On the Pathological Behavior of Adaptive Differential
Evolution on Hybrid Objective Functions
Ryoji TanabeAlex S. Fukunaga

Graduate School of Arts and Sciences
The University of Tokyo
Tokyo, Japan

Graduate School of Arts and Sciences
The University of Tokyo
Tokyo, Japanrt.ryoji.tanabe@gmail.com

fukunaga@idea.c.u-tokyo.ac.jp

ABSTRACT
Most state-of-the-art Diﬀerential Evolution (DE) algorithms
are adaptive DEs with online parameter adaptation. We
investigate the behavior of adaptive DE on a class of hybrid functions, where independent groups of variables areassociated with diﬀerent component objective functions. An
experimental evaluation of 3 state-of-the-art adaptive DEs(JADE, SHADE, jDE) shows that hybrid functions are ”adaptive-DE-hard”. That is, adaptive DEs have significant failure rates on these new functions. In-depth analysis of theadaptive behavior of the DEs reveals that their parameter
adaptation mechanisms behave in a pathological manner on
this class of problems, resulting in over-adaptation for one of
the components of the hybrids and poor overall performance.
Thus, this class of deceptive benchmarks pose a significant
challenge for DE.

Categories and Subject Descriptors
G.1.6 [Mathematics of Computing]: Optimization—Global optimization; I.2.8 [Artificial Intelligence]: Problem
Solving, Control Methods, and Search—Heuristic methods

General Terms
Algorithms

Keywords
Adaptive Diﬀerential Evolution, Parameter Control, Deception, Hybrid Functions

1. INTRODUCTION
Diﬀerential Evolution (DE) is an Evolutionary Algorithm(EA) that was primarily designed for real parameter optimization problems [11]. Despite its relative simplicity, DE
has been shown to be competitive with more complex optimization algorithms, and has been applied to many practical problems [7]. As with other EAs, the search perforPermission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies are notmade or distributed for profit or commercial advantage and that copies bear
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mance of DE algorithms depends on control parameter settings [6, 7, 10]. A standard DE has three main control parameters, which are the population size N , scaling factor F ,and crossover rate CR. However, it is well-known that the
optimal settings of these parameters are problem-dependentand parameter tuning in real-world problem is often infeasible for various reasons. Since this is a significant problem
in practice, adaptive mechanisms for adjusting the DE control parameters on-line during the search process have beenstudied by many researchers [6, 13, 18].
It is well known that when the objective function is unimodal and nonseparable, high values of CR tend to perform
well, while low values of CR tend to be eﬀective for problems that are multimodal and separable.1 In addition, when
the objective is unimodal, moderate values of F work well,
but if the objective is multimodal, high F is desirable inorder to maintain diversity. Thus, adaptive DE algorithms
attempt to adapt control parameters in order to match thecharacteristics of the objective function [6, 13, 18].
EA performance evaluations for black-box optimization
are usually based on a suite of artificial benchmark problems. In recent years, benchmark sets such as the GECCOBBOB [1] and CEC2005 benchmarks [12] have been widelyused in the EA community. However, it has been notedthat these benchmark sets consist mostly of problems whichhave unrealistic, “extreme” characteristics, e.g., all of thevariables are separable, or all of the variables are nonseparable [14]. Many real-world problems have more complex
structure and can not be trivially characterized. For example, problems can be partially separable, where some groupsof variables interact with each other but are independent of
the other groups, i.e., “separability” is not a binary feature.
Yet, previous standard benchmarks focused almost exclusively on such extreme benchmarks, casting some doubt on
the utility of such benchmarks in predicting the performanceof EAs on real-world problems. For example, real-world
problems where all variables interact uniformly and completely with each other (as in the standard Rosenbrock andRotated-Rastrigin functions) are rare. Because of this growing realization, the EA community has recently introduced
benchmark sets that incorporate more complex interactions
between variables [9, 14].
Hybrid objective functions [9] are a class of benchmark
1

In separable functions∑D the solution vector can be decomposed as f (x) =i=1 fi (xi ) and can be each dimension
can be optimized independently. On the other hand, nonseparable functions have dependences between variables andcannot be so easily decomposed and optimized.

functions that have been recently proposed and incorporated
into the CEC2014 single-objective optimization competition
test suite. In a hybrid function, the variables in a solution
vector are partitioned into multiple groups, and each groupis evaluated according to a diﬀerent component evaluation
function. The overall objective function value is the sum of
the component objective values. Unlike previous benchmark
functions where all variables were associated with the samesearch space, each group of variables are associated with asearch space with characteristics that are distinct from thesearch spaces associated with the other variable groups.
In this paper, we show that state-of-the-art adaptive DE
algorithms such as SHADE [13], JADE [18], and jDE [6]perform extremely poorly on a class of hybrid functions [9]composed of 2 components. We show, for example that asimple hybrid function which, in some sense, closely resembles the well know, trivial Sphere function, poses a serious
challenge for adaptive DEs. An in-depth analysis of the failure of DEs on hybrid functions, and show extensive evidence
that adaptive DEs are easily deceived by hybrid functions– they tend to quickly adapt to solve the component of thehybrid function that is easier to make progress on, and are
unable to solve the other component. Thus, this study identifies a class of adaptive-DE-hard problems that confound
adaptive DEs and poses a challenging direction for futurework.

