Improving the Search Performance of SHADE Using
Linear Population Size Reduction
Ryoji Tanabe and Alex S. Fukunaga
Graduate School of Arts and Sciences
The University of Tokyo
Abstract—SHADE is an adaptive DE which incorporates
success-history based parameter adaptation and one of thestate-of-the-art DE algorithms. This paper proposes L-SHADE,
which further extends SHADE with Linear Population Size
Reduction (LPSR), which continually decreases the population
size according to a linear function. We evaluated the performanceof L-SHADE on CEC2014 benchmarks and compared its search
performance with state-of-the-art DE algorithms, as well as
the state-of-the-art restart CMA-ES variants. The experimental
results show that L-SHADE is quite competitive with state-ofthe-art evolutionary algorithms.

I.

I NTRODUCTION

Differential Evolution (DE) is an Evolutionary Algorithm(EA) that was primarily designed for real parameter optimization problems [1]. Despite its relative simplicity, DE has beenshown to be competitive with more complex optimization algorithms, and has been applied to many practical problems [2].
However, the search performance of DE algorithms depends
on control parameter settings [2]. Where a standard DE has
three main control parameters, which are the population size
N , scaling factor F , and crossover rate CR. Unfortunately, it
is well-known that the optimal settings of these parameters
are problem-dependent. Therefore, when applying DE to areal-world problem, it is often necessary to tune the control
parameters in order to obtain the desired results. Since thisis a significant problem in practice, adaptive mechanisms foradjusting the control parameters on-line during the search
process have been studied by many researchers [3]–[6].
Success-history based adaptation is a novel mechanism
for parameter adaptation based on a historical memory of
successful parameter settings that were previously used foundduring the run [7]–[9]. Success-history based adaptation uses
a historical memory MCR , MF which stores a set of CR, F
values that have performed well in the past, and generate new
CR, F pairs by directly sampling the parameter space close to
one of these stored pairs. Success-History based Adaptive DE(SHADE) [7]–[9] is an improved version of JADE [5] whichuses a different parameter adaptation mechanism based on thissuccess-history based adaptation. This paper uses SHADE 1.1[9], the latest version of the SHADE algorithm. SHADE 1.1
dominates the SHADE 1.0 algorithm proposed in two earlier
conference papers [7], [8].
While SHADE 1.0 and SHADE 1.1 automatically adjust
the CR and F parameters, the population size N remains
constant throughout the search. The size of the population
used by EA plays a significant role in controlling the rateof convergence. Small population sizes tend to result in fasterconvergence, but increases the risk of converging to a local

optimum. On the other hand, large population sizes can encourage wider exploration of the search space, but the rateof convergence tends to be slower. The optimal population
size depends on complex interaction of a number of factors,
including the problem to which the EA is being applied, as wellas the values of other control parameters such as the mutation
rate. Thus, adaptive population resizing methods have been an
active area of research (see [10] for a survey).
However, as noted in [10], adaptive population resizing
methods such as GAVaPS [11] tend to replace one control
parameter (population size) with multiple, meta-level control
parameters, and are difficult to use in practice due to theneed to tune the meta-level control parameters. Unlike adaptive
methods for other control parameters such as mutation rateand crossover rate, successful adaptation of population size
has proven difficult. Thus, in recent years, population resizing
methods based on simple, deterministic rules (as opposed
to adaptive schemes) have been proposed. These approaches
have been found to be highly effective for improving EA
performance. Representative examples include IPOP-CMAES [12], GL-25 [13], IPSO [14], Dynamic Population Size
Reduction (DPSR) [15], and Simple Variable Population Sizing(SVPS) [16]. In contrast to adaptive methods which canboth increase or decrease the population size depending on
the state of the search, these deterministic methods either
monotonically increase or decrease the population size, basedon predetermined conditions. This significantly simplifies the
behavior of the population resizing methods.
To further enhance the performance of SHADE 1.1, we
propose L-SHADE, which incorporates Linear Population Size
Reduction (LPSR), a simple deterministic population resizing
method which continuously reduces the population size inaccordance with a linear function. Our LPSR method is a
simplified, special case of SVPS [16] which reduces the
population linearly, and requires only 1 parameter (initial
population sizes). DPSR [15] reduces the population by half
at predetermined intervals and has 2 parameters: the initial
population size and the frequency of the population reduction
rf req which has to be tuned to match the initial population
size as well as the dimensionality of the problem. SVPS is amore general framework in which the shape of the population
size reduction schedule is determined according to control
parameters τ and ρ. In addition to τ and ρ, which influence theshape of the population size curve, SVPS requires the initialand final population size at the end of the run. In contrast
to DPSR, which drastically reduces (halves) the population at
discrete, intervals, SVPS continuously reduces the population
size. Compared with both methods, LPSR is very simple andIndex
MCR
MF

1
MCR,1
MF,1

2
MCR,2
MF,2

...
...
...

H −1
MCR,H−1
MF,H−1

H
MCR,H
MF,H

In Eq. (1), if MCR,ri has been assigned the “terminal value”⊥, CRi is set to 0.

Fig. 1: The historical memory MCR , MF

B. Reproduction of trial vectors by using current-topbest/1/bin

has the advantage in that there are fewer control parameterswhich the user must tune.

After the control parameter values CRi and Fi are assigned
for each individual xi,G , a mutant vector v i,G is generated
from an existing population members by applying the currentto-pbest/1 mutation strategy, which is the mutation strategy
used by JADE [5], and a variant of the traditional currentto-best/1 strategy where the greediness is adjustable using aparameter p:This paper evaluates our proposed method, called LSHADE, on the 30 benchmark functions from the CEC2014
Special Session on Real-Parameter Single Objective Optimization benchmark suite [17]. We show experimentally that
L-SHADE significantly improves upon the performance of
SHADE 1.1 and also outperforms previous DE variants, including JADE [5], CoDE [18], EPSDE [6], SaDE [4], anddynNP-jDE [15]. In addition, we compared L-SHADE withNBIPOP-A CMA-ES [19] and iCMAES-ILS [20] which arestate-of-the-art restart CMA-ES [12] variants and co-winnersof competition on real-parameter single objective Optimization
at CEC 2013. Our results show that L-SHADE is highly
competitive with these CMA-ES variants.
II.

