Constrained Optimisation and Morse Theory 1 ... - Semantic Scholar

Constrained Optimisation and Morse Theory David E. Stewart April 29, 1998 Abstract

In classical Morse theory the number and type (index) of critical points of a smooth function on a manifold are related to topological invariants of that manifold through the Morse inequalities. There the index of a critical point is the number of negative eigenvalues that the Hessian matrix has on that tangent plane. Here de nitions of \critical point" and \index" are given that are suitable for functions on fx 2 M j gi (x)  0; i = 1; : : : ; mg. These de nitions are based on Kuhn{Tucker theory of constrained optimization. The Morse inequalities are proven for this new situation. These results may be extended to smooth manifolds with corners.

1 Introduction This report contains a proof that the Morse inequalities relating the index of critical points of a smooth non-degenerate function f : X ! R to the Betti numbers of X hold for X being a \cornered manifold", provided \critical point" and \index" are suitably re-de ned. This paper was submitted for publication in 1990; however, the result was already known to Guddat, Jongen and Rueckmann [6] (1986). The proof presented here diers considerably from the proof of Guddat et al.; Guddat et al. use a direct argument following the original approach of M. Morse. The proof presented here uses a logarithmic barrier function to approximate the constrained problem. Morse theory ([7]) provides a way of relating critical points of a smooth realvalued function on a manifold to certain topological invariants of that manifold, in particular, the Betti numbers. These Betti numbers can be computed from the homology groups Hn (M ; K ) where K is a eld. The lth Betti number is just the dimension of Hn (M ; K ) considered as a vector space over K . Often I

would like to thank Dr. Norman Dancer of the University of New South Wales, Sydney, Australia and Dr. Bevan Thompson of the Department of Mathematics, University of Queensland, Australia for their helpful suggestions and comments.

1

K = Q, R or C (which all give the same Betti numbers) or a nite eld such as Zp , p prime, is used.

These manifolds are almost always assumed to be without boundary. Sometimes, particular manifolds with boundary can be considered directly by the theory (the so{called \level sets" of the smooth function considered), general boundaries cannot be well accommodated. Somewhat more dicult to accommodate are \cornered" manifolds which are manifolds with a dierential structure, but where each point in the manifold is dieomorphic with an open set of the \octant" Rn+ . (Note that the distinction between \manifolds with boundary" and \cornered manifolds" is lost if \dieomorphic" is replaced by \homeomorphic".) While classical Morse theory ts well with the classical theory of unconstrained optimisation, it does not t well with the theory of constrained optimisation, just as it does not deal well with cornered manifolds. The aim of this paper is to extend classical Morse theory by using the concepts of constrained optimisation theory. The reader may nd it of interest that the following theory was developed by the author while investigating a \homotopy" method for solving constrained optimisation problems. The approach taken in this paper is derived from the following optimisation problem: min f (x) x

subject to gi (x)  0

for i = 1; : : : ; m.

Here f and gi are C 2 real valued functions on Rn . The approach works just as well if we consider x to lie on an n{dimensional manifold M as on Rn . The feasible set is just the set

X = f x 2 M j gi (x)  0 for i = 1; : : : ; m. g

(1)

We aim to show that with the appropriate de nition of \index" for suitable \critical points" that the Morse inequalities hold for the Betti numbers of X . In x2 we propose to replace the notion of \critical points" with \Kuhn{ Tucker points", and in x3, the corresponding de nition of \index" for a Kuhn{ Tucker point is given. In x3 the main result is proven, which is that the Morse inequalities are proven for this new \index". In x4, the ideas developed in this paper are carried over to manifolds with corners. In x5, the non-degeneracy requirement of x2 is shown to be true for a dense subset of functions in C 2 (X ). Finally, in x6 a simple example is given illustrating the use of the results.

