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MATHEMATICAL
AND COMPUTER MODELLING
IOIRNCE~DIRICT" ELSEVIER
Mathematical and Computer Modelling 41 (2005) 523-538 www.elsevier.com/locate/mcm
A Smoothing Broyden-Like Method for the Mixed Complementarity Problems CHANGFENG M A College of Mathematics and Physics Zhejiang Normal University, Zhejiang 321004, P.R. China and Department of Computational Science and Mathematics Guilin University of Electronic Technology Guangxi 541004, P.R. China mac~©gliet, edu. cn
macf 99_cn©yahoo. com. cn
(Received October 2003; acceptedDecember 2003) A b s t r a c t - - T h e mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we propose a new smoothing Broyden-like method for the solution of the mixed complementarity problem by constructing a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. (~) 2005 Elsevier Ltd. All rights reserved. K e y w o r d s - - M i x e d complementarity problem, Smoothing Broyden-like method, Global convergence, Superlinear convergence.
1. I N T R O D U C T I O N Let F : R n ~ R n be a c o n t i n u o u s l y differentiable m a p p i n g a n d X = {1 < x < u} be a n o n e m p t y
closed convex set in R n. The mixed complementarity problem, denoted by MCP(F), is to find a vector x* E X, such that F (x*) T (x - x*) _> 0,
for an x e X,
(1.1)
where li E R U {-co}, ui E R U {+c~}, and li < ui, i = 1 , . . . , n. The mixed complementarity problems are also called the box constrained variational inequality problem [1-4]. Furthermore, if X -- R~_, MCP(F) reduces to the nonlinear complementarity problem, denoted NCP(F), which is to find x E R n, such that x _> 0,
F(x) >_0,
xTF(x)
= 0.
(1.2)
T h i s c o r r e s p o n d s to t h e case t h a t li = 0 a n d ui -- + c ~ for all i -- 1, 2 , . . . , n. A n o t h e r e x t r e m e case of M C P ( F ) is the p r o b l e m w i t h li -- - c ~ , ui = + c o , i = 1, 2 , . . . , n. I n t h e case, M C P ( F ) Supported by Guangxi Science Foundation (No. 6448075) and the National Natural Science Foundation of China (No. 10361003). 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2003.12.013
Typeset by A MS-TEX
524
C. MA
reduces to the following nonlinear equation:
F(x) = O.
(1.3)
Two comprehensive surveys of variational inequality problems and nonlinear complementarity problems are [5] and [6]. The study on iterative methods for solving M C P ( F ) and NCP(F) has been rapidly developed in the last decade. One of the most popular approaches is to reformulate NCP(F) or M C P ( F ) as an equivalent nonsmooth equation so t h a t generalized Newton-type methods can be applied in a way similar to those for smooth equations. It is not difficult to see that, if function F is convex, M C P ( F ) is equivalent to its K K T system F ( x ) + y - z = 0,
x - - 1 >_O,
z >_ O,
( x - l)Tz = O,
x -- u 0,
x-u
< O,
F(x)+y>0,
( x - 0 T ( F ( ~ ) + y ) = 0,
y >_O,
( x - u)Ty = O.
(1.5)
It is easy to see that the above relations can be written as the following system of nonsmooth equation:
(min(F(x) + y,x-l)
)
m i n ( y , - ( x - u))
- 0.
(1.6)
Another nonsmooth equation reformulation of M C P ( F ) is the following equation (see [7]): mid{x - 1, F ( x ) , x - u} = O.
(1.7)
Here the operator mid{a, b, c} stands for the median of three scalars a, b, c E RU{+oo}. For vector u, v, w E (RU {q-oc}) ~, the mid operator is done componentwise. We will focus on system (1.7) in this paper because it does not increase the dimensions of the problem. For N C P ( F ) , (1.7) reduces to the following well-known nonsmooth equation: rain{x, F(x)} = 0.
