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WA3 11:20
Transversality Theorem : A Useful Tool for Establishing Generici ty
S. S. Keerthi, N. K. Sancheti and A. Dattasharma
Department of Computer Science & Automation
Indian Institute of Science, Bangalore-560 012, India
Abstract
We end this section with a list of some notations
used in the paper and an useful lemma on Lebesgue
measure. E" will denote the n-dimensional Euclidean
real space. If 2: E E" then xi =the i-th component o f t , tT =transpose of z,and diag(z) = n x n
diagonal matrix whose i-th diagonal element is zi,
i = 1,.. . , n. If { t lz2,.
, . . , z k }is a sequence of vectors in E" then ( t 'x,2 , . . . ,z k )will denote the vector,
It is shown, via a number of examples in linear algebra and control, optimization, and geometry, that
the transversality theorem of differential topology is
a useful tool for establishing genericity of a property
which is a function of a finite number of real parameters.
1
((z')'l(t?)'I.. . I(rk)T)T
E E". I,, is the n x n identity matrix and det(A) = determinant of the square
matrix, A . C k will denote the class of k-times continuously differentiable functions. If F : A + B where
A and B are open in Ek and E' respectively, and
F E C', then, for a E A, J F ( a ) will denote the Jacobian of F at a ; J F ( a ) is an 1 x k matrix. Also,
if a1 E E k t is a subvector of CY formed by choosing
some specified components of a,then Fa, ( a )will denote the l x 61 matrix of partial derivatives of F with
respect to the variables in cq evaluated at CY. In particular, JF(cu) = F,(a). I f f : E" + E , of(.) and
fza. will respectively denote the n x 1 gradient and
the n x n Hessian o f f evaluated at t . q5 will denote
the null set.
The following well-known result [ll] is useful.
Lemma 1.1. (a) A finite union of measure zero
sets has zero measure. (b) If X and Y are open sets
in some finite dimensional real spaces, 2 c X and 2
has zero measure, then X x Y has zero measure.
Introduction
Consider P ( y ) , aproperty P which is afunction
of y, a real finite-dimensional vector of parameters.
Let y take values in Y , an open set. We will say,
"property P(y) holds for a.a y E Y" ('a.a' denotin
'almost all') to mean that "the set, { y E Y : P y
does not hold} has zero (Lebesgue) measure". If P y
holds for a.a y E Y then we say that P is generic in
Y.
Hirsch and Smale [GI give a definition of genericity
which is different from ours. They call a property
as generic if Y = {y E Y : P ( y ) holds} contain an
open dense set. It is easy to give examples to show
that the two definitions are different, i.e., one does
not imply the other. If a property is_generic by our
definition then is dense because Y\Y is of measure
zero; however Y may n2t contain an open dense set.
On the other hand, if Y con_tains an open dense set,
it does not imply that Y\Y has zero measure. It
appears that our definition is more commonly used
in the literature. Both definitions are reasonable. All
references to genericity in this paper correspond to
our definition.
Suppose X , Y and 2 are open sets in some finitedimensional real spaces, 2 contains the origin, and
there exists a differentiable function F : S x Y + 2
such that, for any given y E Y , P ( y ) holds iff
F ( t , y) = 0 has (or, does not have) a solution, z E X.
In such a situation, transversality theorem [5], or,
parametrized Sard's theorem [lo] as it is otherwise
called, is a very useful tool for establishing that P is
generic in Y. T h e aim of this paper is to illustrate
this point via a nuniber of examples in linear algebra
and control, optimization, and geometry. In many of
these examples the genericity of the property involved
is intuitively obvious and yet, establishing genericity
by direct arguments is hard. In such situations the
transversality theorem is a very useful tool.
The organization of the paper is as follows. In section 2 we state a simple version of the transversality
theorem which suffices for all our examples. Examples from linear algebra and control, optimization,
and geometry are given in sections 3, 4 and 5, respectively. Section 6 contains some concluding remarks.
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I
CH3229-2/Q2/OOOO-OOB6$l.OO Q 1992 IEEE
I
2
Transversality Theorem
Consider a function F : X x Y -+ E N where
X and Y are open sets in EL and EM respectively.
