Reevaluating Exponential Crossover in
Differential Evolution
Ryoji Tanabe and Alex Fukunaga
Graduate School of Arts and Sciences, The University of Tokyo

Abstract. Exponential crossover in Differential Evolution (DE), which
is similar to 1-point crossover in genetic algorithms, continues to be used
today as a default crossover operator for DE. We demonstrate that exponential crossover exploits an unnatural feature of some widely used synthetic benchmarks such as the Rosenbrock function – dependencies between adjacent variables. We show that for standard DE as well as stateof-the-art adaptive DE, exponential crossover performs quite poorly onbenchmarks without this artificial feature. We also show that shuffled exponential crossover, which removes this kind of search bias, significantly
outperforms exponential crossover.

1

Introduction

Differential Evolution (DE) is an Evolutionary Algorithm (EA) that was primarily designed for real parameter optimization problems [16], and has been applied
to many practical problems [14]. A DE population is represented as a set of realparameter vectors xi = (x1 , ..., xD ), i = 1, ..., N , where D is the dimensionality of the target problem, and N is the population size. In each generation t,
a mutant vector v i,t is generated from an existing population member xi,t byapplying some mutation strategy. Then, the mutant vector v i,t , is crossed withthe parent xi,t in order to generate trial vector ui,t . After all of the trial vectors
ui,t , 0 ≤ i ≤ N have been generated, each individual xi,t is compared with its
corresponding trial vector ui,t , keeping the better vector in the population.

Algorithm 1: exponential crossover
ui,t = xi,t , j is randomly selected from [1, D], L = 1;
repeat3
uj,i,t = vj,i,t , j = (j + 1) modulo D, L = L + 1;
4 until rand[0, 1) < CR and L < D;
1
2

The two most common type of crossover in DE are binomial crossover, analogous to uniform crossover in GA’s, and exponential crossover, analogous to
1 or 2 point crossover in GA’s [16]. Binomial crossover is implemented as follows: For each j (j = 1, ..., D), if rand[0, 1) ≤ CR or j = jrand , uj,i,t = vj,i,t .

Otherwise, uj,i,t = xj,i,t , where rand[0, 1) denotes a uniformly selected random
number from [0, 1), and jrand is a decision variable index which is uniformly
randomly selected from [1, D]. CR ∈ [0, 1] is the crossover rate. Exponential
crossover is implemented as shown in Algorithm 1. The choice of crossover typedetermines the distribution of the number of variables that are inherited bychildren (trial vectors in DE terminology), as well as the contiguousness of theinherited variables. Although binomial crossover appears to be more frequently
used in state-of-the-art DEs [2, 18, 23], a number of recent papers have reported
successful usage of exponential crossover [3, 7, 9, 13, 24, 25].
This paper questions the continued usage of exponential crossover in DE.
It has been long known in the GA community that traditional 1-point/2-pointcrossover introduces significant positional biases – interactions between genesthat are positionally far from each other in a genome tend to be disrupted, whileinteractions between genes that are close to each other tend not to be disrupted[4]. This positional bias tends to result in undesirable search behavior in realworld problems, and classical 1-point/2-point crossover, while still introduced in
textbooks, tends not to be used by experienced GA practitioners. Why then, is
exponential crossover, which is quite similar to 1 or 2 point crossover, still usedby the DE community?
We argue that exponential crossover in DE has been overrated because it successfully exploits unnatural dependencies between adjacent variables in widelyused synthetic benchmarks. We show that if the benchmarks are altered to eliminate these unnatural dependencies, then exponential crossover performs very
poorly. We also evaluate shuffled exponential crossover (SEC), a method for
implementing exponential crossover without relying on arbitrary dependencies
between adjacent variables, which was briefly suggested in [14] but to our knowledge has never been evaluated. Although exponential crossover has been one of
the recommended crossover methods since the introduction of DE in 1995 [16],we believe that the use of exponential crossover needs to be carefully reconsideredin light of our experimental results.

