Multicriteria adaptive differential evolution for global numerical optimization

Integrated Computer-Aided Engineering 22 (2015) 103–117 103DOI 10.3233/ICA-150481IOS Press

Multicriteria adaptive differential evolutionfor global numerical optimization

Jixiang Chenga, Gexiang Zhanga,∗, Fabio Caraffinib,c and Ferrante Nerib,c,∗aSchool of Electrical Engineering, Southwest Jiaotong University, Chengdu, Sichuan, ChinabCentre for Computational Intelligence, School of Computer Science and Informatics, De Montfort University, TheGateway, England, UKcDepartment of Mathematical Information Technology, University of Jyväskylä, Agora, Jyväskylä

Abstract. Differential evolution (DE) has become a prevalent tool for global optimization problems since it was proposed in1995. As usual, when applying DE to a specific problem, determining the most proper strategy and its associated parameter val-ues is time-consuming. Moreover, to achieve good performance, DE often requires different strategies combined with differentparameter values at different evolution stages. Thus integrating several strategies in one algorithm and determining the applica-tion rate of each strategy as well as its associated parameter values online become an ad-hoc research topic. This paper proposesa novel DE algorithm, called multicriteria adaptive DE (MADE), for global numerical optimization. In MADE, a multicriteriaadaptation scheme is introduced to determine the trial vector generation strategies and the control parameters of each strategyare separately adjusted according to their most recently successful values. In the multicriteria adaptation scheme, the impacts ofan operator application are measured in terms of exploitation and exploration capabilities and correspondingly a multi-objectivedecision procedure is introduced to aggregate the impacts. Thirty-eight scale numerical optimization problems with various char-acteristics and two real-world problems are applied to test the proposed idea. Results show that MADE is superior or competitiveto six well-known DE variants in terms of solution quality and convergence performance.

Keywords: Continuous optimization, differential evolution, adaptive algorithms, multicriteria adaptive systems, meta-heuristics,evolutionary algorithms, swarm intelligence

1. Introduction

Modern industrial processes and engineering prob-lems often impose the solution of complex optimiza-tion problems, see e.g. [12,25,32,38,52]. Some exam-ples of pioneering studies in the field of structuraloptimization have been reported in [2,57]. More re-cently, the complexity of real-world problems encour-aged computer scientists to define algorithms to han-dle large many variables [1,9], real-time and hardware

∗Corresponding authors: Ferrante Neri, Centre for ComputationalIntelligence, School of Computer Science and Informatics, De Mont-fort University, The Gateway, Leicester LEI 9BH, England, UK. E-mail: [email protected]; Gexiang Zhang, School of Electrical Engi-neering, Southwest Jiaotong University, Chengdu, Sichuan, China.E-mail: [email protected].

limitations [34], and to tackle both design and con-trol issues [3]. Since most real-world problems cannotbe solved by exact methods, meta-heuristics, i.e. algo-rithms which do not require specific hypotheses on theoptimization problem, have been widely diffused [59].

The meta-heuristics can be single-solution or pop-ulation based and are often inspired by diverse meta-phors. Population based meta-heuristics are dividedinto two classes, Evolutionary Algorithms (EAs) andSwarm intelligence, respectively. Modern example aregiven in [10,61], where the algorithms are inspired bya cloud drop and a chemical reaction metaphor, re-spectively. In [6,37] two algorithms are designed in thecontext of multi-objective optimization. In [40], a two-phase genetic algorithm is proposed with reference toa civil engineering problem. In [31,41,75], genetic al-gorithm and particle swarm optimization are applied

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104 J. Cheng et al. / MADE for global numerical optimization

to address transportation problems. Other hybrid ap-proaches which combine global and local search areoften used as they offer a better performance with re-spect to single paradigm based algorithms, see e.g. [35,62]. An hybridization of EAs with fuzzy logic is pro-posed in [58], while hybrid approaches that use a neu-ral system to perform optimization problems have beenproposed in [39,73].

Among the plethora of meta-heuristics, differentialevolution (DE) is a popular option for handling nu-merical optimization problems [48]. In some contextsDE is considered as an EA because of the mutationand crossover operations and in some other contextsis regarded as a swarm intelligence algorithm becauseof one-to-one spawning in the survival selection. TheDE performance is significantly dominated by the trialvector generation strategy composing of mutation andcrossover operators, and its associated parameters in-cluding scaling factor F and crossover rate Cr. So far,many trial vector generation strategies have been de-signed; however, no single one is suitable for all thestages in a problem solving process. Thus it is benefi-cial to dynamically determine strategies by an adaptiveoperator selection (AOS) mechanism [30,43,54].

This paper proposes a novel DE algorithm with fourtrial vector generation strategies, called multicriteriaadaptive DE (MADE). The novelties of MADE in-clude three aspects: (1) a multi-objective aggregationmethod based on two impact measures, fitness and di-versity, is presented to evaluate the effect of an oper-ator application; (2) a diversity impact measure, nor-malized distance to the best individual (NDBI), is de-signed; (3) for each of multiple trial vector generationstrategies, the two control parametersF and Cr are ad-justed by Cauchy and Gaussian distributions, respec-tively, where the distribution parameters are learnedfrom the most recently successful values. Thirty-eightscalable benchmark functions with various character-istics and two real-world problems are employed totest MADE performance. Experimental results demon-strate the superiority and competitiveness of MADEover six state-of-the-art DE variants.

This paper is organized as follows. Section 2 de-scribes basic DE algorithm. Section 3 presents a lit-erature survey on DE and AOS. Section 4 expoundsthe proposed MADE algorithm. Section 5 discusses theexperimental results. Section 6 concludes this paper.

