57
Original scientific paper
 MIDEM Society
Population Ranking Based Differential 
evolution with Simulated Annealing for Circuit 
Optimization
Jernej Olenšek, Árpád Bűrmen
University of Ljubljana, Faculty of Electrical Engineering, Ljubljana, Slovenia
Abstract: Finding the values of circuit parameters for which the resulting circuit satisfies the design requirements can be formulated as 
an optimization problem. This problem is often solved using global optimization algorithms that provide some guarantee the resulting 
solution is the best possible one provided that the algorithm is given sufficient time. Unfortunately, these algorithms are slow and 
require many circuit evaluations. One of the algorithms proposed in our past research is PSADE that combines the favorable properties 
of simulated annealing and differential evolution and was shown to be a fast and reliable tool for solving circuit optimization problems. 
To make PSADE faster we replace the Metropolis criterion for accepting a trial point with one that is based on population ranking. The 
proposed algorithm retains its highly parallel nature. We tested the algorithm on a set of mathematical test functions and on a real-
world circuit optimization problem. The results on the analog circuit sizing case show that the modified algorithm is more efficient and 
reliable than PSADE and some other global optimization methods. 
Keywords: differential evolution; simulated annealing; population ranking; global optimization;, circuit design
Optimizacija vezij z diferencialno evolucijo in 
simuliranim ohlajanjem na osnovi rangiranja 
populacije
Izvleček: Iskanje vrednosti parametrov, pri katerih vezje zadosti načrtovalskim zahtevam, lahko predstavimo kot optimizacijski 
problem, ki ga pogosto rešujemo z globalnimi optimizacijskimi postopki. Ti nam zagotavljajo, da bomo našli najboljšo možno 
rešitev pod pogojem, da ima postopek na razpolago dovolj časa. Na žalost tovrstni postopki zahtevajo veliko simulacij vezja. Eden 
od postopkov, ki smo jih razvili, je postopek PSADE, ki združuje ugodne lastnosti diferencialne evolucije in simuliranega ohlajanja in 
se je izkazal kot hiter in zanesljiv pri optimizaciji vezij. Da bi postopek pospešili, smo zamenjali Metropolisov kriterij za sprejem točk s 
kriterijem, ki temelji na rangu točke znotraj populacije. Dobljeni postopek lahko zelo učinkovito paraleliziramo, saj obdrži vse ugodne 
lastnosti postopka PSADE. Preizkusili smo ga na naboru matematičnih funkcij in na praktičnem primeru iz načrtovalske prakse. Rezultati 
kažejo, da je pri načrtovanju vezij predlagani postopek bolj učinkovit in zanesljiv kot nekatere druge znane globalne optimizacijske 
metode
Ključne besede: diferencialna evolucija; simulirano ohlajanje; rangiranje populacije; globalna optimizacija; načrtovanje vezij
* Corresponding Author’s e-mail:  jernej.olensek@fe.uni-lj.si
Journal of Microelectronics, 
Electronic Components and Materials
Vol. 46, No. 2(2016), 57 – 64
1 Introduction
Choosing the values (parameters) of circuit compo-
nents (also referred to as circuit sizing) with the goal of 
satisfying the design requirements can be formulated 
as an optimization problem by introducing a cost func-
tion (CF) that reflects the quality of the circuit in a real 
number. Finding the best performing circuit reduces to 
finding the minimum of a CF. Unfortunately real-world 
CFs have many local minima. Finding the best local 
minimum is a computationally hard problem that is ad-
dressed with global optimization algorithms. 
Many global optimization algorithms were devised in 
the past. Some of the most successful mimic the evolu-
tion (e.g. [1, 2]) and behavior (e.g. [3, 4]) of living beings 
and physical processes (e.g. [5, 6]). Simulated anneal-
58
J. Olenšek et al; Informacije Midem, Vol. 46, No. 2(2016), 57 – 64
ing (SA) ([7, 8]) was one of the first global optimization 
algorithms drawing its inspiration from the process of 
cooling a metal. Due to its nature this algorithm is ca-
pable of finding a global minimum, albeit with a large 
number of CF evaluations. 
