Variable dimension algorithms: Basic theory, interpretations and

Mathematical Programming 24 (1982) 177-215 North-Holland Publishing Company

VARIABLE DIMENSION ALGORITHMS: BASIC THEORY, INTERPRETATIONS AND EXTENSIONS OF SOME EXISTING METHODS Masakazu KOJIMA Department of Information Sciences, Tokyo Institute of Technology, Tokyo 152, Japan Yoshitsugu YAMAMOTO Institute of Socio-Economic Planning, University of Tsukuba, Ibaraki 305, Japan Received 28 May 1980 Revised manuscript received 18 May 1981 In this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by Van der Laan and Talman. We introduce a new concept 'primal-dual pair of subdivided manifolds' and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms. Our basic model furnishes interpretations to several existing methods: Lemke's algorithm for the linear complementarity problem, its extension to the nonlinear complementarity problem, a variable dimension algorithm on conical subdivisions and Merrill's algorithm. We shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us. A method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed. Key words: Variable Dimension Algorithm, Fixed Point, Subdivided Manifold, Nonlinear Equations.

1. Introduction I n t h e field o f t h e fixed p o i n t a n d c o m p l e m e n t a r i t y t h e o r y (see, f o r e x a m p l e [1, 4, 7, 25, 29]) w h i c h w a s o r i g i n a t e d b y L e m k e a n d H o w s o n [20] a n d S c a r f [26], a class o f n e w a l g o r i t h m s g i v e n b y V a n d e r L a a n a n d T a l m a n [14, 15, 16, 17, 18, 19, 28], R e i s e r [23], T o d d [31], T o d d a n d W r i g h t [34] a n d W r i g h t [35] h a s b e e n attracting considerable attention. They are called variable dimension algorithms. A c o m m o n f e a t u r e o f t h e m is t h a t t h e y s t a r t f r o m a single p o i n t , a z e r o dimensional simplex, and generate a sequence of simplices of varying dimens i o n s to a t t a i n a s i m p l e x w h i c h c o n t a i n s an a p p r o x i m a t i o n of a fixed p o i n t . S o m e n u m e r i c a l r e s u l t s [15, 17] h a v e s u p p o r t e d t h a t t h e v a r i a b l e d i m e n s i o n t e c h n i q u e is v e r y u s e f u l in i n c r e a s i n g c o m p u t a t i o n a l efficiency. T h e a i m o f this p a p e r is to p r o v i d e a b a s i c t h e o r y f o r v a r i a b l e d i m e n s i o n algorithms. We establish foundations on which we can construct a wide class of v a r i a b l e d i m e n s i o n a l g o r i t h m s a n d s t u d y s o m e t y p e s of a l g o r i t h m s o f t h e class. T h r o u g h o u t t h e p a p e r , w e shall e m p l o y t h e unified f r a m e w o r k g i v e n b y E a v e s 177

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[4] (see also Eaves and Scarf [7]) for the fixed point and complementarity theory, which will be outlined in Section 2. In Section 3, we introduce a new concept 'a primal-dual pair of subdivided manifolds', which has a close relation with the dual complex in the theory of piecewise linear topology (see, for example, Rourke and Sanderson [24]) and is a generalization of the complementarity. Then we give fundamental theorems (Theorems 3.2 and 3.3) on a primal-dual pair of subdivided manifolds. In Section 4, combining the fundamental theorems with Eaves' unified framework for the fixed point and complementarity theory, we establish a basic model for variable dimension algorithms. In Section 5 we present four applications of the primal-dual pair of subdivided manifolds and the basic model. In Section 5.1, we give an interpretation of Lemke's algorithm [20] for the linear complementarity problem in terms of the basic model. In Section 5.2 we also give a similar interpretation of an extension of Lemke's algorithm (see Kojima [10]) for the nonlinear complementarity problem and make use of it in designing several variable dimension algorithms for solving systems of equations. In Section 5.3, we give an interpretation of the variable dimension algorithm proposed by Van der Laan and Talman [17] and Todd [3l]. Section 5.4 is concerned with Merrill's algorithm [21]. We derive a special primal-dual pair of subdivided manifolds from a triangulation of R " x [0, 1] and propose an efficient implementation of Merrill's algorithm by utilizing this primal-dual pair. In Section 6 we also present a method for constructing triangulations with continuous refinement of mesh size. The reader might refer to the recent work by Freund [8] who independently developed a unified interpretation of variable dimension algorithms.

2. Preliminaries

We call a convex polyhedral set (the intersection of a finite number of closed halfspaces) in some Euclidean space R k a cell. The dimension of a cell C, denoted by dim C, is defined to be the dimension of aft C, the affine subspace spanned by C. By an m-cell we mean a cell of dimension m. We use the notation relint C and OC for the interior and the boundary of C relative to aft C, respectively. If a cell B is a face of a cell C, we write B < C. Let :g be a finite or countable collection of m-cells in R k. We denote the collection {B : B < C, C ~ M} by ~ . We call M a subdivided m-manifold if

(2.1)

for every pair of B, C ~ J~, either B V~C = 0 or B n C is a c o m m o n face of B and C,

(2.2)

each (m - D-cell of 3~ lies in at most two m-cells of ~ ,

(2.3)

A~ is locally finite; each point x ~ [~1 has a neighborhood (an open

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subset of R k containing x) which intersects with only finitely m a n y cells of 3l. H e r e IMI denotes the union of all m-cells of d/. If M is a subset of R k and M = ]M I for some subdivided m-manifold d/, we say that M is an m-manifold and that d/ is a subdivision of M. Generally, an m-manifold M (m _->1) has m a n y subdivisions. L e t d / a n d d / ' be subdivisions of an m-manifold M C R k. If each C ' E d / ' lies in some C E d / , d/' is said to be a refinement of d/. If in addition all cells of d/' are simplices, d/' is said to be a simplicial refinement of d/ and also a simplicial subdivision of M. Let M be an m-manifold in R k and d/ a subdivision of M. The b o u n d a r y of d/, denoted b y ad/, is defined to be the collection of all (m - l)-cells of M which lie in exactly one m-cell of d/, and a M (the b o u n d a r y of M ) the union of all (m - 1)-cells of ad/, i.e., a M = lad/]. It should be noted that a M is independent of the choice of a subdivision d/ of M (see, for example, E a v e s [4]). By an interval we m e a n a convex subset of the real line R 1 which contains more than one point, and b y the unit circle the set {(xl, x2) ~ R2:x~+ x 2 = 1}. It is well k n o w n that a connected 1-manifold is h o m e o m o r p h i c to either an interval or the unit circle. We call a connected 1-manifold a path if it is h o m e o m o r p h i c to an interval, and a loop otherwise. Let M be an m-manifold in R k and d/ a subdivision of M. A continuous map F from M into some Euclidean space R k is said to be piecewise linear (abbreviated b y PL) on d/ if the restriction of the map F to each C E d/ is linear, i.e., F(Ax + (1 - A)y) = AF(x) + (1 - A)F(y) for e v e r y x, y E C and A E [0, 1]. To clarify the subdivision d/ on which F is P L , we write F : [ d / [ ~ R k. Suppose that d / i s a subdivided (n + D-manifold and that H : l d / [ - ~ R" is PL. We say that c ~ R " is a regular value of the P L map H if C E3~ and c E H ( C ) always imply dim H ( C ) = n. It follows that if c is a regular value of H : l d / I ~ R", then H - l ( c ) = { x E [ d / l : H ( x ) = c} (the inverse image of c under the m a p H ) does not intersect with any C E ~ of dimension less than n. We now state one of the basic theorems in the fixed point and complementarity theory. Theorem 2.1. Let d/ be a subdivided (n + D-manifold, and H : Id/]~ R" a P L map. Suppose that c E R" is a regular value of H. Then H-~(c) is a disjoint union of paths and loops and satisfies the following conditions :

