mathbb{N}}\) that satisfy the conditions \[\varlimsup_{n\to\infty}k(n)p^{-1}(n)/n\leq c_{1},\] where \(c_{1}\in[0,1)\) and \(p_{0}\in[0,1)\) are fixed number.
Note that under the condition \(\mathcal{P}_{k,p}(c_{1})\) the estimation \(\min_{l\in\Delta_{n}(N)}l-k\geq (1-c_{1})np(1+o(1))\), \(n\to\infty\), holds.
Let \[\label{cole01} J_{M,n,k,p,N}:=\sqrt{\frac{1-k/n}{1-k/(np)}}p^{k} \exp\left \{-\dfrac{k^{2}(q+N\sqrt{q}/\sqrt{np})}{2np(1-k(np)^{-1})}\right \},\tag{16}\] in the case \(k/\sqrt{np}\to\infty\), and \[\label{cole101} J_{M,n,k,p,N}:=p^{k} e^{-c_{0}N\sqrt{q}-0.5c_{0}^{2}}\tag{17}\] in the case \(k/\sqrt{np}=c_{0}+o(1)\), \(c_{0}\in[0,\infty)\).
Lemma 5. Suppose that \(k\in\mathcal{P}_{k,p}(c_{1})\), \(p\in(0,p_{0})\) with any fixed \(c_{1}\in[0,1)\), \(N\in\mathbb{N}\), \(p_{0}\in(0,1)\). Let \(k\to\infty\), \(np\to\infty\), \(l\in\Delta_{n}(N)\subseteq [k+1,n]\) following bound holds \[\label{010} \lambda_{M}p^{k}\exp\left \{-\dfrac{k^{2}q}{2np}(1+o(1))\right \}=o(1),\qquad{(5)}\] then for \(n\to\infty\) the following asymptotic relations hold \[\begin{align} \label{1main3} p_{M}^{(\lambda_{M})}(n,l,k,p)\geq \lambda_{M} J_{M,n,k,p,N}(1+o(1)),\\ \label{0main3} p_{M}^{(\lambda_{M})}(n,k,p)\geq \lambda_{M}c_{\Phi}(N)J_{M,n,k,p,N} (1+o(1)). \end{align}\] {#eq: sublabel=eq:1main3,eq:0main3}
If, instead of constraint ?? , the condition \[\label{010e} \lambda_{M}J_{M,n,k,p,N}\geq c_{M}^{(0)}(1+o(1)),\qquad{(6)}\] holds, where \(c_{M}^{(0)}>0\) is an absolute constant, then for \(n\to\infty\) the following asymptotic relations hold \[\begin{align} \label{1main30} &p_{M}^{(\lambda_{M})}(n,l,k,p)\geq 1-e^{-c_{M}^{(0)}}(1+o(1)),\;l\geq np ,\\ \label{0main30} &p_{M}^{(\lambda_{M})}(n,k,p)\geq c_{\Phi}(N)(1-e^{-c_{M}^{(0)}})(1+o(1)). \end{align}\] {#eq: sublabel=eq:1main30,eq:0main30}
Proof. The conditions of the lemma imply that \(p\) is separated from 1. We give an explicit form for the probability of the event \(M^{(\lambda_{M})}(n,l,k,p)\) using the representations 2 and ?? from Lemma 3. Under the conditions of the lemma the conditions \(p_{M}^{(1)}(n,l,k,p)\to0\) are satisfied for \(n\to\infty\). Therefore, for \(n\to\infty\), for all \(l\) satisfying the conditions of the lemma, excluding the requirement that \(np\to\infty\), the following representation holds: \[\begin{align} \label{main3} &&p_{M}^{(\lambda_{M})}(n,l,k,p)=1{-}\exp\{\lambda_{M}\ln(1-p_{M}^{(1)}(n,l,k,p))\}=1 \\ \nonumber &-&\exp\Bigg\{- \lambda_{M} \sqrt{\dfrac{1-k/n}{1-k/l}}\dfrac{R(l)}{R(l-k)}\bigg(\frac{l}{n}\bigg)^{k}e^{-kD\big(\frac{k}{l},\frac{k}{n}\big)}(1+o(1))\Bigg\}. \end{align}\tag{18}\]
Next, we return to the full set of conditions of Lemma. We assume that \(k\in\mathcal{P}_{k,p}(c_{1})\), \(np\to\infty\) and \(l\in\Delta_{n}(N)\subseteq [k+1,n]\). Then the conditions \(l,l-k\to\infty\) are satisfied and the fraction \(R(l)/R(l-k)\) in 18 can be omitted since it will be included in the factor \(1+o(1)\). By analogy, it is obvious that the estimate \(\sqrt{1-k/l}=\sqrt{1-k/np}(1+o(1))\) holds.
The proof of the first part of the lemma is based on the estimates for \(\overline{\pi}:= \sup_{l\in\Delta_{n}(N)}p_{M}^{(1)}(n,l,k,p)\) and \(\underline{\pi}:=\inf_{l\in\Delta_{n}(N)}p_{M}^{(1)}(n,l,k,p)\). The first estimate leads to the relation \(\lambda_{M}\overline{\pi}=o(1)\) if condition ?? is satisfied, and the second estimate is of the form \(\underline{\pi}\geq J_{M,n,k,p,N,\delta_{1}}(1+o(1))\).
Let’s move on to the estimates of \(l^{k}/n^{k}\) and \(D\big(k/l,k/n\big)\).