S UCCESS -H ISTORY BASED A DAPTIVE DE WITH
L INEAR P OPULATION S IZE R EDUCTION

This section describes L-SHADE. We first describe
SHADE 1.1 [9] in Sections II-A to II-E. Then, in Section II-F,
we describe LPSR and propose L-SHADE, which is SHADE
1.1 extended with LPSR.
Similar to other evolutionary algorithms for numerical
optimization, 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. At the beginning of the search, the individual
vectors xi in population are initialized randomly. Then, a
process of trial vector generation and selection are repeateduntil some termination criterion is encountered.
A. Control parameters assignments based on historical memory
As shown in Figure 1, SHADE 1.1 maintains a historicalmemory with H entries for both of the DE control parameters
CR and F , MCR , MF . The scaling factor F ∈ [0, 1] controls
the magnitude of the differential mutation operator and CR ∈[0, 1] is the crossover rate. In the beginning, the contents of
MCR,k , MF,k (k = 1, ..., H) are all initialized to 0.5. In eachgeneration G, the control parameters CRi and Fi used by eachindividual xi are generated by randomly selecting an index ri
from [1, H], and then applying the formulas below:0
if MCR,ri = ⊥.
CRi =
(1)
randni (MCR,ri , 0.1) otherwise(2)
Fi = randci (MF,ri , 0.1)
In case a value for CRi outside of [0, 1] is generated, it isreplaced by the limit value (0 or 1) closest to the generated
value. When Fi > 1, Fi is truncated to 1, and when Fi ≤ 0,
Eq. (2) is repeatedly applied to try to generate a valid value.
These manners are according to the procedure for JADE [5].

v i,G = xi,G + Fi · (xpbest,G − xi,G )+ Fi · (xr1,G − xr2,G )(3)In Eq. (3), individual xpbest,G is randomly selected from the
top N × p (p ∈ [0, 1]) members in generation G. The indicesr1 , r2 are randomly selected from [1, N ] such that they differ
from each other as well as i. The greediness of current-topbest/1 depends on the control parameter p, which trades off
exploitation and exploration (small p behaves more greedily).
For each dimension j, if the mutant vector element vj,i,G isoutside the search range boundaries [xmin, xmax], we appliedjjthe same correction performed in [5]: min
(xj + xj,i,G )/2 if vj,i,G < xminjvj,i,G =(4)(xmax+ xj,i,G )/2 if vj,i,G > xmaxjj
After generating the mutant vector v i,G , it is crossed withthe parent xi,G in order to generate trial vector ui,G . In
SHADE 1.1, Binomial Crossover, which is the most commonlyused crossover operator in DE, is used and implemented asfollows:
vj,i,G if rand[0, 1) ≤ CRi or j = jranduj,i,G =(5)xj,i,G otherwise
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].
C. Survival for next generation
After all of the trial vectors ui,G , 0 ≤ i ≤ N have been
generated, a selection process determines the survivors forthe next generation. The selection operator in standard DE
compares each individual xi,G against its corresponding trial
vector ui,G , keeping the better vector in the population.
ui,G if f (ui,G ) ≤ f (xi,G )xi,G+1 =(6)
xi,G otherwise
D. External archive
In JADE, an optional, external archive is used for maintaining diversity [5]. SHADE 1.1 also adopts an external archive.
Parent vectors xi,G which were worse than the trial vectors
ui,G (and are not selected for survival in the standard DE, Eq.
(6)) are preserved. When the archive is used, xr2,G in Eq. (3)is selected from P ∪ A, the union of the population P and
the archive A. Whenever the size of the archive exceeds thepredefined archive size |A|, randomly selected elements aredeleted to make space for the newly inserted elements.

E. Historical-memory update
In each generation, in Eq. (6), CRi and Fi values thatsucceed in generating a trial vector ui,G which is better thanthe parent individual xi,G are recorded as SCR , SF , and at theend of the generation, the memory contents are updated using
Algorithm 1.
Algorithm 1: Memory update algorithm in SHADE 1.1
1 if SCR 6= ∅ and SF 6= ∅ then2
if MCR,k,G = ⊥ or max(SCR ) = 0 then3
MCR,k,G+1 = ⊥;4else
5MCR,k,G+1 = meanW L (SCR );MF,k,G+1 = meanW L (SF );k + +;
8
If k > H, k = 1;
9 else10
MCR,k,G+1 = MCR,k,G ;11
MF,k,G+1 = MF,k,G ;6

7

Algorithm 2: L-SHADE algorithm// Initialization phaseinit
1 G = 1, NG = N, Archive A = ∅;2 Initialize population P G = (x1,G , ..., xN,G ) randomly;
3 Set all values in MCR , MF to 0.5;// Main loop
4 while The termination criteria are not met do
5
SCR = ∅, SF = ∅;6
for i = 1 to N do
7
ri = Select from [1, H] randomly;
8
If MCR,ri = ⊥, CRi,G = 0. Otherwise
CRi,G = randni (MCR,ri , 0.1);
9Fi,G = randci (MF,ri , 0.1);
10
Generate trial vector ui,G according tocurrent-to-pbest/1/bin;
11
12
13
14
15
16