2

2 Kuhn{Tucker points vs. Critical points Instead of looking for critical points of f we should look for Kuhn{Tucker points; that is, for x in Rn (resp. M ) for which there is a  2 Rm where

rf (x) ? Pmi i rgi (x) = 0 gi (x); i  0 =0

for i = 1; : : : ; m. for i = 1; : : : ; m.

i gi (x) = 0

(2)

I call x a KT point and (x; ) a KT pair. (These are sometimes referred to as Karush{Kuhn{Tucker points/pairs, honouring a more obscure contribution to the theory by Karush in 1939. See, e.g., [4, p. 190].) Note that it is often more convenient P to describe these conditions in terms of the Lagrangian L(x; ) = f (x) ? mi=0 i gi (x). A KT pair then satis es rxL(x; ) = 0. Now KT pairs do not always exist, even if x is a local minimum; this can occur if the gradients of the \active constraints" gi are linearly dependent. So some sort of condition is assumed to guarantee the existence of  given a local optimum x: this is usually referred to as a constraint quali cation in the optimisation literature ([4, p. 191]), ([3, pp. 202{204]). For notational convenience, we may refer to the active set at a point x 2 X : I (x) = f i j gi (x) = 0 g. The constraint quali cation which is assumed throughout is that

frgi (x) j i 2 I (x)g is linearly independent for all x 2 X: (3) Linear independence is said to hold trivially if I (x) = ;. The Morse condition that the Hessian r f (x) be non-singular is replaced 2

by the following non-degeneracy conditions: P  r2 g (x) de nes a non-singular quadratic form on the 1. r2 f (x) ? m i i=1 i \tangent plane"

T (x) = f u 2 Rn j rgi (x)T u = 0 i 2 I (x) g: 2. for each i either i > 0 or gi (x) > 0. Equivalently to (1), one can require that there is a matrix Z , whose columns are linearly independent and span T (x), such that Z T r2x L(x; )Z is non-singular. (Note that Z can be computed by techniques of numerical linear algebra.) Item (2) is called a strict complementarity condition. If this condition is satis ed then x is a non-degenerate KT point and (x; ) a non-degenerate KT pair. Analogously to the assumption that f is a Morse function in Morse theory, we assume that every KT point for f is a non-degenerate KT point. Such a function we call a non-degenerate KT-function. These are, in fact, dense in C 2 (X ), which we will prove in x6. 3

3 De nition of index and the theorem The index of a non-degenerate KT point x is the number of negative eigenvalues of Z T r2x L(x; )Z . This index is clearly independent of the choice of Z . Note that if the feasible set X is compact, then there can only be nitely many nondegenerate KT points. Then we can count the number of KT points of each index: Let l (f; g) be the number of KT points for f , gi with index l, and let l (X ) be the lth Betti number of X for a given eld K . Most of the remainder of this paper will prove the usual Morse inequalities for this situation: Theorem 1 If f , gi 2 C 2 (X ), X given by (1) where M is a smooth manifold of dimension n, and the constraint quali cation (3) holds, and every KT point is non-degenerate, then k X

k X

l=0

l=0

(?1)k?l l (f; g) 

(?1)k?l l (X )

for k = 0; : : : ; n and equality holds above for k = n. Consequently l (f; g)  l (X ) for l = 1; : : : ; n. A proof of this result for classical Morse theory can be found in [7]. Remark 1 To simplify the presentation of the proof, we use the following terminology and notation. The \index" of a symmetric matrix is just the number of negative eigenvalues it has. Also, A  B for symmetric matrices is understood in the following sense: uT Au  uT Bu for all u; a matrix function A(t) ! 1 as t ! a if for any symmetric matrix B , A(t)  B for any t suciently close to a. Proof. The proof is made by suitable \approximating" the constrained optimization problem by an unconstrained problem. Note that by the constraint quali cation (3), X is the closure of fx j gi (x) > 0; i = 1; : : : ; mg. Thus we consider the penalty function

P (x; r) = f (x) ? r

m X i=1

ln(gi (x));

r > 0:

(4)

The standard Morse inequalities will hold for P (
; r) where the Betti numbers are l (X ()) where X () = fx j P (x; r)  g,  > 0 suciently large, rather than for l (X ). A retraction argument will later show that l (X ) = l (X ()) if  > 0 is chosen suciently large. We now need to investigate the relationship between critical points of P (
; r) for small r > 0, and KT points. In particular we will see that there is a one{ to{one correspondence between critical points of P (
; r) and KT points which preserves indexes for suciently small r > 0. 4

Note that a critical point of P (
; r) is a solution of

rx P (x; r) = rf (x) ? r

m rg (x) X i 2 i=1 gi (x)

= 0:

Put ~i = r=gi (x) for any critical point x of P (
; r). Then (x; ~) satis es rf (x) ? Pmi=1 ~i rgi (x) = 0 ~i gi (x) = r i = 1; : : : ; m ~i ; gi (x) > 0:

(5)

Let (xk ; ~k ) be a critical point for P (
; rk ) where rk # 0. As X is compact, there is a convergent subsequence (xk j k 2 K) with limit x^. We now show that x^ is KT point. To do this we need a suitable ^. Now for each k rf (xk ) ? Pmi=1 (~k )i rgi (xk ) = 0 (6) (~k )i gi (xk ) = r: K 0. However, [rg (^x) j K g (^x) > 0 so that (~ ) ! If i 2= I (^x), then gi (xk ) ! i i ki i 2 I (^x)] has full rank by (3), so the Moore{Penrose pseudo-inverse (see [5, pp. 138{140]) of [rgi (xk ) j i 2 I (^x)] approaches the Moore{Penrose pseudo-inverse of [rgi (^x) j i 2 I (^x)] as k ! 1 in K. Since

1 0 X (~k )i rgi (xk )A ; [(~k )i j i 2 I (^x)] = [rgi (xk ) j i 2 I (^x)] @rf (xk ) ? +

i=2I (^x)

K K 1 we nd that ~ ! taking limits k ! k ^ where [^i j i 2 I (^x)] = [rgi (^x) j i 2 I (^x)]+ rf (^x):

By continuity

rf (^x) ? Pmi ^i rgi (^x) = 0 =1

^i gi (x) = 0 ^i ; gi (^x)  0

i = 1; : : : ; m

and so (^x; ^) is a KT pair, and x^ a KT point. Conversely, if x^ is a non-degenerate KT point, we wish to show that for suciently small r > 0 there is a unique nearby critical point of P (
; r). To do this an implicit function argument is applied to the equations in (5). The inequalities in (5) hold for small r > 0 by the strict complementarity of nondegenerate KT points. 5

For the implicit function theorem to be applicable, we need the Jacobian of the equations

rf (x) ? Pmi i rgi (x) = 0 =1

i gi (x) = 0

i = 1; : : : ; m

with respect to (x; ) to be non-singular at (^x; ^). This is shown in Poore and Tiahrt ([8, Thm. 2.1]). Thus we can write solutions to (5) as unique smooth functions (x; ~) = (x(r); ~(r)) for suciently small r > 0, and (x(0); ~(0)) = (^x; ^). Thus there is a one{to-one correspondence between critical points of P (
; r) and KT points of f for suciently small r > 0. We will now see that this correspondence preserves the the index. To compute indices we note that the Hessian of P (
; r) at x is T rx P (x; r) = r f (x) ? Pmi ~i r gi (x) + Pmi ~i rgi (gx)(rx)gi (x) i T P r g ( x ) r g ( x ) i i m ~ ~ : = r L(x; ) +  2

2

2

x

=1

2

i=1 i

=1

2

gi (x)2

Taking (x; ~) = (x(r); ~(r)), we will show that for all suciently small r > 0 the index of r2x P (x(r); r) is equal to the index of Z T r2x L(^x; ^)Z where Z is any matrix whose columns form a basis for T (^x). To do this we nd smooth matrix functions Z (x), Q(x) where the columns of Z (x) span T~(x) = fu j rgi (x)T u = 0; i 2 I (^x)g and [Z (x); Q(x)] is an orthogonal matrix for all x near x^. This can be done by QR factoring [rgi (x) j i 2 I (^x)] (see, e.g., [5, pp. 146{153]). Note that the index of r2x P (x; r) is the same as the index of [Z (x); Q(x)]T r2x P (x; r)[Z (x); Q(x)]. Now note that

 T P Q(x)T r2xP (x; r)Q(x) = Q(x)T r2x L(x; ~) + i=2I (^x) ~i rgi (gx)(rx)g2i (x) Q(x) + T i P r g ( x ) r g ( x ) i i + Q(x)T i2I (^x) ~i g (x)2 Q(x): i

If (x; ~) = (x(r); ~(r)) then the part in square brackets is bounded as r # 0, but as ~i (r) ! ^i > 0 and gi (x(r)) ! 0 as r # 0 for i 2 I (^x), it follows that

Q(x(r))T r2x P (x(r); r)Q(x(r)) ! +1

as r # 0.