The \(l\in\Delta_{n}(N)\) condition allows us to write the inequalities \[\begin{align} &\dfrac{\big(np-N\sqrt{npq}\big)^{k}}{n^{k}}\leq l^{k}/n^{k}\leq\dfrac{\big(np+N\sqrt{npq}\big)^{k}}{n^{k}},\\ &D\bigg(\dfrac{k}{np+N\sqrt{npq}},\dfrac{k}{n}\bigg)\leq D\bigg(\dfrac{k}{l},\dfrac{k}{n}\bigg)\leq D\bigg(\dfrac{k}{np-N\sqrt{npq}},\dfrac{k}{n}\bigg). \end{align}\]
According to the Taylor formula for logarithm function, the following representations are valid in the neighborhood of the point 1 \[\dfrac{\big(np\mp N\sqrt{npq}\big)^{k}}{n^{k}}{=}p^{k}e^{k\ln\left (1\mp \frac{N\sqrt{q}}{\sqrt{np}}\right )} {=}p^{k}\exp\left \{\mp \frac{kN\sqrt{q}}{\sqrt{np}} -\dfrac{0.5kN^{2}q(1{+}o(1))}{np}\right \}.\]
As a result, we get \[\label{as1} p^{k}\exp\left\{- \frac{kN\sqrt{q}}{\sqrt{np}}-\frac{0.5kN^{2}q(1{+}o(1))}{np}\right\}\leq l^{k}/n^{k}\leq p^{k}e^{\frac{kN\sqrt{q}}{\sqrt{np}}}.\tag{19}\]
Using the relations \[\begin{align} \dfrac{k}{np+ N\sqrt{npq}}-\dfrac{k}{n}&\geq&\dfrac{kq}{np}-\dfrac{2kN\sqrt{q}}{3(np)^{3/2}},\\ \dfrac{k}{np- N\sqrt{npq}}-\dfrac{k}{n}&\leq&\dfrac{kq}{np}+\dfrac{kN\sqrt{q}}{(np)^{3/2}}, \end{align}\] where \(N\sqrt{q}/\sqrt{np}\leq0.5\), and the inequalities ?? , we obtain the estimates \[\begin{align} \tag{20} \dfrac{kq}{2np}-\dfrac{kN\sqrt{q}}{3(np)^{3/2}}\leq D\bigg(\dfrac{k}{l},\dfrac{k}{n}\bigg)\leq \dfrac{k(q+N\sqrt{q}/\sqrt{np})}{2(np-k)}\\ \tag{21} = \dfrac{k(q+N\sqrt{q}/\sqrt{np})}{2np(1-k/(np))} \leq c_{1}\dfrac{q+N\sqrt{q}/\sqrt{np}}{2(1-c_{1})}. \end{align}\]
To apply relation 15 in the first part of Lemma, the inequalities \[\label{as5} 0\leq\lambda_{M}\big(l/n\big)^{k} e^{-kD\big(k/l,k/n\big)}\leq\lambda_{M}\overline{\pi}=o(1)\tag{22}\] must be satisfied. A sufficient condition for the last relation to hold is \[\begin{align} \lambda_{M}p^{k}\exp\left \{\frac{kN\sqrt{q}}{\sqrt{np}}-\dfrac{k^{2}q(1+o(1))}{2np}\right\}\\ =\lambda_{M}p^{k}\exp\left \{-\dfrac{k^{2}q(1+o(1))}{2np}\right\}O(1)=o(1). \end{align}\]
Here, we used the fact that the estimates \(\frac{kN\sqrt{q}}{\sqrt{np}}-\frac{k^{2}q(1+o(1))}{2np}=O(1)\) are valid for \(k/\sqrt{np}=O(1)\), \(k/\sqrt{np}=o(k^{2}/(np))\) are valid for \(k/\sqrt{np}\to\infty\) and then the condition 15 .
The statement 22 follows from condition ?? .
If the proof of relation 22 is based on upper estimates for \(p_{M}^{(1)}(n,l,k,p)\), \(l\in\Delta_{n}(N)\), then to prove statement ?? we need these lower estimates. The schemes of their proofs coincide, but when estimating from the above double inequalities, in them it is necessary to take the opposite.
Using inequalities 19 and 20 , we obtain the following estimates \[\begin{align} \nonumber &\Big(\frac{l}{k}\Big)^{k}\exp\Big\{{-}kD\Big(\frac{k}{l},\frac{k}{n}\Big)\Big\}\geq p^{k}\exp\left \{{-}\dfrac{kN\sqrt{q}}{\sqrt{np}}-\dfrac{kN^{2}q(1+o(1))}{2np}\right \} \\ \nonumber &\cdot\;\exp\bigg \{-0.5\dfrac{k^{2}(q+N\sqrt{q}/\sqrt{np})}{np(1-k/(np))}\bigg\}=(1+o(1)) \\ \label{00as3} &\cdot \;p^{k} \exp\left \{-\dfrac{kN\sqrt{q}}{\sqrt{np}}-\dfrac{kN^{2}q}{2np}-\dfrac{k^{2}(q+N\sqrt{q}/\sqrt{np})}{2np(1-k(np)^{-1})}\right \}. \end{align}\tag{23}\]
Note that the rihgt part of estimation 23 is equal to \[\lambda_{M}p^{k}\exp\{-c_{0}N\sqrt{q}-0.5c_{0}^{2}q\}(1+o(1))\] in the case of \(k/\sqrt{np}=c_{0}+o(1)\), \(c_{0}\in[0,\infty)\), and it is equal to \[p^{k} \exp\left \{-\dfrac{k^{2}(q+N\sqrt{q}/\sqrt{np})}{2np(1-k(np)^{-1})}\right \}(1+o(1)),\] in the case of \(k/\sqrt{np}\to\infty\).
Condition 22 , representation 18 with the comments below, and evaluation 23 prove the statement ?? . The latter, taking into account the inequality \[\label{00as} p_{M}^{(\lambda_{M})}(n,k,p)\geq\mathbf{E}[p_{M}^{(\lambda_{M})}(n,\ell,k,p);\ell\in \Delta_{n}(N)],\tag{24}\] the conditional probability formula and relation 9 , leads to statement ?? .
The estimate 23 is valid for \(\ell\in \Delta_{n}(N)\) without additional conditions and can be written as \(\Big(\frac{l}{k}\Big)^{k}\exp\Big\{{-}kD\Big(\frac{k}{l},\frac{k}{n}\Big)\Big\}\geq J_{M,n,k,p,N}\). By condition ?? , for \(l\in\Delta_{n}(N)\) the last enequality and representation 18 entail the statement ?? .