In Algorithm 1, index k (1 ≤ k ≤ H) determines the
position in the memory to update. In generation G, the k-th
element in the memory is updated. At the beginning of thesearch k is initialized to 1. k is incremented whenever a newelement is inserted into the history. If k > H, k is set to
1. In the update algorithm 1, note that when all individuals ingeneration G fail to generate a trial vector which is better thanthe parent, i.e., SCR = SF = ∅, the memory is not updated.
The weighted Lehmer mean meanW L (S) is computed
using the formula below, and as with [21], the amount of fitness
improvement ∆fk is used in order to influence the parameter
adaptation (S refers to either SCR or SF ).
P|S|
wk · Sk2(7)meanW L (S) = Pk=1|S|
k=1 wk · Sk∆fk
wk = P|S |(8)CR
l=1 ∆fl∆fk = |f (uk,G ) − f (xk,G )|(9)
As MCR is updated, if MCR,k,G = ⊥ (where ⊥ denotes
a special, “terminal value”) or max(SCR ) = 0 (i.e., allelements of SCR are 0), MCR,k,G+1 is set to ⊥. Thus, if
MCR is assigned the terminal value ⊥, then MCR will remain
fixed at ⊥ until the end of the search. This has the effectof locking CRi to 0 until the end of the search, causing
the algorithm to enforce a ”change-one-parameter-at-a-time”
policy, which tends to slow down convergence, and is effective
on multimodal problems.
F. Linear Population Size Reduction (LPSR) and L-SHADE
algorithm
As mentioned in Section I, population size reduction has
been shown to be highly effective in improving EA performance [15], [16]. In order to improve the performanceof SHADE 1.1, we incorporate a population size reduction
method for dynamically resizing the population during a DE
run. Laredo et al proposed SVPS [16], a general framework for17
18

for i = 1 to N do
if f (ui,G ) ≤ f (xi,G ) thenxi,G+1 = ui,G ;else
xi,G+1 = xi,G ;if f (ui,G ) < f (xi,G ) thenxi,G → A;
CRi,G → SCR , Fi,G → SF ;24

If necessary, delete randomly selected individuals
from the archive such that the archive size is |A|.
Update memories MCR and MF (Algorithm 1);// Optional LPSR strategy
Calculate NG+1 according to Eq. (10);if NG < NG+1 thenSort individuals in P based on their fitness
values and delete lowest NG − NG+1 members;
Resize archive size |A| according to new |P |;

25

G++;

19
20
21
22
23

flexibly defining a population reduction schedule. We use Linear Population Size Reduction (LPSR), a simple specializationof SVPS which reduces the population linearly as a function of
the number of fitness evaluations. LPSR continuously reduces
the population to match a linear function where the population
size at generation 1 is N init , and the population at the end of
the run is N min . After each generation G, the population sizein the next generation, NG+1 , is computed according to theformula:1 minN− N init
init(10)
NG+1 = round· NFE + N
M AX N F E
N min is set to the smallest possible value such that the evolutionary operators can be applied – in the case of L-SHADE,
N min = 4 because the current-to-pbest mutation operator
Eq. (3) requires 4 individuals. N F E is the current numberof fitness evaluations, and M AX N F E is the maximum
number of fitness evaluations. Whenever NG+1 < NG , the(NG − NG+1 ) worst-ranking individuals are deleted from the
population.
1 Eq. (10) is almost equivalent to SVPS when the SVPS control parameters
are set to τ = 1, ρ = 0. See [16] for a detailed description of SVPS.