Let F E C'. We say that ( z , y ) E X x Y is a regular point of F if J F ( z , y ) has full row rank. Let 0
denote the origin in E N . We say that 0 is a regular value of F if (z,y) is a regular point V ( z , y ) E
F-l(O) = {(r,y) : F ( z , y ) = 0). A weak version of
the transversality theorem of differential topology [.5],
one which is sufficient for all our needs in this paper,
is the following.
Transversality Theorem. Let F : X x Y -+
E N be as given above and 0 be a regular value of F .
Then 0 is a regular value of F ( . , y ) : X -+ EN for a.a
y E Y ;in particular, X(y) = {z E X : F ( z , y ) = 0)
is either 4 or a differentiable manifold of dimension
(t- N ) for a.a y E Y.
Remark 2.1. Let ? = {y E Y : 2 ( y ) is either
4 or a differentiable manifold of dimension ( L - N)}.
Transversality theorem shows that Y\? is a set of
measure zero. Y also enjoys an extra property. It
-
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96
achieve distinctness generically by perturbing only
two of the coefficients, say ( u ' , u j ) , and fixing all the
others, say U k = iikV1c $! { i , j } . In showing this, let us
continue reffering to the vector of coefficients formed
and the 6k as a. There are several cases
from
t o be (cons;
a i , " j d ered.
Case 1 . { i , j } = { n , n - 1 ) . Since the last two
columns of Fa are always linearly independent it follows that
is residual, i.e., it is a countable intersection of open
dense sets.
A useful corollary is the following.
Corollary 2.1. Let F be as in th_e transversality
theorem and suppose L < N . Then X ( y ) = 4 for a.a
yEY.
3
Linear Algebra and Control
Example 3.1. Let m > n , A E E m x " , . bE
E m , and consider the solution of the over determined
system, Ax = b with b acting as a parameter. Let us
set L = n , X = E", M = N = rn, Y = E", and
F ( z , b ) = A x - b . Now, J F = [AI - I"], which has
full row rank. Thus Corollary 2.1 implies that Ax = b
does not have a solution for a.a b E E". Of course
the result could have been directly obtained using the
fact that Ax = b has a solution iff b lies in the column
space of A , a manifold whose dimension is at most n .
Example 3.2. We use Corollary 2.1 t o show that
det(A) # 0 for a.a A E EnX".To set up A as a
vector, let Ai denote the i-th column of A and y =
(A1A
, ' , . . . , A " ) , a vector of length n*. We use the
fact that det(A) = 0 iff 3u E E" which satisfies Au =
0, uTu = 1 . Let us set L = n , X = E", M = n 2 ,
Y = E"', N = n . + 1 , and
has a non-distinct real root}
has measure zero in E 2 .
{ ( a i ,a j ) : p ( s , U )
Case 2. { i , j } n { n ,n - 1 ) has a singleton, k. Take
IC = n - 1 as an example. If a, # 0 then F ( 0 , a ) # 0,
the columns of Fa(s,a) corresponding to ai and a,
are linearly independent V ( s , a ) E F-'(O) and (3.1)
follows. Even if a, = 0, (3.1) follows. To show this,
we only have t o reset Y = { ( a i , a j ) : ~ " - 1 # 0},
an open set in E', apply Corollary 2.1, and note
that{(ai,aj) : a,-l = 0) is a set of measure zero
in E*. If k = n , similar arguments hold. Thus ( 3 . 1 )
holds for case 2 also.
Case3. { i , j } n ( n , n - l } = 4. If, either
#0
or ti, # 0 then F ( 0 , a ) # 0, the colums of Fa corresponding to ai and a, are linearly independent, and
(3.1) follows. If zi, =
= 0, then s = 0 is always a
non-distinct root of p ( s , a ) . In that case we can take
X = E\{O}, an open set in E , apply Corollary 2.1
and obtain that { ( a i , a j ) : p ( s , a ) has a non-distinct
real root which is non-zero} is a set of measure zero
in E'.
