Linear complementarity problem with upper bounds

European Journal of Operational Research 40 (1989) 337-343 North-Holland

337

Theory and Methodology

Linear complementarity problem with upper bounds Hans-Jakob LUTHI E T H Zfirich, Institut fftr Operations Research, ClausiusstraBe 47, 8092 Zfirich, Switzerland

Abstract: The problem addressed in this paper is the linear complementarity problem with upper bounds. First we show, using only combinatorial arguments, that the number of solutions is odd. Secondly a variable dimension algorithm based on complementary pivoting is given to find such a solution. Keywords: Linear complementarity, quadratic programming, variational inequalities

1. Introduction In this paper the following problem will be studied: Find (s, h) ~ R ~ × R" such that s = q + a¢~, ~i=O ~i=di

~

O.<~,.
fl),(2)

si~
=*. s , ) O ,

(3)

O < X i < d i =~ s i = O ,

where d > 0 , q ~ R n, M ~ R nx" are given. Such problem arise in the context of linear complementarity with upper bounds as studied for example by van der Laan and Talman [3]. The problem (1)-(3) can be transformed into a classical LCP in R z" and Lemke's method could be used for solving it. The intend of this paper is to show that a 'variable-dimensional' pivoting technique can be used to obtain the following two results: (1) Using only combinatorial arguments, it will be demonstrated that the number of solutions is odd. The procedure is very similar to the classical proof of the Sperner's Lemma [4].

Revised April 1987; revised May 1988

(2) A construction will be given to find one solution by c o m p l e m e n t a r y pivoting in a variable-dimensional 1 setting. We do not know of any similar algorithm to solve this problem directly, but we note the similarity to the method developed by van der Heyden [2], (see Section 5). Indeed, as is well known, problem (1)-(3) corresponds to a related variational problem for the affine function g ( h ) = - q - M h over the feasible region K = ()t l0 ~< )~ ~< d }. Hence, by a fixedpoint argument we know about the existence of solutions to (1)-(3) [1].

2. Notations and basic concepts For dealing with condition (3) in a complementary way we shall introduce some notations. Let I := {1 . . . . . n } and for any h ~ I denote I h : = (1 .... , h } c I .

1 The notion 'variable-dimension' is used in the sense that in the solution process a sequence of lower-dimensional problems are 'solved' while always the whole matrix is updated.

0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

H.-J. Liithi /Linear complementarityproblem

338

Given a vector x ~ R ~, S ( x ) ~ ( + , 0, - }" denotes the sign of its components, i.e.

S , ( x ) = see(x,). To specify the sign of X ~ K in accordance with (3) let us define the mapping X: K--+ ( + , O, - ) " by

Nondegeneracy assumption. The system (1)-(2) is

called nondegenerate iff the number of nonbasic variables in any solution is at most n. For the following we will always assume, that the system (1)-(2) is nondegenerate. As it is well known from LP-theory this is an important but nevertheless nonrestrictive assumption.

if h i = 0 ,

ai(x)=

0 +

if 0 < h i < d i,

(4)

if h i = di,

Lemma 2.4. I f the system is nondegenerate, then for every (s, X) satisfying (1)-(2), S ~ A impfies S =

A. i = 1 , . . . . n.

Proof. In the proof of Lemma 2.3 we showed that S - < A implies complementarity. But then by counting the result follows. []

Let X ~ { +, 0, - )n be a sign vector, then X°= {e~IlXe=O),

X+=(e~Ilge= X--~ (eEIlXe

+), = -}.

Lemma 2.1. Let ( s , ) , ) be a solution to the system (1), (2). Then, (s, X) satisfies (3) iff

(5)

Up to this point the definitions and results are quiet simple. In the next section we are going to give a combinatorial proof that the number of solutions to (1)-(3) is odd.

3. Existence of solutions

where " ~ " means S+ c A + a n d S - c A - . Proof. If (3) holds for (s, X) then by the definition (4) of A, (5) follows. Assume (5) holds, then Ai(X ) = - implies s i ~< 0, i.e. h i = 0 implies s i <~0 and similarly for the other cases. []

As can n = 1 the Note that The main

easily be seen by inspection, for the case number of solutions to (1)-(3) is odd. we will always assume nondegeneracy. result of this section is:

Theorem 3.1. The number of solutions to (1)-(3) is

For notational convenience for given (s, X) in the sequel we will write (S, A) instead of

odd.