Statement ?? follows from estimations ?? and 24 . ◻
Now we present a simple uniform estimate for \(p_{C}^{(\lambda_{C})}(n,l,k,p,c)\). It can be strengthened, but this requires additional calculations and will not change the asymptotic behavior of its leading term.
Lemma 6. Suppose that \(k\in\mathcal{P}_{k,p}(c_{1})\), \(l\in\Delta_{n}(N)\), \(p\in(0,p_{0})\) with any fixed \(c_{1}\in[0,1)\), \(N\in\mathbb{N}\), \(p_{0}\in(0,1)\). Let \(k\to\infty\), \(np\to\infty\), \(l\in\Delta_{n}(N)\subseteq [k+1,n]\) and the following bound holds \[\label{010c} \lambda_{C}J_{C,n,k,p,c,N}:=\lambda_{C}c^{k}(1-c)^{np-N\sqrt{npq}-k}=o(1),\qquad{(7)}\] then for \(n\to\infty\) the following asymptotic bound holds \[\label{1main03} p_{C}^{(\lambda_{C})}(n,l,k,p,c)\geq \lambda_{C}J_{C,n,k,p,c,N}(1+o(1)).\qquad{(8)}\]
The right-hand side of the expression ?? will be optimal for estimating the average number of fitness evaluations when the following condition is satisfied: \[\label{m3ain3} cp= kn^{-1}(1+N\sqrt{q}/\sqrt{np})^{-1}=kn^{-1}(1+o(1)).\qquad{(9)}\]
In the case \(cp=kn^{-1}(1+o(1))\), the following inequalities hold: \[\begin{align} \label{m1ain3} &e^{-k(1+o(1))/(1-c)}\leq(1-c)^{np}\leq e^{-k(1+o(1))},\\ \label{m01ain3} &e^{-N\sqrt{ck}(1+o(1))/(1-c)}\leq(1-c)^{N\sqrt{npq}}\leq e^{-N\sqrt{ck}(1+o(1))}. \end{align}\] {#eq: sublabel=eq:m1ain3,eq:m01ain3}
Proof. Due to the monotone decrease of the function \((1-c)^{x}\), in view of the representation 3 , and the condition \(\lambda_{C}c^{k}(1-c)^{l-k}= o(1)\) for \(l\in\Delta_{n}\), with \(n\to\infty\) we have \[\begin{align} &&p_{C}^{(\lambda_{C})}(n,l,k,p,c)=1-\exp\big\{\lambda_{C}\ln\big(1-p_{C}^{(1)}(n,l,k,p,c)\big)\big\}\\ &=&1-\exp\big\{-\lambda_{C}c^{k}(1-c)^{l-k}(1+o(1))\big\} \\ &=& \lambda_{C}c^{k}(1-c)^{l-k}(1+o(1))\geq \lambda_{C} c^{k}(1-c)^{np+N\sqrt{npq}-k}(1+o(1)). \end{align}\]
Let’s find the maximum of the function \(c^{k}(1-c)^{np+N\sqrt{npq}-k}\) for \(c\in(0,1)\), which, according to 6 , minimizes one of the factors in the upper bound for the average number of fitness evaluations till reaching the optimum, \(\mathbf{E}[T_{F}]\). Setting its derivative to zero and canceling out the common factors, we obtain \(k(1-c)=c(np+N\sqrt{npq}-k)\) or \[c=k/(np+N\sqrt{npq})= (1+N\sqrt{q}(1+o(1))/\sqrt{np})k/(np) ,\] which proves condition ?? .
In the case \(cp=kn^{-1}(1+o(1))\), taking into account the bounds 10 , we obtain \[e^{-\frac{k(1+o(1))}{1-c}}{\leq}(1-c)^{np}=e^{\ln(1-c)np}{\leq}e^{-cnp}=e^{-k(1+o(1))},\] which proves inequalities ?? . Bounds ?? are proved similarly. ◻
Let’s proceed to the description of the asymptotics of \(p_{\lambda_{M},\lambda_{C}}(n,l,k,p,c)\) under the conditions of lemmas 5 and 6.
Lemma 7. Suppose that \(k\in\mathcal{P}_{k,p}(c_{1})\), \(l\in\Delta_{n}(N)\), \(p\in(0,p_{0})\) with any fixed \(c_{1}\in[0,1)\), \(N\in\mathbb{N}\), \(p_{0}\in(0,1)\). Let \(k\to\infty\), \(np\to\infty\), \(l\in\Delta_{n}(N)\subseteq [k+1,n]\) and bounds ?? and ?? hold, then the following asymptotic bounds hold as \(n\to\infty\): \[\begin{align} \label{main300} p_{\lambda_{M},\lambda_{C}}(n,l,k,p,c)&\geq& \lambda_{M}\lambda_{C}J_{M,n,k,p,N}J_{C,n,k,p,c,N} (1+o(1)), \\ \label{mai1n3} p_{\lambda_{M},\lambda_{C}}(n,k,p,c)&\geq& c_{\Phi}(N) \lambda_{M}\lambda_{C}J_{M,n,k,p,N}J_{C,n,k,p,c,N}(1{+}o(1)). \end{align}\] {#eq: sublabel=eq:main300,eq:mai1n3}
Assuming that \(\lambda_{M}=\lambda_{C}=\lambda\), and condition ?? is satisfied instead of constraint ?? , then \[\begin{align} \label{0m1ain3} &p_{\lambda_{M},\lambda_{C}}(n,l,k,p,c)\geq \lambda(1{-}e^{-c_{M}^{(0)}})(1{-}c)^{np{-}N\sqrt{npq}-k}(1{+}o(1)),\\ \label{m1ai1n3} &p_{\lambda_{M},\lambda_{C}}(n,k,p,c)\geq c_{\Phi}(N) \lambda(1{-}e^{-c_{M}^{(0)}})(1{-}c)^{np{-}N\sqrt{npq}-k}(1{+}o(1)). \end{align}\] {#eq: sublabel=eq:0m1ain3,eq:m1ai1n3}
In view of relations 6 , the assertion of Lemma 7 implies a generalization of Theorem 8 [5] in the case of \(np\to\infty\). The case of uniformly bounded \(np\) must be investigated using other asymptotic methods, in particular, Poisson approximations.