(1.8)
Much effort has been made to construct smoothing approximation functions for approach to the solution of M C P ( F ) or N C P ( F ) in recent years [3,7-12]. This class of algorithms, called smoothing Newton method, is due to Chen, Qi and Sun [7]. In [7], the locally superlinear convergence of a smoothing Newton method is established. Moreover, smoothing quasi-Newton methods have studied by Chen [13]. Chen [13] discusses a locally superlinear convergence property of the smoothing quasi-Newton methods without line search. In this paper, we will construct a new smoothing approximation function and then present a new smoothing Broyden-like method. Under appropriate conditions, we will show global and superlinear convergence of the proposed method. Next we introduce some words about our notation: Let G : R ~ -* R m be continuously differentiable. The VG(x) C R mxn denotes the Jacobian of G at a point x E R n. If m = 1, VG(x) denotes the gradient of G at a point x E R n. If is G : R n --~ R m only local Lipschitzian, we can define Clarke's [14] generalized 3acobian as follows:
OG(x) : = c o n v { H e R ' ~ x ~ ] 3 {x k} G D e : x k --* x and G' (x k) ---* H} ; here D c denotes the set of differentiable points of G and conv S is the convex hull of a set S. If m = 1, we call cOG(x) the generalized gradient of G at x for obvious reasons. Usually, cOG(x) is not easy to compute, especially for m > 1. Based on this reason, we use in this paper a kind of generalized Jacobian for the function G, denoted by cOvG and defined as (see [15]) O c G = O G l ( x ) x COG2(x) x . . . x cOG,,(z),
where Gi(x) is ith component function of G.
A Smoothing Broyden-Like Method
525
Furthermore, we denote by IIxll the Euclidian norm of x if x E R n and by IIAll the spectral norm of a matrix A ~ R ~×'~ which is the induced matrix norm of the Euclidian vector norm. If A E R ~×~ is any given matrix and A4 c_ R n×~ is a nonempty set of matrices, we denote by dist(A, A/I) := i n f s e M IIA - B I I the distance between A and M . The remainder of the paper is organized as follows. In the next section, the mathematical background and some preliminary results are summarized. The algorithm is proposed in detail in Section 3. Sections 4 and 5 are devoted to proving global local superlinear convergence of the algorithm.
2. P R E L I M I N A R I E S In this section, we first introduce the conception of MCP-function. Denote/~3 = (RU {-co}) × R × (R t2 {+c~}). A function ¢ : /~a _.~ R is called an MCP-function if ¢(a, b, c) = 0, a > c is equivalent to mid{a, b, c} = 0, a > c. If we define function ¢ :/~3 _~ R by ¢(a,b,c)
= a + c - v/(a-c)2+x/(b-c) 2,
(2.1)
then it is easy to see that we have ¢(a, b, c) = 2 mid{a, b, c}, where a > c. Hence ¢ is an MCP-function. In view of (1.7), let us define the function ¢(x) = (¢1(x), ¢ 2 ( x ) , . . . , Cn(x)) T, where for each i = 1, 2 , . . . , n, ¢ , ( ~ ) = ¢ ( ~ , - ~,, F , ( z ) , ~, - ~,) (2.2) = 2~, - (l, + ~ ) - ~/(x~ - i, - F , ( ~ ) ) : + v / ( ~ , - u, - F , ( ~ ) ) ~ .
Then M C P ( F ) can be reformulated as the following nonsmooth equation: • (z)=0.
(2.3)
Function ¢, and hence, ~) are not differentiable everywhere but semismooth in the sense of Mifflin [16] and Qi [17] if F is continuously differentiable. Denote
~ ( x ) = { i : x ~ - ~ x i - l ~ } , 72(x)={i:F~(x)<xi-ui}. Then we have { 2F~(x),
¢~(~) =
if i E a(x),
2 ( ~ - l~),
if i e Z l ( x ) u ~ l ( x ) ,
2(x, - ~,),
if i e / 3 2 ( ~ ) u ~2(x).
By using the chain rule for generalized derivatives of Lipschitz functions (see [14]), we have the following expression of Ov~(x) = 0¢1(x) × 0¢2(x) x --. × 0¢n(x) for each i --- 1, 2 , . . . , n:
0¢,(x) =
{
{2VF,(x)},
if i E a(x),
{ ( l + p ) e , + (1-p)VF,(~)},
i f / E ill(x)U~2(x),
{2e,},
i f i e 71(x) U~2(x),
where ei denotes the ith unit vector in R '~ and p E [0, 1].
(2.4)
526
C. MA
Smoothing techniques have attracted much attention in recent years [3,7-12,18-20]. Now let us construct a new smoothing approximation function of M C P ( F ) and discuss its some useful properties. We define the function ~ ( x ) = (¢~(x, . ) , ¢ 2 ( x , . ) , . . . , ¢~(x, #))T as follows: if i • a ( x , # ) U 71(x,#) U 72(x,.),
¢i(x),
2x~ - (l~ + u~) -
( ~ - t, - F~(x)) 2 + V ( ~ ,
- ~ , - F , ( ~ ) ) ~ - ~~-,
2. ¢,(~,.)
=
(2.5)
if i • ~1 (x, . ) , 2 ~ - (t, + ~ ) - x/(z~ - z~ - F~(x))2 + (x~ - z, - F , ( ~ ) ) ~ . 2. + ~"
if i • f~(x, . ) , where ~(x,.)