2. DIFFERENTIAL EVOLUTION
This section briefly describes DE [11]. A DE population is represented as a set of real parameter vectors xi =(x1 , ..., xD ), i = 1, ..., N , where D is the dimensionality ofthe 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 by applying somemutation strategy. v i,t = xr1 ,t + F · (xr2 ,t − xr3 ,t ) is rand/1
mutation strategy which is the most popular mutation strategy. The indices r1 , r2 , r3 are randomly selected from [1, N ]such that they diﬀer from each other as well as i. The parameter F ∈ [0, 1] controls the magnitude of the diﬀerential mutation operator. After generating the mutant vector
v i,t , it is crossed with the parent xi,t in order to generate
trial vector ui,t . Binomial Crossover, the most commonlyused crossover operator in DE, 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), andjrand is a decision variable index which is uniformly randomly selected from [1, D]. CR ∈ [0, 1] is the crossover rate.
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. In Section 3, we say that a generation of trial
vector is successful if this replacement occurs. These process
are repeated until some termination criterion is encountered.

ranking DE method in the CEC2013 Competition on RealParameter Single Objective Optimization [4].
jDE [6] assigns a diﬀerent set of parameter values Fi and
CRi to each xi , which is used for generating the trial vectors. Initially, the parameters for all individuals xi are set
to Fi = 0.5, CRi = 0.9. The control parameter values of
the trial vectors are inherited from their parents. However,each parameter is randomly modified (within a pre-specified
range) with some probability, and modified parameters are
kept for the next generation only when a trial is successful.
JADE [18] has two corresponding, adaptive variables, µCR ,µF . The CR and F associated with each individual are
generated according to a normal/Cauchy distribution withmeans µCR , µF . At the end of each generation, the values ofµCR , µF are updated according to the CR, F pair that resulted in the generation of the successful trial vector in thatgeneration. As the search progresses, µCR , µF should gradually approach the optimal values for the given problem. In
addition to parameter adaptation, JADE also uses a novel
mutation strategy called current-to-pbest/1 and an externalarchive for storing previously generated individuals.
SHADE [13] is an adaptive DE which based on JADE algorithm, but uses a historical memory of successful parameter settings based parameter adaptation scheme. Successhistory based adaptation uses a historical memory MCR , MF
which stores a set of CR, F values that have performed wellin the past, and generate new CR, F pairs by directly sampling the parameter space close to one of these stored pairs.

4.

HYBRID FUNCTIONSIn this paper, we study a simplified, special class of hybrid
functions [9] in which the number of groups is 2, i.e., eachhybrid function is composed of two objective functions f, g
and sum the component objectives to obtain the overall,hybrid objective value.
More specifically: Given a D-dimensional decision variable vector x, in our hybrid functions Hf,g , x is partitioned
into xf (xf = xf1 , ..., xfDf ) and xg (xg = xg1 , ..., xgDg ), thegroup of variables which will be evaluated using f , g respectively. Let r ∈ [0, 1] be the fraction of variables assigned to component objective f . If r is set to high value,
xf is allocated to the high fraction of variables and viceversa. We ensured the integrality of Df , Dg by setting Df =round(D × r), Dg = D − Df . The Df indices corresponding
to xf are randomly selected from [1, D], and the remainder
are assigned to xg .
The objective function value of a candidate individual xis Hf,g (x) = f (xf ) + g(xg ), where f (xf ) and g(xg ) are
the result of evaluating xf and xg according to f and g,
respectively. For example, if f is the Sphere function andg is the Rastrigin function, then the objective value for a
candidate solution vector for the 30-dimensional (D = 30)
hybrid function HSphere,Rastrigin with r = 0.4 would be thesum of the results of evaluating a 12-variable Sphere function
and a 18-variable Rastrigin function.

3. STATE-OF-THE-ART ADAPTIVE DE ALGORITHMS

5.

This section reviews three state-of-the-art adaptive DE
variants, jDE [6], JADE [18], SHADE [13] that we evaluate
in our experimental study. These three algorithms were selected because jDE and JADE are among the most highly
cited adaptive DE algorithms, and SHADE is the hightest

Our study uses hybrid objective functions that combine
pairs of the following five, commonly used benchmark functions: Sphere, Schwefel 1.2, Rosenbrock, Rastrigin, Ackley.
Table 1 shows their properties (unimodal/multimodal, separable/nonseparable). The search spaces were normalized to