Example 3.4. Consider the nonlinear autonomous system described by
Let us study the Jacobian of F ( u , y ) . It is easy to
check that
Since uTu = 1 implies that at least one of the ui
is non-zero, J F ( u , y ) has full row rank V(u,y) E
F-'(O). By Corollary 2.1, for a.a A E EnX"the
system Au = 0, uTu = 1 does not have a solution,
i.e., det(A) # 0.
Example 3.3. We again use Corollary 2.1 to
show that distinctness of the real roots of a polynomial is generic. Let a = ( a l ,a ' , . . . ,a,) and consider
p ( s , U ) = S"
(3.1)
(34
x = f(x)
where f : E" + E" and f E C2. Let us consider a
perturbed system,
i. = f ( x )
+Az +b
(3.3)
where A E EnX"and b E E" denote the parameters denoting the perturbation. Given ( A ,b) consider
the system (3.3). A point, x* is said t o be an equilibrium point of (3.3 if f ( x * ) Ax* b = 0. An
equilibrium point of (3.3), x*, is said to be hyperbolic
if det( fz(x*) A ) # 0. We will use the transversality theorem and Corollary 2.1 t o show that, for a.a
( A ,b ) , each of the equilibrium points is isolated and
hyperbolic. To set up A as a vector, let A' denote the
i-th column of A and a = ( A ' , A 2 , .. . , A " ) , a vector
of length n 2 . Let Y be any open set in Rn2+". Y
denotes the set of all perturbation vectors, ( a , b ) . If
Y is a small set containing the origin, then one expects ( 3 . 2 ) and (3.3) to be close to each other. In this
scenario the following result is useful.
Theorem 3.1. For a.a ( U , b ) E Y , the following
hold: (i) each equilibrium point of (3.3) is isolated;
and (ii) each equilibrium point of (3.3) is hyperbolic.
Proof. We prove (i) using the transversality theorem. Set L = n , X = E" , M = n2 n , Y as defined
above, N = n , and F ( x , U , b ) = f ( x ) Ax b. Since
Fa = In,it follows by the transversality theorem that
+ u1sn-l + . .. + a,.
+
+
+
Given U , p ( . , a ) will have a non-distinct real root iff
p ( . , a ) and p s ( . , a )have a common real root. Let L =
1, X = E , M = n , I' = E", and N = 2. Define F by
Examining the last two columns of
+
+
we see that J F ( s , U ) always has rank two. By Corollary 2.1, { U : p ( s , a ) has a non-distinct real root} has
measure zero in E".
97
+
dimensional manifold. Note that a zero dimensional
manifold is nothing but a set of isolated points. This
proves (i).
We employ Corollary 2.1 to prove (ii). A point
z is a non-hyperbolic equilibrium point of (3.3) iff
3u E E" such that the following are satisfied: ' ( 2 )
A+ 6 = 0, (fz(z)+ A ) . = 0, uTu - 1 = 0. Let
us set L = 272, X = E2", M = n2 n, Y as in the
statement of the theorem, M = 2n 1 and
+
+
+
+
f(z)
F ( z ,U , a ,b ) =
+A t +6
Since uTu = 1, at least one component of
zero, say ui. Now
F ( b , A * , u , ) ( z , u , a , b=)
(
I"
U
is non-
a non-singular matrix. Thus J F ( z , U , a , b ) has full
row rank V ( z , u , u ,6 ) E F-'(O). By Corollary 2.1, for
a.a ( a , b ) E Y, F = 0 does not have a solution and so
(ii) is also proved.
W
4
Optimization
Homotopy is a pow-
Homotopy methods.
zero manifold associated with a homotopy method
is generically regular. Here we will briefly illustrate
the ideas for the fixed point homotopy [2, 121. Let
f : E"
E" be in C2 and consider the solution of
f(z) = 0. Introduce a homotopy variable, X E E and
a parameter, a E Y , an open set in E" to get the homotopy function, F ( z , A, a ) = Af(z) (1 - A ) y - a ) .
