Adaptive multi-population inflationary differential evolution

This paper proposes a multi-population adaptive version of inflationary differential evolution algorithm. Inflationary differential evolution algorithm (IDEA) combines basic differential evolution (DE) with some of the restart and local search mechanisms of Monotonic Basin Hopping (MBH). In the adaptive version presented in this paper, the DE parameters CR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ CR}$$\end{document} and F are automatically adapted together with the size of the local restart bubble and the number of local restarts of MBH. The proposed algorithm implements a simple but effective mechanism to avoid multiple detections of the same local minima. The novel mechanism allows the algorithm to decide whether to start or not a local search. The algorithm has been extensively tested over more than fifty test functions from the competitions of the Congress on Evolutionary Computation (CEC), CEC 2005, CEC 2011 and CEC 2014, and compared against all the algorithms participating in those competitions. For each test function, the paper reports best, worst, median, mean and standard deviation values of the best minimum found by the algorithm. Comparisons with other algorithms participating in the CEC competitions are presented in terms of relative ranking, Wilcoxon tests and success rates. For completeness, the paper presents also the single population adaptive IDEA, that can adapt only CR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{CR}$$\end{document} and F, and shows that this simpler version can outperform the multi-population one if the radius of the restart bubble and the number of restarts are properly chosen.

vergence by combining DE with local optimisation strategies (Qing 2010).
In previous works by the authors, Locatelli and Vasile (2015) and Vasile et al. (2011), it was demonstrated that DE can converge to a fixed point, a level set or a hyperplane that does not contain the global minimum. The collapse of the population to a fixed point or a neighbourhood of a fixed point from which DE cannot escape was one of the motivation for the development of inflationary differential evolution algorithm (Vasile et al. 2011).
IDEA is based on the hybridisation of DE with the restarting procedure of monotonic basin hopping (MBH) (Wales and Doye 1997); it implements both a local restart in the neighbourhood of a local minimum and a global restart in the whole search space. IDEA was shown to give better results than a simple DE, but its performance is dependent upon the parameters controlling both the DE and MBH heuristics (Vasile et al. 2011). These parameters are the crossover probability CR, the differential weight F, the radius of the local restart bubble δ local and the number of local restarts n LR , whose best settings are problem dependent. Different adaptive mechanisms for adjusting CR and F during the search process can be found in the literature, (Brest et al. 2006(Brest et al. , 2013Liu and Lampinen 2005;Omran et al. 2005); a parameter-less adaptive evolutionary algorithm has been presented in Papa (2013). However, no approach has been proposed so far to adapt δ local and n LR . In this paper, we present a simple mechanism to adapt CR and F within a single population and a multi-population strategy to adapt δ local and n LR . The multi-population version of IDEA is in the following called MP-AIDEA (Multi Population Adaptive Inflationary Differential Evolution Algorithm).
The resulting algorithm was extensively tested over 51 test problems from the single objective global optimisation competitions of the Congress on Evolutionary Computation (CEC) 2005, 2011 and 2014. Tests to assess the performance of the algorithm include rankings, Wilcoxon test and success rate. It will be shown that the adaptive version of IDEA always ranks in the first three best algorithms in every competition for every number of dimensions except for the CEC 2014 test set with 30 dimensions. Furthermore, it will be shown that the simple adaptation of CR and F within a single population can outperform the multi-population version with adaptation of δ local and n LR if these two parameters are properly chosen.
This paper extends the work presented in Di Carlo et al. (2015). In Di Carlo et al. (2015), the basic mechanisms that constitute MP-AIDEA were introduced, and the performance of MP-AIDEA was measured only by a relative ranking against other algorithms. This paper provides a more detailed explanation of all the mechanisms and heuristics inside MP-AIDEA; moreover, it presents an extensive empirical assessment of its performance, using several metrics in addition to the relative ranking. As part of this extensive performance evaluation, we compare MP-AIDEA against a number of other algorithms and a single population version of MP-AIDEA with no adaptive local restart. Detailed results obtained for each test functions are also presented, so that the paper can be used as a reference for comparison against other algorithms.
The paper starts stating the problem we are trying to solve in Sect. 2 and briefly introducing the basic principles and fundamental theoretical developments underneath inflationary differential evolution in Sect. 3. The adaptation mechanisms are presented, together with the resulting adaptive multipopulation version of IDEA, MP-AIDEA, in Sect. 5. The test cases are presented in Sect. 6, and the obtained results are presented in Sect. 6.1. Finally, the paper presents the results of all the comparative tests in Sects. 6.2, 6.3 and 6.4. Section 7 concludes the paper.

Problem statement
This paper is concerned with the following class of global minimisation problems with box constraints: (1) with f : B ⊆ R n D → R, n D the number of dimensions and the box B defined by the upper and lower boundaries x lower ≤ x ≤ x upper . In the following, we will use a gradient-based local search algorithm; therefore, we further require that f ∈ C 2 (B). Note, however, that this is not a strict requirement as we can show that the algorithm can work also when a finite number of non-differentiable points exist.