EXPERIMENTS

Table 1: 5 basic functions for hybrid functions
Functions

Properties

SphereSeparable, Unimodal

Schwefel 1.2

Nonseparable, Unimodal

Rosenbrock

Nonseparable, Weakly Multimodal

Rastrigin

Separable, Strongly Multimodal

AckleySeparable, Weakly Multimodal

Table 2: 8 hybrid functions
Type of Hybrid

Heterogeneous

HomogeneousHybrid Functions

Base Functions

HSph,Ras

Sphere & Rastrigin

HSph,Ack

Sphere & Ackley

HSch,Ras

Schwefel 1.2 & Rastrigin

HSch,Ack

Schwefel 1.2 & Ackley

HRos,Ras

Rosenbrock & Rastrigin

HRos,Ack

Rosenbrock & Ackley

HSph,Sch

Sphere & Schwefel 1.2

HRas,Ack

Rastrigin & Ackley[−100, 100]D and the global optima were shifted to a point
where each dimension is uniformly selected from [−80, 80].
See details of these standard functions and their original
feasible regions in [16].
Table 2 shows the 8 hybrid functions that we studied,which combine pairs of the 5 basic functions from Table 1.
We write Hf,g to denotes the hybrid functions constructedfrom f and g (e.g., HSph,Ras is composed of the Sphere andRastrigin functions). The first 6 hybrid functions in Table 2, which we refer to as heterogeneous hybrids, combinefunctions whose characteristics diﬀer significantly from eachother. For example, HSph,Ras combines the unimodal Spherefunction and the multimodal Rastrigin function. In contrast, the remaining 2 functions, which we refer to as homogeneous hybrids, (HSph,Sch , HRas,Ack ) combine 2 similarfunctions – HSph,Sch combines two unimodal functions, andHRas,Ack combines two multimodal functions. The heterogeneous functions are studied in Section 5.1, and the homogeneous functions are studied in Section 5.2.
We studied problems with D = 30, 50 dimensions. The
ratio of variables assigned to each component function, r (seeSection 4), was varied from 0.0 to 1.0 in increments of 0.1.
Each run of the DE continues until either (1) the diﬀerence
between the best-so-far solution and the optimal solution≤ 1e-8, in which case we treat the run as “success”, or (2)the number of objective function calls exceeds D×10, 000, in
which case the run is treated as a “failure”. On each problem,for each r, each algorithm is executed 50 times. The successrate is the number of “successes” (as defined above) divided
by 50.
We evaluated three, state-of-the-art adaptive DE algorithms: SHADE [13], JADE [18] and jDE [6] (see Section
3). In addition, we also evaluate the standard DE [11] forcomparison. The source code for JADE was from [2], minimally modified these programs so that it would work with
our hybrid benchmark functions. The other DE’s (standard

DE, SHADE, and jDE) were implemented by ourselves.2
For each algorithm, we used the control parameter values
that were suggested in the cited, original papers.3 For the
standard DE, we used a typical configuration [6, 18] withpopulation size N = 50 4 , rand/1/bin generation strategy,
F = 0.5. We set CR to 0.1 and 0.9 since these values havebeen shown to work well for separable functions and nonseparable functions respectively [10]. Below, we refer to thestandard DE using CR of 0.1 and 0.9 as DE-CR0.1 and
DE-CR0.9, respectively.

5.1 Results for hybrid functions with dissimilar (heterogeneous) components
Figure 1 shows the success rates of DE algorithms on
the hybrid functions, HSph,Ras , HSph,Ack , HSch,Ras , HSch,Ack ,HRos,Ras , HRos,Ack (30, 50 dimensions) for various allocations (r) of variables among the two components.
We highlight several trends that are evident in Figure 1:• The adaptive DE algorithms (SHADE, JADE, jDE)
tend to have very high success rates when r = 0.0
and 1.0. That is, they are quite capable of solving thepure benchmark functions from which the hybrids arederived. The only exception is the low success rates
for jDE on Schwefel 1.2 and Rosenbrock function for
50 dimensions.
• The success rates for adaptive DE algorithms tend tobe high when r is close to 0, deteriorate steeply forsome particular range of r (most commonly when r =0.5 ∼ 0.9) but recover again at r = 1.0, and most of
the curves in Figure 1 are shaped like a ”V” or ”U”.
• Any degradations in success rates as a function of rthat are present in 30 dimensions are even more pronounced in 50 dimensions. This is expected, since
problems with higher dimensionality are more diﬃcult
for DE algorithms in general.

5.1.1 Discussion
It is well known that in general, unimodal functions areeasier to search than multimodal functions. Thus, on hybrid functions that combine unimodal and multimodal components (HSph,Ras , HSph,Ack , HSch,Ras , HSch,Ack ), one might
expect that as the multimodal component of the hybrid function increases, (i.e., fraction of variables evaluated using themultimodal Rastrigin and Ackley functions increases), theproblems would become monotonically more diﬃcult. However, our results indicate the opposite: success rates tended
to be higher as the multimodal component increased, andthe lowest success rates for SHADE, JADE and jDE occurred when the allocation to unimodal components (Sphere,
Schwefel 1.2) were 0.8 and 0.9. A similar tendency was
shown in HRos,Ack , HRos,Ras , which combines weakly multimodal (Rosenbrock, Ackley) and strongly multimodal (Rastrigin) functions.
2

