A Fuzzy Selection Based Constraint Handling Method
for Multi-objective Optimization of Analog Cells
Bo Liua., Francisco V. Fernándezb., Peng Gaoa. and Georges Gielena.
a.
ESAT-MICAS, Katholieke Universiteit Leuven, Leuven, Belgium
e-mail: georges.gielen@esat.kuleuven.be
b.
IMSE, CSIC and University of Sevilla, Sevilla, Spain
e-mail: pacov@imse.cnm.es
Abstract—In this paper, a constraint handling technique for
multi-objective optimization of analog cells is presented.
Selection-based constraint handling and fuzzy membership
functions are combined to construct a new constraint handling
method, multi-objective fuzzy selection (MOFS). This is
integrated with multi-objective sizing, which has the following
two advantages: (1) enhances the effectiveness and efficiency of
handling specifications distinctly; (2) specializes in solving
multi-objective analog sizing problems mimicking the
designer’s interactive design process, avoiding inflexibility of
crisp constraint sizing methods.
I.
INTRODUCTION
Driven by the increasing complexity of modern analog
integrated circuits (ICs) and the ever-demanding time-tomarket pressures, analog sizing has evolved from a “pencil
and paper” approach to a “simulation and optimization”
approach [1]. Especially in recent years, Pareto-based multiobjective sizing [1]-[8] has received much attention. Analog
sizing problems often require handling multiple noncommensurate and even conflicting goals. By multiobjective sizing, the trade-offs and sensitivity analysis
between the different objectives can be explored.
In multi-objective sizing, performances can either be
set as objectives or constraints. An objective means that the
designer is interested in its trade-offs with all other
objectives in the whole performance space, which contains
much information; while a constraint usually means that the
performance needs to be “larger than” or “smaller than” a
certain value, and no trade-off information is considered.
Usually, only a part of the important performances are
considered as objectives, and the others can be set as
constraints. The reason is that the designer is often not
interested in the trade-offs among all objectives. Moreover,
there are typically other design decisions, such as restricting
the transistors to operate in the saturation region, which are
essentially defined as functional constraints.
Consequently, constraint handling technology is very
important in multi-objective sizing. Although a few multiobjective sizing approaches have appeared in literature [1][8], there are very few works that study the constraint
This work is supported by a special bilateral agreement scholarship of the
Katholieke Universiteit Leuven, Leuven, Belgium. This research was also
supported by the TIC-2532 Project, funded by Consejería de Innovación,
Ciencia y Empresa, Junta de Andalucía, Spain.
handling technology. Hence, effective and efficient
approaches need to be investigated.
Apart from the crisp constraint handling, the designer’s
intentions need to be integrated in. The designer is also
interested in solutions in the neighborhood of the often
arbitrarily chosen constraint boundaries. It is possible that
the designer would like to know if the power vs. area tradeoff curve will be significantly improved when the phase
margin is slightly lower than 60 (assuming the spec is
phase margin > 60 ). If we define the trade-off between the
objectives in the whole performance space as global tradeoff, the trade-off between the neighborhoods of the
constraint boundaries can be defined as local trade-off.
Currently, with crisp constraints, the only way to explore the
local trade-offs is by repeatedly tuning the constraint
boundaries. For example, the designer may first reduce the
60 phase spec to 58 and check the results. If satisfactory,
he / she may further decrease the constraint boundary to 55 ,
which is the lowest requirement and compare the results;
otherwise, 60 will be retained. However, more than one
spec may be involved in the local trade-offs in a real design,
and the combination of them may result in numerous tunings.
To address the above problems, a new constraint
handling method, named multi-objective fuzzy selection
(MOFS), is proposed in this paper, whose purpose is to
reflect the designer’s imprecise intentions, enabling them to
correctly control the local trade-offs in a single run.
The rest of the paper is organized as follows. Section II
elaborates the MOFS method. Section III provides some
practical experiments and comparisons to show the
effectiveness of the proposed approach, and the concluding
remarks are discussed in Section IV.
II.
A.