Theorem 1. Suppose that \(k\in\mathcal{P}_{k,p}(c_{1})\), \(p\in(0,p_{0})\) with any fixed \(c_{1}\in[0,1)\), \(N\in\mathbb{N}\), \(p_{0}\in(0,1)\). Let \(k\to\infty\), \(np\to\infty\) and the conditions ?? and ?? hold, then \[\begin{align} \mathbf{E}[T_{I}]&\leq& \dfrac{(1+o(1))}{c_{\Phi}(N) \lambda_{M}\lambda_{C}J_{M,n,k,p,N}J_{C,n,k,p,c,N}}\\ \mathbf{E}[T_{F}]&\leq& \dfrac{(\lambda_{M}+\lambda_{C})(1+o(1))}{c_{\Phi}(N) \lambda_{M}\lambda_{C}J_{M,n,k,p,N}J_{C,n,k,p,c,N}}. \end{align}\]
If we assume that \(\lambda_{M}=\lambda_{C}=\lambda\), and conditions ?? and ?? are satisfied, then \[\begin{align} \label{PTh3} &\mathbf{E}[T_{I}]\leq\Big(c_{\Phi}(N)\lambda(1-c)^{np-N\sqrt{npq}-k}(1-e^{c_{M}^{(0)}})\Big)^{-1}(1+o(1)),\\ \nonumber &\mathbf{E}[T_{F}]\leq 2\lambda\mathbf{E}[T_{I}]. \end{align}\qquad{(10)}\]
Theorem 2 (Theorem 8 [5]). Let \(k\leq n/4\). Assume that \(p\geq2k/n\) and \(q_{\ell}=\mathbf{P}(\{\ell\geq pn\}\cap \{\ell \leq 2pn\})>0.1\). Then the expected runtime of \((1 + (\lambda, \lambda))\) GA on Jump\(_{k}\) is \[\label{A22BK} \mathbf{E}[T_{I}]\leq\dfrac{4}{q_{l}\min\{1,\lambda_{M}(p/2)^{k}\}\min\{1,\lambda_{C}c^{k}(1-c)^{2pn-k}\}}\qquad{(11)}\] iterations and \[\label{A22BFK} \mathbf{E}[T_{F}]\leq\dfrac{4(\lambda_{M}+\lambda_{C})}{q_{l}\min\{1,\lambda_{M}(p/2)^{k}\}\min\{1,\lambda_{C}c^{k}(1-c)^{2pn-k}\}}\qquad{(12)}\] fitness evaluations if the algorithm starts in the local optimum.
Let’s compare the statement of Theorem 1 above and Theorem 8 from [5] using the example of \(\mathbf{E}[T_{I}]\). The upper bound established in [5] is \[\label{zlpdktnw} \mathbf{E}[T_{I}]\leq\dfrac{4}{q_{l}\min\{1,\lambda_{M}(p/2)^{k}\}\min\{1,\lambda_{C}c^{k}(1-c)^{2pn-k}\}}\tag{25}\]
Let’s note the advantages of our formula ?? in comparison with the one just given in ?? . Instead of \((p/2)^{k}\), we use \(J_{\lambda_{M},n, k,p,N}\), which is almost equivalent to \((p(1+o(1)))^{k}\), and instead of \((1-c)^{2pn-k}\) from Theorem 8 from [5], we use \((1-c)^{np-N\sqrt{npq}-k}\). Instead of the estimate \(\min\{1,a\}\), we use the more precise formula \(1-e^{-a}\). The constant \(4/q_{l}\) corresponds to \(c_{\Phi}^{-1}(N)\) here, which can be chosen arbitrarily close to 1. In addition, all constants in the bounds are given explicitly here. The bounds from Theorem 1 significantly refine their analogs from [5]. Naturally, our upper bounds for \(\lambda_{M}\) and \(\lambda_{C}\), leading to infinitesimal \(p_{\lambda_{M},\lambda_{C}}(n,k,p,c)\) are significantly lower than the analogs from [5].
A significant difference between the studies in [5] and ours is the range for \(l\) in the estimates of \(p_{M}^{(1)}(n,l,k,p)\) and \(p_{C}^{(1)}(n,l,k,p,c)\). In [5], it is \(np\leq l\leq 2np\), while in our study, it \(l\in\Delta_{n}(N)\). It is obvious that if we additionally require the condition \(l\geq np\) in the conditions of Theorem 1, then all components containing the factor \(N\) should be removed from the conditions and statements, and \(c_{\Phi}(N)\) should be replaced with \(0.5c_{\Phi}(N)\). In this version of Theorem, our result remains more precise than Theorem 8 from [5]. The effect of changing the lower bound for \(l\) in the interval \([np-N\sqrt{npq},np]\) is easy to investigate, but we will not do so. We will only note that some of the components in the average estimate increase, while others decrease.
Let’s consider a very simple case of estimates for the means, where the parameter \(k\) is finite and does not depend on the others that depend on \(n\to\infty\). These estimates are obtained using the central limit theorem.
Set \(\Delta:=\sqrt{npq}\) \(B=\Delta_{n}(N_{1},N_{2}):=[np - N_{1}\Delta,np + N_{2}\Delta]\), where \(N_{1},N_{2}\in\mathbb{R}^{+}\). By the de MoivreLaplace Theorem (see e.g. Theorem 5.3.1 from [8]) for \(N_{1},N_{2}=o\big((npq)^{1/6}\big)\) (possibly dependent on \(n\)) we have \[\label{1aa} \mathbf{P}(\ell\in\Delta_{n}(N_{1},N_{2}))\sim \Phi(N_{1})+\Phi(N_{2})-1=: c_{\Phi}(N_{1},N_{2}),\tag{26}\] where \(\Phi(N)\) is the cumulative distribution function of the standard normal random value.