We used version 1.1 of SHADE, available at [3].
However, for jDE, we reduced the population size to 50
because using the population size of 100 suggested in [6],since jDE did not converge within the maximum # of fitness
evaluations used in this study.
4
Although N = 100 is commonly used, we chose 50 for thesame reason as for jDE above.
3

30 dimensions

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6
0.4
0.2

Success rate

1
0.8

0

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6
0.4
0.2
0

0

0.2

0.4
0.6
Fraction of variables

0.8

1

0.2

0.4
0.6
Fraction of variables

0.8

1

0

1
0.8

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.2

Success rate

1
0.8

0.4

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6
0.4
0.2
0

0

0.2

0.4
0.6
Fraction of variables

0.8

1

0.2

0.2

0.4
0.6
Fraction of variables(b) HSph,Ack

30 dimensions

30 dimensions

0.8

1

0

0

Success rate

Success rate

1
0.8

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.4
0.2
0

0.2

0.4
0.6
Fraction of variables

0.8

1

0.2

0.4
0.6
Fraction of variables

50 dimensions
1
0.8

Success rate

1
SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6
0.4
0.2
0

0.8

0.2

0.4
0.6
Fraction of variables(d) HSch,Ack

0.4
0.2

1

0

0.2

0.4
0.6
Fraction of variables

0.8

1

0.8

1

0.8

1

50 dimensions
1

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6
0.4
0.2
0

0

1

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6

50 dimensions

0.8

0.8

0
0

Success rate

0

0.4
0.6
Fraction of variables

30 dimensions

1

0.6

0.2

(c) HSch,Ras

0.8

0.2

1

0
0

(a) HSph,Ras

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.8

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.4

1

0.4

0.4
0.6
Fraction of variables

0.6

0.8
0.6

0.2

50 dimensions

1
Success rate

Success rate

0.2

50 dimensions

0

Success rate

0.4

0.8
0.6

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6

0050 dimensions

Success rate

30 dimensions

1
0.8

Success rate

Success rate

30 dimensions
1
0.8

0.8
0.6

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.4
0.2
0

0

0.2

0.4
0.6
Fraction of variables

0.8

1

0

0.2

(e) HRos,Ras

0.4
0.6
Fraction of variables(f) HRos,Ack

Figure 1: Performance of DE on heterogeneous hybrid functions (HSph,Ras , HSph,Ack , HSch,Ras , HSch,Ack , HRos,Ras , HRos,Ack ).
Success rate (out of 50 runs) is shown as a function of r, the fraction of variables allocated to component function f . Resultsfor 30 and 50 dimensions are shown.

5.2 Results for hybrid functions with similar(homogeneous) components
Section 5.1 showed that on heterogeneous hybrid functions
where the components have significantly diﬀerent search spacecharacteristics (e.g., a unimodal-multimodal hybrid), theperformance of DE tended to degrade dramatically for some
mixture r of the the components. Are hybrid functions,in general, extremely diﬃcult for DE? In this section, we
consider homogeneous hybrid functions where, in contrast
to Section 5.1, the component functions have search space

30 dimensions

1

1

0.8

0.8

Success rate

Success rate

30 dimensions

0.6
SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.4
0.2
0

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6
0.4
0.2
0

0

0.2
0.4
0.6
0.8
Fraction of variables
50 dimensions

1

0

1

1

0.8

0.8

Success rate

Success rate

The results on the HSph,Ras hybrid function (Figure 1(a))
are particularly interesting: While the state-of-the-art adaptive DEs that we selected (SHADE, JADE, jDE) have beenshown to perform well overall on previous benchmark suites,
all of them perform extremely poorly when r = 0.7 ∼ 0.9 –In other words, state-of-the-art DE algorithms perform extremely poorly on a hybrid benchmark which is largely composed of the trivial Sphere function! Furthermore, while DE
is known to perform well on separable functions [8], theHSph,Ras function is a completely separable function that isextremely challenging for adaptive DE. This is the first such
example that we are aware of. Finally (and perhaps mostsurprisingly), the standard DE using CR = 0.1 performed
the best among all of the DEs evaluated on HSph,Ras , significantly outperforming all of the sophisticated, adaptive
DEs.

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6
0.4
0.2
0

0.2
0.4
0.6
0.8
Fraction of variables
50 dimensions

1

SHADE
JADE
jDE
DE-CR0.9
DE-CR0.1

0.6
0.4
0.2
0

0

0.2
0.4
0.6
0.8
Fraction of variables(a) HSph,Sch

1

0

0.2
0.4
0.6
0.8
Fraction of variables

1(b) HRas,Ack

Figure 2: Performance of DE on homogeneous hybrid functions (HSph,Sch , HRas,Ack ). Success rate (out of 50 runs) isshown as a function of r, the fraction of variables allocatedto component function f . Results for 30, 50 dimensions areshown.