MOFS METHOD
Basic constraint handling mechanism
There are some constraint handling methods for multiobjective analog sizing. One of them is that constraints are
not considered in the optimization process, the Pareto-
optimal front (POF) of competing objectives is first built,
and infeasible points are discarded from the POF at the end.
This method suffers from a considerable waste of
computational effort and the density of points in the POF
may be severely degraded. Another method is setting
objectives of infeasible solutions to highly non-optimal
values to prevent it from being selected in the successive
generation [1]. This method finds many difficulties when all
the candidates are infeasible in the first few generations (a
common situation in analog sizing).
The selection-based constraint handling method
proposed by Deb [9] is very effective and robust. This
method is used as the basic constraint handling mechanism
by MOFS.
Given two candidates in the population, there may be,
at most, three situations:
(1)
Both solutions are feasible;
(2)
Both solutions are infeasible;
(3)
One solution is feasible, but the other is not.
Accordingly, the selection rules are:
(1)
Given two feasible solutions, select the one with
the better objective function value;
(2)
Given two infeasible solutions, select the solution
with the smaller constraint violation;
(3)
If one solution is feasible and the other is not, select
the feasible solution.
n
Constraint violation is calculated as
 | min( gi ( x), 0) |
i 1
for {g1 ( x)  0, g 2 ( x)  0,
, g n ( x)  0} .
B. Fuzzy numbers and construction
To avoid repeated tuning of the specification boundaries,
fuzzy sets theory [10] is a possible solution. In this paper,
fuzzy numbers (also called fuzzy sets) are integrated in the
optimization process. A fuzzy number is a set of values, each
having a membership  ( x ) that reflects the degree of
acceptability of the required value.  ( x)  1 means that the
value is fully acceptable, and  ( x)  0 means that the value
is not acceptable at all. The concept and construction of fuzzy
numbers are detailed in [10].
By using fuzzy sets, the specifications of analog sizing
are not “black and white” definitions but vague definitions,
(e.g. 60dB is preferred, but 59dB is also acceptable), which is
much closer to human intentions. A more important thing is
this technology enable the designer to control the local tradeoffs, thus can generate a sizing result capturing the designer’s
intention in one run.
degradations when the POF can effectively be improved. If
there is no improvement to the dominance or distribution,
the relaxation of the constraints is of little sense, because the
new solution is inferior to the solution that does not allow
the tolerance. Thirdly, solutions that have many
performances with  ( x)  0 but close to  ( x)  0 may
appear, but such solutions can hardly be acceptable for the
designer, as the degradations are too large.
To address the third problem, besides the lower and
upper bounds of the fuzzy number, a  -cut [10] is
introduced. A  -cut set is a clear set from the fuzzy set,
which can be expressed as A  { x |  A ( x )  } . Therefore,
there are 4 levels to judge an obtained performance, as
shown in Table 1. With this classification, the designer may
set a positive number m to require more than m
performances to reach the comparatively satisfactory value.
Table 1. Fuzzy identifications of the performances
Level of performances
Judgment
Fully satisfactory
 ( x)  1
Comparatively satisfactory
Allowed but not satisfactory
Not acceptable
 ( x)   AND  ( x)  1
 ( x)  0 AND  ( x)  
 ( x)  0
The selection rule is finally defined as follows.
Considering an analog multi-objective sizing problem with n
constraints, out of which m are required to reach the
comparatively satisfactory level,
(1) Given two solutions with min( i ( x))  0, i  1, , n ,
select the solution with the smaller constraint violation;
(2) Given one solution with min( i ( x))  0, i  1,
,n
and one solution with min( i ( x))  0, select the solution
with min( i ( x))  0 ;
(3) Given two solutions with min( i ( x))  0, i  1,
,n,
compute the number of constraints with i ( x )   (noted
m1
as
,
m2
.
Compute
mA  min(m1 , m2 )
,
mB  max(m1 , m2 ) )
(3.1) if mA
 m , select the solution with the better
dominance;
(3.2) if mB < m, select the solution with the better
n
  ( x) .
i
C. MOFS selection rules
In order to integrate human flexibility in the selection
rules, human intentions are first analyzed, which can be
summarized as follows. Firstly, the relaxation of the
constraints should be small, because the designer wishes to
explore the local trade-offs around the specifications.