TABLE I: The results of L-SHADE on the CEC2014 benchmarks for D = 10, 30, 50 and 100 dimensions. Each column shows
best, worst, median, mean and standard deviation of the error value between the best fitness values found in each run and thetrue optimal value. The maximum number of objective function evaluations is D × 10, 000. All results are based on 51 runs.
D = 10D = 30D = 50
D = 100
Func. Best
Worst Median Mean
Std.
Best Worst Median Mean
Std.
Best Worst Median Mean
Std.
Best Worst Median Mean
Std.
1 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 2.9e+01 9.5e+03 8.2e+02 1.2e+03 1.5e+03 8.0e+04 3.4e+05 1.6e+05 1.7e+05 5.7e+04
2 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00
3 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00
4 0.0e+00 3.5e+01 3.5e+01 2.9e+01 1.3e+01 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 7.2e-01 9.8e+01 9.8e+01 5.9e+01 4.6e+01 1.0e+02 2.1e+02 1.5e+02 1.7e+02 3.1e+01
5 1.5e-01 2.0e+01 2.0e+01 1.4e+01 8.8e+00 2.0e+01 2.0e+01 2.0e+01 2.0e+01 3.7e-02 2.0e+01 2.0e+01 2.0e+01 2.0e+01 4.6e-02 2.0e+01 2.1e+01 2.1e+01 2.1e+01 3.1e-02
6 0.0e+00 8.9e-01 0.0e+00 1.8e-02 1.3e-01 0.0e+00 7.1e-06 0.0e+00 1.4e-07 9.9e-07 4.9e-05 1.8e+00 1.8e-02 2.6e-01 5.2e-01 4.5e+00 1.4e+01 8.8e+00 8.7e+00 2.3e+00
7 0.0e+00 2.5e-02 0.0e+00 3.0e-03 6.5e-03 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00
8 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 3.3e-08 0.0e+00 2.6e-09 7.5e-09 2.4e-03 3.7e-02 9.6e-03 1.1e-02 7.4e-03
9 2.2e-03 5.0e+00 2.0e+00 2.3e+00 8.4e-01 3.3e+00 9.2e+00 7.1e+00 6.8e+00 1.5e+00 5.4e+00 1.5e+01 1.1e+01 1.1e+01 2.1e+00 2.4e+01 4.6e+01 3.4e+01 3.4e+01 5.0e+00
10 0.0e+00 6.2e-02 0.0e+00 8.6e-03 2.2e-02 0.0e+00 6.2e-02 2.1e-02 1.6e-02 1.6e-02 5.4e-02 2.3e-01 1.1e-01 1.2e-01 4.1e-02 1.6e+01 4.2e+01 2.6e+01 2.6e+01 5.8e+00
11 3.9e-01 1.4e+02 1.6e+01 3.2e+01 3.8e+01 8.5e+02 1.7e+03 1.2e+03 1.2e+03 1.8e+02 2.5e+03 3.8e+03 3.3e+03 3.2e+03 3.3e+02 9.2e+03 1.2e+04 1.1e+04 1.1e+04 5.6e+02
12 2.5e-02 1.2e-01 7.0e-02 6.8e-02 1.9e-02 1.1e-01 2.3e-01 1.6e-01 1.6e-01 2.3e-02 1.5e-01 2.7e-01 2.2e-01 2.2e-01 2.8e-02 3.2e-01 5.2e-01 4.4e-01 4.4e-01 4.7e-02
13 1.6e-02 8.7e-02 5.3e-02 5.2e-02 1.5e-02 9.0e-02 1.7e-01 1.2e-01 1.2e-01 1.7e-02 1.1e-01 2.1e-01 1.6e-01 1.6e-01 1.8e-02 1.9e-01 2.9e-01 2.4e-01 2.4e-01 2.1e-02
14 4.5e-02 1.6e-01 7.6e-02 8.1e-02 2.6e-02 1.8e-01 3.0e-01 2.4e-01 2.4e-01 3.0e-02 2.4e-01 3.5e-01 2.9e-01 3.0e-01 2.5e-02 1.0e-01 1.3e-01 1.2e-01 1.2e-01 7.3e-03
15 2.1e-01 5.2e-01 3.7e-01 3.7e-01 6.9e-02 1.6e+00 2.6e+00 2.1e+00 2.1e+00 2.5e-01 4.1e+00 6.2e+00 5.1e+00 5.2e+00 5.1e-01 1.4e+01 1.9e+01 1.6e+01 1.6e+01 1.2e+00
16 4.0e-01 1.7e+00 1.3e+00 1.2e+00 3.0e-01 7.2e+00 9.5e+00 8.6e+00 8.5e+00 4.6e-01 1.6e+01 1.8e+01 1.7e+01 1.7e+01 4.8e-01 3.8e+01 4.0e+01 3.9e+01 3.9e+01 4.8e-01
17 0.0e+00 5.6e+00 6.2e-01 9.8e-01 1.1e+00 4.9e+01 3.3e+02 2.0e+02 1.9e+02 7.5e+01 4.9e+02 2.6e+03 1.2e+03 1.4e+03 5.1e+02 2.5e+03 5.9e+03 4.4e+03 4.4e+03 7.1e+02
18 4.3e-03 1.5e+00 1.6e-01 2.4e-01 3.1e-01 3.5e-01 1.7e+01 5.3e+00 5.9e+00 2.9e+00 6.8e+01 1.4e+02 9.5e+01 9.7e+01 1.4e+01 1.9e+02 2.6e+02 2.3e+02 2.2e+02 1.7e+01
19 1.3e-02 3.8e-01 6.2e-02 7.7e-02 6.4e-02 1.6e+00 4.9e+00 3.9e+00 3.7e+00 6.8e-01 5.4e+00 1.2e+01 7.9e+00 8.3e+00 1.8e+00 9.1e+01 1.0e+02 9.5e+01 9.6e+01 2.3e+00
20 1.4e-02 6.1e-01 9.5e-02 1.8e-01 1.8e-01 8.8e-01 7.6e+00 3.1e+00 3.1e+00 1.5e+00 6.1e+00 2.5e+01 1.4e+01 1.4e+01 4.6e+00 7.2e+01 2.5e+02 1.4e+02 1.5e+02 5.2e+01
21 3.5e-04 1.1e+00 4.3e-01 4.1e-01 3.1e-01 1.8e+00 3.7e+02 3.2e+01 8.7e+01 9.0e+01 2.5e+02 1.0e+03 5.0e+02 5.2e+02 1.5e+02 1.1e+03 3.8e+03 2.2e+03 2.3e+03 5.3e+02
22 3.6e-04 1.1e-01 4.4e-02 4.4e-02 2.8e-02 9.7e+00 1.4e+02 2.5e+01 2.8e+01 1.8e+01 3.2e+01 2.8e+02 6.9e+01 1.1e+02 7.5e+01 6.1e+02 1.4e+03 1.1e+03 1.1e+03 1.9e+02
23 3.3e+02 3.3e+02 3.3e+02 3.3e+02 0.0e+00 3.2e+02 3.2e+02 3.2e+02 3.2e+02 0.0e+00 3.4e+02 3.4e+02 3.4e+02 3.4e+02 4.4e-13 3.5e+02 3.5e+02 3.5e+02 3.5e+02 2.8e-13
24 1.0e+02 1.1e+02 1.1e+02 1.1e+02 2.3e+00 2.2e+02 2.3e+02 2.2e+02 2.2e+02 1.1e+00 2.7e+02 2.8e+02 2.8e+02 2.8e+02 6.6e-01 3.9e+02 4.0e+02 3.9e+02 3.9e+02 2.9e+00
25 1.0e+02 2.0e+02 1.1e+02 1.3e+02 4.0e+01 2.0e+02 2.0e+02 2.0e+02 2.0e+02 5.0e-02 2.0e+02 2.1e+02 2.1e+02 2.1e+02 3.6e-01 2.0e+02 2.0e+02 2.0e+02 2.0e+02 4.0e-13
26 1.0e+02 1.0e+02 1.0e+02 1.0e+02 1.6e-02 1.0e+02 1.0e+02 1.0e+02 1.0e+02 1.6e-02 1.0e+02 2.0e+02 1.0e+02 1.0e+02 1.4e+01 2.0e+02 2.0e+02 2.0e+02 2.0e+02 6.2e-13
27 8.5e-01 4.0e+02 1.5e+00 5.8e+01 1.3e+02 3.0e+02 3.0e+02 3.0e+02 3.0e+02 0.0e+00 3.0e+02 3.9e+02 3.3e+02 3.3e+02 3.0e+01 3.3e+02 4.6e+02 3.7e+02 3.8e+02 3.3e+01
28 3.6e+02 4.7e+02 3.7e+02 3.8e+02 3.2e+01 8.1e+02 8.7e+02 8.4e+02 8.4e+02 1.4e+01 1.1e+03 1.2e+03 1.1e+03 1.1e+03 2.9e+01 2.2e+03 2.4e+03 2.3e+03 2.3e+03 4.6e+01
29 2.2e+02 2.2e+02 2.2e+02 2.2e+02 4.6e-01 7.1e+02 7.4e+02 7.1e+02 7.2e+02 5.1e+00 7.4e+02 9.0e+02 7.9e+02 7.9e+02 2.4e+01 7.0e+02 1.1e+03 7.9e+02 8.0e+02 7.6e+01
30 4.6e+02 5.5e+02 4.6e+02 4.6e+02 1.3e+01 4.8e+02 3.5e+03 1.1e+03 1.2e+03 6.2e+02 7.9e+03 1.0e+04 8.6e+03 8.7e+03 4.1e+02 5.7e+03 1.0e+04 8.3e+03 8.3e+03 9.6e+02