Inflationary differential evolution
This section briefly recalls the working principles of inflationary differential evolution and presents the parameters that the algorithm proposed in this paper adapts. Following the notation introduced in Vasile et al. (2011), we express the general DE process as a discrete dynamical system. The governing equation, for the i-th individual at generation k, is expressed as: with u i,k = e Gx r 1 ,k + (1 − G)x i,k + F(x r 2 ,k − x r 3 ,k ) where G can be either 0 or 1 [with G = 1 corresponding to the DE strategy DE/rand and G = 0 corresponding to the DE strategy DE/current-to-best (Price et al. 2006)]. In Eq.
(3), r 1 , r 2 and r 3 are integer numbers randomly chosen in the population, and e is a mask containing random numbers of 0 and 1 according to: U is a random number taken from a random uniform distribution [0, 1]. The product between e and the term in square brackets in Eq.
(3) has to be intended component-wise. In this work, given u t,i,k , the t-th component of the trial vector u i,k , the following correction is applied to satisfy the box constraints (Zhang and Sanderson 2009): u t,i,k = x t,i,k + x t,lower /2, if u t,i,k < x t,lower x t,i,k + x t,upper /2, if u t,i,k > x t,upper The selection function S is defined as: In the general case in which the indices r 1 , r 2 and r 3 can assume any value, in Vasile et al. (2011) it was demonstrated that the population can converge to a fixed point different from a local minimum or to a level set. Furthermore, in Locatelli and Vasile (2015) it was demonstrated that DE can converge to a hyperplane that does not contain the global minimum. Finally, consider the following proposition.
Proposition 1 Consider the subset Ψ = {x ∈ B : f (x) ≤f } and the superset φ such that: then if the population at iteration k is entirely contained in Ψ it cannot escape from Ψ at any future iteration.
Proof The proof descends from the definition of S. Suppose that a candidate individual x i,k+1 was generated by map (2) then, because of point 3 of the proposition, it would be rejected by the selection operator.
Therefore, when the population contracts within a ball B c ⊆ Ψ of radius ρ l , DE can only converge to a point or a subset within B c . We call ρ l the contraction limit, in the following.
In inflationary differential evolution, the DE heuristics is iterated until the population reaches the contraction limit. A local search is then started from the best individual in the population x best , the corresponding local minimum x LM is saved in an archive of local minima A and the population is restarted in a bubble B R of radius δ local around the local minimum x LM . This mechanisms is borrowed from the basic logic underneath monotonic basin hopping (Wales and Doye 1997). To assess if the contraction condition is satisfied, the maximum distance between all possible combinations of individuals of the population at generation k, ρ (k) , is computed: where N pop is the number of individuals in the population. The contraction is verified when ρ (k) ≤ρρ max , where ρ max = max k ρ (k) is the maximum value of ρ ever recorded until generation k andρ is one of the parameters of the algorithm, the contraction threshold. This contraction criterion is consistent with Proposition 1 under the assumption that ρ l =ρρ max . After a number n LR of such local restarts, without any improvement of the current best solution, the archive A collects all the local minima found so far. At this point, the population is restarted globally in the search space so that every individual is initially at a distance √ n D δ global from the centres of the clusters of the local minima in A. During local restarts, the most important information is preserved in the local minimum. The assumption is that the basin of attraction of that local minimum has already been explored and that exploration led to the convergence of the population to B c . When the population is restarted globally the essential information, all the local minima, is stored in the archive A.
Here the assumption is that IDEA has completely explored a funnel structure resulting in a cluster of minima.
These restart procedures were proven to be very effective in a series of difficult real problems in which the landscape presents multiple funnels (see Vasile et al. 2011 for additional details).
The complete inflationary differential evolution process with trial vector (3) is governed by the following key parameters: N pop , CR and F, G,ρ, δ local , n LR , δ global . From experience, we know that δ global is not a critical parameter in most of the cases while CR, F, δ local and n LR play a significant role and are not always easy to define. The parameters CR and F are applied to update each individual in a population while δ local and n LR are applied to restart the whole population. Therefore, in this paper we propose two adaptation mechanisms, one for CR and F and one for δ local and n LR . In particular, the adaptation mechanisms of CR, F and δ local are such as to result in the definition of numerical values for these parameters, to be used by the algorithm. On the contrary, the use of n LR is replaced by a mechanism that allows the algorithm to decide when to perform a local or global restart, so that the definition of a numerical value for n LR is not required anymore.