Secondly, the designer only accepts some constraint
i 1
(3.3) if mA
 m  mB , select the solution with more
constraints that reach the comparatively satisfactory level.
D. Environmental selection
In multi-objective sizing, not only good convergence is the
optimization goal, but also the diversity of the points in the
POF is important. Thus, environmental selection, that
n
(
  ( x) ). Second, if the distance of an individual in the
i
i 1
obtained population by previous truncation method is less
than a threshold value, it is replaced by the individuals with
n
large
  ( x)
i
. Suppose there are N individuals with
i 1
distances below the threshold, then the N individuals with the
largest aggregated membership function values will be
selected to replace them. It can be seen that the algorithm call
for a balance between distances between non-dominated
solutions and higher acceptability of specifications.
III.
EXPERIMENTAL RESULTS AND COMPARISONS
In this section, we combine MOFS with NSGA-II (but
can be analogously combined with any other multi-objective
optimization engine) and test them with a practical analog
circuit sizing problem: the folded-cascode amplifier in Fig. 1.
The load capacitance is 5 pF, and the technology used is a
0.18  m CMOS process with 1.8V power supply. The
design variables are the transistor widths, the transistor
lengths, and the bias current. All the examples are run on a
Pentium IV PC at 3GHz with 2GB RAM and Linux
operating system. The program is implemented in MATLAB
and HSPICE is used as the performance evaluator.
VDD
Using fuzzy numbers to capture human intentions in
single objective analog circuit sizing has appeared in
literature, such as [12]. But most of them are weight-sumbased, whose drawbacks have been shown in [13]. Therefore,
weighted-sum methods are not considered here, and MOFS
and the selection-based method using crisp constraints are
compared. The population size is 200, and 20 runs with
independent different random numbers in NSGA-II
optimization are performed for each experiment.
A first experiment is performed with the high
performance constraints in the second column of Table 2
and a second experiment with the low constraints in the
fourth column. One typical solution is plotted. The power vs.
area POF are shown in Fig. 2. The high and low
specifications and the average values of the 200 individuals
in the typical solution on each spec are shown in Table 2.
Table 2. Experimental results with crisp constraints
Performances
constraints
average
constraints
DC gain (dB)
60.05
 60
 55
50.03
 50
 40
GBW (MHz)
Phase margin (º)
82.60
 60
 60
Output swing (V)
1.25
 1.2
1
Slew rate
34.00
 30
 20
(V/  s )
dm1
17.64
1
1
dm3
1.81
1
1
1
1
dm5
3.52
1
1
dm7
2.42
dm9
1.56
1
1
dm11
7.56
1
1
Time (s)
1706
average
55.05
40.02
85.72
1.23
22.86
17.58
1.67
4.24
2.23
1.79
8.28
1672
MOFS is then applied. The fuzzy membership
functions used are shown by solid lines in Fig. 3. Notice that
this is an example of fuzzy numbers used in sizing, and that
different designers can use different fuzzy numbers
according to their own preferences.
high specs
low specs
fuzzy specs m=3
fuzzy specs m=4
fuzzy specs m=5
20
18
16
sqrt(area)(um)
considers both dominance and distance between individuals,
is necessary [11]. The truncation method is the most
commonly used approach for environmental selection in
recent years [11]. A typical truncation method proceeds as
follows: (1) Collect all the candidates that compete for the
next generation and rank them on dominance. (2) If the
number of non-dominated solutions exceeds the size of the
population, then the non-dominated solutions are ranked by
distance. The solutions with ranks larger than the population
size are cut off. (3) If the number of non-dominated
solutions is less than the population size, then fill in the
available places with dominated solutions according to their
quality on dominance and distribution.