2

Adjacent Functions: A Common But Unnatural Classof Benchmarks

In black-box optimization, synthetic benchmarks (e.g., the 13 classical functions [22], CEC benchmarks [17, 20], GECCO BBOB [5], and SOCO benchmarks [8]) are often used by EA researchers as proxies for performance on realworld problem instances. Although synthetic benchmark suites are designed inorder to include representatives of many class of real-world problems (e.g. unimodal/multimodal/separable/nonseparable), previous work has pointed out thatbenchmark suites can some pitfalls in using synthetic benchmarks to evaluate
EA’s [11, 15, 20, 21]. One specific issue is the presences of exploitable problem
characteristics that do not arise in real-world problems [11].
One such “exploitable problem characteristic” found in some widely used,
nonseparable synthetic benchmarks is unnatural dependencies between adjacentTable 1: Nonseparable Benchmark Functions
Name
RosenbrockDefinitions
f (z) =

Schwefels 1.2 f (z) =Search Range Properties

PD−1

100(zi+1 − zi2 )2 + (zi − 1)

PD

Pi

i=1

i=1 (

j=1 zj )


2

2

2
PD−1 P2  j−1
Block-rotated
Ri · (zi , zi+1 )f (z) =
i=1
j=1 αEllipsoid [1][−30, 30]DD

Multimodal[−100, 100]

Unimodal[−5, 5]D

Unimodalvariables – variables interacting (exclusively) with other variables that happen
to have similar variable indices. For example, in the Rosenbrock function, acanonical, nonseparable function that has been widely used as an EA benchmark, each term in the summation depends on adjacent variables zi and zi+1
(Table 1). However, there is no particular reason that adjacent variables shouldhave such dependencies in real-world, black-box optimization problems, and suchdependencies are an artifact of synthetic benchmarks.1
This unnatural problem structure can be easily eliminated using the randomization procedure described in [20]. First, the permutation vector P (P =P1 , ..., PD ) is initialized to a random permutation at the beginning of the DE
run. During the DE run, whenever a trial vector x is evaluated, we permute x
using P , resulting in the permuted trial vector x′ = (xP1 , ..., xPD ). Then, x′ isevaluated using the evaluation function, and the result is the fitness score fortrial vector x. This permutation effectively eliminates arbitrary dependencies
between variables with consecutive in the trial vector x – the dependencies are
now between variables with indices that are consecutive in x′ , but exponential
crossover, which operates on x, can not exploit the consecutiveness in x′ .
2.1

Exponential Crossover on Adjacent/Distributed Functions

We evaluate exponential crossover on (1) functions with dependencies between
lexicographically adjacent variables, i.e., standard versions of widely used synthetic benchmarks, and (2) modified versions of functions in (1) where the permutation method described above is used to randomize the variable dependencies
and eliminate the adjacent dependency structure. Following [4], we call functionsof the former class adjacent functions, and functions of the latter class distributed
functions.
We used the 3 nonseparable, adjacent functions in Table 1. The Rosenbrockand Schwefels 1.2 functions are well-known, classical benchmark functions thathave been widely used to evaluate EA’s [22]. The Block-rotated Ellipsoid [1] is apartially separable function designed to only have dependencies between zi and1

Of course, there are real-world problems that can be represented in such a way as
to have dependencies between adjacent variables [1] – we are merely arguing thatthese are not representative of black-box optimization problems.

exponential (A)exponential (D)binomial (D)SEC (D)

105
104

10

20

30 40 50 60
Dimension sizes

70

80

107

106

105

exponential (A)exponential (D)binomial (D)SEC (D)1020

30 40 50 60
Dimension sizes

70(d) Rosenbrock (jDE)80

Avg. # of evaluations / success rate

Avg. # of evaluations / success rate(a) Rosenbrock (DE)
108

107

exponential (A)exponential (D)binomial (D)SEC (D)

106
105
104

10

20

30 40 50 60
Dimension sizes

70

80

Avg. # of evaluations / success rate

106

108

107

106

105

104

exponential (A)exponential (D)binomial (D)SEC (D)1020

30 40 50 60
Dimension sizes

70(b) Schwefels 1.2 (DE)(c) B. L. Ellipsoid (DE)

107

107

106

105

104

exponential (A)exponential (D)binomial (D)SEC (D)1020

30 40 50 60
Dimension sizes

70(e) Schwefels 1.2 (jDE)80

Avg. # of evaluations / success rate

107

Avg. # of evaluations / success rate

Avg. # of evaluations / success rate

108

80

106

105

104

exponential (A)exponential (D)binomial (D)SEC (D)1020

30 40 50 60
Dimension sizes

7080(f) B. L. Ellipsoid (jDE)

Fig. 1: Evaluation of various crossover operators (standard DE and jDE). (A)and (D) stand ”Adjacent” and ”Distributed” respectively. The horizontal axis
represents the dimensionality D, and the vertical axis represents the averagefitness evaluations (for successful runs) divided by success rate.