(2.4)

If C E d / a n d c E H ( C ) , then H-~(c) f3 C is a 1-cell (a line segment, a half line or a line) with rel i n t ( H - l ( c ) f3 C) C rel int C and a ( H - l ( c ) fq C) C OC.

(2.5)

A loop of H-~(c) does not intersect with aid/I.

(2.6)

If a path S of H - l ( c ) is compact, then aS consists of two distinct points in aid/].

Proof. See Section 9 of E a v e s [4].

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3. Primal-dual pair of subdivided manifolds L e t m be a positive integer. We say that (~, 9 ; d ) is a primal-dual pair of subdivided manifolds (abbreviated b y PDM) with degree m if it satisfies the following conditions (3.1)-(3.5): (3.1)

~ and 9 are subdivided manifolds.

(3.2)

For e v e r y X E #, either X d E ~ or X d = 0.

(3.2)'

For e v e r y Y E 0, either yd E ~ or y d = 0.

(3.3)

If Z ~ # U ~ and Za~ O, then (Za) d = Z.

(3.4)

I f X b X 2 E # , X l < X 2 , X~#OandX~#O, t h e n X d < X d.

(3.4)'

If Y~,Y2E~,Yl0 the collection ~ * [ X = {~ ~ ~ * : o - C X , dim o- = k} forms a subdivision of X ( ~ * [ X is itself a subdivided k-manifold). Using this fact and Theorem 3.2, we can easily prove. Theorem 4.1. Let

At = { ~ x Y :

Y~,Y~#O,

crE~*]Y~}.

Then At is a refinement of ££. Letting

H(x, y) = y + F(x)

for every (x, y) E ]M],

we have a P L map H:]At I-o R n. Consider the system of equations

H(x, y) = c,

(x, y) E [Atl.

(4.3)

This system will afford a basic model for a wide class of variable dimension algorithms. We assume that c ~ R" is a regular value of the P L map H. Then H-l(c), the set of solutions to (4.3) turns out to be a disjoint union of paths and loops and satisfies (2.4)-(2.6) (Theorem 2.1). We want to design 5¢ = ( ~ , 9 ; d), ~ * and F : [ ~ * [ ~ R " such that (4.4)

H-l(c) has a known point (x °, y0) E ]Atl,

(4.5)

the connected component S O of H ~(c) which contains (x °, yO)E [At[ forms a path,

(4.6)

S O contains a point (~,)~)E[At[ which serves as an approximate solution to a given problem.

Then we can obtain an approximate solution to the problem by starting from (x 0, yO) and tracing the path S °. Suppose the path S Ointersects with (n + 1)-cells C1, C2 and an n-cell B which is a c o m m o n face of C1 and C2. Then from L e m m a 3.1 we can write B = X x Y, C I = X × X ~ and C 2 = Y a x Y for some X E ~ , Y ~ . W h e n w e are in C1 (or C2) we move along the cell X (or y d ) in the primal manifold [~[ and along X d

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(or Y) in the dual manifold 191. Since dim X + dim Y = n and dim X + dim X d = dimYd+dimY=n+l, we have d i m Y ~ = d i m X + l and d i m Y = d i m X d - l . H e n c e as we traverse C1, B and C2 in this order, the dimension of cells we m o v e along increases b y one in 1~[ while it decreases by one in I 1- Since each (n + 1)-cell of M in C~ has the form cr x X a for some o- E ~ * [ X, the dimension of the sequence of cells ~r E ~ * which we encounter coincides with the dimension of the cell X as long as we are in Ca. This dimension also increases b y one w h e n we m o v e into a new (n + 1)-cell C2. Thus every time we come across a new (n + 1)-cell of ~g in tracing S °, the dimension of cells cr E ~ * encountered in I~*1 vary. We conclude this section by pointing out that the subdivided manifolds ~ and m a y be replaced by complexes. A finite or countable collection ~ of cells in R k is said to be a complex if it satisfies the following conditions: (4.7)

For every pair of B, C E K, either B VI C = 0 or B N C is a c o m m o n face of B and C.

(4.8)

If C E ~ and B < C, then B E ~.

(4.9)

~ is locally finite.

If M is a subdivided manifold, then ~ is a complex. H e n c e we m a y regard that a subdivided manifold is a special case of a complex. If two complexes ~, @ and a dual operator d saisfy conditions (3.2)-(3.5), then we can p r o v e that ( ~ , @ ; d ) defined b y (3.6) forms a subdivided m-manifold. T h e o r e m 3.3 also remains valid. Fig. 2 illustrates an example. H e r e = { x , : i = o , 1 . . . . . 8}, ~={v,:i=o,

1 . . . . . 8},

x~=Y,

( i = 0 , 1 . . . . . 5),

Yq=X,

(i = 0 , 1 . . . . . 5),

Xq=f~

(i = 6,7, 8),

Y~=0

(i = 6,7, 8). Y6

X6

Y5 /Y~ ~ X8 Y8 YO Y1 X1 ---" "---Y4 ~

X4 X5

Y7

x7 p

D

Fig. 2.