Lemma 8. Suppose that \(p=p(n)\in(0,p_{0})\subseteq (0,1)\), \(c\in (0,c_{0})\subseteq (0,1)\), where \(p_{0}\) and \(c_{0}\) any fixed, \(np\to\infty\), then for any \(2\leq k\in\mathbb{N}\) independent of \(n\) the estimates hold \[\begin{align} \label{0PTh1} &\mathbf{E}[T_{I}]\leq\dfrac{c_{\Phi}^{-1}(N_{1},N_{2})(1+o(1))}{\big(1-e^{-\lambda_{M}(p-N_{1}\Delta/n-k)^{k}}\big)\big(1-e^{-\lambda_{C}c^{k}(1-c)^{np+N_{2}\Delta-k}}\big)}, \\ \label{0PTh2} &\mathbf{E}[T_{F}]\leq (\lambda_{M}+\lambda_{C})\mathbf{E}[T_{I}]. \end{align}\] {#eq: sublabel=eq:0PTh1,eq:0PTh2}
Proof. Given the definitions 1 and 3 , by the monotone increasing of \(p_{M}^{(1)}(n,l,k,p)\) decreasing \(p_{C}^{(1)}(n,l,k,p,c)\) with respect to \(l\), we obtain the estimates \(p_{M}^{(1)}(n,l,k,p)\geq p_{M}^{(1)}(n,np-N_{1}\Delta,k,p)\) and \(p_{C}^{(1)}(n,l,k,p,c)\; \geq p_{C}^{(1)}(n,np +N_{2}\Delta,k,p,c)\) for all \(l\in\Delta_{n}(N_{1},N_{2})\). Therefore, taking into account the inequalities 13 , we obtain the ratio \[\begin{align} \nonumber &&p_{\lambda_{M},\lambda_{C}}(n,l,k,p,c)\geq \left (1-e^{-\lambda_{M}p_{M}^{(1)}(n,np-N_{1}\Delta,k,p)(1+o(1))}\right )\\\label{1PTh2} &\cdot&\left (1-e^{-\lambda_{C}p_{C}^{(1)}(n,np +N_{2}\Delta,k,p,c)(1+o(1))}\right). \end{align}\tag{27}\]
Statements ?? and ?? are consequences of relations 6 , 8 and 26 when the values of probabilities corresponding to the lemma statement are substituted. ◻
The following two theorems are analogous to Corollary 9 from [5], but instead of abstract upper bounds on the parameters \(\lambda_{M}\) and \(\lambda_{C}\), we choose their explicit asymptotic representations, and as a result, instead of upper bounds up to an unknown constant, we obtain explicit expressions. It is natural to divide the general case into two: \(k\) is finite (which is the same as \(k\) is uniformly bounded by \(n\)) and \(k\to\infty\).
Theorem 3. Assume that \(2\leq k\in\mathbb{N}\) is fixed, \(\lambda_{M}=\lambda_{C}=\lambda=\lambda_{0}\sqrt{n/k}^{k}\), where \(\lambda_{0}=\lambda_{0}(n)\) and \(\varliminf\limits_{n\to\infty}\lambda_{0}(n)>0\), and \(p=c=\sqrt{k/n}\), then the estimates hold \[\begin{align} \label{0PTh3} &\mathbf{E}[T_{I}]\leq\dfrac{1+o(1)}{\big(1-e^{-\lambda_{0}(1+o(1))}\big)\big(1-\exp\{-e^{-k\lambda_{0}(1+o(1))}\}\big)},\\ \label{0PTh4} &\mathbf{E}[T_{F}]\leq 2\lambda\mathbf{E}[T_{I}]=2\lambda_{0}\sqrt{n/k}^{k}\mathbf{E}[T_{I}]. \end{align}\] {#eq: sublabel=eq:0PTh3,eq:0PTh4}
In the case \(\lambda\in\mathbb{N}\) and \(\lambda_{0} (n)=o(1)\), the estimate ?? is transformed into the inequality \[\label{0PTh03} \mathbf{E}[T_{I}]\leq \dfrac{\sqrt{n/k}^{k}(1+o(1))}{\lambda_{0}\big(1-\exp\{-e^{-1}\}\big)}.\qquad{(13)}\]
Proof. Set \(N_{1}=N_{2}=N=\sqrt[4]{np}\). Then \(\mathbf{P}(l\in\Delta_{n}(\sqrt[4]{np},\sqrt[4]{np}))=1+o(1)\). If the conditions of the theorem are met, taking into account definitions 1 and 3 , it is easy to obtain the following relations \[\begin{align} \lambda_{M}p_{M}^{(1)}(n,l,k,p)&=&\lambda_{0}p^{-k}p^{k}(1+O(N/\sqrt{np}))=\lambda_{0}(1+o(1)),\\ \lambda_{C}p_{C}^{(1)}(n,l,k,p,c)&=&\lambda_{0}(1-c)^{np(1+O(N/\sqrt{np})-k)}\\ &=&\lambda_{0}e^{-cnp(1+o(1))}=\lambda_{0}e^{-k}(1+o(1)), \end{align}\] uniformly over \(l\in\Delta_{n}(\sqrt[4]{np},\sqrt[4]{np})\).
By the latter relations inequalities ?? , ?? and ?? follow from estimates ?? and ?? . ◻
We will not go into detail on optimizing the results of Theorem 1 with respect to the parameters \(\lambda_{M}\) and \(\lambda_{C}\), instead we will use the conditions of Corollary 9 [5] with the additional constraint \(np\to\infty\) and present an analogue of the statement of Corollary 9 from [5].
Let’s introduce the notation for \(\lambda_{M} J_{M,n,k,p,N}\) and \(\lambda_{C}J_{C,n,k,p,c,N}\) in the case \(\lambda_{M}=\lambda_{C}=\lambda=\lambda_{0}\sqrt{n/k}^{k}\) and \(p=c=\sqrt{k/n}\). \[\begin{align} \tag{28} I_{M,n,k,N}&:=&\lambda_{0}\sqrt{n/k}^{k}J_{M,n,k,\sqrt{k/n},N},\\\tag{29} I_{C,n,k,N}&:=&\lambda_{0}\sqrt{n/k}^{k}J_{C,n,k,\sqrt{k/n},\sqrt{k/n},N}. \end{align}\]
In the case \(k/n=O(1)\), the natural convenient condition for asymptotic representations will be \(\sqrt{k/n}=c^{*}_{0}+ o(1)\), where \(c^{*}_{0}\in[0,c_{1}^{2})\) is an absolute constant (depending on \(n\)). In the following lemma, we provide conditions for replacing rough estimates of the type \(e^{f(y)(1+o(1))}\) for expressions from 28 and 29 to \(e^{f(y)}(1+o(1))\) depending on the rate of convergence of \(y=k/n\) to 0.
