Feasible Region Approximation Using Convex Polytop es
Sachin S: Sapatnekar Pravin M: Vaidya S: M: Kang
Department of EE/CprE Department of Computer Science Co ordinated Science Lab oratory
Iowa State University University of Illinois University of Illinois
Ames, IA 50011. Urbana, IL 61801. Urbana, IL 61801.
Abstract A p olytop e can b e thought of as the convex hull of its
vertices. Hence, it is capable of representing any convex
A new technique for p olytop e approximation of the
b o dy in the limiting case where the numberofvertices
feasible region for a design is presented. This metho d
and hence, hyp erplanes is in nite. A satisfactory ap-
is computationally less exp ensive than the simplicial
proximation to the convex b o dy can b e obtained using a
approximation metho d [1]. Results on several circuits
nite number of hyp erplanes.
are presented, and it is shown that the qualityofthe
In this work, we address the problem of feasible region
p olytop e approximation is substantially b etter than
approximation by convex p olytop es. We present a new
an ellipsoidal approximation.
approach that is computationally cheap er than the exist-
I. Introduction
ing approachin[1], and illustrate how p olytop e approxi-
mation is sup erior to ellipsoidal approximation.
While designing a circuit according to certain sp eci -
cations, it is useful to de ne a feasible region as the set of
II. Approximation of the Feasible Region
parameter values for which the design sp eci cations are
satis ed. It is a commonly-made assumption [1, 2] that
A. Introduction
the feasible region is a convex b o dy; while in general, this
n
The feasible region FR , where n is the number of
is not true, it is usually true that the feasible region is
design parameters, is formally de ned as the set of p oints
\nearly convex."
in the design parameter space for which the circuit satis-
An example of a situation in which the concept of the
es all sp eci cations on its b ehavior. It is often assumed
feasible region is useful is the problem of design center-
that F is a convex body. In this work, we preserve that
ing. Realising that pro cess variations may cause param-
assumption in de ning the approachtobetaken, and use
eter values to drift from their nominal values, this prob-
prop erties of convex sets to create an approximation to
lem seeks to select the nominal values of design param-
F . In our implementation though, we showhowwemay
eters in suchaway that the b ehavior of the circuit re-
make allowances for the incorrectness of this assumption.
mains within sp eci cations with the greatest probability.
The simplicial approximation metho d [1] provides a
In other words, the aim of design centering is to ensure
metho d for approximating a feasible region by a p oly-
that the manufacturing yield is maximized. An approach
top e. Each iteration of this metho d involves the solution
that is commonly used to solve this problem [1, 2] approx-
of f + 1 linear programs where f is the numb er of faces of
imates the feasible region by a convex body, and takes the
the p olytop e , a line search, and up dating a convex hull.
design center to b e its exact or estimated center.
Apart from the fact that each linear program solution is
The pro cedure in this pap er can b e used for the approx-
exp ensive, the numb er of linear programs to b e solved in
imation of any convex or \nearly convex" b o dy.
the entire pro cedure is large sp eci cally, k f + 1 , where
Twotyp es of convex b o dies have b een used most com-
k is the numb er of iterations . Moreover, the pro cedure
monly for feasible region approximation:
of up dating the convex hull is also computationally ex-
i An el lipsoid, which is a b o dy de ned by
p ensive. This is b ecause up dating a convex hull consists
T 2 nn n
of identifying hyp erplanes to b e removed and nding the
fy j y y B y y r g; B2R ; y ; y 2 R : 1
c c c
equations of the new hyp erplanes. Obtaining the equation
Ellipsoids are inherently limited in approximating asym-
3
of a hyp erplane, given the vertices, is an O n op eration;
metric regions, and regions with linear edges.
this needs to b e carried out for each new hyp erplane of the
ii A polytope, which is de ned as an intersection of half-
p olytop e while up dating the convex hull in an iteration.