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5. Applications of the basic model 5.1. A linear c o m p l e m e n t a r i t y p r o b l e m and L e m k e ' s a l g o r i t h m

The problem of finding a (z, y) E R 2m which satisfies the condition z>=O,

y=Az+q>=O,

zi-yi=0

( i = 1,2 .... , m )

is called a linear c o m p l e m e n t a r i t y problem (abbreviated b y L C P ) where A and q are an m x m matrix and an m-dimensional column vector, respectively. If q =>0, L C P has a trivial solution (z, y) = (0, q). H e n c e in what follows we shall assume that q ~ 0, i.e., q has at least one negative component. We first transform the L C P into a system of P L equations. L e t M = {1, 2 . . . . . m}. For e v e r y I C M, define Y ( I ) = {y E Rm: y, >=0 (i E I), yj = 0 (j E M - I)},

where M - I = {j E M : j ¢ 5 I } . We m a y allow the case where I = 0; Y ( 0 ) = {0}CR m. Y ( M ) coincides with the nonnegative orthant R~ of R m. E a c h cell Y ( I ) has the dimension # I where # I denotes the n u m b e r of elements in I. Let = ~ = {R~}.

Obviously @ = @ is a subdivided m-manifold consisting of a single m-cell R~' and = ~ = {Y(I): I C M}. For each I C M, define Y ( I ) ~ = Y ( M ~ I).

Then (@, @ ;d) f o r m s a P D M with degree m. Let 5C = (@, @ ;d). Note that every Z E ~ U ~ has a n o n e m p t y dual. Hence, by T h e o r e m 3.3, 5~ has no boundary. It is easily verified that the nonnegativity and c o m p l e m e n t a r i t y conditions z_>O,

y=>O,

zl. yl = 0

( i = 1,2 .... , m )

are equivalent to the condition (z, y) ~ I~I.

H e n c e we can say that a primal-dual pair of subdivided manifolds is a generalization of the ' c o m p l e m e n t a r i t y ' . Using the equivalence of the two conditions above, we obtain a system of P L equations y - A z = q,

(z, y) E

which is equivalent to the LCP.

I el

(5.1)

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W e n o w state L e m k e ' s algorithm f o r solving an L C P in terms of the basic model. L e t M * = {0, 1 . . . . . m}. We shall d e n o t e the (m + 1)-dimensional E u c lidean space with c o o r d i n a t e s Xo, Xi . . . . . xm variables b y Rl+m; R~+m= {x = (xo, x~. . . . . xm)T: X~E R}. F o r e a c h Y C M * , define

E a c h cell X ( J ) has the d i m e n s i o n #J. L e t ~ = {Rl+m}. T h e n ~ is a subdivided (m + 1)-manifold with

Letting

f o r e v e r y J C M * and

for e v e r y I C M, we obtain a P D M (:~, ~ ; c) with degree m + 1. W e shall write e a c h point x in the primal manifold [~1 as x = (t, z) w h e r e t ~ R+ and z E R~. L e t At(~, ~ ; c). Theorem 5.1.

Proof. If B is an m-cell of At, then B = X ( J ) x Y ( I ) for s o m e f C M * and I C M (possibly J = ~ or I = 0) such that J V) I = ~ and # I + # J = m. W e also see that Y ( I ) C ¢ fJ for e v e r y I C M and that X ( J ) c= 0 if and only if 0 f f J. H e n c e , b y T h e o r e m 3.3, OAt is the collection of m-cells X ( J ) x Y ( I ) satisfying the set of conditions

or equivalently (5.2) N o t e that if I and J satisfy (5.2), t h e n X ( J ) × Y ( I ) = {0} × Y ( M ~ I) × Y ( I ) = {0} x Y ( I ) d × Y ( I ) . C o n s e q u e n t l y , we obtain

As a direct c o n s e q u e n c e o f the t h e o r e m , we obtain: T h e o r e m 5.2.

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Consider the system of equations y - A z - et = q,

(t, z, y) E

I~tl,

(5.3)

where e denotes the m-dimensional column vector of ones. Suppose that (t, z, y) is a solution to the system (5.3) and (t, z, y) E 0IAA]. Then, by T h e o r e m 5.2, t = 0 and (z, y) E I 1. H e n c e (z, y) satisfies (5.1) and is a solution to the LCP. We shall denote left side of the system (5.3) by H ( t , z, y) and assume that q is a regular value of the P L map H :l~tl-,R ~. Applying T h e o r e m 2.1, we see that H - l ( q ) , the set of all solutions to the system (5.3), is a disjoint union of paths and loops and satisfies (2.4)-(2.6). Let t o = min{t _- 0: q + et >=0}, Z ° ---- 0 ,

y0 = q + et o, C O= X({0}) × Y ( M ) E At. Then we have t ° > 0 since we have assumed that q ~ 0 and H l(q) n C ° = {(t, 0, q + et): t >=t°}. We are interested in the connected c o m p o n e n t , denoted b y S °, of H - ~ ( q ) which contains H - ~ ( q ) n C °. Since H-~(q) n C Ois a halfline, S Ocan not be a loop, i.e., it must be a path. The subdivided (m + 1)-manifold At consists of 2 m (m + 1)-cells Y ( I ) c × Y ( I ) (I C M ) , so that S Ois a union of finitely many 1-cells. If the path S O intersects with the boundary 01 1 of I tl at some ({, Z, y), then { = 0 (Theorem 5.2) and (5, y) is a solution to the LCP. In this case, tracing the path S O from (t °, z °, y~) in the direction different from the direction of the halfline H I ( q ) n C O= {(t, O, q + et): t >=to}, we attain a solution (2, y) of the LCP. This is what L e m k e ' s algorithm does. In fact, L e m k e ' s algorithm generates a path of the points (t, z, y) satisfying z>=O, zi.yi=O

y=Az+q+et>=O,

t>=O,

( i = 1,2 . . . . . m)

(see, for example, Cottle and Dantzig [2]). By the construction of the (m + 1)manifold I~1, we see that the condition a b o v e is equivalent to the system (5.3) of P L equations. The path generated by L e m k e ' s algorithm is exactly the same as the path S o. 5.2. A n extension o f L e m k e ' s algorithm to a nonlinear complementarity problem and its application to a system o f equations f ( x ) = O.

In Section 5.1 we have given an interpretation of L e m k e ' s algorithm for solving a linear c o m p l e m e n t a r i t y problem in t e r m s of the basic model for

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variable dimension algorithms. The purpose of this subsection is to give a similar interpretation to an extension of L e m k e ' s algorithm [10, Section 7] and to apply the extended L e m k e ' s algorithm to a system of equations f ( x ) = 0 , where f : R ~ ~ R" is continuous. F o r the time being, we shall e m p l o y the same notations as in Section 5.1. The problem of finding a (z, y) ~ R ~m satisfying the conditions z>=O,

y=g(z)>=O,

z~-yi=0

(i~M={1,2

.... ,m})

(5.4)

is called a nonlinear c o m p l e m e n t a r i t y problem (abbreviated b y NCP), where • rtl m g .R+ ~ R " is a continuous function. L e t 3~* be a simplicial subdivision of R+ with bounded mesh size a* = sup sup{IIu - vii: u, v E o'}. Then ~ * = {R+l x o': o" E ~*} is a refinement of ~ and we can m a k e a refinement N of A~ by using ~ * (see T h e o r e m 4.1). Letting G : [ ~ * I - ~ R m be a simplicial approximation of g, we consider the system of P L equations: H(t,z,y)=y-G(z)-et=O