Finally, the overall L-SHADE algorithm is shown in
Algorithm 2. Lines 21–24 implement the LPSR population
reduction scheme used by L-SHADE. If this option is excluded, Algorithm 2 becomes SHADE 1.1 algorithm. Notethat in line 24, the archive size |A| is readjusted, because wefound in preliminary experiments that this resulted in better
performance than setting the archive size to a constant value|A| = N init .
III.

E VALUATING L-SHADE ON THE CEC2014
BENCHMARKSThis section presents an empirical evaluation of L-SHADE.
We evaluated the performance of L-SHADE on the CEC2014
Special Session on Real-Parameter Single Objective Optimization benchmark suite [17] and compared L-SHADE to bothstate-of-the-art DE methods, as well as state-of-the-art restart
CMA-ES methods.
The CEC2014 benchmark set consists of 30 test functions.
For all of the problems, the search space is [−100, 100]D .2
Functions F1 ∼ F3 are unimodal. F4 ∼ F16 are simple
multimodal functions. F17 ∼ F22 are hybrid functions. Finally,F23 ∼ F30 are composite functions which combine multiple
test problems into a complex landscape. See [17] for details.
2 We treat the outside of this search space [−100, 100]D as infeasible.
Note that local optima outside this search space are easily discovered,and have a better objective function values than local optima solutions
inside [−100, 100]D on some functions (in particular, composite functions
F23 ∼ F30 ). This issue has also been found in the CEC2005 benchmarks[22].

The main difference between the CEC 2014 benchmarks
and the previous CEC 2005/2013 benchmarks [23], [24] is
the inclusion of a new set of hybrid functions (F17 ∼ F22 ).
In these hybrid functions, the variables in the solution vector
are randomly partitioned into 3 ∼ 5 groups, and each groupis evaluated using a different function, each with a distinct
structure. This is a kind of partial separability, which ispresent in real world problems such as transportation networks
and circuit theory, image processing, etc. [25]. Tang et. al.
pointed out that most previous benchmark sets did not include
partially separable functions, and consisted of functions where
all variables have uniform features [26]. Taking this into consideration, the CEC2014 benchmarks adopted hybrid functions(F17 ∼ F22 ) in order to more closely approximate real-world
benchmarks.
We performed our evaluation following the guidelines of
the CEC2014 benchmark competition [17]. When the gap
between the values of the best solution found and the optimal
solution was 10−8 or smaller, the error (score) was treatedas 0. For all of the problems the number of dimensions
D = 10, 30, 50, 100, and the maximum number of objectivefunction calls per run was D × 10, 000 (i.e., 100,000, 300,000,
500,000 and 1,000,000 respectively). The number of runs per
problem was 51, and the average performance of these runs
was evaluated.
The results of L-SHADE on D = 10, 30, 50 and 100
dimensions are shown in Table I. Each column shows best,
worst, median, mean and standard deviation of the error value
between the best fitness values found in each run and the true
optimal value.

TABLE III: Algorithm Complexity

TABLE II: The 4 control parameters of L-SHADE.
Parameters
N init

r
r arc
p
H

Ranges{15, 16, ..., 24, 25}{1.0, 1.1, ..., 2.9, 3.0}
{0.05, 0.06, ..., 0.14, 0.15}{2, 3, ..., 9, 10}