2. Background: Differential evolution

The classical DE is designed for solving continu-ous optimization problems. Without loss of general-

ity, in this paper we consider minimizing the objec-tive function f(x), where x = [x1, x2, . . . , xD] ∈ �D

is the decision vector and D is the number of vari-ables. The feasible solution space is constrained by thelower bound L = [l1, l2, . . . , lD] and the upper boundH = [h1, h2, . . . , hD] of the decision vector x.

DE maintains a population of NP vectors with Ddimensions. Each vector represents a potential solu-tion to the problem, called an individual. At generationg = 0, the initial population P 0 = {x01, x02, . . . , x0NP },where x0i = [x0

i,1, x0i,2, . . . , x

0i,D], is uniformly and

randomly generated in the feasible space by

x0i,j = lj + rand(0, 1) · (hj − lj), (1)

where rand(0, 1) is a random variable uniformly dis-tributed in the interval [0,1].

Then mutation, crossover and selection operationsare performed repeatedly on the population until thetermination criterion is satisfied. At generation g, foreach individual xg

i (called target vector) in the currentpopulation, a mutant individual vgi (called donor vec-tor) is generated by

vgi = xgr1 + F · (xgr2 − xg

r3), (2)

where r1, r2, r3 ∈ {1, 2, . . . , NP}\{i} are distinct in-dividual indices; F ∈ (0, 1) is the scale factor.

Following mutation, the donor vector vgi is recom-

bined with the target vector xgi to produce an offspringugi (called trial vector) by a binomial crossover opera-

tor, formulated as

ugi,j =

{vgi,j , if randj(0, 1) � Cr or j = jrand

xgi,j , otherwise

, (3)

where i = 1, 2, . . . , NP ; j = 1, 2, . . . , D; Cr ∈ (0, 1)is the crossover rate; jrand ∈ {1, 2, . . . , D} is a randominteger instantiated once for each i at every generation.The condition j = jrand ensures that ug

i will not beidentical with xgi .

After crossover, selection operation is performed toselect the better one between ug

i and xgi to enter the

next generation, depicted as

xg+1i =

{ugi , if f(ug

i ) � f(xgi )

xgi , otherwise

. (4)

3. Related work

In this section, we review some main work in theliterature related to DE algorithms and AOS technique.

J. Cheng et al. / MADE for global numerical optimization 105

3.1. Main DE improvement approaches

Although DE has become a prevalent optimizationtechnique for various numerical problems [36,48], itstill suffers from prematurity and/or stagnation. Hencea good volume of work has been devoted to over-come its drawbacks mainly from four perspectives: pa-rameter control, operator design, population structure,and hybridization with other meta-heuristics, whichare briefly reviewed hereinafter.

DE performance strongly depends on the values ofF and Cr. Many guidelines of setting appropriatevalues were suggested in the literature. For example,in [26], it was stated that Cr with the value between0.3 and 0.9, and an initial value of 0.6 for F are goodchoices. In contrast, [56] argued that F should be var-ied between 0.4 and 0.95 with an initial value 0.9, andCr would be chosen between 0 and 0.2 or between 0.9and 1 depending on the problem’s characteristics. Asthere is not an agreement on the setting of DE param-eters, even some guidelines provided in the literatureconflict with each other, many parameter control tech-niques have been developed. For example, [15] linearlyreduced F value from the preset maximal value to theminimal one. [42] adapted both F and Cr via a fuzzyknowledge-based system. [5] self-adapted both F andCr at the individual level through controlling two less-sensitive parameters. [27] presented a fitness-based pa-rameter adaptation scheme. In [30,43,54], the parame-ter values were adjusted by learning from the previousevolution experience.

Mutation plays a key role in DE and consequentlylots of work focused on designing new mutation oper-ators. [53] proposed a rotationally invariant arithmeticline recombination operator for handling rotated prob-lems. [20] presented a trigonometric mutation operatorto speed up convergence. [55] employed an opposition-based learning for generation jumping and local im-provement of the best individual to accelerate conver-gence. [74] extended “current-to-best” mutation to ageneralized version called “current-to-pbest”. [13] putforward an improvement of “current-to-best/1” strat-egy by simultaneously using a global mutation modeland a local neighborhood model. [17] introduced aGPBX-α mutation strategy. To enhance the perfor-mance of various mutation strategies, [18] proposed toselect individuals for mutation based on the proximitycharacteristics among individuals, while [64] selectedindividuals for mutation by simultaneously consider-ing their fitness and diversity.

The population structure determining the way in-dividuals share information with each other also has

a certain effect on DE performance. The populationstructures can be classified into static and dynamictypes. For the former one, [72] designed the first DEutilizing structured population, where a random topol-ogy and a random migration strategy were employed.[11,19] investigated migration strategies in DE vari-ants with static population structures. [17] surveyed,designed and compared several static population struc-tures in DE, including unidirectional ring, Torus, hier-archy topology and small-world topology. Moreover,control parameters in the DE with structured popu-lation were discussed in [67,68]. For the latter one,in [66], the subpopulations were grouped into twofamilies. The first one uses fixed subpopulations andevolves like a DE with a statistic population structure,while the second one utilizes a population size reduc-tion strategy.

Hybridization has become an attractive route in al-gorithm design due to its capability for handling quitecomplex problems. In [14,51], DE was hybridized withtwo well-known global optimizers. [70] proposed aDE variant by probabilistically adding a disturbance toeach component of the target vector. [46] combined DEwith three local search approaches by means of a fit-ness diversity logic and studied the benefits and limi-tations of the hybridization. [49] proposed a crossover-based adaptive local search to enhance DE perfor-mance by adaptively adjusting the number of offspringto the neighborhood of parent individuals. [47] pro-posed a scale factor local search DE algorithm employ-ing two local searchers within a self-adaptive scheme,where the local searchers aimed at detecting a propervalue of F for a good individual rather than search-ing for better solutions. [7] investigated a combinationof DE with both a global optimizer and several localsearch methods.