Adaptation mechanisms
Because of the very nature of CR and F, δ local and n LR , the automatic adaptation of CR and F requires only the evaluation of the success of each candidate increment u i,k . On the other hand, the adaptation of δ local and n LR requires the evaluation of the success of the restart of an entire population. Therefore, in this paper it is proposed to extend the working principle of inflationary differential evolution by evolving n pop populations in parallel, where n pop is defined a priori.
Each population adapts its own values of CR and F. We use a stigmergic approach in which the CR and F of each individual are drawn from a joint probability distribution, over a set of possible values of CR and F, that evolves with the population.
All populations are then concurrently adapting δ local and the number of local restarts. More specifically, the adaptation mechanism of the local restart bubble evolves a probability distribution function over a range of possible values of δ local . Each population draws values from that probability distribution and at each local restart increases the probability associated to the value of δ local that led to a transition from one local minimum to another. The range of δ local is also adapted by taking the mean and the minimum distance among the local minima in A.
The number of local restarts, instead, is dictated by the contraction of a population within the basin of attraction of an already identified local minimum. Given a local minimum x LM ∈ A and a list of n best,L M best individuals from which a local search converged to x LM , the size of the basin of attraction of x LM is defined as Each local minimum x LM in A, therefore, is associated to a particular d basin,L M . Figure 1 illustrates this mechanism. Once d basin,L M is estimated, every time the condition ρ (k) m ≤ ρρ m,max is satisfied for population m, if the best individual x best,m is at a distance lower than d basin,L M from x LM , then no local restart is performed but the population is restarted globally in the search space. The number n best,L M is set to 4 in this implementation.
The overall algorithm, called Multi-Population Adaptive Inflationary Differential Evolutionary Algorithm (MP-AIDEA), is described in more detail in the following section.

Multi-population adaptive inflationary differential evolution
MP-AIDEA is described in Algorithm 1. Let n pop be the number of populations and m the index identifying each population. With reference to Algorithm 1, after initialisation of main parameters and functionalities (Algorithm 1, line 1), MP-AIDEA starts by running n pop Differential Evolutions in parallel, one per population (Algorithm 1, line 3). During each evolution process, the parameters F and CR are automatically adapted following the approach presented in Sect. 5.2. When a population m contracts within a ball B c of radiusρ ρ m,max , the evolution of that population is stopped. Once all the populations have contracted, the relative position of the best individual of each population, x best,m with respect to the local minima in A, x LM , is assessed (Algorithm 1, line 7). This step makes use of all the minima found by all populations and, therefore, it has to be regarded as an information sharing mechanism among populations. If the best individual of population m is not within the basin of attraction of any previously detected local minimum (that is, ∀L M : x best,m − x LM > d basin,L M ) then a local search is run (Algorithm 1, line 8) and the resulting local minimum is stored in the archive A (Algorithm 1, line 16). The flag for the local restart, L R m , is set to 1. On the contrary, if the best individual of population m is inside the basin of attraction of a previously detected local minimum, the local search is not performed and L R m is set to 0 (Algorithm 1, line 20). Before running a local or a global restart (Algorithm 1, line 24), the probability distribution associated to δ local and its range are updated (Algorithm 1, line 23). After restarting the population, if the number of maximum function evaluations, n feval,max , is not exceeded, the process restarts from line 2 in Algorithm 1. Each part of Algorithm 1 is explained in more details hereafter.

Initialisation
The steps for the initialisation of MP-AIDEA are presented in

Differential evolution and the adaptation of CR and F
For each population m, a DE process is run (Algorithm 3, line 6), using Equations 2, 3, 4 and 6. The parameter G, in Equation 3, assumes values equal to 0 or 1 with probability 0.5. During the advancement from parents to offspring, each individual of the population is associated to a different value of CR and F, drawn from a distribution CRF is initialised as a uniform distribution with (n D + 1) 2 points in the space CR ∈ [0.1, 0.99] and F ∈ [−0.5, 1] (Algorithm 3, line 1). A Gaussian kernel is then allocated to each node and a probability density function is built by Parzen approach (Minisci and Vasile 2014). The values of CR and F to be associated to the individuals of the population are drawn from this dis-tribution (Algorithm 3, line 4). A change value dd linked to each kernel is initialised to zero (Algorithm 3, line 3) and is used during the advancement of the population from parents to children to adapt CR and F (Algorithm 3, line 8). The adaptation of CR and F is summarised in Algorithm 4 and described in the following.
Algorithm 3 Differential Evolution with adaptive C R and F 1: Regular meshes CR and F with (n D + 1) 2 points (n D is the dimensionality of the problem) in the space C R ∈ [0.
Algorithm 4 Updating the joint distribution CRF For each individual i of each population m, the adaptation mechanism for CR and F is started only if the child is characterised by an objective function value lower than the parent's one, that is f (x m,i ) (Algorithm 4, line 1). If this condition is verified, the difference in objective function between parent and child at subsequent generation, d f m,q (Algorithm 4, lines 5 and 6). This is because F (k m ) m,i produced a bigger decrease in the objective function than CRF (k m ) m,q,2 (as shown by is greater than a given value C RC (Algorithm 4, line 7), (Minisci and Vasile 2014).
The DE stops according to the contraction condition presented in Sect. 3. In order to prevent an excessive use of resources when the population partitions, a fail safe criterion was introduced that stops the DE after 10D generations (Algorithm 3, line 13).