1
0.8
0.6
0.4

Fraction of variables = 0.2

1

Successful-runs
Schwefel 1.2
Rastrigin

1
0.8
0.6
0.4

0.8
0.6

0.4

0.4

0.2
1e+05
Number of evaluations

2e+05

Fraction of variables = 0.8

0
0e+00

1

3e+05

Fraction of variables = 0.8(a) SHADE

3e+04
6e+04
Number of evaluations

9e+04

Fraction of variables = 0.8

0.6
0.4

Failed-runs
Schwefel 1.2
Rastrigin

0.2
0
0e+00

Successful-runs
Failed-runs
Schwefel 1.2
Rastrigin

0.8

0.4

2e+05

0
0e+00

1

0.6

1e+05
Number of evaluations

Fraction of variables = 0.2

0.2
8e+04
2e+05
2e+05
Number of evaluations

0.8

Failed-runs
Schwefel 1.2
Rastrigin

0.2
0
0e+00

1

Successful-runs
Schwefel 1.2
Rastrigin

0.6

0.2
0
0e+00

Fraction of variables = 0.2

0.8

1e+05
Number of evaluations

0.2
2e+05

0
0e+00

(b) JADE

Successful-runs
Failed-runs
Schwefel 1.2
Rastrigin
8e+04
2e+05
2e+05
Number of evaluations

3e+05(c) jDE

Figure 3: CR parameter adaptation history of adaptive DEs for r = 0.2 (top row) and r = 0.8 (bottom row) on HSch,Ras(D = 30).
1Fraction of variables = 0.2

1

Fraction of variables = 0.2

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4
0.2
0
0e+00

1
0.8
0.6

0.4

Successful-runs
Schwefel 1.2
Rastrigin
1e+05
Number of evaluations

2e+05

Fraction of variables = 0.8

0
0e+00

1

Failed-runs
Schwefel 1.2
Rastrigin

Fraction of variables = 0.8

0.2

0.2

0
0e+00

0
0e+00

2e+05

0
0e+00

1
0.8

0.6
0.4

(a) SHADE

0.2
3e+05

0.8

0.4

1e+05
Number of evaluations

8e+04
2e+05
2e+05
Number of evaluations

0.6(b) JADE

3e+04
6e+04
Number of evaluations

9e+04

Fraction of variables = 0.8
Successful-runs
Failed-runs
Schwefel 1.2
Rastrigin

0.4

Failed-runs
Schwefel 1.2
Rastrigin
1e+05
Number of evaluations

Successful-runs
Failed-runs
Schwefel 1.2
Rastrigin

0.4

Successful-runs
Schwefel 1.2
Rastrigin

0.2

Fraction of variables = 0.2

0.2
2e+05

0
0e+00

8e+04
2e+05
2e+05
Number of evaluations

3e+05(c) jDE

Figure 4: F parameter adaptation history of adaptive DEs for r = 0.2 (top row) and r = 0.8 (bottom row) on HSch,Ras(D = 30).

characteristics that are similar to each other. We used two
hybrid functions HSph,Sch and HRas,Ack . HSph,Sch combines
two unimodal functions, and HRas,Ack is a hybrid of two
multimodal functions.
The results for HSph,Sch and HRas,Ack in 30 and 50 dimensions are shown in Figure 2. SHADE, JADE and jDE
all achieve almost perfect success rates for all values of r
on HSph,Sch and HRas,Ack . This result is somewhat counterintuitive, given that solving Rastrigin and Ackley (both ofwhich are multimodal) individually is much more diﬃcult
than solving Rastrigin and Sphere (unimodal) individually,and yet, the hybrid HRas,Ack seems to be much easier thanthe hybrid function HSph,Ras (Figure 1(a)), which stumps allof the adaptive DE algorithms when r is between 0.7-0.9.
This result suggests that while hybrid functions are exceptionally challenging for DE when the component functions
have diﬀerent search space characteristics from each other,hybrids composed of similar components such as HSph,Sch ,HRas,Ack do not result in diﬃcult problems for DE.
One notable counterexample to the claim that heterogeneous hybrids are challenging is in Figure 1(b), which showsthat all methods perform very well on the hybrid HSph,Ack ,regardless of r, despite the fact that Sphere is unimodaland Ackley is multimodal. However, Ackley is only weakly
multimodal, and furthermore, both Sphere and Ackley areseparable, so HSph,Ack is actually similar to a homogeneous
hybrid function such as HSph,Sch and HRas,Ack and therefore
does not pose a challenge for adaptive DE.

5.3 Why do adaptive DEs fail? – Analysis ofadaptive DE parameter adaptation on hybrid functions
In order to understand why DE performance degrades sodramatically in the hybrid function experiments in Section
5.1, we analyze the parameter adaptation behavior for theadaptive DEs (SHADE, JADE, jDE) during the runs for theexperiments in Section 5.1.5
Figures 3 and 4 show the adaptation of the CR, F parameters (see Section 3) by SHADE, JADE, and jDE forr = 0.2 (top row) and r = 0.8 (bottom row) on HSch,Ras (30dimensions). The x-axis is the number of objective function
evaluations. The y-axis shows the proxy values of CR, F
5
We collected this parameter adaptation data by re-runningthe experiments described in Section 5.1 with identical random seeds used in the original experiments and saving all of
the parameter adaptation histories at every generation.

