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A class of linear complementarity problems solvable in polynomial time
A Class of Linear Complementarity Problems Solvable in Polynomial Time Yinyu Ye* Department of Management Sciences College of Business Administration The University of Iowa lowa City, lowa 52242 and Panos M. Pardalos Department of Computer Science The Pennsylvania State University University Park, Pennsylvania
16802
Submitted by Masakaju Kojima
ABSTRACT We describe a “condition” number for the linear complementarity problem (LCP), which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used. Consequently, we develop a class of LCPs solvable in polynomial time. The result suggests that the convexity (or positive semidefiniteness) of the LCP may not be the basic issue that separates LCPs solvable and not solvable in polynomial time.
INTRODUCTION In this paper, we are concerned with the linear complementarity (LCP), that is, to find a pair x, y E R” such that
x=y = 0,
y=Mx+q,
*Research was supported in part by the
1989 College
LINEAR ALGEBRA AND ITS APPLlCATIONS OEIsevier
and
problem
r,y>O,
Summer Grant.
1523-17
(1991)
3
Science Publishing Co., Inc., 1991
655 Avenue of the Americas, New York, NY 10010
0024-3795/91/$3.50
4
YINYU YE AND PANOS M. PARDALOS
where we assume that all components of M and 9 are integers, and fif = 1(x, y>: y = Mx + q, x > 0,and y > 0) is nonempty. Kojima, Megiddo, and Ye [7] have discussed how to transform an LCP with empty Slf to an equivalent LCP with nonempty 1R+. Thus, the last assumption is merely added for simplicity. We also use s1 to denote the feasible region, i.e., fI = {(x, y): y = Mx + 9, x > 0, and y > O}. If M is a 2 matrix, Cottle and Veinott [2] and Mangasarian [lo] showed that the LCP can be solved as a linear program; therefore, it can be solved in polynomial time. Pang and Chandrasekaran [15] also showed that some special LCPs can be solved in n pivots using several pivoting methods. If M is a positive semidefinite matrix, the LCP is simply a convex quadratic programming problem, and it can be solved in polynomial time by the ellipsoid method (Khachiyan [6]), the projective method (e.g., Kapoor and Vaidya [5] and Ye and Tse [24]), the path-following method (e.g., Kojima, Mizuno, and Yoshise [9] and Monteiro and Adler [12]), and the potentialreduction method (e.g., Kojima, Megiddo, and Ye [7], Kojima, Mizuno, and Yoshise [S], and Pardalos, Ye, and Han [17]). The best complexity result for a convex LCP is 0(&L) iterations and O(n”L) total arithmetic where L is the size of the input data of the problem.
operations,
If M is a P matrix, Ye [22] showed that the potential-reduction algorithm of Kojima et al. [7] solves the LCP in O(n”L max(\hl/nB, 1)) iterations and each iteration solves a system of linear equations in at most O(n">arithmetic operations, where A is the least eigenvalue of so-called P-matrix number for M“, that is,
(M + MT)/2,
and 0 > 0 is the
This indicates that Ih(/nO can be used to measure the degree of difficulty for solving the P-matrix LCP (see also a related discussion of Mathias and Pang [ll]). Th e a lg on ‘th m is a polynomial-time algorithm if (Al/n0 is bounded above by a polynomial in L and n. In this paper, we describe a condition number for the genera1 LCP, which characterizes the degree of difficulty for finding its solution when Kojima et al.‘s potential-reduction algorithm is used. Consequently, we develop a new class of LCPs solvable in polynomial time. We show how the condition number varies as the data (M,9) changes. We also show that several existing classes and some examples of LCPs belong to our class. The result again suggests that the convexity (or positive semidefiniteness) of the LCP may not be the basic issue that separates LCPs that are solvable in polynomial time from ones that are not.
A CLASS OF LCP’S SOLVABLE 1.
POTENTIAL REDUCTION
5
IN POLYNOMIAL TIME
FUNCTIONS
AND POTENTIAL
ALGORITHMS
We use the potential
function
4(X, Y) = p ln(rTv)-
iI
lnCxjYj)
j=l
to associate with an interior feasible solution (x, y) with p > n. This function first appeared in Todd and Ye [21], and was first used to develop an 0(n3L) potential-reduction algorithm for linear programming by Ye [23]. Soon after, this function was used for solving the convex LCP [7, 81. Starting from an interior point (r’, y”> with
the potential-reduction solutions
algorithm
(rk, yk} terminating
generates
a sequence
of interior
feasible
at a point such that
4(xk,yk)<-(p-n)L. From the arithmetic-geometric-mean
nln[(xk)‘yk]
inequality,
- j$lln(r,)yJ)aolnn>O.