= { i : x , - ~, + .
~2(x, . ) = {i : x~ - u, - . ~l(x,.)
< F , ( x ) < x~ - l, - . } ,
< F~(x) < x~ - u~ + . } ,
= ( i : & ( ~ ) > x , - l, + A ,
~2(~,.)
= { i : F~(~) < ~, - ~, - . } .
Clearly, ¢i(x, . ) , for every i = 1 , 2 , . . . , n, is continuously differentiable if and only if 0 < . < minl_ 0, we have for all i = 1 , 2 , . . . ,n,
I¢,(x,K) - ¢ , ( x , ff) I < ~ - ( . ' - . ) ,
Vx e R ~ .
Particularly, we have for e v e r y . > 0, 1
I ¢ ~ ( x , . ) - ¢~(x)l < ~ . ,
W c R ~.
Lemma 2.1 shows that ¢~(x,.) --* ¢(x) uniformly as . -~ 0. following. LEMMA 2.2. For any . ' > .
Furthermore, we have the
~ 0, we have
II,~.,(z) - ¢,,.(x)ll O,
v~ II~,,(x) - ~(x)ll < --y--,,
vx e R ~ .
(2.7)
Note that ¢ i ( x , . ) , for M1 i = 1 , 2 , . . . ,n, is differentiable everywhere if 0 < . < minl O, such that for any x • R ~ and # > O, [[Ht,(x ) - H(x)l [ < c#.
(2.9)
lim dist(~YH~(x), o c g ( x ) ) = 0, t~$0
(2.10)
Further, if for any x • R n,
then we say H and H~ satisfy the Jacobian consistency property, where OcH (x) defined by OcH(x) = Ohl(x) × Oh2(x) x ... x Oh,(x). Inequality (2.7) implies that ~ ( x ) approximates if(x) uniformly. Moreover, we can show that ~b(x) and ~ (x) satisfy the Jacobian consistency property if F is continuously differentiable. LEMMA 2.4. Let x • R ~ be arbitrary but fixed. Then functions ~(x) and ~ ( x ) Jacobian consistency property, i.e., limdist(V~b~(x), Oc~?(x)) = 0. #10
satisfy the
(2.11)
PROOF. By (2.8),
xi
{ (1-x~-l~Fi(X))e~+(l+ V¢~(x, # ) =
(l+X ~
u~F~(X))e~+(l
--
li
--
F~(x) ) VF,(x),
-
i • i e
Hence, we get lira V¢~(x, #) = e~ + V F d x ) , t,$0
i •/31(x) U/32(x).
(2.12)
Thus, the assertion follows from (2.4) with p = 0 if i •/31(x) U fl2(x).
3. A L G O R I T H M In this section, we give a detailed description of our smoothing Broyden-like method for the mixed complementarity problem. The algorithm is stated as follows. ALGORITHM 3.1.
(S.0) Choose constants Pl, f12 • (0, 1), 0 < 3' < m i n { 1 / v ~ , P2/v/-n}, 0-1,02 > 0. Choose an initial point Xo • R n and an n x n nonsingular matrix Bo, and positive constant #o _< (3'/2)[[~(Xo)[[ and rio ~ I[~(Xo)[[. Set k := 0. (S.1) If [[~(xk)[[ = 0, stop. Otherwise, solve the following linear equation:
Bkd + ¢b(Xk) = 0
(3.1)
to get dk. (S.2) If
(xk + dk)ll - P2lf ,k (xk)ll < -0-1Hd ]l 2,
(3.2)
then let Ak := 1 and go to (S.4). (S.3) Let As be the maximum number in the set {1, pl, p 2 , . . . , }, such that A = p~ satisfies the line search condition [14~.k (xk + Adk)l[ - II(I)~k(xk)II < -0-2[IdaH 2 + ~k.