However, Z (x(r))T r2x P (x(r); r)Z (x(r)) and Z (x(r))T r2x P (x(r); r)Q(x(r)) both have well-de ned limits as r # 0 since Z (x)T rgi (x) = 0 for all i 2 I (^x). Then we need a lemma to show that for small r > 0 the negative eigenvalues of r2x P (x(r); r) correspond to the negative eigenvalues of Z (x(r))T r2x P (x(r); r)Z (x(r)). This provided below. 6





Lemma 1 If  is a negative eigenvalue of BAT BC where A = AT and C = C T  0, then  is an eigenvalue of A ? B (C ? I )? B T . 1

Proof. Let

 A B  u   u   u  6 0;  < 0: BT C v = v ; v =

Then

Au + Bv = u and B T u + Cv = v: Thus (C ? I )v = ?B T u. Note that u = 0 then implies v = 0 since C ? I is positive de nite. Thus u 6= 0. Substituting for v in the rst equation gives the desired result. 2 Treating [Z (x(r)); Q(x(r))]T r2x P (x(r); r)[Z (x(r)); Q(x(r))] as a block matrix, of the above form, A(r) = Z (x(r))T r2x P (x(r); r)Z (x(r)), B (r) = Z (x(r))T r2x P (x(r); r)Q(x(r)), and C (r) = Q(x(r))T r2x P (x(r); r)Q(x(r)) ! +1 as r # 0. Hence for any negative , A(r) ? B (r)(C (r) ? I )?1 B (r)T ! A(0+ )

as r # 0:

Now A(0+ ) = limr#0 Z (x(r))T r2x P (x(r); r)Z (x(r)) = Z (^x)T r2x L(^x; ^)Z (^x), so by non-degeneracy of (^x; ^), the index of Z (x(r))T r2x P (x(r); r)Z (x(r)) for suf ciently small r > 0 is equal to the index of Z (^x)T r2x L(^x; ^)Z (^x) which is also the index of the KT point x^, as desired. 2 We now show that for suciently large   0, the set X () = f x j P (x; r)   g is a deformation retract of X . Then we have Hn (X (); K ) ' Hn (X ; K ) for any eld K , and the Morse inequalities hold for i (X ). As noted by E.N. Dancer ([2]), it suces to show that there is a smooth vector eld on X that points inwards on @X . Given this vector eld a deformation retraction H : X  [0; 1] ! X can be constructed as follows for suciently large . Let be the vector eld that points inward, and t be the ow that it de nes for t  0. Then if gi (x) = 0 as the ow is inward pointing, (d=dt)gi (t (x)) > 0 at t = 0. By compactness of X and @X , there is a constant a > 0 and an  > 0, such that for t 2 [0; ), (d=dt)gi (t (x)) > a for all x ina an open set containing @X . Then for suciently large  > 0, the trajectory t (x) must meet X (), and meet it transversally. That is, (t (x))T rx P (x; r) 6= 0 when P (t (x); r) = . (In fact, the inner product is negative.) Let (x) be the rst point of contact. As X () is met transversally,  is a smooth function on X nX (), as is the time t (x) at which rst contact is made. For x 2 X (), (x) = x and t (x) = 0, 7

and both  and t are continuous functions. The deformation retraction can then be de ned to be H (x; s) = s t (x) (x): If x 2 X (), H (x; s) = x for all s. Also, H (x; 0) = x, and H (x; 1) = (x) 2 X () for all x 2 X . Thus H is the desired deformation retraction. The smooth inwardly pointing vector eld referred to above can be constructed with the help of a smooth partition of unity of X . Now, for each x 2 X the matrix [rgi (x) j i 2 I (x)] has full rank, and so [rgi (y) j i 2 I (x)] has full rank and a bounded Moore{Penrose pseudo-inverse for all y in a neighbourhood Ux of x. Now let e(x) = [1; : : : ; 1]T 2 RI (x). Then put x (y) = ([rgi (y) j i 2 I (x)]T )+ e(x). (Put x (y) = 0 if I (x) = ;.) For all i 2 I (x), we have rgi (x)T x (x) = 1. Let Vx  Ux be a neighbourhood of x such that for all i 2 I (x), rgi (y)T x (y) > 0 and I (y)  I (x) for all y 2 Vx . As X is compact there is a nite sub-covering fVx1 ; : : : ; VxN g of f Vx j x 2 @X g. Let i : P X ! R be a smooth partition of unity subordinate to fVx1 ; : : : ; VxN g. That is, Ni=1 i (x) = 1 for all x 2 X , i (x)  0 for all i and x and i 2 C 1 (X ) and the support of i lies entirely within Vxi . Then we can de ne the vector eld by (y) =