(S(s), A(X)).

We will prove Theorem 3.1 by induction on the dimension n. For doing so, some further concepts are needed.

Definition 2.2. Let (s, X) be a solution to the system (1), (2); then - s i is called nonbasic iff S i = 0. - ),i is called nonbasic iff A i 4= 0. - Not nonbasic variables are called basic. - (s, X) is called a complementary solution if either s i or Xi are nonbasic for all i ~ 1. Lemma 2.3. Every solution (s, X) to (1)-(3) is

Definitions 3.2. A solution (s, X) to (1, 2) is called 1g-feasible (for k ~ I ) iff (i) stk ~ A lk, ( i i ) A e 4= O, e¢i I t. (Herein Ss denotes the sign vector in ( + , 0, - } IJ I obtained from S by deleting the components i ~ J _CI.)

complementary. Proof. By Lemma 2.1 it suffices to show that (5) implies complementarity. But, let Xi be basic, i.e. A , = 0; thus by (5), S i = 0 and therefore s i is nonbasic. []

First we note immediately that/h-feasible solutions are complementary and under nondegeneracy equality must hold in (i). To complete, we introduce the notion of almost Ik-feasible and almost-complementary solutions.

H.-J. Liithi / Linear complementarityproblem Definitions 3.3. A solution (s, X) to (1), (2) is called almost 1k-feasible (for k ~ I ) iff (i) S~k-~ ~ A ik 1, (ii) A e ~ 0 , e ~ I k, where for k = 1, I ° -= 0 and (i) holds vacuously. It follows that /k-feasible solutions are almost /k-feasible. Furthermore such solutions are almost complementary, i.e. s, or Xg are nonbasic for all but at most one index i ~ I. Lemma 3.4. For any almost-Ik-feasible solution (s, X) exactly one of the following statements holds: (a) Ik-feasible. (b) Complementary but not Ik-feasible, i.e. 0 -~ S k-~ Ak 4: 0. (c) Almost complementary with basic complementary pair (Sk, Xk). Proof. As one obtains by counting, almost I kfeasible solutions are almost complementary or complementary. In case (s, X) is not complementary, then only (c) can hold by nondegeneracy. Now, assume (s, X) to be complementary but not /k-feasible, then 0 =g Sk 4: A k ~ 0, since S k = 0 implies Sk ~ A~. [] 3.5. Let (s, X) be an /k-feasible solution. It is called with respect to I k an @ - v e r t e x iff s k = 0 ( ¢ ~ A k = 0 ) , @ - v e r t e x iff s k < 0 ( ¢~ A k < 0), (~)-vertex iff s k > 0 ( ~, A k > 0).

339

the basic variables s i, i ~ I k are not to be considered!] Exchanging Sp (or Xp) with h k by pivoting, leads to a new tableau with exactly n nonbasic variables. If p = k, the process stops in a @ - v e r t e x ; else there exists a unique nonbasic complementary pair (Sp, Xp), of which one had just become nonbasic. Its complement will now be increased (decreased) to maintain almost /k-feasibility (together with s k ~< 0) on the path: If Xp became nonbasic with Ap = + , then increase sp (Sp >1 0). If ~kp became nonbasic with Ap = - , then decrease sp (Sp <<.0). - If sp became nonbasic, then decrease hp if A e = + , else increase Xp. Note, pivoting is such that feasibility with respect to Sik l ~ Atk-1 and s k <<.0 is maintained on the path keeping Xe, e ~ I k fixed. Using a standard argument a unique path is followed: The process stops, when either case (a) or (b) of L e m m a 3.4 is reached, namely: In case (a). If s k is nonbasic, an @ - v e r t e x had been reached; else Xk has become nonbasic with A k = - , hence a stop in a @ - v e r t e x occurs. In case (b). Since s k <~0 on the path, we must have for the complementary endpoint (s, X ) : A k = + and S k = -

Definition

Definition 3.6. Let (s, X) be an almost-Ik-feasible solution satisfying case (b) of Lemma 3.4. Then (s, X) is called with respect to I k a @ - v e r t e x iff A k = + ( ~ Sk = -- ),