spaces, is given by
In the algorithm presented here, an initial p olytop e is
mn m
P = fx j Ax bg;A 2 R ; b 2 R : 2 up dated by p erforming a set of line searches. After up 1
to 2n line searches, a new p olytop e center has to b e com- k=0
2:5
Initialize P and z
puted, whichisanO n op eration. Thus, the algorithm
0 0
While planes added in the last iteration 6= 0 {
that we present is computationally less complex than sim-
P = P
plicial approximation.
k +1 k
Cho ose orthogonal directions d ; d d
B. Construction of the Approximating Polytope
1 2 n
For i =1ton {
Our algorithm is based on the following well-known the-
Perform a binary search along direction d
i
orem on convex sets [3] that is stated b elow.
from z to nd a b oundary p oint z of F
0 b;i
Supp orting Hyp erplane Theorem : Let C b e a con-
If z exists, and z 2P , and
b;i b;i k +1
vex set and let y b e a b oundary p ointofC . Then there
distance z ,b oundary of P > {
b;i k +1
exists a hyp erplane containing y and containing C in one
Addahyp erplane to the p olytop e
of its closed half-spaces.
P = P [ new hyp erplane
k +1 k +1
De nition : A tangent plane at a p oint z on the b ound-
0
}
ary of a region is given by
Perform a binary search along direction d
i
T T
from z to nd a b oundary p oint z of F
0 b;i
[rg z ] z =[rg z ] z 3
0 0 0
If z exists, and z 2P , and
b;i b;i k +1
distance z ,b oundary of P > {
b;i k +1
where g z 0 is an active constraint at p oint z , and
0
Addahyp erplane to the p olytop e
rg . is the gradient of the function g . .
P = P [ new hyp erplane
k +1 k +1
For a p oint z on the b oundary of a convex set, C ,a
0
}
tangent plane at z satis es the supp orting hyp erplane
0
}
prop erty. In particular, for the constraint g z 0, C is
Set z = center of the up dated p olytop e, P
0 k +1
contained in the half-space
k = k +1;
T T
}
[rg z ] z [rg z ] z : 4
0 0 0
The pro cedure requires a technique for calculating the
The aim of the algorithm is to approximate the feasible
gradient of an active constraint at a b oundary p oint. As
n
region, FR by a p olytop e
in [2], one of several metho ds may b e employed for this
purp ose. If the exact functional form of the constraintis
mn m
P = fz j Az bg;A 2 R ; b 2 R ; 5
known, an expression for the gradientmay b e derived. If
not, metho ds such as the adjoint network technique [5]or
n
formed by the intersection of m half-spaces in R .
nite di erences may b e used to calculate the gradient.
The algorithm b egins with an initial feasible p oint,
n n
III. Experimental Results
z 2F R .Ann-dimensional b ox, fz 2 R j z
0 min
z z g, containing F is chosen as the initial p oly-
i max
A. A Numerical Example
top e P . In each iteration, n orthogonal search directions,
0
Anumerical example in two dimensions is presented
d ; d d are chosen. A binary search is conducted
1 2 n
here to illustrate our technique. The feasible region is
from z to identify a b oundary p oint z of F , for each
0 b;i
represented by the ellipse
direction d .Ifz is relatively deep in the interior of P ,
i b;i
2 1
T
then the tangent plane to F at z is added to the set
b;i
z j z t z t 1 7 F =
1 2
of constraining hyp erplanes in Equation 5 . In practice,
since the feasible region is not strictly convex, it is useful
T
with the ellipse center, t,at[4; 4] . The approximating
to shift the tangent plane awayby a distance factor .In
p olytop e, P , shown in Fig. 1, can b e seen to b e a go o d
other words, if c is the gradient of the active constraint
approximation of the ellipse.
at the b oundary p oint, the hyp erplane added is
B.ATunable Active Filter
The techniques describ ed in Section I I were used to ap-
T T
c z c z 1 6
b;i
proximate the feasible region for a tunable active lter
shown in Fig. 2. The one p ole role-o mo del used for the
where is typically of the order of 0.01.