zi+1 . Here, z = (yP1 , ..., yPD ), y = x − o, and for each function, the locationof the global optimum has been shifted by offset o (o = o1 , ..., oD ), where eachcomponent of o is a uniformly generated random offset. For adjacent functions,the permutation vector P = (1, ..., D), and for distributed functions, P is arandomly generated ordering such as P = (6, 1, 3, ...). The 2 × 2 rotation matrix
Ri is uniformly generated according to the method of [15] and α = 1e+6.
We studied problems with 10 − 80 dimensions. Each DE run continues until
either (1) the difference between the best-so-far and the optimal solution ≤1e-8, in which case we treat the run as a “success”, or (2) the # of objectivefunction calls exceeds 2.0×106, in which case the run is treated as a “failure”. On
each problem, each algorithm is executed 50 times. Following [6], our evaluation
metric is the average # of fitness evaluations in successful runs divided by the# of successes. Small values of this metric indicate a fast and stable search.
We use standard DE [16], as well as the state-of-the-art adaptive DE variants jDE [2], JADE [23], and SHADE [18].2 The standard DE used a population
size of 100 and F = 0.5, and the most commonly used, and rand/1 mutation strategy – this is a standard configuration in the DE literature [2, 23]. In
2

jDE [2], JADE [23], SHADE [18] were originally designed to use binomial crossover;in order to evaluate exponential crossover on state-of-the-art DE’s, we modified theseto use exponential crossover.

addition to exponential crossover, we also ran the experiments with binomial
crossover for comparison. On the standard DE, for each crossover method, foreach benchmark function, and for each dimensionality (D), we use the valueof CR ∈ {0.90, 0.91, ..., 0.99} that yields the best performance according to theperformance metric defined above. For jDE, JADE, and SHADE, which automatically adjusts CR, we use the control parameters recommended in the original
papers on these adaptive methods [2, 18, 23].
The results for standard DE and jDE are shown in Figure 1. Detailed results for JADE and SHADE are in the supplemental material [19] due to spaceconstraints, but the SHADE and JADE results are qualitatively similar to thejDE results. “Shuffled exponential crossover (SEC)” is explained in Section 3.
For cases where all runs failed (success rate for 50 trials = 0), then the data is
not shown. Since binomial crossover behaves almost identically for adjacent anddistributed functions, only the distributed function results are shown.
Figure 1 shows that exponential crossover performs much better on adjacent functions compared to distributed functions. The performance gap increaseas the dimensionality increases. For standard DE, exponential crossover outperforms binomial crossover on all adjacent functions, for all dimensionalities.
In stark contrast, on the distributed functions, the performance of exponential
crossover drops significantly – for all the distributed functions, for all D, exponential crossover performs worse than binomial crossover. In particular, on the
distributed-Rosenbrock and distributed-Schwefels 1.2 functions, for D ≥ 70 dimensions, exponential crossover fails on every single run. The results are similar
for jDE. Aside from the results on the Rosenbrock function (Figure 1(d)), theperformance of exponential crossover on distributed functions is clearly worse
than on the adjacent functions.
These results clearly show that the performance of exponential crossoveron nonseparable function benchmarks such as Rosenbrock and Schwefels 1.2
depends on an arbitrary feature of these synthetic benchmarks – variable dependencies between adjacent variables. Given essentially the same, nonseparable functions without this arbitrary structure, exponential crossover performs
much worse (significantly worse than binomial crossover). Functions such as
Rosenbrock and Schwefels 1.2 have been part of benchmark suites used by toevaluate DE since the original paper introducing DE [16], and continue to be
used today [3, 7, 9, 13, 24, 25]. As a consequence, exponential crossover has been
inaccurately overrated as a DE crossover operator for black-box optimization.
How fragile is exponential crossover to perturbations in the variable index
order? Instead of completely randomizing the variable index order, we investigate the effect of gradually decreasing the dependency between adjacent variableindices by applying n = 1, ..., D randomized swaps to the variable indices. Asn increases, the number of dependencies between adjacent variables decreases.
We ran standard DE on 50-dimensional problems (same setup as in the previous experiment). For each problem, we used the CR value that performed bestfor each n. Figure 2 shows that for all of the functions, exponential crossover
performs best when n = 0, and rapidly degrades as n increases, i.e., exponential1.6×106