Assume (s, X) to be/k-feasible for some k. Fix all the nonbasic variables h e for e ~ I k at their upper or lower bound, respectively. If (s, X) is a @ - v e r t e x with respect to I k then by increasing the nonbasic variable Xk (keeping the other nonbasic variables s i or Xi, i ~ I k fixed) a unique 1-dimensional almost-lk-feasible path is initiated. More precisely: h A is increased until either (a) a basic variable si or X~, i ~ I k, becomes for the first time nonbasic, or (b) k k reaches its upper bound d k, without violating the almost /k-feasibility and s k <~0 on the path. In case (a), because of nondegeneracy, there exists a unique basic variable sp or hp, p ~ I k, which becomes for the first time nonbasic. [Note:

(~)-vertex iff A k = - ( ~ Sk = + ). Summarizing: Starting from a @ - v e r t e x an almost-Ik-feasible path leads either to a uniquely defined (by the above procedure) @ - v e r t e x , @ vertex, or @ - v e r t e x , with respect to I k. Now, assume (s, X) to be an @ - v e r t e x i.e. (i) (s, X) is /k-feasible, and (ii) S k = 0 ( ¢* A k = 0 under nondegeneracy). Two almost-Ik-feasible pathes can be initiated at (s, X): - a ( ~ - p a t h by increasing Sk and maintaining Sik-1 --1 0 on the path, - a C ) - p a t h by decreasing s k and maintaining S1k, -
340

H.-J. Li~thi / Linear complementarityproblem

The (5)-path either ends in another @-vertex, or in a @-vertex, or in a @-vertex with respect to Ik:

@

(5)

- Starting with a (~-vertex. The following pathes are possible:

© @,

(even):

@(5)@,

@

(5) © ,

@

@,

(5) ®.

(5) @

©,

path,

@••

(even).

(5)

A path starting with a @-vertex, leading to an @-vertex, leaving it in the opposite direction from entering:

@

path, @,(odd): @--@path,

- An @ - - @

path:

An C ) - - @

path:

(~ @

Combining all those paths together the following configurations with respect to I k may occur: - A loop of @-vertices:

(5)

(5)

@ (5) @ _(5) @ , ( e v e n ) : @ - - @ p a t h .

-

@

©--©

(odd): © - - ©

Similarly for the (5)-path: Either it ends in another @-vertex, or a ©-vertex or an ©-vertex with respect to Ik:

(5)

@. (5) @

©@@@@@®,

@(5)@.

@

(5)

-

@

An @ - - @

C)

@ (odd).

path:

@ e @ (5) @ (5) @ (even). We indicated the number of /h-feasible solutions on each paths as even or odd depending on the endpoints of the path. This characterization will be the main tool for the Sperner-type proof of Theorem 3.1.

-

@@@@@@@, (even): @ - - @

path,

@©@®@©@, (odd): @ - - @ @

(~

@ (~

path, ©,(odd): (~)--©

( ~ (~) @ (~) (~),(even): ( ~ ) - - ©

path, path.

3.7. The number of Ik-feasible solutions is odd for k = 1 . . . . . n where A j ~ O, for j q~ I k are fixed. T h e o r e m

Proof. By induction on k. For k = l : Fixing A e ~ 0 for e = 2 .... ,n the theorem has to be verified for a line, which can be done by inspection. Assume the theorem to be true for h = k - 1. Then fix A k = + and let V+ be the set of I hfeasible solutions with A k = +. Similarly, let Vbe the set of /h-feasible solutions with A k = By assumption I V+ [ and IV- I are odd. (Note that for j ~ I ~, Aj ~ 0 are fixed.) Consider now the elements of V+ and V-, respectively, with respect to Ik: The elements in

H.-J. Liithi /Linear complementarityproblem V+ are either Ik-feasible and hence are C)'vertices by definition, or are not Ik-feasible and hence @-vertices. Similarly the elements of V- are either @-vertices or @-vertices, (Lemma 3.4). Therefore,

v+=B+uv2, where V+ , VJ, Vb , V~ denote the set of all @ - , @ - , @ - and @-vertices with regard to I k, respectively. Follow now, with respect to I k, all almostIh-feasible pathes leading from a @ - or @-vertex to a C)" or @-vertex. Note that the number of /h-feasible solutions on each path is even. Drop the corresponding vertices connected by such a path from V + to obtain the reduced set Vx+. Similarly drop all vertices from V- corresponding to @ - @ , ( ~ ) - - @ and @ - - @ pathes respectively to obtain the reduced set II1-. Certainly V( and V~- are odd. But they must be equal, since each remaining path starting in a vertex of V~+ must end in a vertex of V1- and vice-versa. By pairing them up an odd number of such pathes is obtained. To prove that each such path contains an odd number of solutions we delineate the possible pathes:

@--@

(odd):

@--@

(odd):

© @--(~

@

®

©

@ @®©@,

341

4. The algorithm The proof of Theorem 3.1 is 'almost'-constructive in the sense that it parallels the classical proof for Sperner's Lemma [4]. Certainly the question arises to find a solution by an algorithm, i.e. to give a totally constructive proof. The algorithm described in this section is based on complementary pivoting and determines an Ik-feasible solution starting at any vertex of the feasible set (2). It is a variable-dimension pivoting scheme, where the pivot is guided by A° of the initial starting point (s o, )to). To implement the algorithm upper-bound pivoting techniques should be used for the variables )t. To describe the algorithm some of the concepts developed in the previous sections must be refined.

Definition 4.1 (A°-adjacency). Let A° be a given sign-vector such that A° 4:0 for i ~ I. An almost/k-feasible solution ( s , ) t ) is called A°-adjacent iff A t=A°~

f o r e ~ I k.

The algorithm will maintain A°-adjacency along the path where A° = A()t°). Let ( s , ) t ) be A°-adjacent and complementary. Define

p:=max{l~{1

..... n + l } l S f - , ~ A , , - ~ } ,

q : = m i n { ! ~ { O ..... n } [ A e = A ° e , e > l } ,

@

@@,@ © @,

(odd) :

® © ® © ® ©@,@©@, © ® © @. @ - - @ (odd): @

(6) (7)

where (p, q) are well defined and evidently, since (s, X) is almost /k-feasible p >/k 1> q. Furthermore: p = (n + 1) if and only if ( s , ) t ) is/"-feasible, hence a solution to (1)-(3); q = 0 if and only if ( s , ) t ) = (s °, ?t°) i.e. the 'initial' solution; p = q if and only if ( s , ) t ) is almost F-feasible and O 4: Sp ~ Al,=g O, A° 4: Ap. The algorithm will be such that p > q.

Therefore, since there is an odd number of pathes and each path contains an odd number of /h-feasible solutions, the total number of I~-feasible solutions must be odd. []

Lemma 4.2. Let ( s , ) t ) be Ao-adjacent and complementary. Assume 0 4, q < p <~n. Then (i) (s, X) is almost IP-feasible with 0 --g Sp ~ Ap = A° 4= 0, (ii) ( s , ) t ) is 1q-feasible with Sq = Aq ~ A °.

Note, that Theorem 3.1 follows from Theorem 3.7 immediately for the special case k = n.

Proof. This is an immediate consequence of the definitions (6), (7) and the remarks above. []

342

H.-J. Liithi / Linear complementarityproblem

Being at a A°-adjacent, complementary state (s,)~) with q < p , based on Lemma 4.2 we define (two) permissible ways to leave this state. If O ~ q < p < ~ n : (a) Follow the unique almost-IP-feasible path maintaining S p ~ - A ° along the path. [Note: (s,)~) is with respect to I p a (~)- or @-vertex.] (b) Follow the unique almost-Iq-feasible p a t h while maintaining S q ~ - A ° q along the path. [Note: (s, ~) is with respect to I q a (~)-, (~)- or @ -vertex.] If O=qp. Therefore ql = p , furthermore pl > ql. Or: (lb) (s 1, )~1) is not /P-feasible, i.e. 0 4: Spa ~ A~ * A °, A ~ = A ° for e > p . Therefore p a = p and q~ < pl. Path (b). For path (b) the complementary endpoint (s 1, k1) is either: (2a) /q-feasible: Sq~ = A~q ~ A °, Ale = a °e , e > q . Therefore p~ > q~ = q. Or:

(2b) Not /q-feasible: 0 ~ S~ ~ Alq = A °, Ale = A °, e > q. Therefore q = pl > ql. Summarizing, we have shown that either path leads to an new Ao-adjacent complementary solution (s ~, )~1) with p ~ > ql. Moreover, we easily deduce from the proof of (ii) that in the new state one permissible path must lead back to the previous state.