values for each algorithm. Since each algorithm has a different adaptation mechanism, appropriate proxy values areshown for each mechanism: (1) for SHADE, we show the median values of the CR, F values stored in historical memory,
MCR , MF , [13]; (2) for JADE, µCR , µF [18], the adaptive
parameters for CR and F , are shown; (3) for jDE, whichassigns a diﬀerent CR and F to each individual i [6], themedian values CRi and Fi are shown.
Each of 50 runs (for each r, for each function, for eachalgorithm) was classified as Successful Runs or Failed Runs,
depending on whether the run achieved the success criteria
defined at the beginning of Section 5 for this study, i.e., error≤ 1e-8. In cases where all 50 runs were failures or all 50 runs
were successes, only one of these lines are shown.
Each figure also shows how proxy values for F and CR
adapt in successful runs on pure Rastrigin and Schwefel 1.2
functions (i.e., r = 0.0, 1.0). The “Schwefel 1.2” and “Rastrigin” lines in Figures 3 and 4 show that on the Schwefel
1.2 function, SHADE, JADE, and jDE adapt CR to highvalues and F to medium values. On the other hand, for the
Rastrigin function, SHADE, JADE, and jDE adapt CR to
low values and F converges to high values. Thus, CR and
F converge to very diﬀerent values depending on the targetfunction (Rastrigin vs. Schwefel 1.2). These results are consistent with previous work which experimentally analyzed
the dynamics of DE parameter adaptation [6, 10], as wellas previous results on parameter adaptation for SHADE,
JADE, and jDE [6, 13, 18].
Now, consider the behavior of adaptive DE on HSch,Ras
when r = 0.2 (high fraction of variables allocated to the
Rastrigin function). From the figures, the adaptive behaviorof CR and F in SHADE and JADE very closely matches
their behavior on the Rastrigin function. jDE also displays
a similar tendency, albeit weaker (possibly due to the way wedefined the proxy metrics as medians). On the other hand,for r = 0.8 (high fraction of variables allocated to Schwefel1.2), the opposite trend can be seen, as the proxies for CR
and F behave similarly to the pure Schwefel 1.2 case. This
leads to the main, qualitative result of our study:
Observation 1. When an adaptive DE is applied to a
hybrid function, the parameter adaptation mechanism tends
to adapt and converge the CR and F parameters so as tomatch the component for which initial improvements are easier. More specifically, the adaptive mechanisms target thecomponent with larger allocation of variables.
Observation 1 holds quite robustly for our class of hybrid
functions. Trends similar to those seen in Figures 3 and 4
were observed for almost all hybrid functions we tested, forvarious values of r.
Since limited space prevents us from showing the equivalents of Figures 3 and 4 for all functions and all r, we instead
rely on a summary, diﬀerence metric which allows to compactly summarize and visualize these trends across all of thehybrid functions, for all r.
Suppose that we run jDE on the Sphere function twice,
and that runs #1 and #2 terminate after 5 and 4 generations, respectively, and that the CR-proxy values (as defined
above in this section) are (0.7, 0.6, 0.5, 0.7, 0.4) for run #1
and (0.9, 0.8, 0.3, 0.5) for run #2. The average behavior of
CR on these two runs is (0.8, 0.7, 0.4, 0.6, 0.4).
Further suppose that jDE is run on Rastrigin twice, withan average CR behavior of (0.4, 0.5, 0.4). The diﬀerential

Algorithm 1: Function δ(pf , pg ) for computing the difference between two average adaptation histories
input : pf , pg : the average adaptation histories on f, g
output: δ: the average distance on overall generations
between pf and pg
tmax = min(|pf |, |pg |), δ = 0;
for t = 1 to tmax do3
δ = δ + |pf,t − pg,t |;124

δ = δ/tmax ;Algorithm 2: Aggregating the average distances (to obtain the data points in Figure 5)
input : pHf,g , pf , pg on all hybrid functions
output: δ Hf,g ,f , δ Hf,g ,g : the average distance on all
hybrid functionsδ Hf,g ,f = 0, δ Hf,g ,g = 0;
for Hf,g ∈ I do3
δ Hf,g ,f = δ Hf,g ,f + δ(pHf,g , pf );
1