N X i=1

i (y) xi (y):

This is clearly smooth. To verify that it points inwards, note that for all i 2 I (x), x 2 V xj , rgi (x)T (x) = PNk=1 k (x)rgi (x)T xk (x)  j (x)rgi (x)T xj (x) > 0: Thus we have a smooth, inwardly pointing vector eld, which permits the construction of a deformation retraction between X and X () as required.

Remark 2 As a side-note, if a manifold is de ned by a system of equations gi (x) = 0 for i = 1; : : : ; m where frgi (x)ji = 1; : : : ; mg is linearly independent for all x, then the Morse index point x is just the number of negative P at ra critical eigenvalues of r f (x) ? m g ( x ) on the tangent space to the manifold i i i 2

=1

2

where  is the vector of Lagrange multipliers. This is shown by K.K. Yun ([9]). Formally, this sits very well with the above result.

4 Extension to cornered manifolds Now suppose that M is a cornered manifold. This means that there is a nite atlas of charts: i : Ui ! Rn+ for i = 1; : : : ; N where Ui is an open set in S M , M = i Ui and i is continuous and has a continuous inverse on i (Ui ). Furthermore, these charts i are compatible as dierentiable functions. That 8

is, if x 2 Ui \ Uj and u = i (x), then j  ?i 1 is smooth (say C 1 ) mapping of i (Ui \ Uj ) to j (Ui \ Uj ). For each chart i : Ui ! Rn+ we assume that for each co-ordinate in Rn+ either (i (x))j = 0 for some x 2 Ui , or inf x2Ui (i (x))j > 0. This can be ensured by adding one to every co-ordinate (i (
))j where this condition is violated. Then the constraint functions are just the (i (
))j functions. We say x 2 Ui  M is a KT point for f 2 C 2 (M ) if i?1 (x) is a KT point for f  ?i 1 with constraint functions gj (y) = yj . Equivalently, x 2 Ui is a KT point for f if for some  2 Rn

rf (x) ? Pnj j r(i (x))j = 0 =1

j (i (x))j = 0 j  0:

j = 1; : : : ; n

The question naturally arises if being a KT point changes with dierent charts. Suppose x 2 Ui \ Uk . Then = i  ?k 1 : k (Ui \ Uk ) ! i (Ui \ Uk ) is a dieomorphism. By permuting co-ordinates we can ensure that fj j i (x)j = 0g = fj j k (x)j = 0g = f1; : : : ; lg. Let (k) be the Lagrange multipliers for k . We can then construct (i) for i . Firstly since (k (x))j ; (i (x))j > 0 we can put (ji) = (jk) = 0 for j > l. For each 1  j  l, since i (x)j = k (x)j = 0 and i (x0 ); k (x0 )  0 for all 0 x 2 M , we see that r (k (x)) must map the \octant" Rl+  Rn?l into itself. Thus the principal l  l matrix of r (k (x)) is, in fact, after possibly another permutation of the co-ordinates, a diagonal matrix with non-negative entries. In fact, the matrix of r has the form D 0 0 A where D is a diagonal matrix. As is a dieomorphism, the diagonal entries of D are strictly positive and A is non-singular. This means that there are numbers j > 0 such that r(i )j (x) = j r(k )j (x) for j = 1; : : : ; l. We can therefore set j(i) = (1=j )(jk) for j = 1; : : : ; l and (ji) = j(k) = 0 for j > l. The index of a KT point in Ui of a cornered manifold is the number of negative eigenvalues of

Z T (r2 f (x) ?

m X j =1

(ji) r2 (i (x))j )Z

where the columns of Z forms a basis for

T (x) = fu j r(i (x))Tj u = 0 for all j where (i (x))j = 0g: By the previous paragraph, the de nition of T (x) does not change if x 2 Ui \ Uk and we replace i with k . 9