(3.3)
528
C. MA
(s.4) Let xk+i := xk + Akdk. (s.5) Update Bk by the Broyden-like update rule Bk+~ = Bk + Ok (yk - Bksk)s-~
ii~kll 2
,
(3.4)
where sk = xk+i - sk and Yk = ~ (x~+i) - ~ (xt¢). The parameter Ok is chosen so that IOk -- 11 < 0 for some constant O E (0, 1) and Bk+i is nonsingular (see [22]). (s.6) If (3.2) holds or 711@(Xk+i)ll < #k, let #k+l := min
[1~ (xk+i)ll, ~#k
•
(3.5)
Otherwise, let #k+i := #k. (S.7) Let 7/k+i := min{(1/2)~k, II@(xk+i)ll} and k := k + 1. Go to (S.1). REMARK 3.2. (1) Since ~?k+l 0, such that for all k 6 / ~ sufficiently large,
C~ll~(xk)ll
_< lldkll 0, then d = 0. So it follows from (3.1) that ~(2) = 0, which contradicts the fact that 0 < # < 11~(2)11. We now consider the case of A = 0, or equivalently, limsuPk(eg)_.~ ~k = 0. By (S.3) of Algorithm 1, when k e /{ is large enough, A~ = Ak/Pl does not satisfy (3.3). Therefore, we have
Ile.(x~ + ~2d~)LI - II~(~k)ll > --~211Akdkll 2 Multiplying both sides by (A~)-l(]]¢p(xk + A~dk)ll + II~n(xk)ll) and then taking the limit as k(e K ) ~ oo yield 2 ~ p ( ~ ) T v ( I ) p ( ~ ) J > 0. (4.24) Since ~(xk) = - V ~ ( x k ) d ~ , by (4.1), taking the limit as k(E /~) --* c~ yields ¢(~) = - V ~ p k (2)~ It then follows from (4.24) that
~/~('~)T'(I)(~) S O.
A Smoothing Broyden-Like Method
533
This together with (3.7) implies (1 _ 1 ~n7 2)[l~(~)l] 2 _< l ] ~ (~)1]2 + 0 _ ¼~?) i1,~(~.)11~ _< I I ~ ( ~ ) l f + II~(~)ll: - ¼ ~
1 -2 = II~(~) - @(~)II= - ~ . + 2@~(~)T@(~)
(1)~ -- ~n/z 1-~+ 2 ~ p ( e ) T ~ ( e )
_< ~V~/2
(4.25)
_< 0,
where the third inequality follows from (2.7). Notice that 0 < ~/ < 1/~/~, that is, 3/4 < 1 - (1/4)n~/2 < 1, hence (4.12) is also a contradiction. So, K must be infinite, and thus, the assertion follows from Lemma 4.3.
5. S U P E R L I N E A R
CONVERGENCE
We turn to in this section analyze the convergence rate of {xk} generated by Algorithm 3.1. For this purpose, we make further the following assumptions. ASSUMPTION C. (i) The sequence {xk } generated by Algorithm 3.1 converges to a solution x* of MCP(F) and V ~ (x*) is nonsingular. (ii) The strict complementarity holds at x*, that is, ~ ( x * ) = ~ ( x * ) -- ~. Let 1
(5.1)
= 2 l 0 and 0 ¢ ~. It is also obvious that ~z(x) --
#2(x) -- Z for every x E 0, and hence, ~(x) is continuously differentiable on 0. In particular, for every x C 0, we have V@(x) = (V¢l(x), V ¢ 2 ( x ) , . . . , V¢2(x)) T with
V¢i(x) =
2VFi(x), 2ei,
i E a(x), i e 7(x),
(5.3)
where 3(x) -- 3z(x) U'y2(x). This means that V(b~(x) - Vq~(x) for every x E U. Moreover, there exists a constant M0 > 0, such that
II~(x) - '~ (x*)ll O, such that v~,~ e O, (5.5) liVe(x) - Vq~(y)H < Lll]x Yl], -
and
IlV~,(x) - V~,(y)H _< nzHx - Y]I,
Vx, y E U.
(5.6)
534
C. MA
LEMMA 5.2. Let Assumption C hold. Then there is a constants M~ > O, such that for every #>0, (5.7) Vz~O. II~.(~) - ~(x)ll -< M I # 3, Paoo~. By (2.2) and (2.5), we have
I¢~(x,g) - @(x)[ _< max
2~
'
E (5.8)
/
~- max
~3
~4
< 2~]xi - us - F~(z)[ 2 < 2--~" Letting M1 = v ~ / 2 g 2, we get (5.7). LEMMA 5.3. Let Assumptions B and C hold, {xk} be generated by Algorithm 3.1, and 5k be defined by (4.17). Then there exist a constant 5 > 0 and an integer k' > O, such that Ak = 1 whenever 5k < 5 and k > k'. PROOF. Since Xk --* x*, there is an index k[, such that xk E 0 for all k > k~. Notice that xk + rsk c U for any T E [0, 1] when k is sufficiently large. Then we have for every k > k~,
IIAk - V~(x*)ll =
IIV~.~(Xk + TSk)-- V~(X*)II d"
=
HV~(xk + TSk) -- V ~ (x*)H dT
_< LI
(5.9)
/01
lit ( z k + l - x*) + (1 - r) (xk - ~*)11 d r
MATHEMATICAL
AND COMPUTER MODELLING
IOIRNCE~DIRICT" ELSEVIER
Mathematical and Computer Modelling 41 (2005) 523-538 www.elsevier.com/locate/mcm
A Smoothing Broyden-Like Method for the Mixed Complementarity Problems CHANGFENG M A College of Mathematics and Physics Zhejiang Normal University, Zhejiang 321004, P.R. China and Department of Computational Science and Mathematics Guilin University of Electronic Technology Guangxi 541004, P.R. China mac~©gliet, edu. cn
macf 99_cn©yahoo. com. cn
(Received October 2003; acceptedDecember 2003) A b s t r a c t - - T h e mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we propose a new smoothing Broyden-like method for the solution of the mixed complementarity problem by constructing a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. (~) 2005 Elsevier Ltd. All rights reserved. K e y w o r d s - - M i x e d complementarity problem, Smoothing Broyden-like method, Global convergence, Superlinear convergence.