Local search and restart mechanisms
After the evolution of all populations has stopped, MP-AIDEA checks if the best individual of each population is inside the basin of attraction of any previously detected local minimum (see Algorithm 1, line 7). If that is not the case, a local search is performed from the best individual and the population is locally restarted within a hypercube with edge equal to 2δ local around the detected local minimum; otherwise, no local search is performed and the population is restarted globally in the whole search space (Algorithm 1, line 24). Prior to the implementation of the restart mechanisms, MP-AIDEA updates the estimation of the size of the basin of attraction of each minimum, the archive A (see Algorithm 1, lines 5 to 22) and the distribution over the possible values of 2δ local (see Algorithm 1, line 23). In the following the identification of the basin of attraction, the estimation of δ local and the two restart mechanisms are described in more details.

Identification of the basin of attraction
In order to mitigate the possibility of running multiple local searches that converge to already discovered local minima, MP-AIDEA estimates for each local minimum in A the radius of the basin of attraction of that local minimum. The radius of the basin of attraction is here defined as the distance d basin,L M from a given local minimum x LM such that if the best individual in population m, x best,m , is at a distance from x LM lower than d basin,L M a local search starting from x best,m would converge to x LM .
The radius d basin,L M is estimated with the simple procedure in Algorithm 1, lines 7 to 19. Once the evolution of all populations has stopped, the distance x best,m − x LM of the best individual, in each population, with respect to all the minima in A is calculated and compared to the d basin,L M associated to each local minimum in A; initially all d basin,L M are set to 0. If the distance x best,m − x LM is grater than d basin,L M a local search is started from x best,m . If the resulting local minimum x (s m ) min,m already belongs to A, the counter i LM is updated and the new estimate of the basin of attraction of ε is set to 10 −3 . If i LM exceeds a given maximum value and x best,m − x LM < d basin,L M ∀ L M no local search and no local restart are performed. The counter i LM is initialised to 1 for every new local minimum and keeps track of the number of times a local minimum is discovered.

Adaptation of ı local
When a population m is locally restarted, individuals are generated by taking a random sample, with Latin Hypercube, within a hypercube with edge equal to 2δ local,m . The dimension δ local,m is drawn from a probability distribution that is progressively updated at every restart. We use a kernel approach with kernels centred in the elements of a vector B (see Algorithm 6) containing a range of possible values of δ local,m . The vector B is initialised, with the procedure presented in Algorithm 5, when all populations performed a local search for the first time and at every global restart. During initialisation the distance between all the local minima in the archive A is computed (Algorithm 5, line 1) and B is initialised with values spanning the interval between the minimum and the mean distance among minima (Algorithm 5, lines 2-3). The mean values instead of the max is used to limit the size of the restart bubble and speed up convergence under the assumption that a local restart needs to lead to the local exploration of the search space. In the experimental tests, it will be shown that this working assumption is generally verified and δ local,m tends to converge to small values. Then, a second vector dd b , with the same number of components of B, is initialised to zero (Algorithm 5, line 4).
During the update phase of δ local,m , MP-AIDEA uses the index s m to keep track of the number of times population m performed a local search and calculates the difference p m between two subsequent local minima (see Algorithm 6, line 5). The value p m is then compared to the elements in dd b and when dd b,q < p m then δ local,m replaces B q , and p m replaces dd b,q (Algorithm 6, lines 7-10). In other words, if the δ local,m used to restart population m led to a local minimum x Compute Cluster local minima in A and compute cluster baricentres x c 7: Initialise vector B using Algorithm 5 9: end if 10: end for

Local and global restart
After the identification of the basin of attraction and the update of the value of δ local , populations undergo a restart process in which a new population is generated either by sampling a neighbourhood of a local minimum (local restart) or by sampling the whole search space (global restart). The two restart procedures are described in Algorithm 7.
The local restart procedure takes the latest identified local minimum x The global restart procedure identifies clusters of local minima with a Fuzzy C-Mean algorithm (Bezdek 1981), computes the centre of each cluster and initialises population m so that each individual is at distance at least √ n D δ global from each of the centres of the clusters (Algorithm 7, lines 6 and 7).
At each local and global restart, the CRF matrix is reinitialised while the vector B is initialised only after every global restart. The motivation for re-initialising CRF at every restart is twofold: on the one hand different values of CR and F might be optimal in different parts of the search space, and on the other hand convergence to the optimal value of CR and F is not always guaranteed. In search spaces with uniform and homogeneous structures, restarting CRF and B might lead to an overhead on the computational cost; therefore, in future implementations we will test the possibility of retaining CRF and B across the restart process.