Default settings
20
2.0
0.1
5

T0

Tuned settings
D = 10D = 30D = 50
D = 100

18
2.6
0.11
6

0.11

T1

T̂ 2

(T̂ 2 − T 1)/T 0

0.21
0.97
2.37
8.71

0.38
1.41
3.04
10.09

1.53
3.95
5.91
12.25

B. Algorithm Complexity
A. Algorithm Parameters
This section describes the parameter setting of L-SHADE
and its execution environment. In this paper, for tuning theparameter settings of L-SHADE, we use ParamILS [27] which
is a versatile and efficient automatic parameter tuner which
can be applied to a wide range of parameterized algorithms,
including optimization algorithms. We used ParamILS because
it has been shown to be highly successful in tuning search andoptimization algorithms [27], and ParamILS is significantly
less labor-intensive compared to manual tuning.
The control parameters of L-SHADE are: (1) initial population size N init , (2) external archive size |A|, (3) historicalmemory size H, (4) p value for current-to-pbest/1 mutation(Eq. (3)). Following the standard practice for setting the
population size in DE [1], we set the initial population size
N init to the dimensionality D of the functions multiplied by
init
init
a parameter rN , i.e., N init = round(D×rN ). Similarly,
init
the external archive size |A| is set to N
multiplied by aparameter rarc , i.e., |A| = round(N init × rarc ). We used thedefault control parameters for the ParamILS parameter tuner.
The max number of parameter configurations evaluated byParamILS (i.e. the number of execution times of L-SHADE)
was set to 1, 000. The objective function used by ParamILS
was the error (difference) between the best fitness values foundin L-SHADE’s run and the true optimal value. Following[20], we tried to avoid overfitting the control parameters to
the CEC2014 benchmarks by using 5 composition functions
F21 ∼ F25 from the CEC2013 benchmarks [24] as the training
problem set for the ParamILS parameter tuner. Note that thesetraining functions are not from the CEC2014 benchmarks (i.e.,
the test problem set). Dimension sizes are set to 10 and 30(i.e. the total number of training problems are 5 × 2 = 10).
Execution of L-SHADE algorithm in this tuning phase was according to the CEC2013 benchmark competition rules, which
are identical with the CEC2014 competition rules describedabove.
Table II shows the search range of control parameters, thedefault parameter settings (i.e. initial parameter configurationof ParamILS), and the best settings found by ParamILS, which
are used as the L-SHADE’s parameter settings in the remainderof this paper. The default parameter settings were based on
preliminary experiments. The parameter tuning process using
ParamILS executed for approximately 4 hours. Due to spaceconstraints, we do not show the comparison results, but LSHADE with the tuned parameter settings by ParamILS isslightly better than one with the default parameter settings.

This section describes the algorithm complexity of our
L-SHADE code as defined in [17]. All experiments were
executed on the following system:•

OS: Ubuntu 12.04 LTS•

CPU: core i7 (2.20GHz)•

RAM: 8GB•

Language: C++•

Compiler: g++ (gcc) with -O3 optimization flag

Table III shows the computed algorithm complexity on 10,30, 50 and 100 dimensions. As defined in [24], T 0 is the time
calculated by running the following test problem:
for i=1:1000000
x=0.55+(double)i; x=x+x; x=x/2; x=x*x;
x=sqrt(x); x=log(x); x=exp(x); x=x/(x+2);
end
T 1 is the time to execute 200, 000 evaluations of benchmark
function F18 by itself with D dimensions, and T 2 is the time
to execute SHADE with 200, 000 evaluations of F18 in D
dimensions. T̂ 2 is the mean T 2 values of 5 runs. According
to Table III, both T 1 and T̂ 2 scaled linearly with the number ofdimensions, as shown by the linear growth of (T̂ 2 − T 1)/T 0.
C. L-SHADE vs. state-of-the-art DE algorithms on the
CEC2014 benchmarks
We compared L-SHADE with the state-of-the-art DE algorithms SHADE 1.1 [9], CoDE [18], EPSDE [6], SaDE [4],JADE [5], and dynNP-jDE [15] on the CEC2014 benchmarks.
The Matlab source code for CoDE, EPSDE, SaDE and JADE
were downloaded from [28], the web site of Q. Zhang, oneof the authors of [18]. These were used for the experiments
in [18], and are based on code originally received from theoriginal authors of CoDE, EPSDE, SaDE and JADE. We
minimally modified these programs so that they would workwith the CEC2014 benchmark codes. We implemented theprograms of SHADE 1.1 and dynNP-jDE by C++ and Eq. (4)is applied as the bound handling method. For each algorithm,
we used the control parameter values that were suggested in
the cited, original papers.
The results for 10, 30, 50 and 100 dimensions are shownin Tables IV. Due to space constraints, we only show theaggregate results of statistical testing (+, −, ≈). In Tables IV,
we show the aggregate results of comparing each algorithm
vs. L-SHADE on 30 functions. For example, according toTABLE IV: Comparison of L-SHADE with state-of-the-art DE
algorithms on the CEC2014 benchmarks (D = 10, 30, 50, 100
dimensions). The aggregate results of statistical testing (+, −, ≈) on
30 functions are shown. The symbols +, −, ≈ indicate that a given algorithm performed significantly better (+), significantly worse (−), or
not significantly different better or worse (≈) compared to L-SHADE
using the Wilcoxon rank-sum test (significantly, p < 0.05). The
maximum number of objective function evaluations is D × 10, 000.
All results are based on 51 runs.
D = 10D = 30D = 50

D = 100

SHADE+ (better)− (worse)≈ (no sig.)

0
16
14

3
18
9

5
19
6

7
17
6

CoDE

+ (better)− (worse)≈ (no sig.)

6
12
12

4
19
7

6
23
1

4
22
4

EPSDE

+ (better)− (worse)≈ (no sig.)

4
20
6

5
22
3

6
21
3

7
23
0

JADE

+ (better)− (worse)≈ (no sig.)

1
20
9

2
22
6

3
23
4

6
20
4

SaDE

+ (better)− (worse)≈ (no sig.)

4
17
9

1
25
4

0
27
3

1
27
2

dynNP-jDE+ (better)− (worse)≈ (no sig.)