3.2. Adaptive operator selection

AOS, dynamically determining which operatorsshould be incorporated into the algorithm and at whichrate an operator should be applied, provides an on-lineautonomous control of the operators through the feed-back of the search process. A framework of how to ap-ply AOS into EAs is shown in Fig. 1. In the frame-work, there are four elements: operator pool, operatorevaluation, operator quality and operator selection. InAOS, a good operator pool should comprise multipleoperators with distinct characteristics and the operatorquality often refers to a credit register for storing thecredit of each operator obtained during the evolution.


... ... ... ...

Fig. 1. Framework of AOS.

In the following we briefly review the work on operatorevaluation and operator selection parts.

Several operator evaluation approaches have beenimplemented and they differ mainly in two aspects:impact evaluation and credit assignment. The impactevaluation refers to evaluating the impact of an op-erator application on the search process. Some sim-ple impact evaluation methods include fitness improve-ment [30], success or fail of the produced offspring toenter the next generation [45] and the rank of the pro-duced offspring [24]. In [28], a diversity improvementbased on the Euclidean distance between the offspringproduced by an operator and the current best individ-ual was applied as the impact measurement when thesearch stagnates. In [45], a Compass measurement, si-multaneously considering fitness improvement, diver-sity variance and execution time, was presented. Afterimpact evaluation, a credit is assigned to each operatorby synthesizing its impact values. Usually, the creditcan be an instantaneous value, an average value, a nor-malized value or an extreme value [21]. [23,24] in-troduced two impact synthesis approaches called area-under-curve and sum-of-ranks. [44] investigated twosynthesis methods based on Pareto dominance andPareto rank concepts.

Operator selection is used to automatically adapt tothe current performance of the operator (operator qual-ity) and decide the best operator(s) for the next applica-tion (or generation). Most operator selection methodsreported in the literature belong to the type of probabil-ity matching (PM) [29]. When many operators are con-sidered, the PM performance will be restricted sinceany ineffective operator always has a small probabilityof being selected. To address the issue, [63] introducedan adaptive pursuit (AP) method inspired from learn-ing automata. The operator probabilities determined by

PM or AP are used to select operator(s) by means ofa roulette wheel or a stochastic uniform sampling pro-cedure. The methods that directly determine the oper-ator, instead of the operator probabilities, are also in-vestigated, such as multi-armed bandits (MAB) [4] andsliding MAB (SLMAB) [22].

4. MADE Algorithm

In this section, we propose a novel DE algorithm,MADE, by designing a multicriteria adaptive trial vec-tor generation strategy selection method and an adap-tive parameter control approach for each trial vec-tor generation strategy. A preliminary study on thisidea, in the context of super-fit adaptation within DEframework was presented in [8]. In what follows, wefirst present the motivation of this work. Subsequently,the multicriteria trial vector generation strategy adap-tation method is exhaustively described. Afterwards,the control parameter adaptation is expounded. Finally,MADE is given and the complexity is briefly analyzed.

4.1. Motivation

The outstanding performance of DE is mainly owingto its special trial vector generation strategy consistingof mutation and crossover. Although there are manytrial vector generation strategies in the literature, onestrategy may perform well only on a particular stage ofa problem-solving process. Actually, it is preferable touse different trial vector generation strategies for dif-ferent stages. To obtain the most satisfactory optimiza-tion performance, there is a feasible path that integratesseveral trial generation strategies into one algorithmand chooses the best strategy by an AOS technique dur-ing the search process.


1. rand/1/bin:

ugi,j =

{xgr1,j

+ F (i) · (xgr2,j

− xgr3,j

), if randj(0, 1) � Cr(i) or j = jrandxgi,j , otherwise (5)

2. rand/2/bin:

ugi,j =

⎧⎨⎩xgr1,j

+ F (i) · (xgr2,j

− xgr3,j

)+F (i) · (xgr4,j

− xgr5,j

),

if randj(0, 1) � Cr(i) or j = jrandxgi,j , otherwise

(6)

3. rand-to-best/2/bin:

ugi,j =

⎧⎨⎩

xgi,j + F (i) · (xg

best,j − xgi,j)+F (i) · (xg

r1,j− xg

r2,j)

+F (i) · (xgr3,j

− xgr4,j

), if randj(0, 1) � Cr(i) or j = jrandxgi,j , otherwise

(7)

4. current-to-rand/1:

ugi = xgi + rand(0, 1) · (xg

r1 − xgi ) + F (i) · (xgr2 − xg

r3) (8)

Among the four elements in AOS, operator evalu-ation is the most important part. To date most opera-tor evaluation methods are principally based on the fit-ness and the diversity information is not well exploited.However, when tackling multi-modal problems, the di-versity is very important and should be considered inthe process of impact evaluation. For example, assumethe algorithm stagnates or traps in at a local optimumand two operators A and B are applied. The operatorA produces a small fitness improvement but makes alarge diversity variance, while the operator B yields arelatively large fitness improvement but a minor diver-sity variance. To help the algorithm jump out of the lo-cal optimum, the operator A should get no less or evenmore reward than B in order to increase its chance ofbeing applied. Moreover, at different evolution stages,different control parameter values for different trialgeneration strategies should be applied. Motivated bythe two considerations, we propose MADE, in whichboth trial vector generation strategies and their controlparameter values are automatically adapted during thealgorithm running. The trial vector generation strate-gies are selected by a multicriteria AOS method, wherethe impacts of an operator application are measured interms of the exploitation and exploration capabilitiesand they are aggregated by a multi-objective decisionprinciple. The control parameter values of each trialvector generation strategy are adapted based on theirmost recently successful applications.