1. I N T R O D U C T I O N Let F : R n ~ R n be a c o n t i n u o u s l y differentiable m a p p i n g a n d X = {1 < x < u} be a n o n e m p t y
closed convex set in R n. The mixed complementarity problem, denoted by MCP(F), is to find a vector x* E X, such that F (x*) T (x - x*) _> 0,
for an x e X,
(1.1)
where li E R U {-co}, ui E R U {+c~}, and li < ui, i = 1 , . . . , n. The mixed complementarity problems are also called the box constrained variational inequality problem [1-4]. Furthermore, if X -- R~_, MCP(F) reduces to the nonlinear complementarity problem, denoted NCP(F), which is to find x E R n, such that x _> 0,
F(x) >_0,
xTF(x)
= 0.
(1.2)
T h i s c o r r e s p o n d s to t h e case t h a t li = 0 a n d ui -- + c ~ for all i -- 1, 2 , . . . , n. A n o t h e r e x t r e m e case of M C P ( F ) is the p r o b l e m w i t h li -- - c ~ , ui = + c o , i = 1, 2 , . . . , n. I n t h e case, M C P ( F ) Supported by Guangxi Science Foundation (No. 6448075) and the National Natural Science Foundation of China (No. 10361003). 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2003.12.013
Typeset by A MS-TEX
524
C. MA
reduces to the following nonlinear equation:
F(x) = O.
(1.3)
Two comprehensive surveys of variational inequality problems and nonlinear complementarity problems are [5] and [6]. The study on iterative methods for solving M C P ( F ) and NCP(F) has been rapidly developed in the last decade. One of the most popular approaches is to reformulate NCP(F) or M C P ( F ) as an equivalent nonsmooth equation so t h a t generalized Newton-type methods can be applied in a way similar to those for smooth equations. It is not difficult to see that, if function F is convex, M C P ( F ) is equivalent to its K K T system F ( x ) + y - z = 0,
x - - 1 >_O,
z >_ O,
( x - l)Tz = O,
x -- u 0,
x-u
< O,
F(x)+y>0,
( x - 0 T ( F ( ~ ) + y ) = 0,
y >_O,
( x - u)Ty = O.
(1.5)
It is easy to see that the above relations can be written as the following system of nonsmooth equation:
(min(F(x) + y,x-l)
)
m i n ( y , - ( x - u))
- 0.
(1.6)
Another nonsmooth equation reformulation of M C P ( F ) is the following equation (see [7]): mid{x - 1, F ( x ) , x - u} = O.
(1.7)
Here the operator mid{a, b, c} stands for the median of three scalars a, b, c E RU{+oo}. For vector u, v, w E (RU {q-oc}) ~, the mid operator is done componentwise. We will focus on system (1.7) in this paper because it does not increase the dimensions of the problem. For N C P ( F ) , (1.7) reduces to the following well-known nonsmooth equation: rain{x, F(x)} = 0.