Due to the so-called “no free lunch” theorems [9] the 
research in the area of global optimization started to 
focus on hybrid algorithms (e.g. [10, 11]). In our past 
research we hybridized SA with differential evolution 
(DE) [12] which resulted in the parallel simulated an-
nealing and differential evolution algorithm (PSADE) 
[14] that exhibited good performance on mathemati-
cal test functions, as well as, real world circuit design 
problems. Due to the SA component of PSADE it can 
mathematically be proven that the algorithm converg-
es to a global minimizer given a sufficiently large num-
ber of CF evaluations. PSADE is highly parallelizable 
which makes it possible to significantly speed up the 
optimization when multiple processors are available. 
PSADE has the same drawbacks as other heuristic op-
timization algorithms. Most notable is the possibility 
of premature convergence to a local minimizer. The 
main cause of this drawback is the Metropolis accept-
ance criterion of SA which often rejects trial points that 
would otherwise lead the algorithm away from a local 
minimizer. Furthermore, the Metropolis acceptance cri-
terion is based on an artificial parameter (temperature) 
that is based on the CF value. Because the CF value can 
differ to a great extent between similar optimization 
problems the acceptance criterion can be misguided 
into rejecting promising points. 
To counter this drawback and simplify the algorithm 
we propose a different approach for accepting points 
based on the solution ranking within the popula-
tion. The rank of the point is determined by sorting 
the points according to their CF value. We also assign 
every individual in the population a separate position. 
In turn, every position is assigned a set of parameters 
for SA and DE. Parameters corresponding to higher 
positions have a greater probability to be used in the 
process of constructing and evaluating a trial point. 
The algorithm tries to align the positions of individu-
als within the population with their ranks. This align-
ment is occasionally broken to increase the probability 
of accepting an inferior solution which may lead the 
algorithm away from a local minimum. We deem the 
proposed algorithm DESAPR (DE and SA with Popula-
tion Ranking). 
The paper is divided as follows. Section 2 outlines the 
proposed approach. The asynchronous parallel ver-
sion of the algorithm is the subject of section 3. The 
optimization results for mathematical test function are 
given in section 4 while section 5 presents the results 
obtained on a real-world circuit optimization problem. 
Section 6 concludes the paper. 
Notation. Vectors are denoted by bold lowercase let-
ters. The i–th component of vector a is denoted by ai. 
Inequalities are applied to vectors component-wise. 
The realization of a uniformly distributed random num-
ber from interval (0,1) is denoted by U(0,1). 
2 The proposed method
The problem subject to optimization can mathemati-
cally be formulated as
 * argmin ( ) S f∈= xx x
:f S → R
 { }: , NS = ∈ ≤ ≤x x L x UR
   (1)
where f is the CF, N is the number of optimization 
variables, and L and U are vectors of lower and upper 
bounds imposed on these variables. The outline of the 
DESAPR is given by Algorithm 1. 
Algorithm 1: DESAPR outline.
1. Initialization.
2. Competition.
3. Selection of parameters.
4. Trial point generation.
5. Trial point evaluation. 
6. Replacement of a point in the population.
7. Local search.
8. If termination condition is not met, go back to 
step 2. 
The population consists of M individuals. It is initialized 
by dividing each of the N parameter ranges given by 
vectors L and U uniformly into M subintervals. For every 
variable the M subintervals are assigned randomly to 
M individuals. The value of a variable for an individual 
is then chosen by randomly selecting a point in the as-
signed subinterval. 
We denote the available positions with i = 0,1, …, M – 1. 
The behavior of the algorithm is determined by the 
weight factor W, crossover probability PX, and random 
step probability distribution width parameter η. A set 
of parameter values is assigned to every one of the M 
available positions. The values of parameters assigned 
to i-th position are 
59
 
0
Wc i
iW W e
−=
, ,0 ,
PXc i
X i XP P e
−=
 1ii eη = −
    (2)
Coefficients cW and cpX are computed from the weight 
factors and crossover probabilities of positions 0 and 
M-1 which are user defined parameters of the algo-
rithm. 