Given U E Y, let W a = {(.,A)
: F ( z , A , a = 0)
denote the zero mani o d of F ( . , .,a). The homotopy
method consists of choosing an a E Y and tracking
W ( a )from the known point r = a, A = 0, until the
hyperplane A = 1 is hit. The value of z when this
occurs is a solution of f (z) = 0.
An important requirement for the homotopy
method to work well is that W ( a ) is a one dimensional manifold. T h e transversality theorem is useful
here. Let L = n+ 1, X = R" x -00, l ) , M = n , Y as
mentioned earlier, and N = n. ince Fa = -(l-A)In,
J F ( z , A,a) has full row rank V(z, & a ) E X x Y .Further, W ( a ) # qi since it contains (a,O). Thus by
the transversality theorem, W ( a )is a one dimensional
manifold for a.a a E Y.
Nonlinear Programming. Consider the problem,
-+
+
I./
s
-
minimize f(z
subject t o h ( z { = 0, g ( z ) 5 0,
(4.1)
where x E E", f : E"
E , h : E"
Em, g :
E"
E', and, f , g , h E C2. Consider the following
perturbation of (4.1):
-+
-+
I'
+
+
+
z) oTz xT Bz
Illinimizeto h z) 7 = 0, g(z) + 6
subject
5 0,
r
6
T ( y ) = {z : h ( z )
0
XiI"
are perturbation parameters. Let p E E" denote
the vector containing the elements of B. Then,
y = (o,p27 , 6 ) E Rn+na+m+' denotes the vector of
perturbation parameters. Let us choose small gi > 0,
define
Y = {y : -gi < yi < gi V i } ,
(4.3)
and restrict y to the set Y so that (4.1) and (4.2) will
be "close" to each other for any y E Y. The question,
'how close are 4.1) and (4.2 ' is an issue of the stability of (4.1) w ich will not e discussed here. See [4]
for results on stability. Our main aim here is to study
the generic relationship between necessary and sufficient conditions of optimality using the transversality
theorem.
For y E Y, define
(4-2)
+ 7 = 0,
g(z)
+ 6 5 0).
(4.4)
If there exists an Z satisfying h(5) = 0, h,(Z) has full
row rank, and g ( Z ) < 0, then it is easy to see that,
if each gi is chosen to be sufficiently small (but still,
positive) then F(y) # 4 Vy E Y, so that (4.2) is not
a null problem.
Let y be fixed, z E F(y)and I ( z ) = { j : g j ( z )
6, = 0 } , the index set of active inequality constraints
a t 2. We say z is a regular point of F(y) if the gradient vectors of hi, g , , 1 5 i 5 m ,j E I(z) at z
are linearly independent. F ( y ) is said to be regular
if every z E T ( y ) is a regular point. Define the Lagrangian function, L as
+
+
+
+
+
L( I , A , p ;y) = f (z) aTz zT Bz AT h ( z ) p T g (z).
The following theorems condense the well-known necessary and sufficient conditions of optimality, due to
Karush, Kuhn and Tucker [7].
KKT Theorem (NC). Let y E Y be fixed and
suppose z is a local minimum of (4.2) as well as a
regular point of F ( y ) . Then 3A E Em and p E E'
such that the following hold : (i) L,(z,A,p;y) = 0;
(ii) p T g ( z ) = 0; (iii) p 2 0; and (iv) L , , ( z , A , p ; y )
is positive semidefinite on T ( z , the tangent space of
active constraints at I, define by
d
T ( z )= { t : h,(z)t = 0, g j , ( z ) t = 0 V j E I ( I ) } .
(4.5)
KKT Theorem (SC). Let y E Y be fixed and,
z E F(y)is such that 3A E Em, E E' and the
following conditions are satisfied : (ip Lz( z,A, p ;y) =
0; (ii) p T g ( z ) = 0; (iii) p, > 0 V j E I ( z ) ; and (iv)
,L,(?, A, p ; y) is positive definite on T ( z ) ,where T(I)
15 as in (4.5). Then z is a local minimum of (4.2).