1.4×106

1.2×106

1.0×106

8.0×105

6.0×105

4.0×105exponential (D)SEC (D)
0

10
20
30
40
50
# of swapped variables(a) Rosenbrock6

1.2×106
1.1×106
1.0×105
9.0×105
8.0×105
7.0×105
6.0×105
5.0×105
4.0×105
3.0×105
2.0×10exponential (D)SEC (D)
0

10
20
30
40
50
# of swapped variables(b) Schwefels 1.2

Avg. # of evaluations / success rate

6

Avg. # of evaluations / success rate

Avg. # of evaluations / success rate

1.8×108.0×105

7.0×105

6.0×105

5.0×105

4.0×105

3.0×105

2.0×105exponential (D)SEC (D)
0

10
20
30
40
50
# of swapped variables(c) B. L. Ellipsoid

Fig. 2: Effect of gradually decreasing dependencies between adjacent variables forstandard DE using exponential crossover and SEC on 50-dimensional problems.
Horizontal axis = # of randomized swaps (n = 1, ..., D); Vertical axis = averagefitness evaluations (for successful runs) divided by success rate.

crossover is quite fragile with respect to perturbations in dependencies between
adjacent variables.

3

Shuffled Exponential Crossover

The experiments in Section 2 showed that exponential crossover performs poorly
on distributed functions. To alleviate this problem, we evaluate shuffled exponential crossover. Caruana et al observed that 1-point crossover in GA’s introduced
a very strong search bias in that variables located close to each other in a linear genome (i.e., variable indices with similar lexicographic values) tended to
be inherited by the same child, while variables located far apart tend to be inherited by different children [4]. To eliminate this bias, they proposed shufflecrossover [4]: First, the variable indices of the parents are randomly shuffled(the same shuffle is applied to both parents). Next, standard 1-point crossoveris applied to the shuffled genomes. Finally, the indices are restored to their preshuffled states.3
Price et al note that exponential crossover in DE is subject to the same bias
as 1-point crossover in GA’s, and suggested that the shuffling mechanism fromshuffle crossover can be added to exponential crossover in order to alleviate thisproblem [14]. In this paper, we call this shuffled exponential crossover (SEC). The
algorithm is shown in Algorithm 2. Recently, Lin et al evaluated a mechanism
called non-consecutive exponential crossover which is in fact, the same as SEC[12]. However, they do not analyze the effect of this mechanism in the context
of dependencies among variables as we do. In fact, [12] claims “in non-separable
ridge functions (Rosenbrock and Schwefels 1.2 function), differential evolution
3

While shuffle crossover is similar to uniform crossover in that the genes that are exchanged are dispersed throughout the genome, the # of genomes that are exchanged
has a very different probability distribution, and there is a different search bias.

algorithms with consecutive crossover are more reliable than those with nonconsecutive crossover” (this is clearly contradicted by our result above).

Algorithm 2: Shuffled Exponential Crossover (SEC)ui,t = xi,t , k = 1, S(= s1 , ..., sD ) is randomly shuffled permutation {1, ..., D};repeat3
j = sk , uj,i,t = vj,i,t , k = k + 1;
4 until rand[0, 1) < CR and k < D;
1

2

Figure 1 shows the results of SEC for standard DE as well as jDE for thesame problems/settings as in Section 2.1.4 Unlike exponential crossover (but,similar to binomial crossover), the performance of SEC is unaffected by whether
the test function is adjacent or distributed; thus, only the distributed function
results are shown. Figure 2 also shows the results for SEC with best CR value
when n = 0 for the same problems/settings as in Section 2.1. The performanceof SEC is clearly shown to be independent of n.
Figure 1 shows that overall, SEC slightly outperforms exponential crossoveron the distributed functions. However, on the distributed Rosenbrock function (60 dimensions), SEC significantly outperforms exponential crossover. Also,on the distributed-Schwefels 1.2 function, SEC is competitive with binomial
crossover, even on 70 ≥ dimensional problems where exponential crossover failedon every single run. These results show that while exponential crossover depends
on the ordering of variables (i.e., whether the function is adjacent or distributed),
SEC, as expected, does not depend on the index positions of the variables andyields a much more stable search performance as a result.

4

Is Exponential Crossover Overrated?