 ( ) 1 0
1
1 lnW
M
Wc M
W
−
−
= −
 
( ) 1 ,0
, 1
1 ln
X
X
P
X M
P
c M
P
−
−
= −
   (3)
Every individual xi is assigned to one of the M available 
positions. Let pi and ri denote the position and the rank 
of i-th individual. The rank of an individual is deter-
mined by ordering the individuals according to the CF 
value fi = f(xi) and assigning ranks from M-1 (for the low-
est CF value) to 0 (for the highest CF value). The goal of 
the competition in step 2 of Algorithm 1 is to align the 
rank of the individuals in the population with their po-
sitions. For this purpose two individuals are randomly 
selected from the population. Let i and j denote their 
indices. They exchange positions if rj > ri and pj < pi. 
This forces individuals with high rank (low CF value) to 
move to high positions. 
In step 3 of Algorithm 1 an individual is selected ran-
domly with probability PS,i. 
 
, 1
0
i
i
r
S i M r
i
eP
e−
=
=
∑
    (4)
Let k denote the position of the selected individual. The 
weight factor (Wk), the crossover probability  (PX,k), and 
the range parameter (ηk) values assigned to this posi-
tion are used in steps 4-7 of Algorithm 1. Individuals 
with high rank are selected with higher probability. Be-
cause higher ranking individuals tend to occupy higher 
positions, parameter values corresponding to higher 
positions are often used (but not always). 
To simplify the notation the search space is transformed 
so that the components of vectors (i.e. individuals) lie 
within [0,1], where 0 and 1 correspond to the lower and 
the upper bound, respectively. 
In step 4 of Algorithm 1 a trial point is generated using 
a modified DE operator [12] and polynomial mutation 
[13]. First a parent (x) and three additional distinct in-
dividuals (u, v, and w) are selected. The DE operator is 
applied component wise with probability PX,k resulting 
in point y‘ with components 
 ( ) ,  ' (0,1) .
otherwise
i k i i X k
i
i
u W v w U P
y
x
 + − ≤
= 

 (5)
If y‘ violates the bounds the components that are out-
side the bounds (i.e. y1‘ ∉ [0,1]) are corrected resulting 
in point y with components defined as 
 
( )
( )
' '
'
'
0 1
0,1 (1 ) 1 .
0,1 0
i i
i i i i
i i i
y y
y x U x y
x U x y
 ≤ ≤
= + ⋅ − >
 − ⋅ <
 (6)
Finally, polynomial mutation is applied to y compo-
nent wise. For every component a random number 
ai = U(0,1) is generated. The trial point z is then com-
puted as
 ( ) ( )
( )( )
( )
1/ 11
1/ 11
1 2 2 2 1 0.5
2 1 2 1 1 otherwise
kk
kk
i i i i
i i
i i i
y
z y
y
ηη
ηη
α α α
α α
++
++
  − − + − > = + 
  + − − − 
 (7)
Let fz denote the CF value corresponding to the trial 
point z. The population along with the trial point is or-
dered according to the CF value. Rank 0 is assigned to 
the point with the highest CF value. Let rz and rx denote 
the ranks of the trial point and the parent, respectively. 
The trial point is accepted into the population (i.e. re-
places the parent point x in step 6 of Algorithm 1) with 
probability 
( )
min 1,
z xr r k
M kP e
−
−
 
=   
    (8)
The acceptance criterion resembles the original Me-
tropolis criterion [7] in the sense that higher ranking 
trial points are accepted with higher probability. A trial 
point ranking higher than the parent point is always ac-
cepted. 
Finally, in step 7 of Algorithm 1 a simple local search 
strategy is performed if the trial point is accepted, the 
parent point is the best point in the population, or with 
some small probability PL (set to 0.05). Local search uses 
one of the points in the population as the origin and 
two additional points for computing a search direction 
(d). All three points are chosen randomly. Two addition-
al points are evaluated along direction d and a quad-
ratic model of the CF is computed. This model is mini-
mized and the resulting point evaluated. The evaluated 
point with the lowest cost function value is the result of 
J. Olenšek et al; Informacije Midem, Vol. 46, No. 2(2016), 57 – 64
60
the local search. The resulting point replaces the parent 
point if its CF value is lower. The complete details of the 
local search can be found in [14]. 
3 Parallel implementation
Suppose one has m parallel processors available. As-
suming most of the computational time is spent for 
evaluating the trial point and for local search, these two 
tasks are outsourced to parallel processors in an asyn-
chronous manner. The main process (master) runs the 
following algorithm: 
Algorithm 2: asynchronous parallel optimization
1. Initialization.
2. If no processor is idle go to step 5. 
3. Perform steps 2-4 of Algorithm 1 (generate a trial 
point).