To show that the index is independent of the co-ordinate chart chosen, we consider a parametrerization of the the set fx0 j (i (x0 ))j = 0 () (i (x))j = 0g in a neighbourhood of x. The tangent plane to this set at x is precisely T (x). We take : Rn?l ! M to be a suitable parameterization nearby, such that (0) = x, and r(0) has full rank. Then for j = 1; : : : ; l, i ((u))j = 0 for all u. Then taking second derivatives, @ 2 (f  ) (0) = X @ 2 f (x) @p (0) @q (0) + X @f (x) @ 2 p (0):

@ur @us

pq

@xp @xq

@ur

@us

p

@xp

@ur @us

Similar formulae hold for the (i )j . Then subtracting the sum of (ji) @ 2 (i  )j =@ur @us (which are zero for all j ) we get

1 0 X X @ (f  ) (0) = @ @ f (x) ?  i @ (i )j (x)A @p (0) @q (0): j @xp @xq @ur @us @xp @xq @ur @us 2

2

( )

pq

2

j

As the matrix Z = [@i =@uj ] has columns which forms a basis for T (x), it follows that the index of

r (f  )(0) = Z T (r f (x) ? 2

2

X j

(ji) r2 (i )j (x))Z

is independent of the parameterization used. Now, the previous section shows that there is a one{to{one correspondence between critical points of

Pi (x; r) = f (x) ? r

n X j =1

ln(i (x)j )

on Ui for suitably small r > 0, and KT points in Ui , that preserves indices. As the index is independent of the chart chosen, the Morse inequalities should hold for the function f on M provided the Kuhn{Tucker index is used.

5 Non-degenerate KT-functions are dense in C 2 For many applications the assumption that f is a non-degenerate KT-function (once the gi are given and satisfy the constraint quali cation) is rather restrictive. In order to extend results for non-degenerate KT-functions to degenerate functions it is common in classical Morse theory to use the following result to justify approximating arbitrary smooth functions by Morse functions, due to Morse: Theorem 2 If M is a compact smooth manifold, then the set of Morse functions in C 2 (M ) is open and dense in C 2 (M ). 10

This is commonly proven via the Parameterised Sard Theorem (PST) ([1, Thm. 2.1]), which we will use here to prove the denseness of non-degenerate KTfunctions in C 2 (X ). Before we state the PST we need a de nition from dierential topology: De nition 1 Let f : M ! Rp where M is a smooth manifold. An point x 2 M is said to be a regular point if rf (x) has full rank. A value y 2 Rp is said to be a regular value if every point which maps to y is a regular point. Note that f ?1 (y) is empty, then y is trivially a regular value. Theorem 3 Let V  Rq , U  Rm be open and let : V  U ! Rp be C r where r > max(0; m ? p). If y 2 Rp is a regular value of  then for almost all a 2 V , y is a regular value of (a;
). The main result of this section is the following: Theorem 4 If M is smooth manifold and gi: M ! R are smooth functions for i = 1; : : : ; m, where the constraint quali cation (3) holds and X = f x 2 M j gi (x)  0; i = 1; : : : ; m g 6= ; is compact, then the set of non-degenerate KT-functions in C 2 (X ) is dense in C 2 (X ). Proof. Let M be the given manifold of dimension n. Let J  f1; : : : ; mg and for any such J write MJ = f x 2 X j I (x) = J g. It is easy to see that MJ is a manifold of dimension n ? jJ j. Let f 2 C 2 (X ). We wish to nd functions f~ 2 C 2 (X ) that are arbitrarily close to f in C 2 (X ). For now, suppose that M is an open set in Rn . The general result will follow the same general outline. Given f 2 C 2 (X ) we wish to nd non-degenerate KTfunctions f 2 C 2 (X ) that are arbitrarily close to f . These functions can, in fact, be nicely parameterised. Put f~: Rn  X ! R to be f~(a; x) = f (x) + aT x: Now consider rx f~: Rn  X ! Rn where rx f~(a; x) = rf (x) + a. We now need to consider the behaviour of f~ on each MJ . For each KT point x 2 MJ we have X rf (x) ? i rgi (x) = 0: (7) i2J