2

4
5

δ Hf,g ,g = δ Hf,g ,g + δ(pHf,g , pg );δ Hf,g ,f = δ Hf,g ,f /|I|, δ Hf,g ,g = δ Hf,g ,g /|I|;values between the CR-history on Sphere and the Rastriginfunction is (0.4, 0.2, 0.0), and the distance between the CRhistory over time is (0.4 + 0.2 + 0.0)/3 = 0.26 . Algorithm 1
describes the computation of average distance in general.
The distance δ(pf , pg ) can be used as a measure of thesimilarity between the parameter adaptation histories pf , pg
on f and g. The smaller the value of δ(pf , pg ), the moresimilar the parameter adaptation histories are, and conversely, the larger δ(pf , pg ) is, the more diﬀerent the parameter adaptation histories. Average distances among histories
among a set of instances I can be computed similarly.
For example, for some particular r, suppose that δ(HSph,Ras ,Sphere)=0.4 and δ(HSch,Ack , Schwefel 1.2) = 0.6. In thiscase, the average value of the distances is (0.4 + 0.6)/2 =0.5. For the same r, if δ(HSph,Ras , Rastrigin) = 0.6 andδ(HSch,Ack , Ackley) = 0.8, then the average value of the distances is (0.6 + 0.8)/2 = 0.7. In particular, we are interested
in distances between the histories on a hybrid function Hf,gwith particular r value and the histories on its component
functions, δ(Hf,g , f ) and δ(Hf,g , g), averaged over all inr
rstances δ Hf,g ,f , δ Hf,g ,g . Algorithm 2 shows the pseudocode
for this overall computation for some particular r.
rr
Figures 5 shows δ Hf,g ,f , δ Hf,g ,g for both DE parameters
CR (top row) and F (bottom row), averaged over all 6 of
the heterogeneous hybrid functions in Table 2 according to
Algorithm 2, for r ∈ 0.1, ..., 0.9. Data for both 30 and 50
dimensions are included in the SHADE and JADE figures(the similarity metric can be computed independently of dimensionality). Since jDE had a success rate of 0 on Schwefel
1.2 for 50-dimensions, we only included data for 30 dimensions. When computing the average histories for the hybrid
function, we include all runs, including successful and failed
runs. On the other hand, for the average histories for the6
If two histories have diﬀerent lengths, the unmatched portions of the longer history are ignored.

CR

5.2e-030.1

0.2

0.3 0.4 0.5 0.6 0.7
Fraction of variables

0.8

1.8e-02
1.3e-027.5e-03
0.1

0.9

0.2

0.3 0.4 0.5 0.6 0.7
Fraction of variables

Distance from f
Distance from g

2.4e-02
1.4e-023.5e-030.1

0.2

0.3 0.4 0.5 0.6 0.7
Fraction of variables(a) SHADE

0.8

0.9

6.4e-03
5.4e-03
Distance from f
Distance from g

4.4e-03
3.4e-032.4e-030.1

0.2

FAvg. Distances

Avg. DistancesF
3.4e-02

Avg. Distances

1.1e-02CR

Distance from f
Distance from g

2.3e-020.8

0.9

6.0e-02

Distance from f
Distance from g

4.7e-023.4e-022.1e-02
7.9e-030.1

0.2

0.3 0.4 0.5 0.6 0.7
Fraction of variables(b) JADE

0.3 0.4 0.5 0.6 0.7
Fraction of variables

0.8

0.9

0.8

0.9

F
Avg. Distances

1.7e-02CR

Distance from f
Distance from g

Avg. Distances

Avg. Distances

2.3e-020.8

0.9

Distance from f
Distance from g

2.8e-022.0e-021.2e-023.5e-030.1

0.2

0.3 0.4 0.5 0.6 0.7
Fraction of variables(c) jDE

Figure 5: Average distances of parameter adaptation history of SHADE, JADE, jDE between heterogeneous hybrid functions(HSph,Ras , HSph,Ack , HSch,Ras , HSch,Ack , HRos,Ras , HRos,Ack ) and its component function f , g.

pure component functions f and g, we used only the successful runs, because we want to compare the behavior on hybrid
functions vs. the “ideal” behavior on the pure functions, andthe average successful behavior is used as an approximation
for this “ideal” behavior.
Figures 5 show that for all of the adaptive DEs. When r islow (Hf,g is mostly allocated to g), the distance between therbehavior on δ Hf,g ,g is small, which means that the parameter adaptation behavior of the DE on Hf,g is quite similar
to its parameter adaptation behavior on g. However, as rr
increases, δ Hf,g ,g grows monotonically, i.e., the DE behaves
diﬀerently on the hybrid than it would on g. At the samertime, as r increases, δ Hf,g ,f decreases, and the DE behaves
more and more like it would on f . This trend can be veryclearly seen for both CR and F .
From the results above, we can see that in a heterogeneous
hybrid function Hf,g composed of two component functions
f , g with diﬀerent search space characteristics, the proxyvalues for CR, F in state-of-the-art adaptive DEs behave
similarly to when the DE is applied to the individual component function f or g which is allocated more variables in
the hybrid function, i.e., the individual component which ismost “similar” to the hybrid function.
Thus, we have shown that Observation 1 consistently holds
across all 3 state-of-the-art adaptive DEs, for all of the heterogeneous, hybrid functions where pathological behavior
was observed in Section 5.1.