 Send it to an idle processor (p) for evaluation. 
 Remember the parent point (x) and the select-
ed parameters position (k) for that processor. 
4. If there are idle processors left return to step3. 
5. Wait until one of the processors (p) finishes its 
task. Collect the results.
6. If p was performing global search point evalua-
tion.
 Perform step 6 of Algorithm 1 (point replace-
ment) using the parameters corresponding to 
position k that was stored for p. 
 If required, start a local search (step 7 of Algo-
rithm 1) on processor p. 
 If p was performing local search.
 Replace the parent point of p if local search 
found a better point. 
7. If termination condition is not met, go back to 
step 2. 
Algorithm 2 performs multiple passes of Algorithm 1 
in parallel and in this way accelerates the evolution of 
the population. 
4 Performance on mathematical test 
functions
DESAPR was implemented within the framework of the 
PyOPUS library [15]. The performance of DESAPR was 
compared to that of PSADE, DE, SA, and JADE [18] on 
13 test functions from [14]. For the sake of comparison, 
30 optimization runs were performed for every func-
tion. We will later use the presented method for analog 
circuit sizing, where CF evaluations can take several 
seconds. Therefore we impose a CF evaluation budget 
of 100000 function evaluations per run to maintain rea-
sonable optimization run times. 
For DESAPR we used fixed parameter values in all ex-
periments: M=20, W0=0.9, WM-1=0.9, PX0=0.9, PXM-1=0.1. 
We made no attempt to fine tune the parameter val-
ues to any specific problem. It is very time consum-
ing especially for real world problems, where every 
CF evaluation can take considerable amount of time. 
The values were chosen based on our experience with 
evolutionary algorithms. High weight factor for DE and 
low crossover probability tend to maintain population 
diversity longer, which is desirable since DESAPR uses 
very small population. Fine tuning the parameters and 
introducing parameter adaptation or evolution could 
lead to even better performance for our method and is 
also subject of our future research. 
For the compared methods, the parameters were se-
lected according to guidelines from the authors of the 
methods. For DE we used DE/rand/bin strategy with 
population size 100, weight factor 0.5 and crossover 
probability 0.9. SA used in our experiments uses only 
two parameters. We set the final temperature and ran-
dom step range parameter to Tmin=1e-10, Rmin=1e-10. 
For PSADE we set population size to 20, Tmin=1e-10, 
Rmin =1e-10, t1 = 0.01 (local step), t2 =0.1 (parameter 
adaptation). For JADE we also followed the author sug-
gestions. We used the version without the archive, as 
suggested by the authors for problems with low di-
mensionality (< 30). We used the population size of 
100, the learning parameter c=0.1 and the percentage 
of points considered as the best in population p=5%.
The results are given in Table 1. 
For every function we chose a target CF value ftarget, that 
lies in the basin of attraction of the global minimum. 
Finding this solution means that global search was suc-
cessful, and any local search procedure can be used to 
quickly find the exact minimum. Not all runs succeed 
in reaching ftarget. We report the success rate and the 
average number (over successful runs) of CF evalua-
tions (#CF) needed to reach ftarget. The average final CF 
error (with respect to the true global minimum) after 
100 000 CF evaluations is listed in the error column. The 
best value of #CF and error are written in boldface if 
there exist a statistically significant difference between 
the best and second best method. 
No method was able to reach ftarget for all functions in all 
runs. JADE and DESAPR outperformed the other meth-
ods so we will focus on them.  Considering final CF val-
ue, JADE outperformed DESAPR on 8 functions, while 
DESAPR was better only on 2 functions. On 3 functions 
there was no statistically significant difference. When 
J. Olenšek et al; Informacije Midem, Vol. 46, No. 2(2016), 57 – 64
61
Table 1: Performance comparison results for 30D mathematical test functions. The function value at the global mini-
mum is denoted by f*. The results of the best performing algorithm are written in boldface if there exist a statistically 
significant difference to the second best method. 