4.2. Multicriteria trial vector generation strategyadaptation

In AOS, when evaluating an operator from both fit-ness and diversity perspectives, two issues arise: (1)how to measure the diversity impact of an operator ap-plication; and (2) how to aggregate the fitness and di-versity impacts. This subsection develops a multicrite-ria AOS to address the two issues. For the convenienceof description, we use NP , K , pgk (k = 1, 2, . . . ,K)to denote population size, operator pool size and theprobability of the kth operator being selected at gener-ation g, satisfying pmin � pgk < 1 and

∑Kk=1 p

gk = 1,

where the introduction of pmin is used to avoid pgk = 0and it is set to a small constant value. Fitness, diver-sity and aggregated impacts of an operator applica-tion are represented as ηgi , τgi , and γg

i , respectively,i = 1, 2, . . . , NP . Assuming that sgk is a set of in-dices of the individuals which use the kth strategy togenerate trial vectors. At the end of each generation, aunique reward rgk (k = 1, 2, . . . ,K) is calculated bythe use of γg

i and sgk and the reward is utilized to updateoperator quality qgk and operator rate pgk.

In AOS, a good operator pool should consist of op-erators with diverse characteristics so that less effec-tive operator for a specific stage can be suppressed andthe effective one can be selected in time. In MADE,the operator pool is constructed by considering thefour trial vector generation strategies, i.e., K = 4,in Eqs (5), (6), (7), and (8). In the equations, xgbest


is the best individual in P g; r1, r2, r3, r4, r5 are dis-tinct individual indices randomly selected from the set{1, 2, . . . , NP}\{i}; F and Cr are the parameters ofDE.

The “rand/1/bin” and “rand/2/bin” strategies bearstrong exploration capabilities and suit for multi-modal problems because of their random properties,i.e., all vectors participating in operation are ran-domly selected. Besides, “rand/2/bin” strategy is atwo-difference-vectors strategy, which produces betterperturbation than the one-difference-vectorbased strat-egy “rand/1/bin”. The strategy “rand-to-best/2/bin” isbased on the best solution found so far and has fast con-vergence characteristic. However, it often suffers fromstagnation or premature convergence when solvingmultimodal problems. The “current-to-rand/1” strat-egy replaces the binomial crossover operator with therotationally invariant arithmetic line recombination togenerate trail vector. Thus it is a rotationally invariantstrategy and fit for rotated problems.

In the sequel, we discuss the operator evaluation andselection. Initially, all strategies have equal probabil-ities to be selected, i.e, pgk = 1/K, k = 1, 2, . . . ,Kand g = 0. At generation g, for each target vector xg

i ,an operator ni ∈ {1, 2, . . . ,K} is chosen accordingto the operator rates and then applied to generate anoffspring xg+1

i . After evaluating all offsprings, the im-pacts of each operator application are evaluated. In thiswork, we use normalized relative fitness improvement(NRFI) and normalized distance to the best individual(NDBI) to measure the fitness impact (ηgi ) and the di-versity impact (τgi ) (i = 1, 2, . . . , NP ), formulated as

ηgi =η̃gi

max{η̃gi , . . . , η̃gNP }, η̃gi =

f(xgi )−f(xg+1i )

f(xgi )−f(xgbest), (9)

τgi =τ̃gi

max{τ̃gi , ..., τ̃gNP }, τ̃gi = ‖xg

i − xgbest‖ . (10)

Here, NRFI measures the degree of fitness improve-ment and NDBI weights the potential of escaping froma local optimum. These two measurements can wellreflect the exploitation and exploration capabilities ofan operator application, which are shown in Fig. 2by using two fitness landscape illustrations. Ideally, agood operator application is the one with both largeexploitation and exploration capabilities. In fact theyare often a pair of conflicts. For instance, as shown inFig. 2, the individual on the left bottom (the end pointof left arrow) shows a low exploitation capability butbears a high exploration capability, while the individ-ual on the right bottom (the end point of right arrow)

indicates a high exploitation capability but a low ex-ploration capability. To measure the overall impact γg

i

(i = 1, 2, . . . , NP ), we consider the fitness and di-versity impacts as two objectives and use the multi-objective decision principle to aggregate them. Herewe introduce three aggregation methods listed as fol-lows.

– Weighted-sum (WS):

γgi = α · ηgi + (1− α) · τgi , (11)

where α ∈ [0, 1] weights the relative importanceof ηgi and τgi .

– Number-of-dominating-impacts (NDGI):

γgi = |{(ηgj , τgj )|(ηgi , τgi )�(ηgj , τ

gj ), j∈hg

i }|, (12)

where | · | denotes the cardinality of a set;a � b means that vector a = (a1, . . . , ak)dominates vector b = (b1, . . . , bk), i.e., ∀i ∈{1, . . . , k}, ai � bi ∧ ∃i ∈ {1, . . . , k}, ai > bi;hgi = {⋃K

k=1 sgk}\sgni

denotes the set of indices ofthe target vectors which use the operator differingfrom the operator ni used by xg

i .– Number-of-dominated-impacts (NDDI):

γgi = max{γ̃g

j , . . . , γ̃gNP } − γ̃g

i (13)

γ̃gi = |{(ηgj , τgj )|(ηgi , τgi ) (ηgj , τ

gj ), j ∈ hg

i }|,where a b means that the vector a is dominatedby the vector b, i.e., b � a; hg

i is the same as inEq. (12).

Following operator evaluation, we use PM me-thod [29] to update operator probabilities based on theaggregated impacts. First, the credit of each operator inthe current generation is calculated by

rgk =

∑ni∈sgk

γgi

|sgk|, (14)

where k = 1, . . . ,K . Then the operator quality is up-dated by an additive relaxation mechanism, i.e.,

qgk = (1− β) · qgk + β · rgk, (15)

where β ∈ [0, 1] is the adaptation rate; q0k is initial-ized with 1. Finally, the selection probability of eachoperator is updated by

pg+1k = pmin + (1−K · pmin)

qgk∑Kk=1 q

gk

. (16)

According to Eq. (16), when an operator has not re-ceived a reward for a long time, its selection probabil-ity converges to pmin rather than zero, which makessure that a seldom selected operator will not disappearfrom the operator pool.