Computational complexity
The computational complexity of MP-AIDEA is defined by the three main sets of operations: -Restart mechanisms. The cost of the local restart procedure is limited to the generation of n pop N pop individuals, while the global restart has a cost associated also to clustering, which is O = (n 2 LM n D n iter ) (Bezdek 1981), where n iter is the number of iterations for the clustering, and one associated to the verification that the new population is far from the clusters, which is O(N pop n LM ) (see line 7 of Algorithm 7).
Overall when n pop N pop < n D the dominant algorithmic cost is the local search while the adaptation of CR and F becomes more expensive for large and numerous populations. Since in the experimental test cases we will use N pop = n D and n pop = 4 the overall algorithmic complexity remains O(n 3 D ).    The settings of MP-AIDEA were maintained constant for all problems within a particular test set and were changed going from one test set to another. This is in line with the way all the other algorithms competed. Table 4 summarises the parameters and settings used for the CEC 2005, CEC 2011 and CEC 2014 test functions. More details about the chosen parameters will be given in Sect. 6.1.
The ranking of the algorithms participating in every competition was adjusted to account only for their performance on the selected subset of differentiable functions.
It will be shown that all metrics lead to similar conclusions: MP-AIDEA ranks among the first four algorithms, if not first, in all three test sets and for all dimensions. We will also show that MP-AIDEA can detect previously undiscovered minima on some particularly difficult functions.
The current implementation of MP-AIDEA can be found open source at https://github.com/strath-ace/smart-o2c together with the benchmark of test cases.

Test sets
This section briefly describes each test set, the settings of MP-AIDEA and metric 1 for all test sets.   2005 competition were not included in the test set because non-differentiable. The number of populations in MP-AIDEA was set to n pop = 4 and the number of individuals in each population was set to N pop = n D . The number of populations to be deployed on a particular problem depends on the type and complexity of that problem, and the available number of function evaluations. We tested MP-AIDEA with different numbers of populations from 1 to 4 (results using MP-AIDEA with one population are presented in Sect. 6.2). Results showed that MP-AIDEA with 4 populations performs consistently well on all benchmarks, and, thus, we decided to present our findings for n pop = 4. The contraction limit was set toρ = 0.2 and the global restart distance was set to δ global = 0.1 (Table 4). In line with the metrics presented at the CEC 2005 competition, Tables 5 and 6 reports the difference, in the objective value, between the result obtained with MP-AIDEA and the known global minimum. Table 7 reports the best objective function error values obtained by all the algorithms participating in the CEC 2005 competition and MP-AIDEA for functions 13 and 16 and n D = 10. According to the CEC 2005 specifications, the accuracy level for the detection of the global minimum is 10 −2 for these functions. MP-AIDEA is able to identify the global minimum of both functions 13 and 16. Previously only EvLib (Becker 2005) succeeded in identifying the global minimum of function 13 and no other algorithm managed to find the global minimum of function 16.

CEC 2011 test set
Following the rules of the CEC 2011 competition (Das and Suganthan 2010), MP-AIDEA was run for n feval,max = 150000 function evaluations on the CEC2011 test set. The experiments were repeated for n runs = 25 independent runs. Test functions with equality and inequality constraints were not included in the tests. The number of populations n pop  was set to 4 and the number of individuals in each population was set to N pop = 30 regardless of the dimensionality of the problem. The contraction limit and the global restart distance were set, respectively, toρ = 0.2 and δ global = 0.1 (Table  4). Table 8 reports the best, worst, median, mean objective function found by MP-AIDEA and the associated standard deviation.

CEC 2014 test set
In line with the rules of the CEC 2014 competition (Liang et al. 2013), MP-AIDEA was applied to the solution of the functions in the CEC 2014 test set in dimension n D = 10, 30, 50 and 100, with maximum number of function evaluations n feval,max = 10000n D . The experiments were repeated for n runs = 51 independent runs. Non-differentiable functions 6, 12, 19, 22, 26, 27, 29 and 30 were not included in the test set (see Table 3). The number of populations was set to n pop = 4 and the number of individuals in each population was set to N pop = n D . The contraction limit and the global restart distance were set, respectively, toρ = 0.2 and δ global = 0.1 (Table 4). Tables 9 and 10 report the difference between the objective value found by MP-AIDEA and the known global minimum. In agreement to the guidelines of the competition error values smaller than 10 −8 are reported as zero, (Liang et al. 2013). Table 11 reports the best objective function values obtained by all the algorithms participating in the competition and MP-AIDEA for functions 9, 10, 11 and 15 in 10 dimensions. MP-AIDEA finds the global minimum of function 11, unlike all the other competing algorithms, and gives good results for the other functions.

Ranking
In this section, MP-AIDEA is ranked against a group of algorithms participating in each CEC competition. The rankings include those algorithms that reported their results in a paper and MP-AIDEA with two different settings: n pop = 4 and N pop = n D . This settings will be indicated as "MP-AIDEA" in the following and corresponds to the settings that was used to generate the results in Sect. 6.1. n pop = 1, N pop = 4n D ; MP-AIDEA adapts CR and F but uses fixed values for δ local and n LR . In particular, n LR = 10 and δ local = 0.1, unless otherwise specified. This settings will be indicated as "MP-AIDEA, n pop = 1" in the following.
The ranking method follows the rules of the CEC 2011 competition, . All algorithms are ranked on the basis of the best and mean values of the objective function obtained over a certain number of runs. The following procedure is used to obtain the ranking: -for each function, algorithms are ranked according to the best objective value; -for each function, algorithms are ranked according to the mean objective value; -the ranking for the best and mean objective values of a particular algorithm are added up over all the problems to get the absolute ranking.
In the following, the rankings obtained for the CEC 2005, CEC 2011 and CEC 2014 test sets are presented.