(t,z,y)ElJC[,

where e is an m-dimensional column vector of ones. We here assume that zero is a regular value of H. Then it is k n o w n that H-l(0) is a disjoint union of paths and loops. L e t t o = min{t ->_O: G(O) + et >- O} = max{O,-G,(O) . . . . . -G.(O)}, Z ° ---- O ,

y0 = G(0) + et °, C O= X({0}) × Y ( M ) ~ At, then we obtain that H-~(0) N C O= {(t°+ 0, z °, y0+ 0e): 0 ->_0}. In the following we shall present some properties of the connected c o m p o n e n t S o of H-~(0) that contains H-~(0) n C °. L e m m a 5.3. (i) S o is a path homeomorphic to either (-o% O] or (-0% +oo). (ii) I f S o is homeomorphic to (-0% 0], then S o has an endpoint (0, z, y) such that (z, y) is a solution to (5.4) with g replaced by its simplicial approximation G.

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(iii) If S O is homeomorphic to ( - ~ , +~), then

Ilzll-~ +~ along the path.

Proof. (i) is trivial because S o has an unbounded ray H-l(0) N C o (see T h e o r e m 2.1). As we have shown in T h e o r e m 5.2, ol~l--01~l--{0I×l~l. Hence the endpoint of S Om u s t be (0, z, y) for some (z, y) E [~1, which implies (ii). In order to p r o v e (iii) assume on the contrary that there exists a real number such that

Ilzll- c holds for any (t, z, y) E S °. L e t U = {(t, z, y) E INI: Ilztl----C}, then U meets only finitely m a n y cells of N. H e n c e the 1-manifold S Oconsists of finitely m a n y 1-cells each of which is either a half line or a line segment. (Note that S Ocannot contain a line.) S o is assumed to be h o m e o m o r p h i c to (-oo, +~), so that S O must contain exactly two distinct half lines. One is H-l(0) tq C O and the other is written as H-l(0) n C = {(t* + O,~t, z* + Oziz, y* + 0ay): 0 > 0} for

some C E X

with C # C °, (t*, z*, y*) E R 1+2m and nonzero (At, Az, A y ) E z* + OAz is bounded. H e n c e

R +l+2m We have that ziz = 0, since

y* + 0Ay = G(z*) + e(t* + OAt)

for every 0 =>0.

(5.5)

We also h a v e that (z*)T(y * + 0Ay) = 0

for every 0 >=0.

(5.6)

It follows f r o m (5.5) that Ay = ezat. This implies that At > 0 since otherwise (At, Az, Ay) = 0. H e n c e for sufficiently large 0 > 0, we have y* + 0ziy > 0. Therefore f r o m (5.5) and (5.6) Z* = 0 = Z0,

y* = G ( 0 ) + et*, and we obtain c = C °. This is a contradiction. In what follows we shall explain how to convert the problem of finding a zero of a continuous function f : R " ~ R " to an N C P . L e t x ° E R n be an arbitrary point and B be an n x m (n = m) matrix satisfying the following two assuraptions (5.7) and (5.8): (5.7)

R n = {Bz + x°: z E R7}.

(5.8)

The system of linear equations

y+BTu=e,

O < = y ~ R m and

uER"

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is n o n d e g e n e r a t e , i.e., e v e r y solution (y, u) ~ R m+" to the s y s t e m has at least m n o n z e r o c o m p o n e n t s . It follows directly f r o m (5.7) that rank B = n,

(5.9)

and f r o m (5.8) that if (y, u) E R m+" is a solution to the s y s t e m a b o v e , t h e n y has at m o s t n zero c o m p o n e n t s . W h e n the pair consisting of an n × m matrix B and e E R m does not satisfy (5.8), we can slightly perturb the e ~ R m s u c h that the resultant pair satisfies (5.8); the p e r t u r b e d e c a n n o t be the v e c t o r of ones, but there will be no trouble in our discussion if all the c o m p o n e n t s of e ~ R m are positive. This matrix B is a generalization of the matrices used b y V a n der L a a n and T a l m a n [15] in making subdivisions of R ~ for their variable d i m e n s i o n algorithms. W e c a n e m p l o y various matrices as B. T y p i c a l e x a m p l e s are -e

i ......

6-i-----e-J

N o w define a c o n t i n u o u s f u n c t i o n w :R~' ~ R " - as w(z)

=

f ( B z + x°),

and consider the following N C P : z>=O,

y=BTw(z)>=O,

zi • y~ = 0

(i ~ M).

(5.10)

T h u s a solution to (5.10) is related to a zero of f in the following m a n n e r . L e m m a 5.4. x ~ R n is a zero o f f if and o n l y if there e x i s t s a s o l u t i o n (z, y) E R 2~ to N C P (5.10) s u c h t h a t x = B z + x °. Proof. T h e ' o n l y if' part is trivial b e c a u s e y 'if' part first o b s e r v e that there is a v e c t o r d f a c t b y the a s s u m p t i o n (5.7) there exists - B e + x ° = B z + x °. T h u s w e h a v e a v e c t o r solution to N C P (5.10), then

= B T w ( z ) = BT.f(x) = 0. To s h o w the E R m such that B d = 0 and d > 0. In a n o n n e g a t i v e v e c t o r z satisfying d = e + z. S u p p o s e (z, y) ~ R TM is a

o 0 imply that

B T f ( B z + x °) = O. U s i n g (5.9) w e h a v e f ( x ) = f ( B z + x °) = O.

In order to find an a p p r o x i m a t e solution to (5.10) w e c o n s i d e r the s y s t e m of