(a) NRFI (b) NDBI

Fig. 2. Fitness landscape illustrations of the impact evaluation of operator applications.

4.3. Control parameter adaptation

In MADE, we adapt the parameter values of eachstrategy based on their recently successful values.Specifically, let sFk and sCrk denote the sets of mostrecent LP values of F and Cr associated with the kthstrategy which generates vectors entering the next gen-eration. At each generation g, for each target vector xgithat selects the kth strategy, the scale factor F (i) andcrossover rate Cr(i) are sampled by

F (i) = randc(μFk, 0.1), (17)

Cr(i) = randn(μCrk, 0.1), (18)

where randc(μFk, 0.1) is the Cauchy distribution witha local parameter μFk and a scale parameter 0.1;randn(μCrk, 0.1) is the normal distribution with meanμCrk and standard deviation 0.1. The parameters areregenerated if their values are out of the range (0,1).

In Eqs (17) and (18), the local parameter μFk andthe mean μCrk are initialized as 0.5. At the end ofeach generation, if |sFk| < LP , the values of μFk

and μCrk remain unchanged, otherwise old values ofsFk and sCrk are first removed to guarantee |sFk| =|sCrk| = LP . Then they are updated by

μFk =∑

F∈sFk

F 2/∑

F∈sFk

F , (19)

μCrk =∑

Cr∈sCrkCr

/LP. (20)

4.4. MADE

By combining the multicriteria strategy adaptationscheme using one of the three aggregation methods,WS, NDGI and NDDI, and the parameter adaptationscheme, three MADE versions are developed and rep-resented as MADE/WS, MADE/NDGI and MADE/NDDI, respectively. The pseudocode of MADE algo-rithm is illustrated in Fig. 3.

The main computational complexity of the adapta-tion schemes in MADE lies in the calculation of theimpacts ηgi and τgi according to Eqs (9) and (10), andthe aggregated impact γg

i according to Eq. (11) or (12)or (13). In general, the calculation of ηgi and τgi in-volves a sorting procedure to find the maximal valuefor normalization, hence it has an average complexityof O(NP logNP ). For whole population, the WS ag-gregation has a complexity of O(NP ); while NDGIand NDDI involves a dominance checking procedure,which has a complexity of O(NP 2). Therefore, theoverall complexities of three MADE variants are aboutO(NP logNP )+O(NP ), O(NP logNP )+O(NP 2)and O(NP logNP ) +O(NP 2), respectively.

5. Numerical experiments and results

5.1. Test functions

To verify MADE performance, 38 scalable bench-mark functions, including 11 unimodal functions (f01


Input: NP, LP, α(if any), β , pmin1: g = 0, K = 42: Generate initial population P0 and evaluate it3: Set q0

k = 1, p0k = 0.25, μFk = μCrk = 0.5, sFk =

sCrk =∅, k = 1, . . . ,44: while termination criterion is not satisfied do5: for i = 1 to NP do6: Select a strategy k according to pk for xg

i7: Determine F(i) and Cr(i) for xg

i8: Generate ug

i using kth strategy and parametervalues F(i) and Cr(i)

9: Evaluate ugi as f (ug

i )10: if f (ug

i )≤ f (xgi ) then

11: xg+1i = ug

i ; sFk = sFk ∪ {F(i)}; sCrk =sCrk ∪{Cr(i)}

12: else13: xg+1

i = xgi

14: end if15: end for16: for k = 1 to NP do17: Calculate impacts ηg

i and τgi

18: Calculate aggregated impact γgi

19: end for20: for k = 1 to K do21: Calculate reward rg

k22: Update quality qg

k23: Update probability pg

k24: if |sFk| ≥ LP then25: Remove old values from sFk and sCrk so

that |sFk|= |sCrk|= LP26: Update μFk and μCrk27: end if28: end for29: g = g+130: end while

Fig. 3. The pseudocode of MADE algorithm.

−f04, f06, f07, f14 − f18), and 27 multimodal func-tions (f05, f08 − f13, f19 − f38), are selected as thetest suit. The functions f01 − f13 are the first 13 scal-able functions in [71] and functions f14 − f38 are the25 instances proposed in CEC2005 [60]. In the exper-iments, we test all functions with 30 and 50 dimen-sions. For each algorithm and each test function, 25 in-dependent runs are conducted using the maximal num-ber of function evaluations (MaxNFEs) and tolerancevalues as the termination criteria. The MaxNFEs andtolerance values for each problem are given in Table 1,where most of the values are suggested by [60,71].All the algorithms considered in this paper are imple-

Table 1Tolerance and MaxNFEs for all test functions

Functions Dimension Tolerance MaxNFEsf01, f06, f10, f12, f13 30 1E-8 150,000f02, f08, f11 30 1E-8 200,000Kf09 30 1E-8 300,000f03 − f05 30 1E-8 500,000f07, f14 − f38 30 1E-2 300,000f01 − f06, f08 − f13 50 1E-8 500,000f07, f14 − f38 50 1E-2 500,000

mented on the platform MATLAB 7.6a and executedon a HP work station with Intel Xeon 2.93 GHz pro-cessor, 12GB RAM and Windows 7 OS.

5.2. Performance metrics

Five performance metrics are utilized to evaluate thealgorithm performance, described as follows.

– Error: The error in a run is defined as the differ-ence between the final solution f(x) and the op-timum f(x�), where f(x�) of each function wasprovided in [60,71].