CEC 2005 test set
The rankings obtained for n D = 10, n D = 30 and n D = 50 are reported in Table 12. Only the competing algorithms that reported in their paper also the results obtained for the hybrid functions of the CEC 2005 competition (Table 1) are considered. Results show that, for n D = 10 and n D = 30, MP-AIDEA with adaptation of δ local and n LR is ranked first, while for n D = 50 results are better when using MP-AIDEA with non-adapted δ local = 0.1 and n LR = 10. In any case, both settings outperform the winning algorithm of the competition CEC 2005.     CoEVO (Pošik 2005) 16 flexGA (Alonso et al. 2005) 17 ES (Costa 2005)

CEC 2011 test set
The results obtained on the CEC 2011 test set are reported in Table 13. MP-AIDEA ranks first if problem 13 (the Cassini 2 Spacecraft Trajectory Optimisation Problem) is excluded from the test set and second if it is included. The reason can be found in Fig. 2. Figure 2 shows the convergence profile of the best solutions found by MP-AIDEA and GA-MPC, the best algorithm of the competition, on function 13 for an increasing number of function evaluations (greater than the limit prescribed by the CEC 2011 competition). The results for GA-MPC are obtained using the code available online (http://www3.ntu.edu.sg/home/ epnsugan/index_files/CEC11-RWP/CEC11-RWP.htm).
On this test problem, GA-MPC converges very rapidly to a local minimum but then stagnates. On the contrary,  MP-AIDEA has a slower convergence for the first 200,000 function evaluations but then progressively finds better and better minima as the number of function evaluations increases. This demonstrates that in a realistic scenario in which function evaluations are not arbitrarily limited, MP-AIDEA would provide better results than the algorithm that won the competition. Results in Table 13 shows that MP-AIDEA with adaptation of δ local and n LR performs better than MP-AIDEA with fixed values of δ local and n LR . The adaptation history of δ local is shown in Fig. 3 for each of the four populations on test functions 12 and 13 and for 600,000 function evaluations.

CEC 2014 test set
The ranking results for the CEC 2014 test set are reported in Table 14. MP-AIDEA with one population is tested in this    Fig. 4. These figures show how the adaptation of δ local is effective when a sufficient number of adaptation steps can be performed within the limit of the maximum number of function evaluation (300,000 in this case). For function 25, for example, the adaptation steps are only 7, while they are 11 for function 17 and 18 for function 9. In these two cases δ local converges to 0.1 and 0.04, respectively.  The performance of MP-AIDEA for the 30D functions of the CEC 2014 test set is further investigated to test the dependence of the results upon the two non-adapted parameters,ρ and δ global . Table 15 shows the raking obtained when varyinḡ ρ and δ global .
Case B of Table 15 shows the ranking obtained when usinḡ ρ = 0.3 instead thanρ = 0.2. Comparing the results in Table 15 with those in Table 14, it is possible to see that MP-AIDEA performs better usingρ = 0.3 rather thanρ = 0.2, moving from the fourth to the third position in the ranking. At the same time, there is no significant dependence upon the value of δ global , as shown by Cases C and D in Table 15, where δ global is changed from its nominal value of 0.1 to 0.2 and 0.3.

Wilcoxon test
The Wilcoxon rank sum test is a nonparametric test for two populations when samples are independent. In this case, the two populations of samples are, for each problem, the n runs values of the objective function obtained by MP-AIDEA and by another algorithms participating in the CEC 2011 and CEC 2014 competitions. No test is performed for the CEC2005 test set, since for no one of the algorithms participating in the CEC 2005 competition the code is available on-line.
The Wilcoxon test is realised using the Matlab ® function ranksum. ranksum tests the null hypothesis that data from two entries x and y are samples from continuous distributions with equal medians. Results from ranksum are presented in the following as values of p and h. p, ranging from 0 to 1, is the probability of observing a test statistic as or more extreme than the observed value under the null hypothesis. h is a logical value, where h = 1 indicates rejection of the null hypothesis at the 100α % significance level while h = 0 indicates a failure to reject the null hypothesis at the 100α % significance level, where α is 0.05. When h = 1, the null hypothesis that distributions x and y have equal medians is rejected, and additional test are conducted to assess which one of the two distributions has lower median. In order to do so, three types of tests are realised using ranksum for the two distributions x and y: -Two-sided hypothesis test: the alternative hypothesis states that x and y have different medians. Two distributions with equal medians will give as results p B = 1 and h B = 0 (failure to reject the null hypothesis that x and y have equal medians), while two distributions with different medians will give as results p B = 0 and h B = 1 (rejection of the null hypothesis that x and y have equal medians). If the two-sided hypothesis test finds that the two distributions have equal medians ( p B = 1 and h B = 0), no further test is conducted. Otherwise, the left-tailed and right-tailed hypothesis test are conducted. -Left-tailed hypothesis test: the alternative hypothesis states that the median of x is lower than the median of y. If x has median greater than the median of y, results will be p L = 1 and h L = 0 (failure to reject the hypothesis that x has median greater than y) while if x has median lower than y results will be p L = 0 and h L = 1 (rejection of the hypothesis that x has median greater than y). -Right-tailed hypothesis test: the alternative hypothesis states that the median of x is greater than the median of y. If x has median lower than the median of y, results will be p R = 1 and h R = 0 (failure to reject the hypothesis that x has median lower than y) while if x has median greater than y results will be p R = 0 and h R = 1 (rejection of the hypothesis that x has median lower than y).
If x is the distribution of results of MP-AIDEA and y the distribution of results given by another algorithm, the possible results obtained from the ranksum tests are summarised in Table 16. Case 1 in Table 16 (h B = 0) represents a situation in which the distribution of results from MP-AIDEA and a competing algorithm have equal median (failure to reject the hypothesis that x has median lower than y). Case 2 (h B =1, h L =0 and h R =1) represents a situation in which the median of MP-AIDEA is greater than the median of the other algorithm (rejection of the null hypothesis that x and y have equal medians, failure to reject the hypothesis that x has median greater than y, rejection of the hypothesis that x has median lower than y). Case 3 (h B =1, h L =1 and h R =0) represents instead a situation in which the median of MP-AIDEA is lower than the median of the other algorithm (rejection of the null hypothesis that x and y have equal medians, rejection of the hypothesis that x has median greater than y, failure to reject the hypothesis that x has median lower than y). In the following, test functions with results corresponding to cases 1 and 3 are shown in bold (MP-AIDEA has median equal or lower than the competing algorithm). For case 3 results with p B < 5 · 10 −2 , p L < 5 · 10 −2 and p R > 9.5 · 10 −1 are considered significant. Analogously, the competing algorithm has median lower than MP-AIDEA if p B < 5 · 10 −2 , p L > 9.5 · 10 −1 and p R < 5 · 10 −2 .