Let ZJ (x) be a smooth matrix function of x on MJ such that ZJ (x) has orthonormal columns which span the null space of [ rgi (x) j i 2 J ]T . (This can be obtained from the QR-factorisation of [ rgi (x) j i 2 J ]T .) Then (7) is equivalent to requiring that ZJ (x)T rf (x) = 0 2 Rn?jJ j. The corresponding  vector can be computed using [ i j i 2 J ] = ?[ rgi (x) j i 2 J ]+ rf (x), and i = 0 for all i 2= J . For each J  f1; : : : ; mg, we apply the PST to the function ZJ (x)T rx f~(a; x) = ZJ (x)T rf (x) + ZJ (x)T a: 11

To check that this has 0 as a regular value as a function of (a; x), we note that its reduced Jacobian matrix is [ZJ (x)T ; ZJ (x)T r2x f (x)ZJ (x)], which has full rank for all (a; x). Thus by the PST, for almost every a, every KT point x of f~(a;
) in MJ has ZJ (x)T r2x f~(a; x)ZJ (x) non-singular. In order for a KT point to be non-degenerate we also need i 6= 0 for all i 2 J . As [i j i 2 J ] = ?[rgi (x) j i 2 J ]+ rx f~(a; x) = ?[rgi (x) j i 2 J ]+ (rf (x) + a) and [rgi (x) j i 2 J ] has rank jJ j it follows that, for almost all a 2 Rn, i 6= 0 for all i 2 J at a KT point. Combining this result with that of the previous paragraph we see that for almost all a 2 Rn , every KT point on MJ is nondegenerate. As there are only nitely many J  f1; : : : ; mg it follows that for almost all a 2 Rn the function f~(a;
) is a non-degenerate KT-function. Thus the theorem holds if M is an open set in Rn. If M is a given n-dimensional manifold, we can construct a nite collection of sets Wi  Vi , i = 1; : : : ; N that covers X where the Wi and Vi are dieomorphic to balls in Rn. Let i : Vi ! i (Vi )  Rn be a suitable family of dieomorphisms. Then we can construct a collection of smooth functions i : Vi ! R, i = 1; : : : ; N , where i (x)  0 for i = 1; : : : ; N , and i = 1 on Wi . We now set f~: (Rn )N  M ! R to be

f~(a1 ; a2 ; : : : ; aN ; x) = f (x) +

N X i=1

i (x) aTi i (x):

The proof for the case where M is an open subset of Rn can be applied to i (Vi ), i = 1; : : : ; N with f replaced by f  i?1 . If x 2 Wi , Jacobian of rx f~(a1 ; : : : ; aN ; x) with respect to ai is r( i?1 )( i?1 (x)) which is non-singular, as i is a dieomorphism. So the Jacobian with respect to (a1 ; : : : ; aN ; x) has full rank. Thus, by the PST, for almost every a = (a1 ; : : : ; aN ) 2 (Rn )N the function f~(a; i?1 (
)) is a non-degenerate KT-function. Hence, for almost all a 2 (Rn )N , every KT point of f~(a;
) in Wi is non-degenerate. As there are only nitely many Wi to consider, it follows that for almost all a 2 (Rn )N the function f~(a;
) is a non-degenerate KT-function. The denseness of nondegenerate KT-functions in C 2 (X ) follows immediately. 2 The corresponding result for manifolds with corners can be proven in a similar fashion. Note that the set of non-degenerate KT-functions in C 2 (X ) is open, as well as being dense.

6 A simple example Before the example of how this result may be used, we note here that the result of Yun ([9]) of uniqueness of Kuhn{Tucker points on a disk under certain assumptions follows quite easily from the Morse inequalities. 12