– Successful rate (SR): The SR is calculated as theratio of the number of successful runs. A suc-cessful run is the independent run where the errorreaches the tolerance when the algorithm stops.

– Successful performance (SP): The SP is definedas SP = (SNFEs)/SR, where SNFEs is the aver-age number of function evaluations required bysuccessful runs to reach the tolerance. A lowerSP value indicates a better combination betweenspeed and consistency of the algorithm.

– Statistic Test: Wilcoxon’s signed-rank test [69] isemployed to check whether there is a significantdifference between two algorithms on one prob-lem by the use of errors over total runs.

– Empirical cumulative probability distribution fun-ction (ECDF): ECDF of normalized mean valuesregarding the errors and SP are employed to illus-trate the overall performance of an algorithm. Thecurve of ECDF that reaches the top of the graphis regarded as the best algorithm [18].

5.3. Comparisons of operator adaptation methods

To investigate the effectiveness of the proposedmulticriteria strategy adaptation schemes, we comparethree MADE algorithms, MADE/WS, MADE/NDGIand MADE/NDDI, against three adaptive DE (ADE)algorithms using different adaptation methods, ADE/U, ADE/R and ADE/F. In ADE/U, each operator is se-lected with equal probability during the whole evolu-


Table 2Comparisons of the number of functions where three MADE versions produce results better than (#Better), equivalent to (#Equal), and worsethan (#Worse) three ADE algorithms regarding the mean error (MError), the successful performance (SP) and the successful rate (SR)

ADE/U ADE/R ADE/F#Better #Equal #Worse #Better #Equal #Worse #Better #Equal #Worse

MADE/WS MError 22 10 6 17 14 7 22 11 5SP 16 19 3 15 19 4 14 20 4SR 7 31 0 5 33 0 5 32 1

MADE/NDGI MError 22 12 4 22 10 6 21 13 4SP 17 18 3 15 18 5 17 18 3SR 8 30 0 7 31 0 7 31 0

MADE/NDDI MError 18 14 6 20 9 9 18 13 7SP 15 19 4 13 19 6 15 19 4SR 6 31 1 5 31 2 4 32 2

tion process. In ADE/R, each target vector randomlyselects an operator from the operator pool. In ADE/F,the fitness-based operator adaptation method is used,that is, γg

i = ηgi . All functions with 30 dimensions areused to conduct this experiment. For fair comparisons,we assign the same parameter values for all algorithms:NP = 50, β = 0.7, LP = 50, pmin = 0.02. ForMADE/WS, the weight factor α is set to 0.5.

Due to space limitation, we provide the detail re-sults in a supplemental file.1 Table 2 summarizes thenumbers of functions where each MADE version pro-duces the results better than, equivalent to and worsethan those of each ADE version regarding mean er-rors (MError), SP and SR. The results show that eachMADE version produces overall better results thanthree ADE algorithms. Among the three MADE algo-rithms, according to the capability of producing betterresults than ADE versions, MADE/WS and MADE/NDGI have similar performance and perform slightlybetter than MADE/NDGI. As MADE/WS introducesan extra parameter α, we consider MADE/NDGI asour final MADE algorithm and use it to conduct therest experiments.

5.4. Comparisons of MADE with other DE variants

In this subsection, we intend to compare MADEwith six well-known DE variants, i.e., j-DifferentialEvolution (jDE) [5], Self-adaptive Differential Evo-lution (SDE) [50], J-Adaptive Differential Evolution(JADE) [74], Self-adaptive Differential Evolution(SaDE) [54], Ensemble of Parameters and StrategiesDifferential Evolution (EPSDE) [43] and Composite

1The supplemental file which includes some more detail results ofthe numerical experiments can be obtained from http://www.nicsg.net/portal.php?mod=view&aid=198.

Differential Evolution (CoDE) [65] on all functionswith 30 and 50 dimensions. In the experiments, the pa-rameter setting for each of the six algorithms is set ex-actly the same as that in its corresponding paper. Theparameter setting for MADE is the same as in the pre-vious experiment, i.e., NP = 50, β = 0.7, LP =50, pmin = 0.02.

Due to space limitation, we also provide the meanand standard deviations of the errors in the supplemen-tal file. According to the results, MADE, jDE, SDE,JADE, SaDE, EPSDE and CoDE obtain 22, 15, 8,19, 10, 8 and 15 best or second best results for theproblems with 30 dimensions, respectively, and con-sequently MADE is the best one. For 50 dimensions,MADE has similar performance to JADE but is stillbetter than jDE, SDE, SaDE, EPSDE and CoDE ac-cording to the numbers, 19, 16, 9, 19, 5, 4 and 14, ofthe best or second best results obtained by the seven al-gorithms. The statistic test is performed and the resultsare summarized in Table 3. According to the results,the numbers of functions where MADE significantlyoutperforms jDE, SDE, SaDE, EPSDE and CoDE arehigher than the numbers of functions where MADEproduces significantly worse performance, but there isan exception that MADE is slightly inferior to JADEfor the functions with 50 dimensions.