CEC 2011 test set
For the CEC 2011 test set, we limited the comparison against the two top algorithms GA-MPC and DE-Λ, for which the code is available online (http://www3.ntu.edu. sg/home/epnsugan/index_files/CEC11-RWP/CEC11-RWP. htm; http://uk.mathworks.com/matlabcentral/fileexchange/ 39217-hybrid-differential-evolution-algorithm-with-adapti ve-crossover-mechanism/content/DE_TCRparam.m). The outcome of the Wilcoxon test for the comparison of MP-AIDEA against GA-MPC, the winning algorithm of the CEC2011 competition, can be found in Table 17 for all the functions in the test set in Table 2.
The comparison of MP-AIDEA with GA-MPC shows that the median of MP-AIDEA is lower than the median of GA-MPC (Case 3) for functions 2, 5, 6 and 7, while it is higher  (Case 2) for functions 1, 3 and 13. Results for functions 10 and 12 are not significant enough to obtain a clear indication.
The outcome of the Wilcoxon test for the comparison of MP-AIDEA with DE-Λ is reported in Table 18.
The comparison of MP-AIDEA with DE-Λ (Table 18) shows that the median of MP-AIDEA is lower than the median of DE-Λ for functions 3, 5, 6, 10, 12 and 13. Results for the remaining functions 1, 2 and 7 are not significant enough to obtain a clear indication. Table 19 summarises the outcome of the Wilcoxon tests for the CEC 2011 test set. The table reports the number of functions for which the median of MP-AIDEA is lower, equal or higher than the median of the competing algorithm. The results in Table 19 show that MP-AIDEA clearly outperforms DE-Λ and has median lower than GA-MPC for 4 test functions.    (Tables 24, 25 , 26, 27, 28, 29, 30, 31). A summary of the obtained results is given in Table 20. Table 20 shows the number of function for which Case 1, 2 or 3 in Table 16 are verified and the number of functions for which the results are not significant enough to judge, for n D equal to 10, 30, 50 and 100.
For n D = 10, the median of MP-AIDEA is lower than the one of UMOEAs in 11 cases, while in 3 cases the medians are equal and in 4 cases the median of UMOEAs is lower than the median of MP-AIDEA. In 4 cases (functions 10, 17, 20 and 21), the results are not significant enough. For n D = 30 and n D = 100 the median of MP-AIDEA is lower than the median of UMOEAs in 9 cases and the median of UMOEAs is lower than the one of MP-AIDEA for other 9 functions. For 4 functions, the results are not significant enough to obtain a clear indication. The median of MP-AIDEA is lower than the one of UMOEAs in 11 cases for n D = 50.
As regards the comparison with L-SHADE, MP-AIDEA has lower median for a number of functions greater than L-SHADE only for n D = 10 (9 functions).
In all dimension but n D = 50, the number of functions for which the median of MP-AIDEA is lower than the median of MVMO is greater than the number of functions for which the median of MVMO is lower than the median of MP-AIDEA. In all the cases, MP-AIDEA has median lower than CMLSP for the majority of the tested functions.
Summarizing, results of the Wilcoxon test show that MP-AIDEA clearly outperforms CMLSP for all the values of n D , gives similar or slightly better results than UMOEAs and MVMO while is outperformed by L-SHADE for n D = 30, n D = 50 and n D = 100.