In the second case above (usually occurs in multiobjective sizing), a possible problem is that though some
points are within the population size, their distances are also
small. Thus, even when selected to the next generation, the
points have not enough contribution to the distribution. On
the other hand, there is potential to improve total membership
function values under such condition. A revision has been
made as follows. First, all the non-dominated solutions are
ranked according to the aggregated membership function
14
12
10
8
0.5
0.55
0.6
0.65
0.7
power(mw)
0.75
0.8
0.85
Fig. 2. POF curves in the MOFS experiments
m3A
m4A
mbpA
Vcm
ibb
m11A
m1A
m10A
m2A
vip
vout
vin
Vcm
m9A
m8A
m7A
m6A
cl
mbnA
m5A
VSS
Fig. 1 Folded-cascode amplifier
The specifications and the average values of the
obtained solutions in one typical run are shown in Table 3.
A diacritical tilde over a spec, n , means a fuzzy number
around n, as defined in Fig. 3. The satisfactory levels are
58dB, 46MHz, 57 , 1.1V, 26 V /  s for DC gain, GBW,
phase margin, output swing and slew rate, respectively.
Parameter m is set to 3, 4 and 5 for the above 5 fuzzy
constraints on performances, which means that at least m of
the 5 constraints should meet the satisfactory value.
(a)
(b)
0.5
0
0
0
20 40
GBW
(e)
6070
0.5
0
0 20 40 60 8090
phase margin
IV.
1
membership
membership
1
0.5
0
0.5
0
20 40 60 80
DC gain
(d)
the 200 points in the POF) is 51.21dB, 46.04MHz, 85.16 ,
1.13V, 30.21 V /  s . Compared DC gain and GBW with the
4th column in Table 3, we can find the revision of fuzzy
numbers directed the sizing.
1
membership
1
membership
membership
1
(c)
0
0.5 1 1.51.8
output swing
0.5
0
0
20
40 50
slew rate
Fig. 3. Fuzzy membership functions in the MOFS experiments
Specs
m
DC gain
(dB)
GBW
(MHz)
Phase
margin (º)
Output
swing (V)
Slew rate
(V/  s )
dm1
dm3
dm5
dm7
dm9
dm11
Time (s)
opposite idea, as shown by the dotted line in Fig. 3, he / she
can revise the fuzzy numbers easily to direct the multiobjective sizing. The comparatively satisfactory value is set
to 53dB for DC gain and 46MHz for GBW. Parameter m is
set to 4. This experiment result of a typical run (average of
Table 3. Experimental results with fuzzy constraints
constraints
Average 1
Average 2
Average 3
3
4
5
 60
57.08
58.13
58.12
 50
42.03
42.01
46.03
 60
85.43
82.45
83.51
 1.2
1.24
1.18
1.23
 30
28.30
26.25
30.05
1
1
1
1
1
1
17.51
1.63
5.38
2.53
1.37
8.72
1761
17.66
2.03
3.67
1.66
1.41
9.10
1805
17.65
1.94
3.88
2.16
1.43
8.34
1836
It can be seen from Fig. 2 that the advantage of MOFS
is obvious. For example, when m=5, all the specs reach the
comparatively satisfactory value, which are very near to the
high requirements, but the POF is much better than that of
the high requirements. When m=3, we can see that the POF
is comparable to that of using the low requirements. But at
least three out of the five specs reached the comparatively
satisfactory level. The relaxations are reliable and reflect the
designer’s intention by the fuzzy sets they constructed and
the comparatively satisfactory value as well as the m value
they set. To sum up, MOFS can automatically recognize the
useful relaxation and control the degree of them according to
the designer’s intentions, so the local trade-offs can be
explored successfully in one sizing process, and no
additional spec tuning is necessary.
In the following, we show how the designer controls
the local trade-off based on his / her intentions by revising
the fuzzy sets. In the past experiments, we assumed that the
GBW has a comparatively large acceptable region and DC
gain has a smaller one. But if another designer has the
CONCLUSIONS
In this paper, the MOFS method has been proposed for
constraint handling in multi-objective analog integrated
circuit sizing problems. Thanks to the selection-based
method and the fuzzy sets theory, MOFS can automatically
recognize the useful relaxation and control the degree of
them according to the designer’s intentions. Thus, the local
trade-off can be achieved successfully in one sizing process,
avoiding tedious manual tunings.
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