5
op erational ampli er presumes a dc gain of 2 10 , and
A similar pro cedure is carried out along the direction
a 3-dB bandwidth of 12 rad. The designable parame-
d . Once all of the hyp erplanes have b een generated, the
i
ters are considered to b e [G ;G ;C ;C ]. The transfer
1 4 1 2
center of the new p olytop e is calculated, using a metho d
function for this lter is given by
describ ed in [4]. Then z is reset to b e this center, and
0
v the ab ove pro cess is rep eated.
2
F = : 8
The pseudo-co de describing the algorithm is as follows:
v g
Table 1 : Yield for a Tunable Active Filter An initial feasible p oint [80.2 0, 5.42 m0, 0.72856 F,
0.72856 F] is provided, as in [2].
Parameter Variances Actual Yield Polytop e Yield
[1:5; 1:5; 1:5; 1:5] 46.6 43.4
[1:5; 1:5; 2:0; 0:5] 47.2 45.2
[1:5; 1:8; 1:2; 0:6] 52.8 51.2
[1:5; 1:5; 1:5; 0:5] 53.6 53.4
[1:0; 1:2; 1:0; 0:6] 69.2 68.2
[0:5; 0:8; 1:0; 0:8] 75.0 74.0
[1:0; 0:4; 1:0; 0:6] 84.2 84.4
[0:5; 0:8; 0:7; 0:6] 86.4 86.0
Fig. 1 : Approximation of an ellipsoid by a p olytop e.
Fig. 2 : A tunable active lter.
The sp eci cations for the lter are as follows:
F 0:5 at 90 Hz
F 0:5 at 92 Hz
1:0 F 1:21 at 100 Hz 9
F 0:5 at 108 Hz
F 0:5 at 110 Hz
Yield estimation for various values of parameter vari-
ances ab out the p oint [83.13 0, 5.423 m0, 0.716047 F,
Fig. 3 a : A high-pass lter.
0.769181 F] was carried out. The results in [2] for the
Fig. 3 b : Band diagram for the high pass lter.
same circuit, for yield estimation ab out the same p oint,
show that the p ercentage yield using the approximating
C. A High Pass Filter
ellipsoid as the feasible region is as muchas12points
This is an example of a high-pass lter [2], whose circuit
away from the actual yield.
diagram and sp eci cations are shown in Fig. 3. The fre-
For various sets of parameter variances expressed as a
quencies of interest are f170; 350; 440; 630; 680; 990; 1800g
p ercentage of the nominal values ab ove , Table 1 shows
Hz, which corresp ond to seven constraints. For this ex-
the yield using the approximating p olytop e as the feasible
ample, the reference frequency, ! , is 990 Hz.
0
region in comparison with the actual circuit yield. The
The parameter I is de ned as follows:
loss
yield is calculated on the basis on a Monte Carlo simu-
lation of 500 p oints. It can b e seen that the results of
V
1
I j! = 20 log dB 10 j!
loss 10
our technique are within ab out 3 p ercentage p oints of the
V
2
actual yield, which is a marked improvementover [2].
I j! = I j! I j! dB : 11
loss loss loss 0
The rst exp eriment to ok the design parameters as Table 2 : Yield for a High Pass Filter
[C ;C ;C ;C ]. The initial feasible p ointwas taken to 4 Parameters
1 3 4 5
b e [11:1nF; 12:9nF; 34:3nF; 97:3nF], as in [2]. Table 2
shows the results of yield estimation ab out the p oint
Parameter Variances Actual Yield Polytop e Yield
[10:37nF; 13:28nF; 34:63nF; 87:84nF]. The yield gures
for the approximating p olytop e are seen to b e close to
the actual yield, in contrast with [2], where they are o
[15; 15; 15; 15] 55.4 54.8
byasmany as 26 p ercentage p oints.