Past evaluations of DE crossover operators on standard benchmark test suites
need to be reconsidered in light of our analysis of the interaction between adjacent functions and exponential crossover. Functions with dependencies between
adjacent variables (e.g., Rosenbrock, Schaffer F 7, Whitley’s composite functions [21]) are included in widely used benchmark suites. Table 2 shows thenumber of adjacent functions out of the total number of functions in the typical benchmarks. The larger the number of adjacent functions in the benchmark
suite, the more favorable the suite is for methods that exploit the adjacent structure, such as DE with exponential crossover. In particular, note that 10 out of 19
functions in the recent Soft Computing Journal (SOCO) benchmarks [8] are adjacent functions, making it particularly vulnerable to exploitation by exponential
crossover. In fact, all 7 of the DE algorithms submitted to the SOCO special issue
evaluation used exponential crossover. It would seem that due to the presence of
4

Results for JADE and SHADE are in the supplemental material [19].

Table 2: # of Adjacent Functions in Standard Benchmark Suites
Benchmarks# of adjacent functions

Function

13 classical [22]2 / 13F3 , F5

CEC 2005 [17]4 / 25F2 , F4 , F6 , F13

CEC 2010 [20]2 / 20F19 , F20

GECCO BBOB [5]1 / 24F8

SOCO benchmarks [8]10 / 19

F3 , F8 , F9 , F11 , F12 , F13 , F14 , F16 , F17 , F18

Table 3: SEC vs. binomial and exponential crossover on the CEC2014 benchmark
functions [10] for D = 50 (Wilcoxon rank-sum test significance threshold p < 0.05).
exponential

binomial

DE jDE JADE SHADE DE jDE JADE SHADE
vs. SEC# of better
Wilcoxon rank-sum# of worse(significance: p < 0.05) # of no sig.

1
8
21

0
2
28

0
3
27

0
5
25

12
9
9

18
5
7

19
7
4

13
6
11

adjacent functions, exponential crossover has been overrated in previous evaluations (assuming that benchmark suites are supposed to model true “black-box”
scenarios where there is no a priori reason to believe that adjacent variables havedependencies).
How do DE crossover operators compare on a benchmark set that does notcontain any adjacent functions? The recently proposed CEC2014 benchmark
set [10] consisting of 30 problems, does not include any adjacent functions.
We evaluated SEC vs exponential crossover vs binomial crossover on DE, jDE,
JADE, and SHADE on the CEC2014 benchmarks (in 10, 30, and 50 dimensions),
following the evaluation methodology specified in the CEC2014 benchmark competition [10]. The overall results for D = 50 dimensions are shown in Table 3.5
As shown in Table 3, for a diverse benchmark set, SEC outperforms exponential
crossover for all DE algorithms. Binomial crossover performs best for all DE
algorithms.

5

Conclusion

This paper showed that exponential crossover, one of the standard crossover
methods in DE, implicitly exploits an unnatural structure found in some syn5

The results for D = 10, 30 are in the supplemental data [19].

thetic benchmark problems, including some widely used, nonseparable functions(Rosenbrock and Schwefels 1.2), where there are strong dependencies between
variables with consecutive indices. We showed that exponential crossover performs significantly worse if we slightly perturbing these classical benchmarks
to remove these arbitrary, lexicographic dependencies, i.e., after artificial dependencies between adjacent variables are removed, the “true” performance of
exponential crossover appears to be significantly worse than previously believed.
We believe that for synthetic benchmarks [5, 8, 17, 20, 22], the performance of
exponential crossover has been overrated [3, 7, 9, 13, 24, 25]. Thus, the suitability of exponential crossover should be reevaluated in light of our results. We
showed that SEC, which does not implicitly assume sequential dependencies between variables and does not have the same search bias as exponential crossover,significantly outperforms exponential crossover overall, and is competitive withbinomial crossover. While SEC was suggested by [14], and also proposed by [12],our work is the first to identify the specific weaknesses described above for exponential crossover and show that shuffling results in improved overall performance.
Although there is still no clear criteria to determine whether binomial crossoveror SEC should be used for a particular problem (a direction for future work),
we believe that we have presented sufficient evidence to suggest that plain, exponential crossover should no longer be considered as an appropriate, default
crossover operator for DE. SEC (or binomial crossover) should be used insteadof exponential crossover, unless there is some prior knowledge that there are
dependencies between consecutive variables.
As we showed for exponential crossover in DE, algorithms that (intentionallyor unintentionally) exploit this dependency can appear to perform much better
than they would actually perform on real-world problems without this artificial
structure. As shown in Section 2, randomizing the lexicographic positions of thevariables for all benchmark functions, as suggested by [20] is a simple method
for avoiding this benchmarking pitfall, and we believe that black-box benchmark suites should apply this randomization to avoid unintentionally biasing
the evaluation results.

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