5.3.1 Discussion
In general, most adaptive DEs that have been proposed todate, including SHADE, JADE, and jDE, operate according
to the principle that if the current set of control parametersetting S = (CR, F ) successfully generates trial vectors thatare more fit than their parents, then the next set of control
parameters to be applied should be somehow based on or
influenced by S. In other words, adaptive strategies tend tooperate based on a meta-greedy assumption.
On our class of hybrid functions, this meta-greedy approach leads to failure. In a hybrid with 2 component functions, the component CH with a higher fraction of variables
contributes more to the objective function than the othercomponent with lower allocation, CL , because the overall

objective value is the sum of the component objective values. Our results suggest that there is a tendency for adaptive DEs to quickly adapt to achieve good performance on
CH , neglecting CL . After optimizing the CH component,
the adaptive DE is unable to re-adapt to so that the CLcomponent can be optimized.
This explains the pathological behaviors by adaptive DEs
observed in Section 5.1, Figure 1(c): Although SHADE,
JADE, and jDE had success rates close to 1.0 on HSch,Ras
when r = 0.2, the success rate was almost 0.0 when r = 0.8.
This is most likely due to the meta-greedy nature of theadaptive DE described above. The components of HSch,Ras ,Schwefel 1.2 and Rastrigin, diﬀer significantly. Schwefel 1.2
is unimodal, and nonseparable, while Rastrigin is multimodal and separable, and as noted earlier, parameter settings which are appropriate for one are ineﬀective for theother. Figures 3 and 4 show that the DEs have adapted
entirely for one or the other – this means that they havebecome poorly adapted for the other component.
When r = 0.8, the DEs have adapted to solve Schwefel
1.2, a unimodal, nonseparable function. However, these parameter settings are very poorly suited for Rastrigin, a multimodal, separable function, and it is likely that the DEs
fail because their control parameters (which are adapted fora unimodal function) promote rapid descent into local minima, where the search becomes trapped.
A similar conclusion can be reached regarding the failureof adaptive DE on HRos,Ras in Figure 1(e). Although both
Rosenbrock and Rastrigin are multimodal, Rosenbrock is
nonseparable (requiring high CR values), while Rastrigin is
separable (requiring low CR values), and overadaptation for
one function prevents success on the other component of thehybrid objective. In general, in heterogeneous hybrid functions that include Rastrigin or Ackley, when the allocation ofvariables to the multimodal Rastrigin and Ackley functions
are too small, the adaptive DEs can not suﬃciently adapt tothe multimodal component, resulting on poor success rates
on the hybrid functions.
In contrast, the high success rates for r = 0.2 can be
explained as follows. Adaptation for the multimodal Rastrigin function does not cripple the DE’s ability to solve
unimodal problems such as the Schwefel 1.2. Although con-vergence may be slowed compared to parameter settings thatare tuned for a unimodal function, the lack of local minimameans that the search algorithm will eventually succeed.

6. CONCLUSION
We investigated the behavior of DE on a class of hybrid
functions [9], where groups of variables are assigned to diﬀerent component functions. We showed that state-of-the-art
adaptive DEs (SHADE [13], JADE [18], jDE [6]) all perform
very poorly when the components of the hybrid functions areheterogeneous, with diﬀerent search space characteristics.
The main contribution of this study is an in-depth analysis of adaptive DE control parameter behavior on hybrid
functions, showing that adaptive DE fails on these hybrid
functions because their parameter adaptation mechanisms
are fundamentally mismatched with the structure of hybrid
functions. DE control parameter adaptation tends to be “deceived” into over-adapting for one of the components of thehybrid, at the expense of adaptation for the other component.
Previous analysis of parameter adaptation in DE has been
limited to plotting control parameter values [6, 18]. Our
analysis of DE parameter adaptation introduces a new technique for summarizing and visualizing parameter adaptation(the parameter history distance metric described in Section
5.3) which provided a very clear understanding of when andhow DE succeeds and fails in the class of hybrid functions.
We believe that this kind of aggregated distance can beapplied to understand the adaptive behavior of DE (andother algorithms with adaptive parameters) for other types
of functions, e.g. multi-objective optimization.
This work suggests that a parameter adaptation mechanism that seeks to optimize its behavior for a single, targetfunction is a major design limitation when faced with a hybrid objective function. Our preliminary eﬀorts to overcome
this limitation by, e.g., associating a diﬀerent set of (CR, F )parameters for each variable, have not been successful so far,and this seems to be a nontrivial problem. Therefore, designing methods that are eﬀective on hybrid objective functions
is a challenging direction for future research. One possible
direction would be to investigate crossover methods that explicitly seek to identify and exploit dependencies between
variables in order to automatically partition and handle thecomponents of the hybrid objective, e.g., [17].
While this study was limited to hybrid functions with 2
component functions, preliminary experiments suggest thatsimilar issues arise on hybrids with more components, andwe are currently extending this study to hybrids with 3+
components. Finally, while this study focused on understanding the diﬃculty that hybrid functions pose for adaptive DE, heterogeneous hybrid functions are challenging forother kinds of non-adaptive, state-of-the-art DE. CoDE, awell-known non-adaptive DE [15] also fails on these functions (see Supplemental Data [5]). Understanding the reason why hybrid functions are diﬃcult for CoDE, as well as
evaluation of hybrid functions on other classes of DE is anavenue for future work.

7. ACKNOWLEDGMENTS
This work was supported by JSPS KAKENHI Grants
2324002 and 25330253.

8.

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