Algorithm 4.3. Step 0 (Initialization). Choose )0 with X°, = 0 or d i for i = 1 ..... n. Determine s o .'= q + MA °. Let ( s , ) , ) . ' = (s o, )~0), (S, A ) . ' (S °, A°), A°: = A()~°) (fixed). Go to Step 1. Step 1 (Determine p). p ,= m a x ( l ~ ( 1 , . . . , n + 1} ISI,-~ ~ A , , - ~ } . If p = n + 1, stop. (A solution is obtained.) Else go to Step 2. Step 2 [Path (a)]. Follow the unique almost-/pfeasible path maintaining Sp ~ - A ° e along the path until the next complementary point (s ~, )~1)is reached. (s,)~):= (s 1, hi). If (s,)~) is /P-feasible go to Step 1. Else go to Step 3. Step 3 (Determine q). Let q,=min(lE

(0 . . . . . n } I A e = A °, e > l } .

Go to Step 4. Step 4 [Path (b)]. Follow the unique almost-I qfeasible path maintaining Sq-< -A°q on the path until the next complementary point (s l, )d) is reached. (s,)~):= (s 1, )~1). If (s, ~) is /q-feasible go to Step 1. Else go to Step 3.

Lemma 4.4. Algorithm 4.3 solves problem (1)-(3) in a finite number of steps. Proof. The algorithm always leaves a A 0-adjacent

complementary state on the other permissible path it had entered. Since each state has at most two permissible ways to proceed, but the starting solution exactly one, the algorithm finally must end with p = n + 1, i.e. with a /"-feasible solution. [This is the standard Lemke-type argument.] [] A simple graphical example for a 2-dimensional problem shall help to illustrate the algorithm (Figure 1). (-,

Step O. Choose )0 = (0, 0): A ° = ( - , - ) , S o = + ) [Point (0)]. Step 1. p = 2.

H.-J. Lfithi / Linear complementarity problem

T o i m p l e m e n t Algorithm 4.3 an u p p e r - b o u n d pivoting technique is suitable.

~2 X2

343

= d2

5. Final remarks s2:0 /2

-

~l

=0

kI = dI ,

Figure 1

Step 2. [Path (a)]: Maintain s e > 0 on the path. Increase Xz ~ S l becomes nonbasic. Increase )'1 --+ ~z becomes nonbasic and (s 1, )~1) is complem e n t a r y (Point 1). A 1 = (0, - ) , S 1 = (0, + ) , (s 1, X1) is not I2-feasible --+ Step 3. Step 3. q = 1. Step 4. Since A ] = - : s 1 > 0 on the path; i.e. increase s I --+ Xl becomes nonbasic and hence a c o m p l e m e n t a r y point is reached [Point (2)]. A 2 = ( + , - ) , S 2 = ( + , + ) . Point (2) is/q-feasible ~ go to Step 1. Step 1. p =- 2. Step 2. Maintain s 2 >/0 on the /Z-path: Increase X 2 -+ s 2 becomes nonbasic, hence a complementary point is reached (Point 3): A 3 = ( + , 0), S 3 = ( + , 0). Point (3) is I2-feasible, go to Step 1. Step 1. p = 3, stop with point (3).

T h o u g h this study did not focus on the classical LCP, it is noteworth to m e n t i o n that the L C P ( - q / - M ) is obtained b y letting di ~ ~ for all c o m p o n e n t s i. It is left to further investigations to delineate those classes of matrices which can be 'solved' using the pivoting techniques described herein for suitable chosen d > 0. In particular we note that the algorithm started at ~ = 0 for the ' u n b o u n d e d ' case is identical to the one developed by van der H e y d e n [2], given that no ' u n b o u n d e d line' exists (Definition 3.1 in [2]).

References [1] Eaves, B.C., "On the basic theorem of complementarity", Mathematical Programming 1 (1971) 68-75. [2] Van der Heyden, U, "A variable dimension algorithm for the linear complementarity problem", Mathematical Programming 19 (1980) 328-346. [3] Van der Laan, G., and Talman, A.J.J., "An algorithm for the linear complementarity problem with upper and lower bounds", Technical Report. [4] Sperner, E., "Neuer Beweis f'ur die Invarianz der Dimensionszahlen und des Gebietes", Abhandlung Mathematisches Seminar, Unioersitiit Hamburg 6 (1928) 265-272.