Thus.
and an exact solution to LCP can be obtained
in 0(n3)
additional operations
M. To achieve a potential reduction, Kojima et al. used the scaled gradient projection method. The gradient vector of the potential function with respect to x is V&, = i y - X-‘e,
6
YINYU YE AND PANOS M. PARDALOS
and the one with respect
to y is
v+y= g,
-Y-b,
where A = rry, X = diag(x), Y = diag( y), and e is the vector of all ones. Now, let us solve the following linear program subject to an ellipsoid constraint at the k th iteration: minimize
vC$%k
subject
6y = M&x,
to
8X
+
and denote by 8X and Sy its minimal
v&k
&j
solutions.
Then, we have
where (
, Ak =(rk)ryk, Let rk+l
PYk( Ak
e’
MW)-
$Xk(yk+
xk - rk) -
e I
and Xk (Yk) designates the diagonal matrix of xk (yk). = xk +6X and y k+l = yk + So. Then it can be verified that
~(xx”,yx”)-6(xk,Yk)~-BllPkli+~
(P+& I.
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL Therefore,
TIME
choosing
p=
llPkll
min
i
p+2’2
1
-
1
I
<---, 2
we have
&(Xk+I,yk+‘)-
c$(xk,yk) Q
-
“(llPkl12),
(1)
where
if
llpk02 ,< (P +2)2/4,
otherwise.
2.
A CONDITION
NUMBER
FOR THE
LCP
Naturally, IlpkI12 can be used to measure the potential kth iteration of the potential-reduction algorithm. Let
reduction
at the
g(x,y) = $xY -e and H(x,y)=21-(XMr-Y)(Ys+MXW)-1(A4X-Y). It can be verified
that H(x, y) is positive semidefinite
(PSD).
llpkl12 = gT(Xk,yk)H(zk,yk)g(Xk,~k). Let us use Ilg(x, y>lI”, to denote gT(x, y)H(x, the condition number for the LCP (M, q> as y(M,4)=inf((lg(r,y)119:rTy>2-L,
Note that
(2)
y>g(x, y>. Then, we define
#(x,y)~O(pL),and(x,y)En+). (3)
8
YINYU YE AND PANOS M. PARDALOS
The quantity ]lg(x, y)l]i first appeared in Pardalos and Ye [16] for the row-sufficient matrix M of Cottle, Pang, and Venkateswaran [I]. Our condition number is different from the one defined in Kojima et al. [7], where
y(M)
= (‘-
n
n)” inf{h(M,X,Y):(x,y)
and ACM, X, Y) is the smallest
eigenvalue
~a+}
of the matrix
(I+Y-~MX)(I+XM~Y-“MX)-~(I+XM~Y-~). Of course,
the two capture
similar aspects of positivity
related
to the matrix
H(x, y). We present
a sequence
PROPOSITION 1.
of propositions
for y(M, 9).
Let p 2 2 n. Then, for M a diagonal and PSD matrix,
and any 9 E R”, Y(M,~) Proof. If MX2MT)-l(MX component
M is diagonal and PSD, then I -(XMT - YxY2 + - Y) is diagonal. It is also PSD, since the jth diagonal
is
1_
(Mjjxj-
Yj)”=
yj’ + M;x; Therefore.
z n.
2MjjxjYj y,? + Mj”jx;
>0.
for all (x, y) E CI+ and p 2 2%
PROPOSITION 2 (Kojima,
Megiddo,
for M a PSD matrix and any 9 E R”,
and Ye [7]).
Let p > 2n + \lzn. Then,
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL
PROPOSITION 3 (Ye [22]).
9
TIME
Let p > 3n + \/2n. Then, for M a P matrix and
any q E R”,
where A is the least eigenvalue number of MT, i.e.,
of (M + MT )/2,
and
PROPOSITION 4. lit p > n and be fixed. Then, matrix and {(x, y> E R+: 4(x, y) Q O(pL)} bounded,
Both Pang [14] and Pardalos
Proof.