190

M. Kofima and Y. Yamamoto/ Variable dimension algorithms

P L equations H ( t , z, y) = y - B T W ( z ) - et = 0

(t, z, y) ~ I•1,

(5.1 r,

where W:I~*I--,R n is a P L approximation of w on ~ * , i.e., W ( z ) = ~ Xiw(v ~) i=0

for each z E or = co{v °, v ~. . . . . vm}E ~ * with z=~X,v', i=0

hi_->0

~X,=

1,

i=0

(i = 0 , 1. . . . . m),

where v °, v 1. . . . . v m are the distinct vertices of ~r and c o X means the convex hull of X. As described in the f o r m e r part of this section, if we assume that the connected c o m p o n e n t S o of H ~(0) containing the initial unbounded ray H-l(0) ;3 C o is h o m e o m o r p h i c to (-0% 0], we can obtain an a p p r o x i m a t e solution to (5.1), so that an approximate zero of f. The reader might have the criticism that the a b o v e a p p r o a c h for approximating a solution to f ( x ) = 0 increases the dimension of variable vectors to be handled; the dimension m of the variable vector z a p p e a r e d in (5.10) (or (5.11)) is always greater than the dimension n of the variable v e c t o r x of the system f ( x ) = 0. The following lemma, however, reveals that when we are in the middle of computation of the path S °, the nonnegative variable vector z E R m has at most n nonzero c o m p o n e n t s ; hence z E R "~ lies in some faces of R ? with dimension at most n. F u r t h e r m o r e we shall show that the expression of x by z is unique in the middle of the computation. These facts should be utilized in making an efficient c o m p u t e r code for generating the path S O. L e m m a 5.5. S u p p o s e (t ~, z ~, y~) ~ H-~(0) and t ~> O. L e t 2 = B z ~, K ~= {i: z~i > 0} and k = #K~. (i) z ~ is a m i n i m u m solution to a linear program: minimize

eT z,

subject to B z = 2, z~0. (ii) k 0 and 2 = B z 2, then z ~ = z 2. Proof. L e t y l = y~/t l,

al=_W(z~)/t

~,

M. Kojima and Y. Yamamoto/ Variable dimension algorithms

191

and t ~ = : i : y ' , = 0}. Then we have that B z ~ = £,

z ~:> 0,

~1 + BT/~ = e,

~1--__0,

0 = (:¢l)~zl = eTz I -- "2Ta t.

By the duality theorem for linear programs (see, for instance, Dantzig [1]), we obtain (i) and that ~1 is a solution to the dual of the above linear program. By the assumption (5.8), we also see that Bx, has rank # I 1. Since K * C I ~, we have shown (ii). The assertion (iii) is readily seen if we note that the optimal solution to the linear program in (i) is unique if the dual problem is nondegenerate (see the assumption (5.8)). The following corollary is immediate from L e m m a 5.5.

Corollary 5.6. I f (t, z, y) E H-'(0), t > 0 and Ilzll-~ ÷~, then IIBzll-~ ÷~. N e x t theorem shows that Merrill's condition [21] guarantees homeomorphic to ( - ~ , 0].

S O to be

Theorem 5.7. A s s u m e t h a t f o r a n y 8o > 0 there exist Ix > 0 a n d £ E R " s u c h t h a t IIx - x~l-> ~ a n d Uv - xll 0. T h e n S o is h o m e o m o r p h i c to (-o~, 0]. Proof. Suppose on the contrary that S o is homeomorphic to (-o~, +~). L e t 80 -- IIBIIS*. By L e m m a 5.3 (iii) and Corollary 5.6, we have (t, z, y) E S Osuch that

t>0,

z-0,

B T W ( z ) + et >- O,

IIBzll-->

~,

z T ( B T W ( z ) + et) = O, (z -- ~)Te > O,

where ~ E R ~ is a vector satisfying ~ = B 2 + x °. L e t o- be a simplex of ~ * containing z, z' be an arbitrary vertex of ~r, x = B z + x ° and x ' = B z ' + x °. Then by the assumption, we obtain (z - ~)TBTw(z')

= ( x --

~)T/(x') > 0

M. Koi.ima and Y. Yamamoto/ Variable dimension algorithms

192

b e c a u s e ]Ix - x']l < IIBII, Ifz - z'll implies that

80. This, t o g e t h e r with t > 0 and (z

-

~)T e > O,

0 < (Z -- f f . ) T ( B T W ( z ) q- et) = ( - - z . ) T ( B T w ( z ) q- et).

This is c o n t r a r y to ~ --- 0 and B T W ( z ) + et >=O. W h e n we c h o o s e [ / , - I ] for the m a t r i x B, the m e t h o d for solving f ( x ) = 0 w h i c h we h a v e stated has a close relation with the variable d i m e n s i o n algorithms given b y T o d d [32] and V a n der L a a n and T a l m a n [15] (see also Reiser [23]). To see this we shall t r a n s f o r m the s y s t e m (5.11) to a s y s t e m of P L equations w h o s e solution set is t r a c e d b y those algorithms. S u p p o s e (t, z, y) ~ H-~(0), then, as s h o w n in L e m m a 5.5, the c o l u m n v e c t o r s of BK are linearly i n d e p e n d e n t , w h e r e K = {i: z~ > 0}. If follows that z~ • z,+~ = 0 for a n y i ~ N = {1, 2 . . . . . n}. Define a v e c t o r s E R" as 1

s~=

-1 0

if zi > 0,

ifz,+i>0, otherwise,

and X ( s ) = {x + x ° E R " : slxl >= 0 (i E N ) , xi = 0 w h e n e v e r si = 0}, Y ( s ) = {y ~ R " : s,y, --- IlYlI~ w h e n e v e r s i s 0}.

Then we have x = B z + x ° E rel int X ( s ) .

(5.12)

B y the c o m p l e m e n t a r i t y condition Yi • z~ = 0 (i E M ) we also h a v e Wi(z)=-t->_0

if s~ = - 1

IW,(z)l-< t

(i.e.,z~>0), (i.e., z,+~ > 0),

otherwise,

w h i c h is e q u i v a l e n t to W(z) ~ - Y(s).

(5.13)

On the other h a n d a s s u m e R:+" is subdivided b y K1 (see T o d d [29]). Since zi • z,+~ = 0 (i E N ) , there is a partition (/, J ) of N such that z~Z

= { z E R +2, : zi = 0 (i E I ) , z,+~ = 0 (i U J)}.

Since Z is also subdivided b y K1 of d i m e n s i o n n, we can find an n s i m p l e x ~'=co{v°,v~,...,v"}CZ containing z, w h e r e v ° is an integer point in Z , v k ( k = l , 2 .... , n ) is written as V k = V k - l + e ~ ( k ~ b y using a p e r m u t a t i o n of {1, 2 . . . . . 2n} such that {or(l) . . . . . 7r(n)} = J U { n + i: i E I}, e ~ d e n o t e s the ith unit vector.

M. Kojima and Y. Yamamoto/ Variable dimension algorithms

193

Define u ~ = A v k,

~(k)=

~r

n

otherwise,

where A = [ I , I ] of size n x 2 n , then u ° is also an integer point of R+, u k = u k-~ + e ¢(k~ and ~ is a permutation of {1, 2 .... , n}. This means that A r is a simplex of the triangulation K~ of R+. Furthermore let S be an n x n diagonal matrix having nonzero components sii=

{ 11 i f i E J , _ ifi~I,

then we can see that B'r = SA'r,

which implies that B~- is a simplex of K ' o f R n (see Todd [29]). T h e r e f o r e the simplicial approximation W(z) of w ( z ) with respect to K~ C R 2n is identical to the simplicial approximation F ( x ) of f(x) with respect to K ' + x ° C R n as long as the path S O is traced. Combining this result, (5.12) and (5.13), we can conclude that if K1 and K ' + x ° are employed to approximate w and f, respectively, (5.11) is equivalent to the system y + F ( x ) = O,

x E X(s), y E Y(s).