A point z E F ( y ) which satisfies conditions (i)(iii) of KKT Theorm (NC) will be called as a K A T
point. A corresponding A and p will be referred to as
Lagrange multipliers associated with z. Following is
the main result that we will prove using the transversality theorem.
Theorem 4.1. For a.a y E Y , z E T ( y ) is a local
minimum of (4.2) iff it satisfies conditions (i)-(iv) of
KKT Theorem (NC).
The proof of Theorem 4.1 is based on the following
lemma.
L e m m a 4.2. For 8.8 y E Y the following properties hold:
J Ujln+
portant for our analysis here, T = ( L Z z u ) ~ =
euTl and e = unit vector in E" with ei = 0, i # j and
e j = 1. T is non-singular since det(T) = 2(uj)" # 0.
Thus F ( a ,y, B J , u j ) is non-singular. Hence J F has
full row rank V(z, A , U , y) E F-'(0). By Corollary 2.1
W
we have (c).
Proof of Theorem 4.1. Let Y = {y E Y : properties (a)-(c) of Lemma 4.3 hold}. By Lemma 4.2 and
p a r t l a ) of Lemma 1.1, Y\P has zero measure. Let
y E Y and consider (4.2). By part (a) of Lemma 4.2,
conditions (i -(iv) of KKT Theorem (NC) are necessary for a oca1 minimum. Let us show that they
are also sufficient. Part (b) of Lemma 4.2 and condition (iii) of KKT Theorem (NC) imply condition
(iii) of KKT Theorem (SC). Also, part (c) of Lemma
4.2 and condition (iv) of K K T Theorem (NC) imply
condition (iv of KKT Theorem (SC). Thus by KKT
Theorem (S ), conditions (i)-(iv) of KKT Theorem
(KC) are also sufficient for a local minimum.
W
Degeneracy in Linear Programming. We
will show that the absence of degenerate basic feasible solutions in linear programs is, in some sense,
generic. Let A E Emxnhave full row rank and consider
F ( b ) = { z E E" :
= b,z 2 0},
the feasible set of a linear program in standard form
looked a t as a function of b E E". A point t is said
to be a basic feasible solution (BFS) of F(b) if : (i)
z E F ( b ) ;(ii) at least ( n - rn) components of z are
zero; and 111) the columns of A corresponding to the
non-zero e ements of t are linearly independent. A
BFS, z is said to be degenerate if more than ( n - m )
of its components are zero.
Theorem 4.3. F(b) does not have any degenerate BFS for a.a b E E m .
Proof. Let I c { 1,. . . ,n } be any index set of
cardinality ( m- 1) such that the columns of A correspnd-g to the indices in I are linearly independent.
be the matrix whose columns are
Let A e
the columns of A corresponding to the indices in I .
=b
By the result of Example 3.1 it follows that
does not have a solution for a.a b E E". By part (a)
of Lemma 1.1 and the fact that the number of index
w
sets of the type I is finite, the theorem follows.
if 3: is a KKT point of (4.2) and, A, p are
Lagrange multipliers associated with z, then
X i # 0 V 1 5 i 5 rn, and p j # 0 V j E I ( z ) ;
and,
-
for every KKT point, 2: and any Lagrange multipliers X and p associated with it, Lzz(z, X,p; y)
is non-singular.
4.
Proof. Let us say that E , z E F ( y belong to the
same active set type if { j : g 3 ( x )+ = 0) = { j :
gj.(z) + 6, = 0}, i.e., the index set of active inequalities is the same for 2: and z. Since 1, the number of
inequalities is finite, t,he number of active set types
is also finite. Because of this and the fact that the
kind of perturbations for equalities and inequalities
is similar (i.e., each equality or inequality has a separate perturbation variable yi or 6, associated with
it) it is sufficient if we prove the lemma for the problem with equalities only. (Note that the arguments
for the 'equalities only' c s e can be repeated for each
active set type and the resulting conclusions can be
combined using Lemma 1.1.) Thus, for the rest of
the proof given below, we assume no inequalities are
present and take y = (a,
0,y).