f~
6 is the P-matrix
M a row-suficient
and Ye [16] showed that for any
(x, y> E Cl+
Moreover,
for all (x,y) E Cl+, xTy > 2-L, and c$(x, y) < O(pL)
=pln(xTy)-
i
we have
ln(xjyj)
j=l
=(p-n+l)ln(xTy)+(n-l)ln(xTy)-
C j#i
>(p-n+l)ln(XTy)+(n-l)ln(Xry-riyi) -
C
lnCxjYj)
-WxiYi)
j#i
a(p-n+l)ln(xTy)+(fz-l)ln(n-I)-ln(X,y,) >,-(p-n+l)L+(n-l)ln(n-1)-ln(riy,),
ln(xjyj)-ln(x,y,)
YINYU YE AND PANOS M. PARDALOS
10
where
i E { 1,2,. . . , n). Thus,
ln(xjyi)>-(p-n+l)L+(n-l)In(n-l)-O(pL)=-O(pL), that is, xi yi is bounded
away from zero by e --OCPL)for all i. Since {(x, y>E there must exist a positive number E,
R+ : 4(x, y) < O(pLN is bounded, independent of (x, y>, such that xi > E and
yi >E,
for all (r, y) such that rry 2 2-L,
i=1,2
,...,
$(x, y) f O(PL),
12, and (r, y) E a+.
There-
fore,
> inf{llg(x,y)llZ:x > Ee, y > Fe, 4,(x, y) 6 O(pL),
and (x, y> E a}
> 0. The last inequality
holds because
the inf is taken in a compact
set where
Ilg(x, y>[[L is always positive. Note that the
+(r, y) G O(pL)
boundedness
of
n implies
((x, y) E a:
boundedness of {(x, y) E a+: We now derive
that
rTy d 0Cp.L /(p - n)). Hence,
rry < O(pL/(p
- n)))
guarantees
the
4(x, y) G O(PL)I.
The potential-reduction algorithm with p = e(n) > n solves THEOREM 1. the LCP for which y(M, q) > 0 in O(nL /a(y(M, 4))) iterations and each iteration solves a system of linear equations in at most 0(n3) operations, where LY(.) is defined in (1).
Proof. Since LR+ is nonempty, by solving a linear program in polynomial time, we can find an interior feasible point (x0, y”X each component of which is greater than 2-= and less than 2L. The resulting point has an initial potential value less than O(nL). Due to (11, (21, and (31, the potential function is reduced by O(&y(M, 9))) a t each iteration. Hence, in the total of O(nL/cu(y(M,q))) iterations we have 4(rk,yk)< -(p - n)L and (xk)‘yk <2-L.
W
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL COROLLARY
1.
TIME
11
An instance (M, 9) of the LCP is solvable in polynomial
time if y(M, 9) > 0 and if 1/ y(M, 9) is bounded and n.
above by a polynomial
in L
The condition number y(M, 9) represents the degree of diflkulty for the potential-reduction algorithm in solving the LCP (M, 9). The larger the condition number, the easier the LCP. We know that some LCPs are very hard, and some are easy. Here, the condition number builds a connection from easy LCPs to hard LCPs. In other words, the corresponding degree of difficulty shifts continuously from easy to hard LCPs. We feel that such a condition number will be an important criterion in analyzing the complexity of algorithms for optimization problems.
3.
A CLASS OF LCP’S SOLVABLE
We now further LEMMA1.
IN POLYNOMIAL
study IIpkII by first introducing
TIME
the following
lemma.
IIpklI < 1 implies yk+MT.rrk>O,
xk-rrk>O,
2n-J2n
2n+\/2n
and
Akk
P
+ MTak)+(yk)T(~k
Ak,
- .rrk).
Proof. The proof is by contradiction. Let 5 = yk + MTrk 7rk. It is obvious that if ij 2 0 or X > 0, then
llpkl12 > I. On the other hand, as is developed
in Ye [22], we have
and X = x k -
YINYU YE AND PANOS M. PARDALOS
12 Hence,
the following
must
be true:
that is, 2n+J2n
zn-Jzn Akk<<
Ak P
P Zi can be further
expressed
.
as E = 2Ak - +,&
Thus,
we can prove
PROPOSITION5.
the following
propositions.
Denote by 2’
the set (x,y)Efl+
(77:xTy-qT57
that also satisfy x - 7~ > 0 and y + MTrr > 0) , andlet
2,’
beemptyforan
LCP (M,y). r(M,9)
Proof.
The proof
PROPOSITION6.
directly
results
Then, for p>2n+&, 21.
from (2) and Lemma
1.
Let
{Tr:~~y-9%>o~of-some
(x,y)
En+
that also satisfy x - r > 0 and y + MT~ > 0} be empty for an LCP (M, 9). Then, for n < p B 2n - 6,
13
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL TIME Proof.
The proof again results from (2) and Lemma
1.
l
Now, let + is nonempty
#=((M,q):fi
and C’
is empty).