6
16
8

3
20
7

5
20
5

7
18
5

vs. L-SHADE

the Wilcoxon test on the column D = 10 of Table IV,
which summarizes the experimental results on D = 10, CoDE
performed (significantly, p < 0.05) better than L-SHADE on
6 functions and worse than L-SHADE on 12 functions.
As shown in the tables counting the number of +, −, and≈ results, L-SHADE clearly has the best overall performanceon these 30 problems for all D ∈ {10, 30, 50}. These resultsshows that L-SHADE, the proposed methods, significantly
outperform previous state-of-the-art DE variants overall on the
CEC2014 benchmarks. This was to be expected since our previous work [7] shows that SHADE can also perform better thanthese methods. Also, note that the search performance of LSHADE is significantly better than SHADE 1.1, showing that
LPSR successfully contributed to L-SHADE’s performance.
D. Comparing L-SHADE with state-of-the-art restart CMA-ES
variants
Currently, restart CMA-ES methods based on IPOP-CMAES [12] are considered the state-of-the-art methods for single objective, real parameter optimization problems – restart
CMA-ES variants have won the CEC2005 and 2013 competitions [23], [24], as well as the GECCO2009/2010/2012/2013
BBOB [29]. On the other hand, it is widely believed that DE
methods perform significantly worse than restart CMA-ES [2].
Our results in Section III-C indicate that the performance of
L-SHADE is quite competitive compared to previous DE algorithms. A natural question is whether L-SHADE, which seems
to establish a new state-of-the-art for DE, is competitive withrestart CMA-ES. Thus, this section presents an comparison of
L-SHADE with state-of-the-art restart CMA-ES variants.
The evaluation includes NBIPOP-A CMA-ES [19] andiCMAES-ILS [20], sophisticated restart CMA-ES variants
based on IPOP-CMA-ES [12], which tied for first place inthe recent Special Session & Competition on Real-Parameter
Single Objective Optimization held at CEC-2013. NBIPOPA CMA-ES [19], which is improved variants of BIPOP-CMAES [30], adaptively allocates resources among the two components (IPOP-CMA-ES and multistart CMA-ES). iCMAESILS [20] is a hybrid method which combines IPOP-CMA-ES
and an iterated local search method. iCMAES-ILS separates
the search in two components, and initially allocates computational resources evenly between the two. After some time has
passed, the best-so-far solutions found by these components
is compared, and the better method is assigned all of theremaining time. Both algorithms have several characteristics
in common: (1) they are based on CMA-ES, (2) restarts
are performed, (3) compared to L-SHADE, the algorithms
are extremely complex. iCMAES-ILS and NBIPOP-A CMAES allocate computational resources among multiple searchmethods (and multiple control parameter sets) in an effort tomake the overall algorithm more robust and avoid the risk offailure, i.e., these can be seen as instances of an algorithm
portfolio approach [31]. All of these features are in directcontrast to L-SHADE.
In addition to an overall comparison, it is interesting tocompare L-SHADE with CMA-ES variants on specific subclasses of problems in the CEC2014 benchmark set, especiallythe new hybrid functions. We classified the 30 CEC2014
benchmarks into 4 groups: 3 unimodal functions, 13 simple
multimodal functions, 6 hybrid functions, 8 composition functions, and evaluated the performance on both groups separately.
The results for D ∈ {10, 30, 50, 100} dimensions on these4 groups and the overall results on all 30 functions areshown in Table V. Due to space, the table only shows theaggregate results of comparing each algorithm vs. L-SHADE
using the Wilcoxon rank-sum test (significantly, p < 0.05). The
source code for iCMAES-ILS and NBIPOP-A CMA-ES3 , were
downloaded from [32], the web site of CEC2013 competitions.
For L-SHADE, we use the data from Section III-C.
We observe the following: On 3 unimodal functions,
L-SHADE performs similarly to NBIPOP-A CMA-ES andiCMAES-ILS for 10, 30 dimensions, but L-SHADE is outperformed by both methods on one function (F1 ) for 50,
100 dimensions. On the 13 simple, multimodal functions,
L-SHADE performs better than NBIPOP-A CMA-ES on 10,
30 dimensions and comparably to iCMAES-ILS on 10 dimensions, but both methods outperform L-SHADE on higher
dimension size. On the 6 hybrid functions, L-SHADE clearly
outperforms NBIPOP-A CMA-ES and iCMAES-ILS for all
dimensions. This results indicate that L-SHADE is able to outperform state-of-the-art restart CMA-ES variants on real-world
problem which variables in the solution vector might have nonuniform features. On the 8 composition functions, L-SHADE
outperforms NBIPOP-A CMA-ES for 30, 50, 100 dimensions.
While L-SHADE performs better than iCMAES-ILS for 10,
50 dimensions, L-SHADE is worse for 30, 100 dimensions.
Overall on these 30 benchmark functions, L-SHADE performs
better than NBIPOP-A CMA-ES on 10, 30, 50 dimensions,
3 Since the search range in the program code of NBIPOP- CMA-ES is setAto [−∞, ∞]D , we changed this range to [−100, 100]D , to adhere to the
CEC2014 competition rules.

TABLE V: Comparison of L-SHADE with NBIPOP-ACMA-ES and iCMAES-ILS on the CEC2014 benchmarks for D ∈ {10, 30, 50, 100}on 4 groups (3 unimodal functions F1 ∼ F3 (first column), 13 simple multimodal functions F4 ∼ F16 (second column), 6 hybrid functions
F17 ∼ F22 (third column), 8 composition functions F23 ∼ F30 (fourth column)) and the overall results on all 30 functions (final column).