(a) Let L = R , X = E", AP = n + n 2 + + , Y as in
the statement of the lemma, i.e., as in (4.3), N = rn,
and define F by F ( z , y ) = h ( z ) y. Since Fy = I,,
J F always has full row rank. By the transversality
theorem F ( y ) is regular for a.a y E Y .
(b) Take any i such that 1 5 i 5 in, and look at
the possibility of X i becoming zero. Let L = n m ,
X = E"+", M and Y as in the proof of part (a) given
above, N = n + m + 1 , and define F by
I
2
+
5..
+
LZ = f:
F(z1 A , Y) =
+ cr + ( B + BT)z + h:X
h ( z )+ Y
A*
Now, F(a,y,A,)
is an upper triangular matrix with all
diagonal elemnts equal to 1. Thus J F always has full
row rank. By Corollary 2.1, A, # 0 for a.a y E Y .
Combining this result for all i and using part (a) of
Lemma 1 . 1 we obtain part (b) of Lemma 4.2.
(c) Here we use the fact that L,, is singular iff
3u E E" such that Lz,u = 0, u T u - 1 = 0. Let
L = n+rn+n, X =
M and Y as mentioned
earlier, N = n rn n + 1, and define F by
+ +
Geometry
5
In the field of Computational Geometry, it is a
common practice to make generic assumptions which
help avoid troublesome situations that require special
case by case analysis. In the literature these assumptions ate usually called as general position assumptions. In most papers the genericity of such assumptions is never established because it is intuitively obvious. In this section we take up three general position
assumptions and establish their genericity using the
transversality theorem.
Example 5.1. In the construction of the Voronoi
diagram of points in E" [9, 31 the following
general position assumption is usually made :
{J?, . . . , z k }(]E 2 n 2), a given set of distinct data
points in En have the property that there does not
exist a sphere which contains ( n + 2 or more of the
data points on it. We will use Corol ary 2.1 to show
Consider any ( z , X , u , y ) such that F ( z , X , u , y ) = 0.
Since u T u - 1 = 0, at least one component of U is
non-zero, say uj # 0. Let BJ =the j-tli column of
B , which is a part of p, which in turn is a part of y.
At (I,A, U , y), the sub-Jacobian, F(a,y,BJ,uJ)
has the
following structure:
+
I
99
earlier, N = (n
z E E , i = 1 , . . . , I C . It is sufficient if we demonstrate the above for IC = n + 2 only because: the
number of subsets of {z', . . . ,z
'
} with cardinality
equal to ( n 2) is finite; the ideas for {z', .. . , z " + ~ }
can be repeated for each of these subsets; and, the
genericity result for each of the subsets can be combined using Lemma 1.1 to yield the genericity result
for {z', ...,2
'
)
.
Let us consider the existence of a sphere, S =
{ y : (z - c ) ~ ( z- c) = r 2 ) which contains the
z', i = 1 ,... , n + 2 , on it. Let L = n + 1, X = E"+',
M = n(n 2), y = (z', . . .,z"+'), Y = { y : y =
(z',. . . ,z " + ~and
) the zi are distinct}, and N = n+2.
Note that Y is open in E M . Define F as follows.
For i = l , . .. , n 2, define the i-th component of
F , Fi as & ( c , r , y ) = [(z' - c ) ~ ( z-~c) - r2]/2. Let
(c,r,y) E F-'(O). .Since the z' are distinct, r > 0.
This means that I' - c # 0 V i . F y ( c ,r, y) is a block
diagonal matrix with (2' - c ) ~ i, = 1, .. .,n 2, as
the diagonal blocks, and so it has full row rank. Our
result then follows by Corollary 2.1.
Example 5.2. Consider ( n + 2 ) surfaces, {Ci} in
E" represented by
+ 2)(n + 2) and F defined by
+
Consider A', the Jacobian off' with respect to wi=
(q*,p',z'). A' has the following structure:
+
Ai=
+
Ci = {Z E E" : hi(.) = 0 )
+
+
.T
z
+ q'..