As pointed out by Stone [20], the description of 3 is technically similar to those of Eaves’s class [3] and Garcia’s class [4]. These two classes and some others have been extensively studied. It may not be possible in polynomial time to tell if an LCP 04, y) 1s an element of 9 (this is also true for some other LCP classes published so far). However, the coproblem, to tell that an LCP (M,q) is not in 9, can be solved in polynomial time. We can simply run the potential-reduction algorithm for the LCP. In polynomial time the algorithm either gives the solution or concludes that (M,9) is not in 9. The coproblem is actually more important in practice. In the following, we present a dual interpretation using the Lagrangian multiplier+ For (M, 9) E ((M, 9): Isol (M, 9)1> 1) (the class where at least one LCP solution exists), the LCP can be represented as an optimization problem with known zero optimal value: (P)
minimize
XTY
subject to
(x,Y)E1R={(x,y):y=Mx+q,x,y~O}.
A dual to this problem
CD)
can be written
as
maximize
9ra
subject to
(“,y,“)E{(x,y,~):x-rr~o,
- xTy
y+MrTf>,O,(X,y)~~). C + being empty means region of(D), the objective the objective value of(P) is of LCPs satisfying the weak 4.
SOME EXISTING
that for all (x, y, rr) in the interior of the feasible value of (D) is less than or equal to zero. Since always greater than or equal to zero, 9 is a class duality condition for (P) and (D).
CLASSES
BELONG
TO THE NEW CLASS
We see that the new class 9 has the same bound on the condition number as the PSD class, that is, y(M,q)> 1. Various other classes of LCPs can be found in Eaves [3], Murty [13], Saigal [18], and Stone [19]. Here, we list several existing classes of LCPs that belong to 9.
YINYU YE AND PANOS M. PARDALOS
14 (1)
M is positive semidefinite
and q is arbitrary.
We have, if C’
is not
empty, 0 <
(x - T)~( y f M%) =
x'y -
9% - rTMTr,
which implies x T y-qTrr>rTMTr>OO,
a contradiction. (2) M is copositive and q > 0. xTy - q%
We have
= xTMx + qT( x - T).
Thus, x > 0 and x - rr > 0 implies xTy - yT.rr 3 0, that is, 2’ (3) M- ’ is copositive and M- ‘q < 0. We have
xTy
-
qTr = ~r’M-~y - (M-‘qf(
Thus, y > 0 and y + MTr > 0 implies
is empty.
y + MTr).
xTy - qTrr >, 0, that is, I!$’ is empty.
Although a trivial solution may exist for the last two classes (e.g., x = 0 and y = q for the second class), our computational experience indicates that the potential-reduction algorithm usually converges to a nontrivial solution if multiple solutions exist. For example, let
M=((:
Then the potential-reduction
-i)
and
algorithm
IX=
M=(y
constantly
and
from virtually any interior starting and y = q. Another example:
1:)
y=(i).
generates
the solution
y=
point,
avoiding
and
y=(i).
the trivial
solution
x= 0
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL Again, the algorithm constantly
generates
and
X=
TIME
the solution
y=
from virtually any interior starting point. Both problems
iterations, although the second is a nonconvex certainly deserves further research. Another nonconvex LCP also belongs to 9:
M=(i
-:)
and
15
q=(
converge
problem.
This
in a few property
1:).
since x = (3 ljT is an interior feasible This is because R + is nonempty, point; and Z-’ is empty, since xi - x2 > 1, x, - 7Tl> 0, x2 - 772 > 0, x, - xg -1.+7r,+27r,>O,and2~,-1-7r,>Oimply xTy -
qT77 =
x’( Mx + q) - qTz-
= r,( x1 - X2) +2x,x,
- Xi - x2 + 7r, + 7ra
=rf+x,x,-2x,-X2,+1+(r,-x,-1+x,+2n-,)+(x,-7r,)
> xi2 + rlxz -2x,
which indicates
5.
- Xp + I
that Z + is empty.
CONCLUDING As a by-product,
REMARKS we have
In fact, any LCP (M, q> with y(M, q) > 0 belongs to {(M, q): Isol(M, q)l z 1). Furthermore, if y(M, q) > 0 for all q E R”, then M E 9, the matrix class where the LCP (M, q) has at least one solution for all q E R”. How to
16
YINYU
YE AND PANOS
M. PARDALOS
calculate y(M, q) or a lower bound for y(M, q) in polynomial time is a further research topic. At this time, the condition number y(M, q) and the class 9 are only partially understood, and hence 9 has not been shown to be a particularly large class. Nevertheless, the above analysis illustrates that the condition number in the potential-reduction algorithm may lead researchers to the study of new and different classes of LCPs that are solvable in polynomial time. Furthermore, it is possible that many existing classes of LCPs will find a more meaningful and useful characterization through the analysis of various LCP potential-reduction algorithms. The authors wish to thank the referees for their suggestions which improved the paper.
and comments,
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Received 7 July 1989; final manuscript accepted 24 May 1990
algorithm (1989).
for convex
16802
Submitted by Masakaju Kojima
ABSTRACT We describe a “condition” number for the linear complementarity problem (LCP), which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used. Consequently, we develop a class of LCPs solvable in polynomial time. The result suggests that the convexity (or positive semidefiniteness) of the LCP may not be the basic issue that separates LCPs solvable and not solvable in polynomial time.