(1.8)
Much effort has been made to construct smoothing approximation functions for approach to the solution of M C P ( F ) or N C P ( F ) in recent years [3,7-12]. This class of algorithms, called smoothing Newton method, is due to Chen, Qi and Sun [7]. In [7], the locally superlinear convergence of a smoothing Newton method is established. Moreover, smoothing quasi-Newton methods have studied by Chen [13]. Chen [13] discusses a locally superlinear convergence property of the smoothing quasi-Newton methods without line search. In this paper, we will construct a new smoothing approximation function and then present a new smoothing Broyden-like method. Under appropriate conditions, we will show global and superlinear convergence of the proposed method. Next we introduce some words about our notation: Let G : R ~ -* R m be continuously differentiable. The VG(x) C R mxn denotes the Jacobian of G at a point x E R n. If m = 1, VG(x) denotes the gradient of G at a point x E R n. If is G : R n --~ R m only local Lipschitzian, we can define Clarke's [14] generalized 3acobian as follows:
OG(x) : = c o n v { H e R ' ~ x ~ ] 3 {x k} G D e : x k --* x and G' (x k) ---* H} ; here D c denotes the set of differentiable points of G and conv S is the convex hull of a set S. If m = 1, we call cOG(x) the generalized gradient of G at x for obvious reasons. Usually, cOG(x) is not easy to compute, especially for m > 1. Based on this reason, we use in this paper a kind of generalized Jacobian for the function G, denoted by cOvG and defined as (see [15]) O c G = O G l ( x ) x COG2(x) x . . . x cOG,,(z),
where Gi(x) is ith component function of G.
A Smoothing Broyden-Like Method
525
Furthermore, we denote by IIxll the Euclidian norm of x if x E R n and by IIAll the spectral norm of a matrix A ~ R ~×'~ which is the induced matrix norm of the Euclidian vector norm. If A E R ~×~ is any given matrix and A4 c_ R n×~ is a nonempty set of matrices, we denote by dist(A, A/I) := i n f s e M IIA - B I I the distance between A and M . The remainder of the paper is organized as follows. In the next section, the mathematical background and some preliminary results are summarized. The algorithm is proposed in detail in Section 3. Sections 4 and 5 are devoted to proving global local superlinear convergence of the algorithm.
2. P R E L I M I N A R I E S In this section, we first introduce the conception of MCP-function. Denote/~3 = (RU {-co}) × R × (R t2 {+c~}). A function ¢ : /~a _.~ R is called an MCP-function if ¢(a, b, c) = 0, a > c is equivalent to mid{a, b, c} = 0, a > c. If we define function ¢ :/~3 _~ R by ¢(a,b,c)
= a + c - v/(a-c)2+x/(b-c) 2,
(2.1)
then it is easy to see that we have ¢(a, b, c) = 2 mid{a, b, c}, where a > c. Hence ¢ is an MCP-function. In view of (1.7), let us define the function ¢(x) = (¢1(x), ¢ 2 ( x ) , . . . , Cn(x)) T, where for each i = 1, 2 , . . . , n, ¢ , ( ~ ) = ¢ ( ~ , - ~,, F , ( z ) , ~, - ~,) (2.2) = 2~, - (l, + ~ ) - ~/(x~ - i, - F , ( ~ ) ) : + v / ( ~ , - u, - F , ( ~ ) ) ~ .
Then M C P ( F ) can be reformulated as the following nonsmooth equation: • (z)=0.
(2.3)
Function ¢, and hence, ~) are not differentiable everywhere but semismooth in the sense of Mifflin [16] and Qi [17] if F is continuously differentiable. Denote
~ ( x ) = { i : x ~ - ~ x i - l ~ } , 72(x)={i:F~(x)<xi-ui}. Then we have { 2F~(x),
¢~(~) =
if i E a(x),
2 ( ~ - l~),
if i e Z l ( x ) u ~ l ( x ) ,
2(x, - ~,),
if i e / 3 2 ( ~ ) u ~2(x).
By using the chain rule for generalized derivatives of Lipschitz functions (see [14]), we have the following expression of Ov~(x) = 0¢1(x) × 0¢2(x) x --. × 0¢n(x) for each i --- 1, 2 , . . . , n:
0¢,(x) =
{
{2VF,(x)},
if i E a(x),
{ ( l + p ) e , + (1-p)VF,(~)},
i f / E ill(x)U~2(x),
{2e,},
i f i e 71(x) U~2(x),
where ei denotes the ith unit vector in R '~ and p E [0, 1].
(2.4)
526
C. MA
Smoothing techniques have attracted much attention in recent years [3,7-12,18-20]. Now let us construct a new smoothing approximation function of M C P ( F ) and discuss its some useful properties. We define the function ~ ( x ) = (¢~(x, . ) , ¢ 2 ( x , . ) , . . . , ¢~(x, #))T as follows: if i • a ( x , # ) U 71(x,#) U 72(x,.),
¢i(x),
2x~ - (l~ + u~) -
( ~ - t, - F~(x)) 2 + V ( ~ ,
- ~ , - F , ( ~ ) ) ~ - ~~-,
2. ¢,(~,.)
=
(2.5)
if i • ~1 (x, . ) , 2 ~ - (t, + ~ ) - x/(z~ - z~ - F~(x))2 + (x~ - z, - F , ( ~ ) ) ~ . 2. + ~"
if i • f~(x, . ) , where ~(x,.)