Groupsvs. L-SHADE(Wilcoxon rank-sum, p < 0.05)

NBIPOP-A CMA-ES

iCMAES-ILS

D = 10D = 30D = 50

D = 100

D = 10D = 30D = 50

D = 100

3 Unimodal
Functions+ (better)− (worse)≈ (no sig.)

0
0
3

0
0
3

1
0
2

1
0
2

0
0
3

0
0
3

1
0
2

1
0
2

13 Simple
Multimodal
Functions+ (better)− (worse)≈ (no sig.)

5
7
1

4
5
4

6
4
3

6
4
3

6
6
1

6
4
3

9
3
1

7
4
2

6 Hybrid
Functions+ (better)− (worse)≈ (no sig.)

0
6
0

0
6
0

0
6
0

1
5
0

0
6
0

0
6
0

0
5
1

1
4
1

8 Composition
Functions+ (better)− (worse)≈ (no sig.)

3
3
2

4
1
3

3
2
3

3
2
3

5
3
0

2
3
3

3
2
3

3
4
1

30 AllFunctions+ (better)− (worse)≈ (no sig.)

8
16
6

8
12
10

10
12
8

11
11
8

11
15
4

8
13
9

13
10
7

12
12
6

and comparably for 100 dimensions. L-SHADE outperforms
iCMAES-ILS on 10, 30 dimensions, worse than iCMAES-ILS
for 50 dimensions, and comparably to iCMAES-ILS on 100
dimensions.
Overall, we have demonstrated that contrary to current
conventional wisdom (c.f. [2]), we have shown that a DE
approach can be quite competitive with state-of-the-art restart
CMA-ES variants.
E. Comparison between LPSR and DPSR
This section compares LPSR with DPSR [15], to see whichmechanism is better suited as a population resizing mechanism
for the SHADE 1.1 framework. We compare L-SHADE with
D-SHADE, a variant of SHADE 1.1 which uses DPSR insteadof LPSR. In D-SHADE, the linear population size reduction
mechanism of L-SHADE for lines 21–24 in Algorithm 2 isreplaced by DPSR strategy (See [15]). The parameter settingsof D-SHADE is tuned by ParamILS same as L-SHADE. Table
VI shows the search range of control parameters, the default
parameter settings, and the best settings found by ParamILS,
which are used as the D-SHADE’s parameter settings. Where,
the tuning scenario of ParamILS is same as Section III-A.
Interestingly, as shown in Table II and VI, the tuned parameter
init
values of rN , rarc and p are very similar between LSHADE and D-SHADE.
Table VII shows the aggregate results of the comparison
between L-SHADE and D-SHADE for D = 10, 30, 50, and100 dimensions. L-SHADE clearly outperforms D-SHADE for
10 and 30 dimensions. However, L-SHADE performs similarly
to D-SHADE for D = 50 and is outperformed for 100
dimensions. Thus, the best population reduction strategy for
SHADE 1.1 appears to depend on the dimensionality of the
problem. A more detailed evaluation of population reduction
strategies is an avenue for future work. However, comparedwith DPSR, LPSR has one definite advantage in that LPSR isvery simple and has fewer control parameters which the user
must tune.

TABLE VI: The 5 control parameters of D-SHADE.
Parameters
N initrr arc
pH
rf reqRangesDefault settings

Tuned settings{15, 16, ..., 24, 25}{1.0, 1.1, ..., 2.9, 3.0}
{0.05, 0.06, ..., 0.14, 0.15}{2, 3, ..., 9, 10}{4, 5, ..., 9, 10}

20
2.0
0.1
5
4

18
2.5
0.11
9
7

TABLE VII: Comparison of L-SHADE with D-SHADE on the
CEC2014 benchmarks.
vs. L-SHADE
D-SHADE+ (better)− (worse)≈ (no sig.)IV.

D = 10D = 30D = 50

D = 100

2
13
15

3
12
15

8
8
14

13
8
9

C ONCLUSIONSThis paper proposes L-SHADE, which extends SHADE 1.1[9] with Linear Population Size Reduction (LPSR). LPSR is a
simplified, special case of Simple Variable Population Sizing[16] which reduces the population linearly, and only requires
initial population size and population reduction frequency
as user-defined parameters. We evaluated the search performance of L-SHADE on the 30 benchmark functions from the
CEC2014 Special Session on Real-Parameter Single Objective
Optimization benchmark suite [17]. The experiments were performed for 10, 30, 50 and 100 dimensions. The experimental
results showed that L-SHADE significantly improves upon
the performance of SHADE 1.1, and outperforms previous
DE variants, including JADE [5], CoDE [18], EPSDE [6],SaDE [4], and dynNP-jDE [15]. In addition, we compared LSHADE with NBIPOP-A CMA-ES [19] and iCMAES-ILS [20]which are state-of-the-art restart CMA-ES [12] variants and cowinners of the CEC 2013 competition on real-parameter single
objective Optimization. The results showed that L-SHADE ishighly competitive with these CMA-ES variants, especially, on
the hybrid functions F17 ∼ F22 where subgroups of variables
are associated with functions with different properties.

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In spite of the successful results of L-SHADE on F17 ∼F22 in the CEC2014 benchmarks, our recent study [33] showsthat on hybrid functions where the components have significantly different search space characteristics unlike F17 ∼ F22(e.g., a unimodal-multimodal or separable-nonseparable hybrid), the performance of adaptive DE including SHADE 1.1,
JADE and jDE tended to degrade dramatically depending on
the fraction of variables allocated to each component in the
hybrid function. Therefore, designing adaptive DE algorithms
that are effective on hybrid objective functions is a challenging
task.
Our results indicate that the simple LPSR method incorporated into L-SHADE is also quite powerful, resulting insignificant improvement over SHADE 1.1. However, deterministic population resizing methods such as DPSR, SVPS
and LPSR often assume an optimization scenario with the
maximum number of fitness evaluations. While this is astandard benchmarking methodology in the literature as well ascompetitions, real-world applications do not necessarily follow
this model. In an “anytime optimization” setting where thebest-so-far solution is requested at some a priori unknown time,
SHADE 1.1, which does not make any assumptions about thetermination condition, may outperform L-SHADE. Evaluationof such alternative settings, as well as application of LPSR toother DE variants, are directions for future work.

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