Lemma 9. Suppose that \(k\in\mathcal{P}_{k,p}(c_{1})\), \(p\in(0,p_{0})\) with any fixed \(c_{1}\in[0,1)\), \(N\in\mathbb{N}\), \(p_{0}\in(0,1)\). Let \(\lambda_{M}=\lambda_{C}=\lambda=\lambda_{0}\sqrt{n/k}^{k}\) and \(c=p=\sqrt{k/n}\), \(k\to\infty\), then in the case \(k/n=c_{*}=c_{*}(n)\), where \(0<\varliminf\limits_{n\to\infty}c_{*}\) and \(\varlimsup\limits_{n\to\infty}c_{*}<c_{1}^{2}\), the represetations take place \[\begin{align} \label{M1es1} I_{M,n,k,N}&=&\lambda_{0}\sqrt{1+\sqrt{c_{*}}} \exp\left \{-0.5c_{*}^{3/2}n - 0.5\sqrt{n}c_{*}^{5/4}N/\sqrt{q} \right \},\\ \label{M1es2} I_{C,n,k,N}&=&\lambda_{0}\exp\left \{\ln(1-c_{*})\big((\sqrt{c_{*}}-c_{*}) n+N\sqrt[4]c_{*}\sqrt{q}\sqrt{n}\big)\right \}. \end{align}\] {#eq: sublabel=eq:M1es1,eq:M1es2}
If \(k/n=o(1)\), then we have the following estimations \[\begin{align} \label{M1est2}I_{M,n,k,N}&=&\lambda_{0}\exp\left \{-0.5\sqrt{k^{3}/n}-0.5N \sqrt[4]{k^{5}/(n^{3}q^{2})}\right \}(1+o(1)),\\\label{M1est2b} I_{C,n,k,N}&\geq&\lambda_{0}\exp\left\{-k-N\sqrt[4]{k^{3}/n}\sqrt{q}+\sqrt{k^{3}/n}\right\}. \end{align}\] {#eq: sublabel=eq:M1est2,eq:M1est2b} in the case \(k^{3}/n\to\infty\), and \[\label{M1est2a} I_{M,n,k,N}=(1-\exp\{-\lambda_{0}e^{-c_{0}}\})(1+o(1)),\;\;I_{C,n,k,N}\geq \lambda_{0}e^{-k-Nc_{0}\sqrt{q}+c_{0}^{2}},\qquad{(14)}\] in the case \(\sqrt[4]{k^{3}/n}=c_{0}+o(1)\), \(c_{0}\in[0,\infty)\),
A rough version of the ratios ?? and ?? is the estimate \[\label{Mest2a} I_{M,n,k,N}I_{C,n,k,N}\geq \lambda_{0}^{2}e^{-k(1+o(1))}.\qquad{(15)}\]
Proof. By the definitions of \(I_{B,n,k,N}\) and \(I_{C,n,k,N}\) after substituting \(\lambda_{M}=\lambda_{C}=\lambda=\sqrt{n/k}^{k}\) and \(p=c=\sqrt{k/n}\) into ones, taking into account the identity \[\label{est2} \sqrt{\dfrac{1-k/n}{1-p}}=\sqrt{\dfrac{1-k/n}{1-\sqrt{k/n}}}=\sqrt{1+\sqrt{k/n}},\tag{30}\] we have the statements ?? and ?? .
If \(k/n=o(1)\) and \(k^{3}/n\to\infty\), then \(\ln\big(1-\sqrt{k/n}\big)\leq -\sqrt{k/n}\) and \(\big(1-\sqrt{k/n}\big)^{-1}=1+O\big(\sqrt{k/n}\big)\), and by analogy with the previous proof the statements ?? and ?? hold.
Natural modification of the last proof inte case \(\sqrt[4]{k^{3}/n}=c_{0}+o(1)\), \(c_{0}\in[0,\infty)\) leads to the statement ?? .
Statement ?? immediately follows from inequalities ?? and ?? . ◻
Let’s present a special case of Theorem 1 where \(p=c=\sqrt{k/n}\) and \(\lambda_{M}=\lambda_{C}=\lambda=\lambda_{0}\sqrt{n/k}^{k}\), based on the results of Lemma 9. We are focusing on the case \(I_{M,n,k,N}=o(1)\) and \(I_{C,n,k,N}=o(1)\) case, and if at least one these conditions is violated, then the average estimates are obtained using Lemma 4.
Theorem 4. Suppose that \(k\in\mathcal{P}_{k,p}(c_{1})\), \(p\in(0,p_{0})\) with any fixed \(c_{1}\in[0,1)\), \(N\in\mathbb{N}\), \(p_{0}\in(0,1)\). Let \(c=p=\sqrt{k/n}\), \(\lambda_{M}=\lambda_{C}=\lambda=\lambda_{0}\sqrt{n/k}^{k}\), \(k\to\infty\), then, in the case \(I_{M,n,k,N}=o(1)\) and \(I_{C,n,k,N}=o(1)\), the estimates hold \[\label{DT2} \mathbf{E}[T_{I}] \leq c_{\Phi}^{-1}(N)I_{M,n,k,N}^{-1}I_{C,n,k,N}^{-1}(1+o(1)),\;\;\mathbf{E}[T_{F}]\leq 2\lambda_{0}\sqrt{n/k}^{k}\mathbf{E}[T_{I}],\qquad{(16)}\] where for \(I_{M,n,k,N}\) and \(I_{C,n,k,N}\), we need to apply estimates ?? – ?? from Lemma 9.