Function f* ftarget #CF Success rate [%] error Algorithm
f1
Sphere
0 10-10 39388
62347
NA
NA
34000
100
100
0
0
100
9.74 * 10-17
5.32 * 10-12
5.20 * 10-7
7.09 * 10-6
0.0
DESAPR
PSADE
DE
SA
JADE
f2
Schwefel 2.22
0 0.1 14477
47562
58954
78691
16020
100
100
100
76
100
5.19 * 10-7
4.91 * 10-2
1.73 * 10-4
4.30 * 10-3
0.0
DESAPR
PSADE
DE
SA
JADE
f3
Schwefel 1.2
0 15 37693
67121
NA
61051
26334
100
100
0
100
100
7.11 * 10-3
0.811
29.01
7.38 * 10-4
4.81 * 10-9
DESAPR
PSADE
DE
SA
JADE
f4
Schwefel 2.21
0 0.1 35228
71209
94510
73522
88247
100
100
36
100
63
4.07 * 10-5
0.073
2.94 * 10-1
1.18 * 10-3
9.22 * 10-2
DESAPR
PSADE
DE
SA
JADE
f5
Rosenbrock
0 30 18919
36599
57851
63210
17418
100
100
100
70
100
16.52
21.46
21.11
92.07
4.28
DESAPR
PSADE
DE
SA
JADE
f6
Step
0 0 11149
16151
42566
61547
10862
100
100
100
100
100
0
0
0
0
0
DESAPR
PSADE
DE
SA
JADE
f7
Noisy quartic
0 0.02 30999
36211
79544
58044
16225
100
100
57
73
  100
7.84 * 10-13
2.31 * 10-3
2.58 * 10-2
1.41 * 10-2
1.96 * 10-3
DESAPR
PSADE
DE
SA
JADE
f8
Schwefel 2.26
-418.982887*30
= -12569.486618
-12569.45 22049
36955
NA
NA
77722
97
93
0
0
97
3.95
7.90
7.59 * 103
7.11 * 102
3.95
DESAPR
PSADE
DE
SA
JADE
f9
Rastrigin
0 0.1 30697
81588
NA
NA
77644
100
100
0
0
100
6.83 * 10-13
8.19 * 10-3
144.23
6.31
1.60 * 10-4
DESAPR
PSADE
DE
SA
JADE
f10
Ackley
0 10-4 31255
55982
96542
NA
27122
100
100
83
0
100
2.27 * 10-7
1.93 * 10-5
8.14 * 10-4
0.71
4.44 * 10-16
DESAPR
PSADE
DE
SA
JADE
f11
Griewank
0 10-9 42281
77545
NA
NA
35385
100
100
0
0
100
1.37 * 10-13
7.88 * 10-9
4.13 * 10-8
2.10 * 10-2
5.55 * 10-17
DESAPR
PSADE
DE
SA
JADE
J. Olenšek et al; Informacije Midem, Vol. 46, No. 2(2016), 57 – 64
62
comparing the speed, JADE was faster on 7 functions, 
while DESAPR was better on 4. JADE displays a good 
fine tuning capabilities and fast convergence on uni-
modal and well behaved functions.  However on the 
most difficult f8 and f9, that have many local minima, 
DESAPR was significantly better than JADE, regarding 
both the speed and the final solution quality. JADE and 
DESAPR also exhibit similar success rate. 
5 Performance on a real-world design 
problem
We tested DESAPR by sizing a Miller operational trans 
conductance amplifier (OTA) (Figure 1)[16] across a 
large number of corner points. The bias current was set 
to 100µA. The performance measures and the design 
requirements are listed in Table 2. 
Every corner point is a combination of temperature 
(0oC, 25 oC, and 100 oC), operating voltage (1.7V, 1.8V, 
and 2.0V), and CMOS corner model (average, worst 
power, worst speed, worst one, and worst zero). Let 
  denote the set of 45 corner points obtained in this 
manner. Every design requirement must be met for all 
45 corner points from set  . The CF is a function of the 
design parameters (xD). It is constructed as a sum of 
contributions (CFi) corresponding to individual perfor-
mance measures. 