[15; 18; 18; 12] 57.6 54.4
[C ;C ;C ;C ;C ;L ;L ]were used as design parame-
1 2 3 4 5 1 2 [15; 10; 15; 10] 65.0 64.8
ters for the second exp eriment. The initial feasible p oint
[10; 15; 10; 15] 71.8 66.4
[8; 12; 10; 10] 79.8 79.0
was taken, as in [2], as [11.65 nF,10.47 nF,13.99 nF,39.93
[5; 15; 10; 5] 80.2 78.8
nF,99.4 nF,3.988 H,2.685 H]. The yield gures for this
[10; 10; 10; 10] 81.4 81.6
more complex circuit are also almost always within a few
[9; 9; 9; 9] 86.4 88.4
p ercentage p oints of the actual yield, as against [2], where
[10; 5; 5; 15] 92.6 89.8
the discrepancy is as much as 34 p ercentage p oints.
The interpretations of the columns of Tables 2 and 3
are the same as those in Table 1.
Table 3 : Yield for a High Pass Filter
IV. Conclusion
7 Parameters
In this pap er, a new approach to feasible region ap-
proximation has b een presented, using a pro cedure that is
Parameter Variances Actual Yield Polytop e Yield
computationally less exp ensive than the existing metho d.
Exp erimental results show that in contrast to the re-
[10; 10; 10; 10; 10; 10; 10] 29.8 32.4
sults of ellipsoidal approximation in [2], this technique
[5; 10; 5; 10; 5; 10; 5] 41.0 44.2
provides a go o d approximation to the feasible region for
[8; 8; 8; 8; 8; 8; 8] 44.0 48.2
the same example circuits, with yield estimates taken
[5; 5; 5; 5; 10; 10; 10] 45.2 47.8
ab out the same p oints.
[10; 10; 10; 10; 5; 5; 5] 48.8 50.0
The three examples in [2] are all simulated here, with
[9; 10; 8; 10; 5; 4; 6] 50.6 52.6
the following results :
[8; 8; 8; 8; 5; 5; 5] 60.6 62.4
[4; 4; 8; 8; 8; 4; 4] 71.8 74.2
a. For the tunable active lter, our results were within
[5; 5; 5; 5; 5; 5; 5] 72.6 76.6
ab out 3 p ercentage p oints of the actual yield, as
[4; 4; 4; 5; 3; 4; 4] 83.2 84.4
against a discrepancy of upto 12 p ercentage p oints
[2; 2; 2; 5; 2; 4; 4] 86.4 86.6
in [2].
[4; 2; 4; 5; 3; 2; 4] 94.1 92.8
b. For the high pass lter, with four parameters, our
technique calculates the yield within an accuracy of
References
5 p ercentage p oints, whereas [2] shows errors of up
[1] S. W. Director and G. D. Hachtel, \The simplicial approx-
to 26 p ercentage p oints.
imation approach to design centering," IEEE Trans. Cir-
cuits and Systems,vol. CAS-24, no. 7, pp. 363{372, 1977.
c. For the high pass lter, with seven design parameters,
[2] H. L. Ab del-Malek and A.-K. S. O. Hassan, \The ellip-
the yield estimates are accurate to within 4 p ercent-
soidal technique for design centering and region approx-
age p oints, as against 34 p oints in [2].
imation," IEEE Trans. Computer-Aided Design,vol. 10,
Thus, wehave illustrated that p olytop e approximation
pp. 1006{1014, Aug. 1991.
is b etter than ellipsoidal approximation since the highly
[3] D. G. Luenb erger, Linear and Nonlinear Programming.
symmetric ellipsoidal shap e is inherently incapable of ap-
Addison-Wesley, 1984.
proximating less symmetric b o dies. Also, as shown in
[4] S. S. Sapatnekar, P.M.Vaidya, and V. B. Rao, \A con-
Section I I, our metho d is computationally less exp ensive
vex programming approach to transistor sizing for CMOS
than existing approaches.
circuits," Proc. Int. Conf. on Computer-Aided Design,
pp. 482{485, Nov. 1991.
[5] L. O. Chua and P. M. Lin, Computer-AidedAnalysis of
Electronic Circuits: Algorithms and Computational Tech-
niques. Prentice-Hall, 1975.