(5.14)

In the case where B = [I, - e l we can also show in the similar way that (5.11) is equivalent to the system y + F ( x ) = O,

x E X ( I ) , y ~ Y ( N * ~ I)

(5.15)

are employed to make simplicial approvided K I C R n+l and K ~ + x ° C R " proximations of w and f, respectively, where X ( I ) and Y ( I ) are cones which will be defined in (5.17) and (5.18) by using the set of vectors in (5.25) in the next subsection. The system of P L equations (5.14) and (5.15) were implicitly used by Van der Laan and Talman [15], and Todd [32] in their variable dimension algorithms. 5.3. A variable dimension algorithm on conical subdivisions

We give an interpretation in terms of the basic model to the method originally proposed by Van der Laan and Talman [17] for computing Brouwer's fixed points on the standard simplex { x~Rn:x>--O

and

~xi=l}~=l

and then extended by Van der Laan and Talman [18] and Todd [31] to a problem

M. Kojima and Y. Yamamoto[ Variable dimension algorithms

194

of finding fixed points on R". T h e interpretation here is based on K o j i m a [11]. For simplicity of discussions, we shall confine ourselves to a system of equations f(x) = 0, (5.16) where f : R n ~ R" is a continuous function. Obviously the problem of finding a fixed point of g : R " ~ R " can be c o n v e r t e d into the system (5.16) if we set f(x) = g(x) - x for all x ~ R". We first construct two subdivisions ~ and ~ of R" by n + 1 polyhedral cones. Let p~, d ~ (i = 0,1, ... , n) be 2 . ( n + l ) nonzero vectors in R". L e t N * = {0, 1. . . . . n}. For each I { N * , define

X(I)=(~A,p'+x°:A,>=O

(i E I ) }

(5.17)

and

Y(I)={~hidi:hi>=O

(/El)}.

(5.18)

Specifically, X ( 0 ) = { x °} and

Y(0)={0}.

Here x ° is a point in R" which will serve as the initial point of our algorithm. Define ~9 = {X(I): I C N * , # I = n} and = ( Y ( I ) : I C N * , # l = n). We shall impose the following two conditions on the collections ~ and 9 : (5.19)

Each z E ~ U ~ is an n-cell.

(5.20)

3~

and

~ are subdivisions of R".

Note that the condition (5.19) can be stated as for every i C N * the set of n vectors pJ ( j E N * - { i } ) and the set of n vectors d j ( j E N * - { i } ) are both linearly independent. We see that # = {X(I): I C N*},

= {Y(I): t c x*}. For each I C N * , define -# X(I)a={Y(No-I)

if#I_->l, if # I = O,

y(i)d = { X ( N o ~ I)

if # I _-> 1, if # I = O.

M. Kojima and Y. Yamamoto[Variable dimension algorithms

195

Then (~, @; d) forms a P D M with degree n + 1. Fig. 1 shows an example where we take n = 2. L e t ~ = (~, 9 ; d>. T h e o r e m 5.8.

~={xx{o} (x~),{x°}xY (Y~)}. C o r o l l a r y 5.9.

al l

--

R n x {0} U {x °} x R".

The p r o o f of the theorem is quite similar to the p r o o f of T h e o r e m 5.1 and is omitted here. The corollary is a direct consequence of the theorem. L e t ~ * be a simplicial refinement of ~ with a bounded mesh size 8" and let ~t = { c r x Y: Y E ~ ,

Y a ¢ O, o~E ~*1 yd}.

Then d~ is a refinement of ~. Using the simplicial subdivision ~ * of R n, we approximate f : R" -* R" b y a P L map F :[~*]---> R n ; for each x E cr E ~ * with

x = 2 A i v i, 2 A , = l , i =0

k,----O

( i = 0 , 1 .... , n )

i =0

where v °, v ~. . . . . v" are distinct vertices of or, define

F(x) = 2 )td(v'). i=0

If the mesh size 8* of the simplicial subdivision ~ * of R ~ is sufficiently small, we will have F ( x ) - f ( x ) for all x E Rn. Let H ( x , y) = y + F(x)

for all (x, y) E I ~ I,

and consider the system of P L equations

H(x, y) = 0,

(x, y) ~ I~l.

(5.21)

We shall assume that 0 is a regular value of the P L map H : I ~ [ ~ R " . Then H - ' ( 0 ) , the solution set to the system (5.21) is a disjoint union of paths and loops and satisfies (2.4)-(2.6). L e t y0 = _F(x o) Since ~ is a subdivision of R" and 0 is supposed to be a regular value of H, we can find a unique n-cell Yo E ~ which contains y0. L e t

Xo= yd. Obviously, dim Xo = 1 and

{x°}< Xo= Vod.

196

M. K o j i m a and Y. Y a m a m o t o [ Variable dimension algorithms

T h u s w e obtain

(x °, y°)E{x°}x Y o < X o x Y0 E J/, (x 0, y0) e n-'(0). F u r t h e r m o r e , b y C o r o l l a r y 5.9, we see that (x °, yO) lies in the b o u n d a r y of I~1 = pel. H e n c e the c o n n e c t e d c o m p o n e n t , d e n o t e d b y S °, of H ~(0) containing (x 0, yO) f o r m s a path. S u p p o s e that the path S o is b o u n d e d . It is easily verified that H-~(0)f-I C is b o u n d e d f o r e v e r y C E ://. W e also see that any n o n e m p t y b o u n d e d set in intersects with finitely m a n y (n + 1)-cells of A/. H e n c e S O consists of finitely m a n y line s e g m e n t s , and S O is c o m p a c t . B y T h e o r e m 2.1, the path S O starting f r o m (x °, y ° ) E 01 1 t e r m i n a t e s at s o m e (x l, y l ) E 01 1 w h i c h is distinct f r o m (x 0, y0). W e shall show that y ~ = - F ( x l ) = 0 ; hence x ~ gives an a p p r o x i m a t e solution to f(x) = 0. Since (x ~, y~)E alJ/[, b y T h e o r e m 5.2, we have either (x ~, y l ) E { x ° } x Y

for s o m e Y E ~

(5.22)

(x ~,y~)~Xx{O}

for s o m e X E ~ .