If \(k^{3}/n\to\infty\), then we have the estimates \[\label{DT02} \mathbf{E}[T_{I}] \leq \lambda_{0}^{-2}e^{k(1+o(1))},\;\;\mathbf{E}[T_{F}]\leq 2\lambda_{0}\sqrt{n/k}^{k}\mathbf{E}[T_{I}],\qquad{(17)}\] in the case \(k/n=o(1)\), and \[\begin{align} \mathbf{E}[T_{I}] &\leq& \lambda_{0}^{-2}\exp\left\{k{+}N\sqrt[4]{k^{3}/n}\sqrt{q}{-}0.5\sqrt{k^{3}/n}\right\}(1+o(1)),\\ \mathbf{E}[T_{F}]&\leq& 2\lambda_{0}\sqrt{n/k}^{k}\mathbf{E}[T_{I}], \end{align}\] in the case \(k^{5}=o(n^{3})\).
If we replace condition \(I_{M,n,k,N}=o(1)\) with condition ?? , we get the estimates \[\label{pamjkivq} \mathbf{E}[T_{I}] \leq \frac{(1+o(1))}{c_{\Phi}(N)I_{C,n,k,N}\left(1-e^{-c_{M}^{(0)}}\right )},\;\;\mathbf{E}[T_{F}]\leq 2\lambda_{0}\sqrt{n/k}^{k}\mathbf{E}[T_{I}],\qquad{(18)}\] where \(I_{C,n,k,N}\) is estimated in Lemma 9.
Proof. All statements of the theorem are obtained taking into account the relations and 15 when substituting expressions from Lemma 9 into statements from theorem 1. ◻
This theorem clarifies corollary 9 [5], where instead of \(e^{k(1+o(1))}\) and its details from the evaluation of ?? are worth \(e^{O(k)}\). We describe here the case of \(\lambda \sim \lambda_{0}\sqrt{n/k}^{k}\) for \(\lambda_{0}\) that are not bounded above, whereas in [5] they are bounded by \(2^{n}\).
We note that our theorems give an explicit form of upper bounds for the averages, which essentially depend on the main parameters depending on \(n\). In the work [5], similar estimates are given without dependence on the main parameters and with accuracy to unknown bounded constants. Our estimates are more precise than the ones in [5], and their quality can be naturally checked in computational experiments. The general approach without an explicit form of the asymptotics of the constants does not allow to estimate the quality of heterogeneous estimates. Of course, the transition to a logarithmic scale removes almost all problems, but this will be a very rough estimate of the quality of the estimates.
Here is a Corollary of Theorem 1, which is analogous to Corollary 10 in [5]. Our results are obtained under less restrictive conditions, and the constants in the estimates are explicitly given. We exclude the case of fixed (or uniformly bounded) \(k\). In this case, the statement of Corollary 10 is of the form \(\mathbf{E}[T_{F}]=o(n^{k})\). More general result in this case can be easily obtained using a natural modification of Theorem 3. It is not difficult with the help of Lemma 4 to investigate the cases \(\varlimsup\limits_{n\to\infty}\lambda_{M}J_{M,n,k,p,N}>0\) or \(\varlimsup\limits_{n\to\infty}\lambda_{C}J_{C,n,k,p,c,N}>0\) , but we omit them, to simplify the proof. To simplify the formulation of the corollary and its proof, we have excluded the case \(\varlimsup\limits_{n\to\infty}c(n)>0\).
The fundamental difference in the statement of our corollary and Corollary 10 [5] is the replacement of the expressions comparing two sequences of the type \(u(n)=o(w(n))\) and \(u(n)=O(w(n))\) by the functions \(z(n)=u(n)/w(n)\) with the corresponding restrictions on \(z(n)\). This allows us to obtain explicit estimates in terms of \(z(n)\) instead of crude estimates such as \(O(\cdot)\).
Corollary 1. Let the first part of Theorem 1 hold, \(k=k(n) \in\mathbb{N}\), \(k\to\infty\) and \(k=o(n)\). Assume that \(p=p(n)=\nu(n)k/n=\nu k/n\), where \(\nu(n)\to\infty\), \(c=c(n)=\rho(n)k/n=\rho k/n=o(1)\), where \(\rho(n)\to\infty\), and \(pc =\sigma(n)k/n=\sigma k/n\), where \(\sigma(n)=O(1)\). In addition, \(\lambda_{M}p^{k}=\tau_{1}(k/(nc))^{k}\), \(\lambda_{C}c^{k}=\tau_{2}(k/(np))^{k}\), where \(\tau_{i}=\tau_{i}(n)\to\infty\), \(i=1,2\).
Then we have the estimates \[\label{er1} \mathbf{E}[T_{I}]\leq \dfrac{\sigma^{2k}}{\tau_{1}\tau_{2}}\exp\left \{\dfrac{kq(1+o(1))}{2\nu}+k\sigma(1+o(1))\right\}(1+o(1)),\qquad{(19)}\] in the cases \(\varlimsup\limits_{n\to\infty}k/\nu=0\) or \(\varliminf\limits_{n\to\infty}k/\nu=\infty\), and \[\label{er2} \mathbf{E}[T_{I}]\leq c_{\Phi}^{-1}(N)\dfrac{\sigma^{2k}}{\tau_{1}\tau_{2}}\exp\{c_{0}N\sqrt{q}+0.5c_{0}^{2}+k\sigma(1+o(1))\}(1+o(1)),\qquad{(20)}\] in the case \(\sqrt{k/\nu}=c_{0}+o(1)\), \(c_{0}\in[0,\infty)\), and it is fixed.
The expected number of fitness evaluations until the \((1 + (\lambda,\lambda))\) GA reaches the global optimum starting from the local optimum of Jump\(_{k}\) is \[\label{er3} \mathbf{E}[T_{F}]\leq \dfrac{\tau_{1}+\tau_{2}}{\sigma^{k}}\mathbf{E}[T_{I}].\qquad{(21)}\]
Proof. Conditions ?? and ?? are analogous to the inequalities \(\alpha\leq 1\) and \(\beta\leq 1\) from Corollary 10 [5]. From the representations 16 , 17 and ?? , the estimates \[\begin{align} \tag{31} J_{M,n,k,p,N}=\left\{ \begin{array}{ll} p^{k}\exp\left \{\frac{-kq(1+o(1))}{2\nu}\right \}(1+o(1)), &\text{if\;} k/\nu\to\infty,\\ p^{k} e^{-c_{0}N\sqrt{q}-0.5c_{0}^{2}}(1+o(1)), &\text{if\;} \sqrt{k/\nu}=c_{0}+o(1), \end{array} \right .\\ \tag{32} J_{C,n,k,p,c,N}=c^{k}e^{-k\sigma(1+o(1))}.\qquad\qquad\qquad\qquad \end{align}\] are valid under the conditions of the corollary.