INTRODUCTION In this paper, we are concerned with the linear complementarity (LCP), that is, to find a pair x, y E R” such that
x=y = 0,
y=Mx+q,
*Research was supported in part by the
1989 College
LINEAR ALGEBRA AND ITS APPLlCATIONS OEIsevier
and
problem
r,y>O,
Summer Grant.
1523-17
(1991)
3
Science Publishing Co., Inc., 1991
655 Avenue of the Americas, New York, NY 10010
0024-3795/91/$3.50
4
YINYU YE AND PANOS M. PARDALOS
where we assume that all components of M and 9 are integers, and fif = 1(x, y>: y = Mx + q, x > 0,and y > 0) is nonempty. Kojima, Megiddo, and Ye [7] have discussed how to transform an LCP with empty Slf to an equivalent LCP with nonempty 1R+. Thus, the last assumption is merely added for simplicity. We also use s1 to denote the feasible region, i.e., fI = {(x, y): y = Mx + 9, x > 0, and y > O}. If M is a 2 matrix, Cottle and Veinott [2] and Mangasarian [lo] showed that the LCP can be solved as a linear program; therefore, it can be solved in polynomial time. Pang and Chandrasekaran [15] also showed that some special LCPs can be solved in n pivots using several pivoting methods. If M is a positive semidefinite matrix, the LCP is simply a convex quadratic programming problem, and it can be solved in polynomial time by the ellipsoid method (Khachiyan [6]), the projective method (e.g., Kapoor and Vaidya [5] and Ye and Tse [24]), the path-following method (e.g., Kojima, Mizuno, and Yoshise [9] and Monteiro and Adler [12]), and the potentialreduction method (e.g., Kojima, Megiddo, and Ye [7], Kojima, Mizuno, and Yoshise [S], and Pardalos, Ye, and Han [17]). The best complexity result for a convex LCP is 0(&L) iterations and O(n”L) total arithmetic where L is the size of the input data of the problem.
operations,
If M is a P matrix, Ye [22] showed that the potential-reduction algorithm of Kojima et al. [7] solves the LCP in O(n”L max(\hl/nB, 1)) iterations and each iteration solves a system of linear equations in at most O(n">arithmetic operations, where A is the least eigenvalue of so-called P-matrix number for M“, that is,
(M + MT)/2,
and 0 > 0 is the
This indicates that Ih(/nO can be used to measure the degree of difficulty for solving the P-matrix LCP (see also a related discussion of Mathias and Pang [ll]). Th e a lg on ‘th m is a polynomial-time algorithm if (Al/n0 is bounded above by a polynomial in L and n. In this paper, we describe a condition number for the genera1 LCP, which characterizes the degree of difficulty for finding its solution when Kojima et al.‘s potential-reduction algorithm is used. Consequently, we develop a new class of LCPs solvable in polynomial time. We show how the condition number varies as the data (M,9) changes. We also show that several existing classes and some examples of LCPs belong to our class. The result again suggests that the convexity (or positive semidefiniteness) of the LCP may not be the basic issue that separates LCPs that are solvable in polynomial time from ones that are not.
A CLASS OF LCP’S SOLVABLE 1.
POTENTIAL REDUCTION
5
IN POLYNOMIAL TIME
FUNCTIONS
AND POTENTIAL
ALGORITHMS
We use the potential
function
4(X, Y) = p ln(rTv)-
iI
lnCxjYj)
j=l
to associate with an interior feasible solution (x, y) with p > n. This function first appeared in Todd and Ye [21], and was first used to develop an 0(n3L) potential-reduction algorithm for linear programming by Ye [23]. Soon after, this function was used for solving the convex LCP [7, 81. Starting from an interior point (r’, y”> with
the potential-reduction solutions
algorithm
(rk, yk} terminating
generates
a sequence
of interior
feasible
at a point such that
4(xk,yk)<-(p-n)L. From the arithmetic-geometric-mean
nln[(xk)‘yk]
inequality,
- j$lln(r,)yJ)aolnn>O.