Through using MaxNFEs and tolerance for eachproblem in Table 1 as the termination criteria, we com-pare the convergence performance of the seven algo-rithms in terms of SP, SR and the rank based on SPvalues. The detail results are also given in the supple-ment file. Noted that only the functions successfullysolved at least once out of all the runs of seven algo-rithms are considered; hence 20 and 18 functions areconsidered for 30 and 50 dimensions, respectively. Ac-cording to the results, of the seven algorithms, MADEachieves the highest numbers, 14 and 9, of the best andthe second best SP over the functions with 30 and 50


Table 3Summary of the statistic test result.“#–”, “#+” and “#≈” denotes the numbers of the functions on which MADE performs significantly betterthan, significantly worse than and significantly equivalent to its competitors, respectively

Dimension Statistics jDE SDE JADE SaDE EPSDE CoDE30 #– 17 17 11 9 13 13

#+ 5 6 9 7 6 11#≈ 16 15 18 22 19 14

50 #– 13 19 12 21 22 12#+ 9 5 15 4 6 10#≈ 16 14 11 14 10 16

100

102

104

106

108

1010

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized mean errors

Em

piric

al d

istr

ibut

ion

over

all

func

tions

jDESDEJADESaDEEPSDECoDEMADE

Fig. 4. ECDF of normalized mean errors over all functions.

dimensions, respectively, and obtains the lowest aver-age rank values, which implies that MADE possessesthe best combination between speed and consistency.The results also show that MADE is superior to jDE,SDE and SaDE, and competitive to JADE, EPSDE andCoDE in terms of the average SR values for the prob-lems under consideration.

To illustrate the overall performance difference ofseven algorithms, we plot ECDF of normalized valuesof the mean errors and SP. Figure 4 presents the ECDFof normalized mean errors over 76 test cases. Figure 5illustrates the ECDF of normalized SP over the func-tions which is successfully solved at least once out ofall the runs of seven algorithms. From Fig. 4, we ob-serve that MADE is above jDE, SDE, SaDE, EPSDEand CoDE, but slightly below JADE. Figure 5 showthat MADE is on the top of the rest six algorithms andis the only one algorithm that reaches the unity. There-fore, we can conclude that MADE is superior or com-petitive to other six algorithms.

5.5. Adaptive characteristics analysis

Of the 38 functions considered, there is a pair ofmultimodal functions, f09 and f23, and the latter is the

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized SP

Em

piric

al d

istr

ibut

ion

over

all

func

tions

jDESDEJADESaDEEPSDECoDEMADE

Fig. 5. ECDF of normalized SP over the functions successfullysolved at least once out of all the runs of seven algorithms.

0 50000 100000 150000 200000 250000 3000000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

NFEs

Sel

ectio

n pr

obab

ility

rand/1/binrand/2/binrand−to−best/2/bincurrent−to−rand/1

Fig. 6. Evolution of pk on f09 in 30 dimensions.

former’s shifted rotated version. So we choose them toinvestigate the adaptive characteristics of MADE. Weplot the evolution trends of the selection probabilitypk, the parameters μFk and μCrk of each operator inFigs 6, 7, 8, 9, 10, and 11. In general, the figures indi-cate that different values of pk, μFk and μCrk are pre-


0 50000 100000 150000 200000 250000 300000

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

NFEs

Sel

ectio

n pr

obab

ility


Fig. 7. Evolution of pk on f23 in 30 dimensions.

0 50000 100000 150000 200000 250000 3000000.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

NFEs

F


µ

Fig. 8. Evolution of µFk on f09 in 30 dimensions.

ferred for different stages. To be specific, we can makethe following analysis.

As shown in Figs 6 and 7, the proposed strategyadaptation scheme can well detect appropriate opera-tors for the problems with different characteristics. Forinstance, for the multi-modal and non-rotated functionf09, the operator with good exploration capability ispreferred and the operator with a rotationally invari-ant feature should be suppressed, which is exactly re-flected in Fig. 6. We can also observe that the selectionprobabilities of four operators remain unchanged aftera period of adaptive adjustment. This may due to thereasons that none of the four operators is able to findbetter solutions or the impacts of the four operators onthe evolution process are nearly equivalent. In contrast,for the rotated multi-modal function f23, the rotation-ally invariant operator is preferred and should have alarge selection rate. As shown in Fig. 7, the probability

0 50000 100000 150000 200000 250000 3000000.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

NFEs

F


µ

Fig. 9. Evolution of µFk on f23 in 30 dimensions.

of “current-to-rand/1” is low in the early stage, then itquickly increases to a value higher than those of otherthree operators; while the selection rate of “rand-to-best/2/bin” quickly decreases because it is an exploita-tive operator and not well suitable for the rotated prob-lem.

On the other hand, the introduced parameter adap-tation scheme can detect appropriate parameter valuesfor each operator. As we know, to obtain a strong ex-ploration capability, the explorative operators shouldbe assigned as large μFk and μCrk values, while theexploitative operators should be allocated with rela-tively small μFk and μCrk values to gain a good ex-ploitation capability. The expectation is achieved in theexperiments shown in Figs 8–11.

5.6. Real world problem I: Lennard-Jones Potentialminimization

In order to prove the viability of the proposed ap-proach over real-world problems, MADE has beentested over the Lennard-Jones Potential (LJP) problem.The LJP problem is widely used in physics for rep-resenting the interaction energy between non-bondingparticles in a fluid. In this case, according to the de-scription given in CEC2011 [16], we refer to the min-imization problem of the potential energy of a set ofatoms by locating them within the three-dimensionalspace. In order to determine their position and evalu-ate the relative LJP 3 parameters, i.e. coordinates alongthe axes x, y and z, for each atom need to be stored inthe fitness function. In this study, atomic cluster of 5and 10 atoms have been considered, resulting into 15and 30-dimensional problems, respectively. Each algo-


Table 4Average fitness value ± standard deviation and statistic comparison (reference: MADE-NDGI) for MADE-NDGI against MADE-WS andMADE-NDDI on problem 2 from CEC2011 (LJP) [16] in 15 and 30 dimensions

MADE-NDGI MADE-WS MADE-NDDILJP (15D) −9.10e+ 00± 8.87e− 15 −9.09e+ 00 ± 8.83e− 02 ≈ −9.10e+ 00 ± 8.18e− 02 +LJP (30D) −2.66e+ 01 ± 8.81e− 01 −2.66e+ 01± 8.64e− 01≈ −2.64e+ 01 ± 1.15e+ 00 ≈