= { i : x , - ~, + .
~2(x, . ) = {i : x~ - u, - . ~l(x,.)
< F , ( x ) < x~ - l, - . } ,
< F~(x) < x~ - u~ + . } ,
= ( i : & ( ~ ) > x , - l, + A ,
~2(~,.)
= { i : F~(~) < ~, - ~, - . } .
Clearly, ¢i(x, . ) , for every i = 1 , 2 , . . . , n, is continuously differentiable if and only if 0 < . < minl_ 0, we have for all i = 1 , 2 , . . . ,n,
I¢,(x,K) - ¢ , ( x , ff) I < ~ - ( . ' - . ) ,
Vx e R ~ .
Particularly, we have for e v e r y . > 0, 1
I ¢ ~ ( x , . ) - ¢~(x)l < ~ . ,
W c R ~.
Lemma 2.1 shows that ¢~(x,.) --* ¢(x) uniformly as . -~ 0. following. LEMMA 2.2. For any . ' > .
Furthermore, we have the
~ 0, we have
II,~.,(z) - ¢,,.(x)ll O,
v~ II~,,(x) - ~(x)ll < --y--,,
vx e R ~ .
(2.7)
Note that ¢ i ( x , . ) , for M1 i = 1 , 2 , . . . ,n, is differentiable everywhere if 0 < . < minl O, such that for any x • R ~ and # > O, [[Ht,(x ) - H(x)l [ < c#.
(2.9)
lim dist(~YH~(x), o c g ( x ) ) = 0, t~$0
(2.10)
Further, if for any x • R n,
then we say H and H~ satisfy the Jacobian consistency property, where OcH (x) defined by OcH(x) = Ohl(x) × Oh2(x) x ... x Oh,(x). Inequality (2.7) implies that ~ ( x ) approximates if(x) uniformly. Moreover, we can show that ~b(x) and ~ (x) satisfy the Jacobian consistency property if F is continuously differentiable. LEMMA 2.4. Let x • R ~ be arbitrary but fixed. Then functions ~(x) and ~ ( x ) Jacobian consistency property, i.e., limdist(V~b~(x), Oc~?(x)) = 0. #10
satisfy the
(2.11)
PROOF. By (2.8),
xi
{ (1-x~-l~Fi(X))e~+(l+ V¢~(x, # ) =
(l+X ~
u~F~(X))e~+(l
--
li
--
F~(x) ) VF,(x),
-
i • i e
Hence, we get lira V¢~(x, #) = e~ + V F d x ) , t,$0
i •/31(x) U/32(x).
(2.12)
Thus, the assertion follows from (2.4) with p = 0 if i •/31(x) U fl2(x).
3. A L G O R I T H M In this section, we give a detailed description of our smoothing Broyden-like method for the mixed complementarity problem. The algorithm is stated as follows. ALGORITHM 3.1.
(S.0) Choose constants Pl, f12 • (0, 1), 0 < 3' < m i n { 1 / v ~ , P2/v/-n}, 0-1,02 > 0. Choose an initial point Xo • R n and an n x n nonsingular matrix Bo, and positive constant #o _< (3'/2)[[~(Xo)[[ and rio ~ I[~(Xo)[[. Set k := 0. (S.1) If [[~(xk)[[ = 0, stop. Otherwise, solve the following linear equation:
Bkd + ¢b(Xk) = 0
(3.1)
to get dk. (S.2) If
(xk + dk)ll - P2lf ,k (xk)ll < -0-1Hd ]l 2,
(3.2)
then let Ak := 1 and go to (S.4). (S.3) Let As be the maximum number in the set {1, pl, p 2 , . . . , }, such that A = p~ satisfies the line search condition [14~.k (xk + Adk)l[ - II(I)~k(xk)II < -0-2[IdaH 2 + ~k.