Let y = (p', q' , p 2 ,q 2 , . . . ,P:+~,q"+.2) E E("+')("+'),
the vector of all perturbation variables, belong to
some open set, Y.If Y is some small open set containing the origin then for each i, the perturbed surface
ei defined by
Ci =
{X
E E"
: h i ( Z , p ' , q ' ) = 0)
+
will be "close" to Ci for any given y E Y.
Let S = {z : (z - c ) ~ ( z- c) - r2 = 0) denote a sphere which touches the surfaces C1, . . . , Cn+2
at ? I , . . . ,z"+2, respectively. s touches Z;i at zi iff
3a' E E such that
-
f i ( Z i , ai, e, ? - , p i q
, i)
-
= 0,
where
f'=
(
(.'-cy
+
+
+p'
"
x
+
where hi : E" + E and hi E C2,i = 1 , . . . , n 2. We
will show that generically there will not exist a sphere
with positive radius which will touch all the (n 2)
surfaces. This property is useful in the construction
of generalized Voronoi diagrams of nonlinear objects.
To establish this property we consider the following
perturbation of hi for each i = 1 , . . . , n 2:
hi(.)
1
0 -1,
0
0
where the x denotes matrix functions which are
unimportant. Since r > 0, z' - c is a non-zero vector.
Thus A' has full row rank V(z, y) E F - ' ( 0 ) . Furthermore, if we set w = (wl,.
.. ,w"+~),then A, the Jacobian of F with respect to w , is a block diagonal matrix with A', . ..An+2 as the diagonal blocks. Because
each A' has full row rank, A also has full row rank.
Thus, JF(z,y) has full row rank V(z,y) E F-'(0).
By Corollary 2.1 F ( z , y) = 0 does not have a solution
for a.a y E Y and our result that for a.a y E Y there
does not exist a sphere with positive radius which
i = 1,: . . ,n 2, follows.
touches each of 3,
If r = 0 is allowed then it corresponds to the condition that all the
have a common point. By a slight
change of the earlier arguments it can be shown that
generically this also cannot occur. In fact it can be
shown that, generically even (n 1) surfaces will not
intersect.
Example 5.3. We consider a general position assumption used in robot motion planning [$]. Suppose P', . . . , P' are k non-intersecting convex polygons in E' representing obstacles, and Po be a convex polygon which denotes the moving object. For
i = O , l , . . . , I C , let: ni = the number of vertices of P';
and Pi be represented by {vi$' E E2 : j = 1,.. ., n i } ,
the set of vertices-of Pi given in clockwise order. A
convex polygon, Po is said to be a homolhelic copy
of Po if 32 E E' such that
= Po { z ! , where
denotes the Cartheodory-Minkowski set sum. Two
convex polygons are said to be touching if they intersect but their interiors do not.
Leven and Sharir [SI have given an algorithm for
moving P o from any initial position t o any final POsition by translation without colliding with any of
the obstacle polygons. To simplify the description of
their algorithm they make the following general position assumption: there does not exist a homothetic
copy of Po which touches more than two members
of {P', . . . , P'}. Here we will show that the above
assumption is generic.
Let us first give a precise statement of our re, E, M =
sult. Let y' = ( v i J , . ..,vi#"'), i = o , ~...,
2 E;"=,
n i , y = (yo, y', . . . , y'), and
+
-h i ( z , P '., q '.) =
(
hi
Po
+
I' = {y E E M : P'is a convex polygon with ni
vertices V i = 0 . 1 . . . . k. and
and vX,
hi is the gradient of hi with respect to 2'. Let
L = (n + l)(n + 3), z = (z', a', . . . , t"+', an+2,c,r),
.
100
“Finding Zeroes of Maps: Homotopy Methods that are Constructive with Probability one”,
Math. Comput., vol. 32, pp. 887-899, 1978.
this note the following: the condition, ‘ P is a convex
polygon with ni vertices’ corresponds to the requirement that the vertices of P’ are distinct and that, for
each three consecutive vertices of P’, say, { a , 6 , c } ,
b lies strictly to the left of the directed line from a
to c ; the condition, P’, n P j = 4 corresponds to the
: a E
requirement that d ( P ’ , P J )= min {[laP’, P.E P i } > 0; and, each of these requirements can
be written as a strict inequality of the form p ( y ) > 0
or p ( y ) # 0 where p is a continuous function of y. Our
main result here is that, for a.a y E Y there does not
exist a homothetic copy of Po which touches more
than two members of { P’ , . . . , Pk}.