 ( ) ( )D i DCF CF= ∑x x     (9)
Let pi(xD), gi, and ni denote the worst value of a perfor-
mance measure across corners from set i ⊆  , the 
corresponding goal, and the corresponding norm, re-
spectively. A contribution of a performance measure to 
the CF for which an upper bound is imposed (design 
requirement of the form pi(xD) ≤ gi) is computed as [17]
 
( )
6
( ) (the design requirement is not met)
( )
( )10 otherwise
i D i
i D i
i
i D
i D i
i
p g p g
n
CF
p g
n
−
− ≥=  − ⋅

x x
x
x  (10)
For performance measures with design requirements 
of the form pi(xD) ≥ gi the roles of pi(xD) and gi in equa-
tion (10) are exchanged. By default the norm is equal 
to the goal. If the goal is 0, the norm is set to 1. An ex-
ception is the circuit area for which the norm is set to 
100µm2. Constructing the cost function in this manner 
penalizes designs that fail to satisfy the design require-
ments with a positive CF contribution while rewarding 
designs that exceed the design requirements with a 
small negative CF contribution.
Figure 1: Miller OTA. 
The optimizer tries to find the minimum of the CF by 
tuning the 13 design parameters (11 transistor channel 
widths and lengths, the resistance of R and the capaci-
tance of C). The optimizer stops as soon as all design 
requirements are satisfied. 
Table 2: Design requirements and circuit analyses from 
which the performance of the Miller OTA is evaluated.
Performance measure Goal Required 
analyses
Supply current [µA] ≤ 200 op
Vgs overdrive [V] ≥ 0.0 op 
Vds overdrive [V] ≥ 0.1 op
Output swing [V] ≥ 1.2 dc 
Gain [dB] ≥ 60 ac 
UGBW [MHz] ≥ 30 ac
Phase margin [o] ≥ 50 ac
PSRR Vdd [dB] ≥ 65 ac, acvdd
f12
Penalty 1
0 10-10 39953
52650
NA
NA
32804
100
100
7
0
100
4.70 * 10-16
3.31 * 10-16
1.71 * 10-9
4.94 * 10-8
3.77 * 10-32
DESAPR
PSADE
DE
SA
JADE
f13
Penalty 2
0 10-10 43895
54261
NA
NA
34960
100
100
0
0
100
1.65 * 10-14
5.74 * 10-15
3.59 * 10-7
1.01 * 10-6
4.39 * 10-30
DESAPR
PSADE
DE
SA
JADE
J. Olenšek et al; Informacije Midem, Vol. 46, No. 2(2016), 57 – 64
63
PSRR Vss [dB] ≥ 65 ac, acvss
CMRR [dB] ≥ 90 ac, accom
Settling time (up) [ns] ≤ 100 tran
Settling time (down) [ns] ≤ 100 tran
Overshoot (up) [%] ≤ 10 tran
Overshoot (down) [%] ≤ 10 tran
Slew rate (up) [V/µs] ≥ 10 translew
Slew rate (down) [V/µs] ≥ 10 translew
Circuit area [µm2] ≤ 1000 -
Because evaluating one point in the design space re-
quires the circuit to be evaluated across all corner 
points from   a strategy for reducing the number of 
circuit evaluations was used. The circuit was optimized 
in multiple passes where the solution of k–th pass (xD,k) 
was used as the initial point for pass k +1. In the be-
ginning of k–th pass all performance measures were 
evaluated across all corners from   at the initial point 
xD,k-1. Let ic ∈  denote the corner point where the i–th 
performance measure reached its worst value. If this 
worst performance did not satisfy the corresponding 
design requirement corner point ci was added to i . If 
no corner was added to any of the corner point subsets, 
no further passes were performed. If the resulting circuit 
satisfied all design requirements the run was deemed as 
successful. If, however, a corner point was added to any 
of the corner point subsets, the CF was minimized using 
an optimization algorithm starting from xD,k-1 which re-
sulted in point xD,k. In most cases the final corner subsets 
contained only a handful of corners where the circuit 
exhibited its worst performance. Therefore the number 
of circuit evaluations was much lower compared to the 
brute force approach where every point in the design 
space is evaluated across all 45 corners. 