(3.3)
528
C. MA
(s.4) Let xk+i := xk + Akdk. (s.5) Update Bk by the Broyden-like update rule Bk+~ = Bk + Ok (yk - Bksk)s-~
ii~kll 2
,
(3.4)
where sk = xk+i - sk and Yk = ~ (x~+i) - ~ (xt¢). The parameter Ok is chosen so that IOk -- 11 < 0 for some constant O E (0, 1) and Bk+i is nonsingular (see [22]). (s.6) If (3.2) holds or 711@(Xk+i)ll < #k, let #k+l := min
[1~ (xk+i)ll, ~#k
•
(3.5)
Otherwise, let #k+i := #k. (S.7) Let 7/k+i := min{(1/2)~k, II@(xk+i)ll} and k := k + 1. Go to (S.1). REMARK 3.2. (1) Since ~?k+l 0, such that for all k 6 / ~ sufficiently large,
C~ll~(xk)ll
_< lldkll 0, then d = 0. So it follows from (3.1) that ~(2) = 0, which contradicts the fact that 0 < # < 11~(2)11. We now consider the case of A = 0, or equivalently, limsuPk(eg)_.~ ~k = 0. By (S.3) of Algorithm 1, when k e /{ is large enough, A~ = Ak/Pl does not satisfy (3.3). Therefore, we have
Ile.(x~ + ~2d~)LI - II~(~k)ll > --~211Akdkll 2 Multiplying both sides by (A~)-l(]]¢p(xk + A~dk)ll + II~n(xk)ll) and then taking the limit as k(e K ) ~ oo yield 2 ~ p ( ~ ) T v ( I ) p ( ~ ) J > 0. (4.24) Since ~(xk) = - V ~ ( x k ) d ~ , by (4.1), taking the limit as k(E /~) --* c~ yields ¢(~) = - V ~ p k (2)~ It then follows from (4.24) that
~/~('~)T'(I)(~) S O.
A Smoothing Broyden-Like Method
533
This together with (3.7) implies (1 _ 1 ~n7 2)[l~(~)l] 2 _< l ] ~ (~)1]2 + 0 _ ¼~?) i1,~(~.)11~ _< I I ~ ( ~ ) l f + II~(~)ll: - ¼ ~
1 -2 = II~(~) - @(~)II= - ~ . + 2@~(~)T@(~)
(1)~ -- ~n/z 1-~+ 2 ~ p ( e ) T ~ ( e )
_< ~V~/2
(4.25)
_< 0,
where the third inequality follows from (2.7). Notice that 0 < ~/ < 1/~/~, that is, 3/4 < 1 - (1/4)n~/2 < 1, hence (4.12) is also a contradiction. So, K must be infinite, and thus, the assertion follows from Lemma 4.3.
5. S U P E R L I N E A R
CONVERGENCE
We turn to in this section analyze the convergence rate of {xk} generated by Algorithm 3.1. For this purpose, we make further the following assumptions. ASSUMPTION C. (i) The sequence {xk } generated by Algorithm 3.1 converges to a solution x* of MCP(F) and V ~ (x*) is nonsingular. (ii) The strict complementarity holds at x*, that is, ~ ( x * ) = ~ ( x * ) -- ~. Let 1
(5.1)
= 2 l 0 and 0 ¢ ~. It is also obvious that ~z(x) --
#2(x) -- Z for every x E 0, and hence, ~(x) is continuously differentiable on 0. In particular, for every x C 0, we have V@(x) = (V¢l(x), V ¢ 2 ( x ) , . . . , V¢2(x)) T with
V¢i(x) =
2VFi(x), 2ei,
i E a(x), i e 7(x),
(5.3)
where 3(x) -- 3z(x) U'y2(x). This means that V(b~(x) - Vq~(x) for every x E U. Moreover, there exists a constant M0 > 0, such that
II~(x) - '~ (x*)ll O, such that v~,~ e O, (5.5) liVe(x) - Vq~(y)H < Lll]x Yl], -
and
IlV~,(x) - V~,(y)H _< nzHx - Y]I,
Vx, y E U.
(5.6)
534
C. MA
LEMMA 5.2. Let Assumption C hold. Then there is a constants M~ > O, such that for every #>0, (5.7) Vz~O. II~.(~) - ~(x)ll -< M I # 3, Paoo~. By (2.2) and (2.5), we have
I¢~(x,g) - @(x)[ _< max
2~
'
E (5.8)
/
~- max
~3
~4
< 2~]xi - us - F~(z)[ 2 < 2--~" Letting M1 = v ~ / 2 g 2, we get (5.7). LEMMA 5.3. Let Assumptions B and C hold, {xk} be generated by Algorithm 3.1, and 5k be defined by (4.17). Then there exist a constant 5 > 0 and an integer k' > O, such that Ak = 1 whenever 5k < 5 and k > k'. PROOF. Since Xk --* x*, there is an index k[, such that xk E 0 for all k > k~. Notice that xk + rsk c U for any T E [0, 1] when k is sufficiently large. Then we have for every k > k~,
IIAk - V~(x*)ll =
IIV~.~(Xk + TSk)-- V~(X*)II d"
=
HV~(xk + TSk) -- V ~ (x*)H dT
_< LI
(5.9)
/01
lit ( z k + l - x*) + (1 - r) (xk - ~*)11 d r