Thus.
and an exact solution to LCP can be obtained
in 0(n3)
additional operations
M. To achieve a potential reduction, Kojima et al. used the scaled gradient projection method. The gradient vector of the potential function with respect to x is V&, = i y - X-‘e,
6
YINYU YE AND PANOS M. PARDALOS
and the one with respect
to y is
v+y= g,
-Y-b,
where A = rry, X = diag(x), Y = diag( y), and e is the vector of all ones. Now, let us solve the following linear program subject to an ellipsoid constraint at the k th iteration: minimize
vC$%k
subject
6y = M&x,
to
8X
+
and denote by 8X and Sy its minimal
v&k
&j
solutions.
Then, we have
where (
, Ak =(rk)ryk, Let rk+l
PYk( Ak
e’
MW)-
$Xk(yk+
xk - rk) -
e I
and Xk (Yk) designates the diagonal matrix of xk (yk). = xk +6X and y k+l = yk + So. Then it can be verified that
~(xx”,yx”)-6(xk,Yk)~-BllPkli+~
(P+& I.
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL Therefore,
TIME
choosing
p=
llPkll
min
i
p+2’2
1
-
1
I
<---, 2
we have
&(Xk+I,yk+‘)-
c$(xk,yk) Q
-
“(llPkl12),
(1)
where
if
llpk02 ,< (P +2)2/4,
otherwise.
2.
A CONDITION
NUMBER
FOR THE
LCP
Naturally, IlpkI12 can be used to measure the potential kth iteration of the potential-reduction algorithm. Let
reduction
at the
g(x,y) = $xY -e and H(x,y)=21-(XMr-Y)(Ys+MXW)-1(A4X-Y). It can be verified
that H(x, y) is positive semidefinite
(PSD).
llpkl12 = gT(Xk,yk)H(zk,yk)g(Xk,~k). Let us use Ilg(x, y>lI”, to denote gT(x, y)H(x, the condition number for the LCP (M, q> as y(M,4)=inf((lg(r,y)119:rTy>2-L,
Note that
(2)
y>g(x, y>. Then, we define
#(x,y)~O(pL),and(x,y)En+). (3)
8
YINYU YE AND PANOS M. PARDALOS
The quantity ]lg(x, y)l]i first appeared in Pardalos and Ye [16] for the row-sufficient matrix M of Cottle, Pang, and Venkateswaran [I]. Our condition number is different from the one defined in Kojima et al. [7], where
y(M)
= (‘-
n
n)” inf{h(M,X,Y):(x,y)
and ACM, X, Y) is the smallest
eigenvalue
~a+}
of the matrix
(I+Y-~MX)(I+XM~Y-“MX)-~(I+XM~Y-~). Of course,
the two capture
similar aspects of positivity
related
to the matrix
H(x, y). We present
a sequence
PROPOSITION 1.
of propositions
for y(M, 9).
Let p 2 2 n. Then, for M a diagonal and PSD matrix,
and any 9 E R”, Y(M,~) Proof. If MX2MT)-l(MX component
M is diagonal and PSD, then I -(XMT - YxY2 + - Y) is diagonal. It is also PSD, since the jth diagonal
is
1_
(Mjjxj-
Yj)”=
yj’ + M;x; Therefore.
z n.
2MjjxjYj y,? + Mj”jx;
>0.
for all (x, y) E CI+ and p 2 2%
PROPOSITION 2 (Kojima,
Megiddo,
for M a PSD matrix and any 9 E R”,
and Ye [7]).
Let p > 2n + \lzn. Then,
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL
PROPOSITION 3 (Ye [22]).
9
TIME
Let p > 3n + \/2n. Then, for M a P matrix and
any q E R”,
where A is the least eigenvalue number of MT, i.e.,
of (M + MT )/2,
and
PROPOSITION 4. lit p > n and be fixed. Then, matrix and {(x, y> E R+: 4(x, y) Q O(pL)} bounded,
Both Pang [14] and Pardalos
Proof.
f~
6 is the P-matrix
M a row-suficient
and Ye [16] showed that for any
(x, y> E Cl+
Moreover,
for all (x,y) E Cl+, xTy > 2-L, and c$(x, y) < O(pL)
=pln(xTy)-
i
we have
ln(xjyj)
j=l
=(p-n+l)ln(xTy)+(n-l)ln(xTy)-
C j#i
>(p-n+l)ln(XTy)+(n-l)ln(Xry-riyi) -
C
lnCxjYj)
-WxiYi)
j#i
a(p-n+l)ln(xTy)+(fz-l)ln(n-I)-ln(X,y,) >,-(p-n+l)L+(n-l)ln(n-1)-ln(riy,),
ln(xjyj)-ln(x,y,)
YINYU YE AND PANOS M. PARDALOS
10
where
i E { 1,2,. . . , n). Thus,
ln(xjyi)>-(p-n+l)L+(n-l)In(n-l)-O(pL)=-O(pL), that is, xi yi is bounded
away from zero by e --OCPL)for all i. Since {(x, y>E there must exist a positive number E,
R+ : 4(x, y) < O(pLN is bounded, independent of (x, y>, such that xi > E and
yi >E,
for all (r, y) such that rry 2 2-L,
i=1,2
,...,
$(x, y) f O(PL),
12, and (r, y) E a+.