(5.23)

or

If (5.22) o c c u r s , then x ~ = x ° and y~ = - F ( x J) b e c a u s e (x ~, y ~) satisfies (5.21). This contradicts to (x 1, y~) # (x °, yO). T h u s we m u s t have (5.23), which t o g e t h e r with (x ~, y~)E S ° C H-~(0) implies 0 = y~ = -F(x~). T h e r e f o r e , tracing the path S O we can attain an a p p r o x i m a t e solution to f(x)= 0. Replacing x ° b y x ~ and refining the primal subdivided m a n i f o l d into finer simplices, we can apply the s a m e p r o c e d u r e to get an a p p r o x i m a t e solution to f(x) = 0 with higher a c c u r a c y . In order to ensure that the path S Ois b o u n d e d we need an a p p r o p r i a t e condition. O b v i o u s l y , if we c h o o s e a n o t h e r set of 2 • (n + 1) v e c t o r s p~, d ~ (i = 0, l .... , n), the resultant s y s t e m (5.21) of P L equations yields a n o t h e r path S O which is generally different f r o m the original one. In w h a t follows, we a s s u m e that the set of 2 • (n + 1) v e c t o r s p~, d ~ (i = 0, 1 . . . . . n) has b e e n c h o s e n such that (p i)TdJ > 0

(i # j).

(5.24)

F o r e x a m p l e , we m a y take pO= _

ei i=1

pi = e i

d o=

(i = 1, 2, ..., n) ei

(5.25)

i-I

d i = ~ e j - ( n + l ) e i (i=1,2 ..... n), j-1

w h e r e e i d e n o t e s the ith unit v e c t o r of R". In this case we can e m p l o y the triangulation K1 and J1 (see |29]) with a n y m e s h size as ~ * . It follows f r o m (5.24)

M. Kojima and Y. Yamamoto/ Variable dimension algorithms

197

that (X--x0)Ty >0

f o r e v e r y (x, y) ~ Pel with ]Ix - x°l} > 0 and Ily}l> 0. H e n c e we can find a positive n u m b e r a such that (x - x°)Ty -----allX -- X°II • [[yllfor e v e r y (x, y) E WI.

(5.26)

T h e a b o v e choice (5.25) of the 2 . (n + 1) v e c t o r s p~, d ~ (i = 0, 1 . . . . . n) and the t h e o r e m below, w h i c h s h o w s that Merrill's condition [21] in T h e o r e m 5.7 also guarantees the b o u n d e d n e s s of S °, are essentially due to the p a p e r [31] by Todd. T h e o r e m 5.10. A s s u m e that (5.26) holds f o r s o m e p o s i t i v e n u m b e r a, and the s a m e condition as in T h e o r e m 5.7. T h e n S O is bounded. Proof. L e t 30 = 6", the m e s h size of 3~*, and 3' = max{/x, IIx° - 21[/a }. Since e v e r y (x, y ) E S O satisfies y = - F ( x ) , if the set T = {x E R " : (x, y) E S O f o r s o m e y E R"} is b o u n d e d then so is S °. A s s u m e on the c o n t r a r y that T is u n b o u n d e d . T h e n there is an (2, Y) E S O such that

Iix - x°ll = v ~ ~. It follows f r o m (2, Y) E S Othat

y = -F(2)

(2, 9) ~ WI.

L e t a be an n-simplex of ~ * w h i c h contains 2. T h e n , by the definition of F : l ~ * l ~ R " , we h a v e

F ( ~ ) = ~] x J ( v ' ) , i=0

IIv' - Ell y(q2) if and only if X ( q l) < X(q2). Prool. It suffices to show the 'only if' part. Suppose that Y(q~)> y(q2). T h e n it follows f r o m the observation in the p r o o f a b o v e that there exists I C Ie(q ~) such that q~=q',-lorq',+l

(iEICIe(q~)),

q~ = q',

(i~ ,).

H e n c e we immediately have X ( q 1)< X(q2). Define two c h e c k e r b o a r d subdivisions of R" as = {X(q): q E R ~, every c o m p o n e n t of q is an odd integer}, = {Y(q): q E R", every c o m p o n e n t of q is an even integer}, then ~ and g are subdivided manifolds of dimension n and = {X(q): q ~ R", q is an integer vector}, = { g ( q ) : q E R", q is an integer vector}.

202

M. Kojima and Y. Yamamoto[ Variable dimension algorithms

T h e r e f o r e defining the dual o p e r a t o r d as (X(q))d = y ( q ) ,

(y(q))d = X(q),

for X ( q ) E ~ and Y ( q ) ~ 2 , we h a v e a P D M ( ~ , @ ; d) with degree n. If we t o o k K1 (see, for e x a m p l e , T o d d [29]) instead of J1, we could obtain in the similar w a y as a b o v e a n o t h e r P D M ( ~ ' , @'; c) with degree n: X ' ( q ; I) = {x @ R": xi = q~(i ~ I), q~ 0 . T h e n we can find a s i m p l e x cr = co{v °, v ~. . . . . v " } ~ J~ and n o n n e g a t i v e n u m b e r s ;to, ;~ . . . . . an such that

t = ~ A.i, i=O

Z = k A-iVi. i=0

Hence

zlt = ~ (,xdt)v i i=O

( a d o = 1,

a.dt >=o (i = o, 1 . . . . . n).

i=0

B y the definition we h a v e that

F~(t, z) = t F ( z / t ) i tF(~=o(,Xlt)v

= k i=0

hiF(vi) •

(5.30)

204

M. Kojima and Y. Yamamoto[ Variable dimension algorithms

This implies that F ~ is linear on each cell of Jl. Let ~ = {cone(I): X E~}, then ~1 satisfies the conditions (2.1) and (2.2) in Section 2, for subdivided manifolds, but not the local finiteness (2.3). Define the dual operator c as [ ~( 1c)o]n e

=X d

foreveryXE~,

{0} c = 0,

yc = cone(yld)

for every Y ~ 9 ,

then the triplet ( ~ , 9 ; c) satisfies the conditions for a P D M except that ~ is not a subdivided manifold. Let 21 = ( ~ , 9 ; c), then we have the following lemma. Lemma 5.13. 21 is a subdivided manifold o f dimension n + 1 and al2l I = {O} x R". Proof. It is sufficient to show the local finiteness, i.e., each point of ]M] has a neighborhood which intersects with only finitely many cells of 21. Let z = (x, y ) E ]211, where x ~ 1~11, Y ~ [@l. Since @ has the local finiteness, there is a neighborhood V of y which meets only finitely many cells of 9. Let U be an arbitrary neighborhood of x. Then W = U x ' V is the desired neighborhood of z. Now letting H i ( t , z, y) -- y + (A-IF~(t, z) - x°),

(5.31)

we consider the system of P L equations H~(t, z, y) -- 0

(t, z, y) ~ 1211.