For our simple application we consider a nonlinear Complementarity Problem: Find x 2 Rn such that x  0; rf (x)  0; xT rf (x): (8) Theorem 5 If f 2 C 2(Rn ) and @f=@xi(x) > 0 for all i and x 2 Rn+ with kxk  M > 0, then there exists a solution to (8). Furthermore, if f in a dense, open subset of C 2 (Rn ), the number of solutions to (8) is odd. Proof. We rst assume that f is a non-degenerate KT function with respect to gi (x) = xi , i = 1; : : : ; n. The set ofp all such functions is the dense, open subset of C 2 (Rn ). Let gn+1 (x) = (M= n) ? eT x where e = [1; : : : ; 1]T . Then for p constraint functions g1 ; : : : ; gn+1 the feasible set is X = fx 2 Rn+ j eT x  M= ng. The Kuhn{Tucker pairs (x; ; n+1 ) satisfy rf (x) ?  + n+1 e = 0 xi i = 0 i = 1; : : : ; n n+1 (M=pn ? eT x) = 0 xi ; i  0: p Now n+1 = 0 as otherwise eT x = M= n > 0 and thus kxk  M ; then as xj > 0 for some j we have j = 0 so

@f (x) +  = 0: n+1 @xj This is impossible as n+1  0 and @f=@xj (x) > 0 whenever kxk  M . Thus rf (x) =  and substituting rf (x) for  in the complementarity con-

ditions for Kuhn{Tucker pairs gives the desired result. Thus Kuhn{Tucker pairs correspond to solutions of the complementarity problem. We now show that the number of solutions must be odd: Since n X

n X

l=0

l=0

(?1)l l (f; g) =

(?1)l l (X );

taking residues modulo 2 and noting that 0 (X ) = 1 and l (X ) = 0 for l > 0 we see that n X l (f; g)  1 (mod 2): l=0

The quantity on the left hand-side is the total number of solutions to (8), which is therefore odd. This proves the second part of the theorem. The rst part then follows by considering arbitrarily close non-degenerate KT functions f~k : Rn ! R with solutions xk to (8) with k ! 1. As X is compact there must be a convergent subsequence with limit x which is a solution of (8) by continuity. 2 13

7 Conclusions In this paper classical Morse theory has been extended to deal with manifolds with corners by means of Kuhn{Tucker theory. The theory is developed most naturally in the context of a constrained \optimization" problem min f (x) x

subject to gi (x)  0; i = 1; : : : ; m:

Instead of looking for a critical point of f (that is, rf (x) = 0) we look for Kuhn{Tucker pairs (x; ) where

rf (x) ? Pmi i rgi (x) = 0 =1

i gi (x) = 0 i = 1; : : : ; m i ; gi (x)  0 i = 1; : : : ; m:

P

2 A Kuhn{Tucker pair is non-degenerate if the Hessian matrix r2 f (x)? m i=1 i r gi (x) is non-singular and for each i either i > 0 or gi (x) > 0; for such a pair, we de ne its index to be the number of negative eigenvalues of the above Hessian matrix. If every Kuhn{Tucker pair for the functions f , gi is non-degenerate (and the set of such f , given the gi , is open and dense in C 2 ) then the Morse inequalities hold where the Betti numbers used are those for X = f x j gi (x)  0; i = 1; : : : ; m g. This has enabled a simple proof of the existence of solutions to a nonlinear Complementarity Problem under some weak global conditions.

References [1] Chow, S.-N., Mallet-Paret, J. and Yorke, J.A. Finding zeroes of maps: Homotopy methods that are constructive with probability one. Mathematics of Computation, 32, 887{899, (1978). [2] Dancer, E.N. Private communication, March 1990 [3] Fletcher, R. \Practical Methods of Optimization", John Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore. 2nd Edition. 1987 [4] Franklin, J. \Methods of Mathematical Economics: Linear and Nonlinear Programming, FixedPoint Theorems", Springer{Verlag, New York, Heidelberg, Berlin. Undergraduate Texts in Mathematics series. 1980. [5] Golub, G. and van Loan, C. \Matrix Computations", North Oxford Academic, Oxford, 1983 [6] Guddat, J., Jongen, H. Th., and Rueckmann, J. On stability and stationary points in nonlinear optimization. J. Austral. Math. Soc. Ser. B, 28, 36{56, (1986). 14

[7] Milnor, J. \Morse Theory", Princeton University Press, Princeton. Annals of Mathematics Studies #51. 1963. [8] Poore, A.B. and Tiahrt, C. Bifurcation problems in nonlinear parametric programming. Mathematical Programming, 39, 189{205, (1987). [9] Yun, K.K. Uniqueness conditions for Kuhn{Tucker points on a disk. Journal Optimization Theory and Applications, 44, 701{721, (1984).

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