Success rate
In this section, we present the success rate of MP-AIDEA and the top performing algorithms on the test sets CEC 2011 and CEC 2014. As for the Wilcoxon test no algorithm participating in the CEC 2005 was included in the comparison due to the lack of availability of the source code. The computation of the success rate S R is reported in Algorithm 5 for a generic algorithm AG and a generic problem min f where n is the number of runs (Vasile et al. 2011). In Algorithm 5,x (AG, i) denotes the lowest minimum observed during the i-th run of the algorithm AG. The quantity f global is the known global minimum of the function and tol f is a prescribed tolerance with respect to f global . The index j sr represents the number of times algorithm AG generates values lower or equal than f global + tol f . For each test set, we report also the total number of problems in which each of the tested algorithms has the best success rate.

CEC 2011 test set
For the calculation of the success rate on the test set CEC 2011, we consider the following algorithms: MP-AIDEA with 4 populations (MP-AIDEA), adaptive δ local and local restart, MP-AIDEA with one population, n LR = 10 and δ local = 0.1 (MP-AIDEA*), GA-MPC and DE-Λ. Table 21 shows the obtained values of S R and the value of tol f used for  Highest success rates for each function are shown in bold. and their total is reported at the bottom of the table for each value of n D . MP-AIDEA* represents MP-AIDEA with settings n pop = 1 and δ local = 0.1  if δ f < tol f then 5: j sr = j sr + 1 6: end if 7: end for 8: S R = j sr /n runs each function and shows that MP-AIDEA outperforms all the other algorithms on most of the functions. The result against GA-MPC would be even better if a higher number of function evaluation was considered, as explained in Sect. 6.2.2.

CEC 2014 test set
For the comparison on the test set CEC 2014, we considered the following algorithms: MP-AIDEA, MP-AIDEA*, UMOEAs, CLMSP, L-SHADE and MVMO (http://web. mysites.ntu.edu.sg/epnsugan/PublicSite/Shared%20Docum ents/Forms/AllItems.aspx). The values of the success rates for all tested algorithms are shown in Tables 22 and 23, together with the associated values of tol f . The total number of problems for which an algorithm yields the best success rate is also reported.
For all dimensions, MP-AIDEA compares very well against the other algorithms. In low dimension, the full adaptive settings is the most competitive while as the number of dimensions increases the single population version with δ local = 0.1 results the most successful algorithm. These results are in line with the results in Sect. 6.2.3 and confirm the position of MP-AIDEA in the ranking.

Conclusions
This paper presented MP-AIDEA, an adaptive version of inflationary differential evolution which automatically adapts the two key parameters of differential evolution, CR, F, the size of the restart bubble δ local and the number of local restarts n LR . The adaptation of the number of local restarts is implemented through a mechanism that mitigates the possibility to detect the same local minimum multiple times. This mechanism allows MP-AIDEA to automatically identify when to switch from a local to a global restart of the population.
MP-AIDEA was tested on a total of 51 problems, taken from three CEC competitions, grouped in three test sets (named CEC 2015, CEC 2011and CEC 2014 and compared against 53 algorithms that participated in those three competitions. Four different metrics were presented to assess the performance of MP-AIDEA. Results demonstrated that MP-AIDEA ranks first in the CEC 2015 outperforming all the other algorithms for all problem dimensionalities. On the CEC 2011 test set, MP-AIDEA ranks second, after GA-MPC, if we restrict the number of function evaluations to the one prescribed by the competition. However, it was demonstrated that, in problem 13, an increase in the number of function evaluations does not provide any improvement of the objective value returned by GA-MPC but greatly improves the result of MP-AIDEA. It was noted, in fact, that GA-MPC has a fast convergence but then tends to stagnate. On the contrary, the convergence profile of MP-AIDEA is slower but, thanks to the restart mechanism, achieves better objective values. In this test set, in particular, the adaptation of the local restart neighbourhood was shown to be effective providing compet-itive results compared to the settings of MP-AIDEA with a single population and predefined values of δ local and number of restarts. This is confirmed by the Wilcoxon test, and the success rate.
On the test set CEC 2014, results are not equally satisfactory for all dimensions. MP-AIDEA is in the top three algorithms except in dimension 30. When the number of populations is reduced to one and δ local = 1, MP-AIDEA outperforms all other algorithms in dimension 50 and 100.
One part of the problem is the extra effort required by the multi-population adaptive algorithm to identify the correct value of δ local . However, another part of the problem was found in the contraction limit. This is in line with the theoretical findings by the authors who demonstrated that DE can converge to a level set in the general case. Furthermore, it was noted that the populations can naturally partition and form clusters that independently converge to separate points. This slow rate of convergence affects the restart and local search mechanisms and the associated adaptation machinery. Since the current implementation uses a synchronous restart and adaptation of δ local and n LR , the number of restarts might be limited by the fact that the evolution of all populations has to come to a stop before any of them can be restarted. Future work will be dedicated to improve these aspects of the algorithm.