(5.32)

We can see the relation between the solutions to (5.29) and (5.32): Lemma 5.14. (i) ( a , x , y) with a > 0 is a solution to (5.29) if and only if (t, z, y) = ((1 - ct)/a, (1 - a ) x / a , y) is a solution to (5.32). (ii) L e t { ( t * + O A t , z * + O A z , y * + 0 z l y ) : 0 = > 0 } is an u n b o u n d e d ray in the solution set of (5.32) such that Ay = O. Then (a, x, y) = (0, ziz/At, y*) is a solution to (5.29). Proof. Since (i) is clear, we shall only show (ii). Let Y E ~ be a cell which contains y*. From the assumption we see that the ray R = {(t*+ OAt, z * + OAz): 0 =>0} stays within a cell y c = c o n e ( 1 / X ) E ~1. Obviously At -->0. If At were zero, then Az ~ 0 and the ray R would intersect infinitely many cells of ~ 1, which is a contradiction. Hence we have that At > 0 and R lies within

M. Kojima and Y. Yamamoto/ Variable dimension algorithms

cone(1/~r) C ye, where (r C.Tt]X. F Lon c o n e ( l h r ) that

205

T h e r e f o r e it follows from the linearity of

0 = F~(t * + At, z* + a z ) - F~(t *, z*) = FI(At, Az) = A t F ( A z / A t ) .

It is clear that (At, n z ) E cone(x~), or equivalently Az/At E X = y d . This implies that (Az/At, y*) E [~']. Let (t °, z °, yO) ~ aid/] be a solution to (5.32), then we can see that (t °, z °) = 0 E R n÷~, and from L e m m a 5.13 and the assumption that A is nonsingular, yO= x 0. Assume that zero is a regular value of H ~, then the connected component S Oof solutions to (5.32) having (t °, z °, yO) forms a path. Suppose S O is homeomorphic to [ 0, 1] and let (t ~, z ~, y~) ~ (t °, z °, yo) be an endpoint. Then (t ~, z ~, y~)~ aiM[, which implies that (t ~, z j, y~)--(t °, z °, yO). This is a contradiction. Thus S O is homeomorphic to [0, +~). Let us further assume that T = { Y E Rn: (t,z, y ) E S °}

is bounded. Then S o intersects with only finitely many cells of ~ , so that S o contains a half line, which is written as {(t* + OAt, z* + OAz, y* + 0Ay): 0 ~ 0}. We immediately have Ay = 0. Therefore from L e m m a 5.14, we have an approximate solution /1z/At to f ( x ) = O. It follows from (5.30) and (5.31) that the system of equations considered in the computation procedure of tracing the path S Ois as follows: k

A ( y - x °) + ~. ;tJ(v') = O, i-O

CO{ '1)0, V l . . . . . l) k} ~ J1 [ X ( q ) ,

y @ Y(q), A~_->0 ( i = 0 , 1 . . . . . k),

(5.33)

where q E R n is an integer vector. Comparing with the original P L system H ( x , t ) = 0, we notice an important feature of the system (5.33). That is, the simplicial subdivision J~ which we use in the system (5.33) is the Union Jack triangulation with the dimension n, which refines ~, while the original system has required a triangulation with dimension n + 1. Furthermore, the constraint y E Y ( q ) in the system (5.33) can be handled by the ordinary bounded variable technique for the simplex method, so that we don't need any simplicial refinement of 9 , either. Hence we can save a large amount of pivoting operations in the artificial level, the level at t = 1. This feature has a close relation with a technique which has been developed by T o d d [30, 33] for saving pivoting operations in general fixed point algorithms and can be effectively utilized in Merrill's algorithm to increase the computational efficiency. It is also important that X ( q ) , Y ( q ) and the Union Jack triangulation J~ in

M. Kofima and Y. Yamamoto/ Variable dimension algorithms

206

(5.33) can be r e p l a c e d with o X ( q ) , o Y ( q ) and a simplicial refinement ~ * of p ~ f o r a n y p > 0. H e r e o X = {px: x ~ X } for a subset X o f R" and OSg = { o X : X E ~ } for a collection of subsets of R". If we take a smaller 0 or a finer refinement ~ * of O~, an a p p r o x i m a t e solution to f ( x ) = 0 c o m p u t e d by the algorithm will have higher a c c u r a c y . The following t h e o r e m e n s u r e s that the algorithm will s u c c e e d in finding an a p p r o x i m a t e solution to f ( x ) = 0. This t h e o r e m is closely related to L e m m a 3.2, C h a p t e r V I I I in T o d d [29]. T h e o r e m 5.15. A s s u m e that A is positive definite and the s a m e condition as in Theorem 5.7. Then T = {y E R~: (t, z, y) E S °} is bounded. Proof. L e t "q = min{xXAx: x E R ~, Ilx[[= 1}. Since A is positive definite, rl > 0. L e t ~* be the m e s h size of ~ * , 80 = max{pX/n,, 6"}, and /x' = max{/x + ~0, IIAIl(~o + llx ° - 211)/~}. S u p p o s e on the c o n t r a r y that T is u n b o u n d e d . T h e n we can find (t-, L f ) E S O such that 27e x ° and

1127- 211 >

p

(5.34)

/n.

Thus t- > 0, b e c a u s e otherwise we h a v e 2 = 0 and 27 = x °, a contradction. H e n c e if we let ~ = ~[{, we have

F ' ( { , ;~) = {F(g) = t- ~ ,Xtf(vi), i=0

w h e r e v °, v ~. . . . . v" are the vertices of an n-simplex o f ~ * containing X and ~--~Aiv i=0

i,

~-~Xi=l,

Ai=>0

( i = - 0 , 1 . . . . . n).

i=0

L e t Y ( q ) be a cell of ~ containing 27. Since 27E Y ( q ) implies that ~ E X ( q ) , we h a v e that I1~ - 2711X1

0

2o

2-1

\

2-2 b

oi

Fig. 6. (a) rk = 2,

>x~ Sk

=

1 for all k. (b) rk =

Sk

=

2 for all k.

r ~ ~*(k - 1) [ ~k ,X(q) and ~ E N*(k) I gk-lY(q) such that

Hence the pivoting rule for ~ turns out to be an appropriate combination of the pivoting rule for ~* and the dual operation. See Mizuno [22] for the complete pivoting rule.

7. Concluding remarks In this paper we have confined ourselves to studying simplicial variable dimension algorithms. It is also possible to establish a unified framework for continuation version of variable dimension algorithms by only considering the system of equations

h(x, y) = y + f(x) = c,

(x, y) ~ I~el

instead of the system (4.4). Under an appropriate regularity condition and differentiability of f we can prove that the solution set of this system is a disjoint union of paths and loops which are smooth on each cell of Af. Therefore we can apply the well-known predictor-corrector method to trace the piecewise smooth path of solutions. See Kojima [12] for further discussions.

214

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