Population based algorithms were started from a given 
initial point xD,k by replacing one member of the ini-
tial population with xD,k. Optimization was stopped as 
soon as all design requirements were satisfied across 
the corresponding corner point sets i  or if the number 
of evaluated circuits exceeded 50000.  Four optimiza-
tion algorithms were tested: differential evolution (DE), 
PSADE [14], DESAPR and JADE. SA was not included 
in this test because its performance on mathematical 
test functions considering the average final CF value 
was significantly worse than the performance of DE. 
Due to the stochastic nature of the tested algorithms 
the circuit was optimized 10 times on a cluster of 100 
processors. For every algorithm the final CF value, the 
run time, and the number of performed circuit analyses 
was recorded. The minimal, the maximal, and the aver-
age results are listed in Table 3.
In terms of the final CF value DE and JADE failed to 
find a circuit satisfying all design requirements, despite 
many more CF evaluations. DESAPR and PSADE both 
succeeded in finding such a circuit in all 10 optimiza-
tions. The final CF value found by PSADE was slightly 
better, although this is not relevant because the opti-
mization was stopped as soon as a circuit satisfying all 
design requirements was found and there was no real 
competition between the algorithms in terms of find-
ing the best possible circuit. In terms of computational 
time DESAPR was on average two times faster than 
PSADE. The same can be said about the number of cir-
cuit analyses. 
6 Conclusion
Finding the values of the circuit’s design parameters is 
an important task in analog design automation. Global 
optimization algorithms are often selected for this task 
due to their ability to find the best possible circuit. Un-
Table 3: Performance comparison results for the Miller OTA. The best results (CF value and time) are written in bold-
face. 
CF Time [s] # op # dc # ac # acvss # acvdd # accom # tran # translew
DE Min 1.76e-02 7.42e+03 597390 132382 999350 221375 241473 261571 528424 264396
Max 4.97e-01 2.95e+04 2358660 381896 5036254 2105616 2280094 2396099 2029708 1133701
Avg 8.66e-02 1.96e+04 1453050.4 271884.2 3049903.2 1004943.8 1080426.9 1134624.7 1247288.0 670108.5
PSADE Min -3.99e-05 6.59e+02 34722 9180 45739 9180 12955 14976 39943 21951
Max -3.50e-05 3.59e+03 180440 42717 365833 112328 95558 119591 259453 124762
Avg -3.77e-05 1.92e+03 107708.4 24171.2 172718.2 48279.7 46172.2 55005.7 136731.1 66726.0
DESAPR Min -3.96e-05 4.08e+02 19222 5750 31718 6225 8090 8090 24758 13626
Max -3.47e-05 1.24e+03 69940 15284 117381 29909 35803 39533 75463 54541
Avg -3.70e-05 9.24e+02 46279.4 10833.1 71826.0 19131.2 20380.1 23858.7 53771.4 32792.2
JADE Min 8.67E-03 2.55E+05 3806920 688720 10603220 5299470 5299470 5299470 5989420 3018020
Max 5.74E-02 4.08E+05 8380226 1280426 27604226 11204410 11204410 11062510 11054026 5803526
Avg 2.78E-02 3.06E+05 5.70e+06 9.55e+05 1.77e+07 7.56e+06 7.59e+06 7.51e+06 8.00e+06 4.23e+06
J. Olenšek et al; Informacije Midem, Vol. 46, No. 2(2016), 57 – 64
64
fortunately these algorithms are quite slow. We pro-
pose a modification of the PSADE global optimization 
algorithm that replaces the original Metropolis crite-
rion of simulated annealing with a ranking based cri-
terion. By replacing the acceptance criterion we hope 
to avoid situations where the algorithm gets caught in 
the neighborhood of a local minimum. The proposed 
algorithm (DESAPR) is highly parallelizable. We tested 
the algorithm on a set of mathematical test functions 
and on a real-world circuit design problem. The results 
confirmed, that DESAPR is an efficient and reliable op-
timizer for analog circuit sizing problem. 
7 Acknowledgements
The research was co-funded by the Ministry of Educa-
tion, Science, and Sport (Ministrstvo za Šolstvo, Zna-
nost in Šport) of the Republic of Slovenia through the 
programme P2-0246 Algorithms and optimization 
methods in telecommunications.
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Arrived: 18. 12. 2015
Accepted: 13. 06. 2016
J. Olenšek et al; Informacije Midem, Vol. 46, No. 2(2016), 57 – 64