There-
fore,
> inf{llg(x,y)llZ:x > Ee, y > Fe, 4,(x, y) 6 O(pL),
and (x, y> E a}
> 0. The last inequality
holds because
the inf is taken in a compact
set where
Ilg(x, y>[[L is always positive. Note that the
+(r, y) G O(pL)
boundedness
of
n implies
((x, y) E a:
boundedness of {(x, y) E a+: We now derive
that
rTy d 0Cp.L /(p - n)). Hence,
rry < O(pL/(p
- n)))
guarantees
the
4(x, y) G O(PL)I.
The potential-reduction algorithm with p = e(n) > n solves THEOREM 1. the LCP for which y(M, q) > 0 in O(nL /a(y(M, 4))) iterations and each iteration solves a system of linear equations in at most 0(n3) operations, where LY(.) is defined in (1).
Proof. Since LR+ is nonempty, by solving a linear program in polynomial time, we can find an interior feasible point (x0, y”X each component of which is greater than 2-= and less than 2L. The resulting point has an initial potential value less than O(nL). Due to (11, (21, and (31, the potential function is reduced by O(&y(M, 9))) a t each iteration. Hence, in the total of O(nL/cu(y(M,q))) iterations we have 4(rk,yk)< -(p - n)L and (xk)‘yk <2-L.
W
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL COROLLARY
1.
TIME
11
An instance (M, 9) of the LCP is solvable in polynomial
time if y(M, 9) > 0 and if 1/ y(M, 9) is bounded and n.
above by a polynomial
in L
The condition number y(M, 9) represents the degree of diflkulty for the potential-reduction algorithm in solving the LCP (M, 9). The larger the condition number, the easier the LCP. We know that some LCPs are very hard, and some are easy. Here, the condition number builds a connection from easy LCPs to hard LCPs. In other words, the corresponding degree of difficulty shifts continuously from easy to hard LCPs. We feel that such a condition number will be an important criterion in analyzing the complexity of algorithms for optimization problems.
3.
A CLASS OF LCP’S SOLVABLE
We now further LEMMA1.
IN POLYNOMIAL
study IIpkII by first introducing
TIME
the following
lemma.
IIpklI < 1 implies yk+MT.rrk>O,
xk-rrk>O,
2n-J2n
2n+\/2n
and
Akk
P
+ MTak)+(yk)T(~k
Ak,
- .rrk).
Proof. The proof is by contradiction. Let 5 = yk + MTrk 7rk. It is obvious that if ij 2 0 or X > 0, then
llpkl12 > I. On the other hand, as is developed
in Ye [22], we have
and X = x k -
YINYU YE AND PANOS M. PARDALOS
12 Hence,
the following
must
be true:
that is, 2n+J2n
zn-Jzn Akk<<
Ak P
P Zi can be further
expressed
.
as E = 2Ak - +,&
Thus,
we can prove
PROPOSITION5.
the following
propositions.
Denote by 2’
the set (x,y)Efl+
(77:xTy-qT57
that also satisfy x - 7~ > 0 and y + MTrr > 0) , andlet
2,’
beemptyforan
LCP (M,y). r(M,9)
Proof.
The proof
PROPOSITION6.
directly
results
Then, for p>2n+&, 21.
from (2) and Lemma
1.
Let
{Tr:~~y-9%>o~of-some
(x,y)
En+
that also satisfy x - r > 0 and y + MT~ > 0} be empty for an LCP (M, 9). Then, for n < p B 2n - 6,
13
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL TIME Proof.
The proof again results from (2) and Lemma
1.
l
Now, let + is nonempty
#=((M,q):fi
and C’
is empty).