By definitions \(\lambda_{M}=\tau_{1}/\sigma^{k}\) and \(\lambda_{C}=\tau_{2}/\sigma^{k}\).
Therefore, under the representations 31 and 32 the estimates hold \[\begin{align} \nonumber p_{\lambda_{M},\lambda_{C}}(n,l,k,p,c) \qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ \geq\left\{ \begin{array}{l} \dfrac{\tau_{1}\tau_{2}}{\sigma^{2k}}\exp\left \{\dfrac{-kq(1+o(1))}{2\nu}-k\sigma(1+o(1))\right\}(1+o(1)),\\ \dfrac{\tau_{1}\tau_{2}}{\sigma^{2k}}\exp\{-c_{0}N\sqrt{q}-0.5c_{0}^{2}-k\sigma(1+o(1))\}(1+o(1)). \end{array} \right . \end{align}\]
To complete the corollary proof we need to use the representations 6 . ◻
Here is a consequence of Theorem 1, which is analogous to Theorem 11 in [5]. Here, we will also exclude the simple case of \(\varlimsup\limits_{n\to\infty}k(n)<\infty\), which is studied by other standard methods.
Theorem 5. Let \(k\in\mathbb{N}\), \(k\to\infty\) and \(k\leq n/4\). Assume that \(p=\lambda/n\), \(c=1/\lambda\) and \(\lambda_{M}=\lambda_{C}=\lambda\) for some \(\lambda\in\mathbb{N}\), \(2k\leq \lambda \leq n\). Then the expected runtime of \((1+(\lambda,\lambda))\) GA on Jump\(_k\) is
Then the estimates hold \[\begin{align} \nonumber \mathbf{E}[T_{I}]\leq \lambda^{-2} n^{k} \qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ \cdot\left\{ \begin{array}{ll} \sqrt{\frac{1-c_{1}^{*}}{1-k/n}} \exp\left \{\frac{kc_{1}^{*}q(1+o(1))}{2(1-c_{1}^{*})}-1+c_{1}^{*}\right \}(1+o(1)),&\frac{k}{\lambda}\sim c_{1}^{*}\in(0,1),\\ c_{\Phi}^{-1}(N)e^{c_{0}N\sqrt{q}+0.5c_{0}^{2}-1}(1+o(1)),& \frac{k}{\sqrt{\lambda}}\sim c_{0}^{*}>0\\ \exp\{0.5k^{2}\lambda^{-1}(1+o(1))-1\}(1+o(1)),& \text{otherwise}. \end{array} \right . \end{align}\]
Proof. From the representations 16 , 17 and ?? , the estimates \[\begin{align} J_{M,n,k,p,N}=\left\{ \begin{array}{ll} (\lambda/n)^{k}(1+o(1)) , &\text{if\;} \frac{k}{\sqrt{\lambda}}=o(1),\\ \sqrt{\frac{1-k/n}{1-k/\lambda}}(\lambda/n)^{k} \exp\left \{-\frac{k^{2}q(1+o(1))}{2\lambda(1-k\lambda^{-1})}\right \}, &\text{if\;} \frac{k}{\sqrt{\lambda}}\to\infty,\\ (\lambda/n)^{k} e^{-c_{0}^{*}N\sqrt{q}-0.5(c_{0}^{*})^{2}}(1+o(1)), &\text{if\;} \frac{k}{\sqrt{\lambda}}=c_{0}^{*}+o(1), \end{array} \right .\\ J_{C,n,k,p,c,N}=\lambda^{-k}e^{1-k/\lambda}(1+o(1)).\qquad\qquad\qquad\qquad \end{align}\] are valid under the conditions of the theorem.
Therefore, under the representations 31 and 32 the estimates hold \[\begin{align} \nonumber p_{\lambda_{M},\lambda_{C}}(n,l,k,p,c)\geq \lambda^{2} n^{-k} \qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ \cdot\left\{ \begin{array}{ll} \sqrt{\frac{1-k/n}{1-c_{1}^{*}}} \exp\left \{-\frac{kc_{1}^{*}q(1+o(1))}{2(1-c_{1}^{*})}+1-c_{1}^{*}\right \}(1+o(1)),&\frac{k}{\lambda}\sim c_{1}^{*}\in(0,1),\\ e^{-c_{0}N\sqrt{q}-0.5c_{0}^{2}+1}(1+o(1)),& \frac{k}{\sqrt{\lambda}}\sim c_{0}^{*}>0\\ \exp\{-0.5k^{2}\lambda^{-1}(1+o(1))+1\}(1+o(1)),& \text{in other cases}. \end{array} \right . \end{align}\]
To complete the theorem proof we need to use the representations 6 . ◻
The runtime analysis of the \((1+(\lambda,\lambda))~GA\) on \(\text{\sc Jump}_k\) fitness functions from the prospective of the de MoivreLaplace Theorem. We assumed that the \((1+(\lambda,\lambda))~GA\) has tunable parameters of mutation rate, crossover bias, and two intermediate population sizes, and it starts from a local optima of \(\text{\sc Jump}_k\) fitness function, and that \(np\to \infty\) as \(n\to \infty\). The main result of this work (Theorem 1) is a refined upper bound on the escape time from the local optimum, known from [5]. Different cases of asymptotic behaviour of parameters \(k, p, c\) are considered (Theorem 4). Computational experiment performed in [1] for some values of parameter \(k\) indicates agreement of our theoretical upper bound on the runtime with experimental results. The proposed bounds may be used for further optimization of the algorithm’s parameters.