Table 5Average fitness value ± standard deviation and statistic comparison (reference: MADE-NDGI) for MADE-NDGI against SADE, jDE and JADEon problem 2 from CEC2011 (LJP) [16] in 15 and 30 dimensions

MADE-NDGI SADE jDE JADELJP (15D) −9.10e+ 00 ± 8.87e− 15 −9.10e+ 00 ± 9.34e− 15 + −9.09e+ 00 ± 5.31e− 02 + −9.00e+ 00 ± 9.36e− 02 +LJP (30D) −2.66e+ 01 ± 8.81e− 01 −2.57e+ 01 ± 2.85e+ 00 ≈ −1.63e+ 01 ± 1.40e+ 00 + −2.38e+ 01 ± 8.64e− 01 +

Table 6Average error ± standard deviation and statistic comparison (reference: MADE-WS) for MADE-WS against MADE varaints, SADE and jDEon Parameter Estimation for FM Sound Waves) [16] in 6 dimensions

MADE-WS SADE jDE8.81e− 01 ± 2.47e+ 00 1.74e+ 00,± 3.51e+ 00 – 2.03e+ 00 ± 2.75e+ 00 +

MADE-NDGI MADE-NDDI9.04e− 01 ± 2.49e+ 00 ≈ 1.01e+ 00 ± 2.65e+ 00 ≈

0 50000 100000 150000 200000 250000 3000000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

NFEs

Cr

rand/1/binrand/2/binrand−to−best/2/bin

µ

Fig. 10. Evolution of µCrk on f09 in 30 dimensions.

rithm under examination has been run 50 times witha computational budget of 5000 × D functional calls.The parameters of the algorithms are the same usedabove. Numerical results are given in Tables 4 and 5.Table 4 shows a cross-comparison among the variousMADE versions illustrated above. Table 5 displays nu-merical results for the most promising DE variants, i.e.SADE, jDE and JADE against the most promising ver-sion of MADE, that is MADE-NDGI. Results are re-ported in terms of mean values ± the correspondingstandard deviation. Furthermore, in order to strengthenthe statistical significance, the result of the Wilcoxontest is reported [69]. A “+” indicates that MADE sig-nificantly outperforms its competitor, a “–” indicatesthat MADE is significantly outperformed while a “≈”

0 50000 100000 150000 200000 250000 3000000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

NFEs

Cr

rand/1/binrand/2/binrand−to−best/2/bin

µ

Fig. 11. Evolution of µCrk on f23 in 30 dimensions.

indicates that the two algorithms have a comparableperformance.

It is interesting to note how the MADE frameworkscan robustly handle different levels of dimensionalitywithout a degradation of the performance with respectto the other methods.

5.7. Real world problem II: Parameter estimation forfrequency-modulated sound waves

The proposed MADE has also been tested over animportant engineering problem with a particular ref-erence to signal processing, that is the parameter esti-mation of Frequency-Modulated (FM) Sound Waves.


Briefly the problem consists of tuning the parame-ters of a synthesizer in order to generate a signal thatmatches as much a possible a target signal. The equa-tions of generated and target signals are given in (21)and (22), respectively.

y (t) = (21)

a1 sin (ω1tθ + a2 sin (ω2tθ + a3 sin (ω3tθ)))

y0 (t) = (22)

1 sin (5tθ − 1.5 sin (4.8tθ + 2 sin (4.9tθ)))

where θ = 2π100 . The optimization problem can be seen

as the search of parameters x = {a1, a2, a3, ω1, ω2,ω3} within the interval [−6.4, 6.35] such that the func-tion in Eq. (23) is minimized.

f (x) =100∑t=1

(y (t)− y0 (t))2 (23)

A thorough description of the problem can be foundin [33], see also [16]. Numerical results of this experi-ment are reported in Table 6.

It can be observed that, as the fitness of this applica-tion problem is the composition of multiple quadraticfunctions, the resulting problem appears highly com-plex and multimodal. All the MADE variants appearvery promising also in this case confirming the capa-bility of the proposed scheme to handle multimodali-ties. The WS variant is for this specific application themost promising.

6. Concluding remarks

This paper presented a novel algorithm, MADE, thatcan adaptively determine suitable operators and theirproper parameter values for different global numeri-cal optimization problems. In MADE, the trial vec-tor generation strategies are decided by a multicriteriaadaption scheme, in which the impacts of an operatorapplication are measured in terms of exploitation andexploration capabilities and then the impacts are ag-gregated by using a multi-objective decision principle;and the control parameters of each strategy are sepa-rately adapted based on their most recently successfulvalues. An extensive experiments were carried out andthe results suggest that MADE is superior or competi-tive to six state-of-the-art DE variants in terms of solu-tion quality and convergence performance.

Possible future directions following this work mayforward three aspects: (1) the diversity impact evalua-

tion measurement and the impact aggregation methodmay be further improved; (2) the hybridization of theproposed operator evaluation method with other oper-ator selection techniques, such as AP [63], MAB [4]and SLMAB [22], is worth investigating; and (3) it issignificant to use the idea of MADE to design some al-gorithms for solving other kinds of problems, such asconstrained optimization problems.

Acknowledgements

The work of JC and GZ is supported by the NationalNatural Science Foundation of China (61170016, 61373047), the Program for New Century Excellent Tal-ents in University (NCET-11-0715) and SWJTU sup-ported project (SWJTU12CX008), the program ofChina Scholarship Council (201307000022) and the2014 Doctoral Innovation Funds of Southwest Jiao-tong University. The work of FN is supported by theAcademy of Finland, Akatemiatutkija 130660, “Algo-rithmic Design Issues in Memetic Computing”.

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Multicriteria adaptive differential evolution for global numerical optimization

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