As pointed out by Stone [20], the description of 3 is technically similar to those of Eaves’s class [3] and Garcia’s class [4]. These two classes and some others have been extensively studied. It may not be possible in polynomial time to tell if an LCP 04, y) 1s an element of 9 (this is also true for some other LCP classes published so far). However, the coproblem, to tell that an LCP (M,q) is not in 9, can be solved in polynomial time. We can simply run the potential-reduction algorithm for the LCP. In polynomial time the algorithm either gives the solution or concludes that (M,9) is not in 9. The coproblem is actually more important in practice. In the following, we present a dual interpretation using the Lagrangian multiplier+ For (M, 9) E ((M, 9): Isol (M, 9)1> 1) (the class where at least one LCP solution exists), the LCP can be represented as an optimization problem with known zero optimal value: (P)
minimize
XTY
subject to
(x,Y)E1R={(x,y):y=Mx+q,x,y~O}.
A dual to this problem
CD)
can be written
as
maximize
9ra
subject to
(“,y,“)E{(x,y,~):x-rr~o,
- xTy
y+MrTf>,O,(X,y)~~). C + being empty means region of(D), the objective the objective value of(P) is of LCPs satisfying the weak 4.
SOME EXISTING
that for all (x, y, rr) in the interior of the feasible value of (D) is less than or equal to zero. Since always greater than or equal to zero, 9 is a class duality condition for (P) and (D).
CLASSES
BELONG
TO THE NEW CLASS
We see that the new class 9 has the same bound on the condition number as the PSD class, that is, y(M,q)> 1. Various other classes of LCPs can be found in Eaves [3], Murty [13], Saigal [18], and Stone [19]. Here, we list several existing classes of LCPs that belong to 9.
YINYU YE AND PANOS M. PARDALOS
14 (1)
M is positive semidefinite
and q is arbitrary.
We have, if C’
is not
empty, 0 <
(x - T)~( y f M%) =
x'y -
9% - rTMTr,
which implies x T y-qTrr>rTMTr>OO,
a contradiction. (2) M is copositive and q > 0. xTy - q%
We have
= xTMx + qT( x - T).
Thus, x > 0 and x - rr > 0 implies xTy - yT.rr 3 0, that is, 2’ (3) M- ’ is copositive and M- ‘q < 0. We have
xTy
-
qTr = ~r’M-~y - (M-‘qf(
Thus, y > 0 and y + MTr > 0 implies
is empty.
y + MTr).
xTy - qTrr >, 0, that is, I!$’ is empty.
Although a trivial solution may exist for the last two classes (e.g., x = 0 and y = q for the second class), our computational experience indicates that the potential-reduction algorithm usually converges to a nontrivial solution if multiple solutions exist. For example, let
M=((:
Then the potential-reduction
-i)
and
algorithm
IX=
M=(y
constantly
and
from virtually any interior starting and y = q. Another example:
1:)
y=(i).
generates
the solution
y=
point,
avoiding
and
y=(i).
the trivial
solution
x= 0
A CLASS OF LCP’S SOLVABLE IN POLYNOMIAL Again, the algorithm constantly
generates
and
X=
TIME
the solution
y=
from virtually any interior starting point. Both problems
iterations, although the second is a nonconvex certainly deserves further research. Another nonconvex LCP also belongs to 9:
M=(i
-:)
and
15
q=(
converge
problem.
This
in a few property
1:).
since x = (3 ljT is an interior feasible This is because R + is nonempty, point; and Z-’ is empty, since xi - x2 > 1, x, - 7Tl> 0, x2 - 772 > 0, x, - xg -1.+7r,+27r,>O,and2~,-1-7r,>Oimply xTy -
qT77 =
x’( Mx + q) - qTz-
= r,( x1 - X2) +2x,x,
- Xi - x2 + 7r, + 7ra
=rf+x,x,-2x,-X2,+1+(r,-x,-1+x,+2n-,)+(x,-7r,)
> xi2 + rlxz -2x,
which indicates
5.
- Xp + I
that Z + is empty.
CONCLUDING As a by-product,
REMARKS we have
In fact, any LCP (M, q> with y(M, q) > 0 belongs to {(M, q): Isol(M, q)l z 1). Furthermore, if y(M, q) > 0 for all q E R”, then M E 9, the matrix class where the LCP (M, q) has at least one solution for all q E R”. How to
16
YINYU
YE AND PANOS
M. PARDALOS
calculate y(M, q) or a lower bound for y(M, q) in polynomial time is a further research topic. At this time, the condition number y(M, q) and the class 9 are only partially understood, and hence 9 has not been shown to be a particularly large class. Nevertheless, the above analysis illustrates that the condition number in the potential-reduction algorithm may lead researchers to the study of new and different classes of LCPs that are solvable in polynomial time. Furthermore, it is possible that many existing classes of LCPs will find a more meaningful and useful characterization through the analysis of various LCP potential-reduction algorithms. The authors wish to thank the referees for their suggestions which improved the paper.
and comments,
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