Let PO = Po + { z } be a homothetic copy of Po.
If PO touches some Pi, i # 0, it will involve a vertex of one of {PolP i } lying on an edge (line segment
joining two consecutive vertices) of the other. Let us
consider the occurence of three such touches. There
are serveral ways in which the three touches can occur. We will take one difficult case and demonstrate
below, genericity of the non-occurence of it. Repeating the ideas for each of the cases, which are finite in
number, and using Lemma 1.1 then yields the overall
result that we wish to prove.
The case that we consider is the following. Let
{ a ,b, c} be three consecutive vertices of Po such that:
a vertex, d of Pi lies on the edge joining a + z , b t ;
a vertex, e of P* lies on the edge joining b z , c z ;
and, b z lies on the edge joining two consecutive
vertices, f and g of P 3 . Note that ( a , b , c , d, e , f , g )
is a subvector of y, the vector of data for the motion
planning problem. Introduce three real variables a,
P and y,set L = 5, x = ( z , a , P , y J , A‘ = E5,N = 6,
define F by F ( z , y ) = ( d - Z - a ( b - Z ) , e - b - P ( Z b ) , b - f - y ( g - f ), where (ZlxlZ) = a+t,b+z,c+z),
and note that I , for any given y, t e three touches
occur then F ( x , y ) = 0 has a solution, I. Thus if we
show that F ( x , y ) = 0 does not have a solution for
a.a y E Y , then the case we consider (i.e., 0 5 a 5
1 , 0 5 /? 5 1, 0 5 y
1) also cannot occur for
is an upper
a.a y E Y. It is easy to see that F(d,e,b)
triangular matrix all of whose diagonal elements are
unity. T h u s J F has full row rank and, by Corollary
2.1, F ( x , y ) = 0 h a s no solution for a.a y E Y .
-
[3] H. Edelsbrunner, Algorithms in Combinatorial
Geometry, Springer Verlag, 1987.
[4] A. V. Fiacco, Introduction t o Sensitivity and Stability Analysis in Non-linear Programming, Academic Press, New York, 1983.
[5] V. Guillemin and A. Pollack, Diflerential Topology, Prentice Hall Inc., New Jersey, 1974.
-
+
+
[GI M . W. Hirsch and S. Smale, Diflerential Equations, Dynamical Systems and Linear Algebra,
Academic Press, New York, 1974.
[7] D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed. , Addison-Wesley, Reading,
Mass, 1984.
[8] D. Leven and M. Shark, “Planning a Purely
Translational Motion for a Convex Object
in Two Dimensional Space Using Generalized
Voronoi Diagrams”, Discrete and Computational Geometry, vol. 2, pp 9-31, 1987.
[9] F. P. Preparata and M. I. Shamos, Computational Geomet y - A n Introduction, Springer Verlag, New York, 1985.
+
+
[lo] S. L. Richter and R. A. De Carlo, “Continution Methods: Theory and Applications”, IEEE
Trans. Auto. Control, vol. AC-28, pp 6GO-665,
1983.
-
--
[ll] H . L. Royden, Real Analysis, The McMillan Co.,
New York, 1963.
6
t)
[12] L. T. Watson, A globally convergent algorithm
for com.puting fixed points of C2 maps, Appl.
Math. Comput., Vol. 5, 1979, pp 297-311.
<
6
Conclusion
We have demonstrated, by way of examples
in linear algebra and control, optimization, and geometry, that the transversality theorem of differential topology is a very useful tool for establishing the
genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. Some of
the results derived in this paper (e.g., Example 5.2)
are non-trivial to establish by direct means, wit,hout
resorting to the use of the transversality theorem.
References
[l] R. A. Abraham and J . Robbin, Transversal Mnppings and Flows, Benjamin, New York ,1976.
101