An interative scheme for complementarity problems

An iterative scheme for complementarity problems M U H A M M A D A S L A M N O O R and S. Z A R A E Mathematics Department, College o f Science, King Saud University, Riyadh, Saudi Arabia

It is shown that the complementarity problems can be formulated as a fixed point problem. This formulation is used to suggest new algorithms for both linear and non-linear complementarity problems. Convergence analysis of the approximate solution is discussed. Furthermore, it is shown that there are certain examples of linear complementarity problems which are solved by the new proposed algorithm, but are unsolvable by the algorithms of Mangasarian and Van Bokhoven. Key Words: complementarity problems, algorithm, convergence, fixed point problem, iterative methods 1. INTRODUCTION

2. PRELIMINARIES

Complementarity theory is a relatively new area of mathematical programming, which has received much attention during the last two decades. There are two classes of numerical methods for solving the complementarity problem, the direct methods which are based on the pivoting and the iterative methods which produce a sequence of iterates (trial solutions) and converge to a solution. In many applications of linear complementarity problems to free boundary value problems and partial differential equations, the underlying matrices are often large, sparse and specially structured 1'2 The direct methods of linear complementarity theory seem to be inefficient when applied to these problems. This fact has stimulated much investigation of alternative approaches for solving the linear complementarity problems.2-4 Recently iterative methods have been found very useful for solving many large-scale linear complementarity problems arising from applications. Most of these methods are based on extensions of their counterparts for solving square systems of linear equations. Among the most widely used iterative methods for solving the system of linear equations in the family of successive over-relaxation (SOR) methods. There are mainly two algorithms for solving the linear complementarity problems, which are due to Mangasarian 3 and Van Bokhoven.s There are some examples of linear complementarity problem, which cannot be solved by these algorithms. This fact has motivated us to modify Van Bokhoven's algorithm. In this paper we investigate and propose an iterative method following the idea of Van Bokhoven.s We study the unified framework for the convergence of the proposed iterative method for both the linear and non-linear complementarity problems. As a special case, we obtain the results of Ahn 6 and Gana. 7 The plan of the paper is as follows: in Section 2, we review the preliminary results along with the basic definitions and algorithms. The convergence theory is discussed in Section 3 both for linear and non-linear complementarity problems. In Section 4, examples of linear complementarity problems are given which can be solved by the modified algorithm, but not by the algorithm of Mangasarian or Van Bokhoven.

We denote the inner product and norm on R n by (., .) and II.11, respectively. Given a continuous mapping T from R n into itself, we consider the problem of finding u such that

Accepted June 1986. Discussion closes February 1987.

0264-682X/86/030221--04 $2.00 © 1986 Computational MechanicsPublications

u ~ O,Tu >~O, (u, T u ) = O

(1)

Problems of the type (1) are known as complementarity problems. If the mapping T is non-linear, then problem (1) is called the non-linear complementarity problem, while if the mapping T is an affine transformation of the form T:u ~ M u + q, M E R n×n and q E R n, then problem (1) is equivalent to u >t O,Mu + q >I O, (u,Mu + q) = 0

(2)

which is known as linear complementarity problem. When M is symmetric, then it is well known that the KarushKuhn-Tucker optimality conditions of the quadratic progra program: min f(u) = 12(Mu, u) + (q, u)

(3)

u~O

are equivalent to the linear complementarity problem (2). The iterative methods for solving the linear complementarity problem (2) are a modification of the iterative methods for solving linear systems of equations. In 1977, Mangasarian 3 proposed a general and unified iterative method for solving (2), when M is a symmetric matrix. Mangasarian 's algorithm 2.1s Let Uo be any arbitrary non-negative vector, 0 < k < 1, and p > 0. Generate the sequence (u n) as follows: Un+1 = (1 -- k) u n + ~ [un -- pE(Mu n + q + L (Un+1 Un)]+; n = 0, 1, 2 . . . . -

-

where E is a positive definite diagonal matrix and L is the strictly lower or strictly upper triangular part of M. Mangasarian 3 proved the convergence of this algorithm relying on the descent of the function (3), which is shown to be a non-increasing function of the sequence of iterates. We also note that Mangasarian's algorithm reduces to Cryer's algorithm, s if ) , = 1 , E = D -t, where D is the diagonal part of M. Ahn 6'9 studied the convergence properties of Mangasarian's algorithm for the non-symmetric matrix M.

EngineeringAnalysis, 1986, Vol. 3, No. 4

221

An iterative scheme for complementarity problems." M. Aslam Noor and S. Zarae Sometimes, it is useful to write the linear complementarity problem (2) in the form:

v = M u + q >~O, u >~O, ( u , v ) = 0

(4)

In 1980 Van Bokhovens using the change of variables: v=lxl--x

{ (Ix"l+x")+q+L(x,,+l--x,,)}]; --pE M 2

and u = l x l + x

(5)

where Ix I = ( I x l l , . . . , Ix,,I), proposed and studied the following algorithm:

n = O, 1,2 . . . .

(10)

where p > 0 is a constant, )t > O, a relaxation parameter, E E R nx,,, a positive diagonal matrix and L is the strictly lower or upper triangular part of M. 3. C O N V E R G E N C E A N A L Y S I S

Van Bokhoven's algorithm 2.2 s Given any real vector x0, compute {x,,} + x,,+, = (1 - )t) Ix,,__l+ xn + )t [Ix,, I X n 2 L 2 pE -

-

-

2

In this section, we study the conditions under which the approximate solution obtained from (10) converges to the exact solution of the linear complementarity problem (4). To prove the convergence of the iterates Xn+a, we need the following:

(M(Ix,, I + x , , ) + q -1

+ L(lxn+x l + x , , + l - Ix,, [ - x , , ) } / A

(6)

Ix,,+l --x,, I <<-(21--)toE IL [)-I I 21-- )tpE(M--L ) I

where E is a positive diagonal matrix, L is the strictly lower triangular part of M, p > 0 and 0 < )t < 1. The convergence properties of this algorithm has been studied by Gana7 where he showed that the sequence (x,,} generated by the iterative scheme (6) converges to the exact solution of (4) for the non-symmetric matrix M. We have found that there are examples of the linear complementarity problem (2), which cannot be solved by the algorithms of Mangasarian and Van Bokhoven. However, if we make some modification in the change of variables (5) and the original algorithm of Van Bokhoven, these problems can be solved. We here consider the following change of variables: u-

Ixl+x - 2

Theorem 3.1 Let (x,,.l} and {x n} be the sequences generated by (10), then

x

Ix,,+,--x I <~(2I--)toE IL

PE~M(lXl+x]+

2t

\---5--J

Proof." From the iterative scheme (10), we have:

(7)

Ixl+x - 2

P_EIM(IXI+x ~ } + q 2t \ 2 /

+ L (Xn+ 1 -- x,,) -- L (x,, -- Xn_l) I y

(8)

(9)

for some constant O > 0 and E E R ,,xn, a positive diagonal matrix. Based on the above observations, we suggest and analyse a new algorithm for solving the linear complementarity problem (4). This modified algorithm is compatible with the algorithms of Mangasarian and Van Bokhoven.

Modified algorithm 2.3 Given Xo, compute x n by the following iterate:

Ix,,l+x,, x,,+~ =(1 - ) t ) ~-- Ix,,I + x . 2

222

Engineering Analysis, 1986, Vol. 3, No. 4

Ix,, I-Ix,,-x I+x,,-x,,_~ 2

+~[(Ixn I-'Xn-ll+Xn-X,,-1)

=

It is worth mentioning that x and Ixl can be determined uniquely by the relations x = u -- t2p Ev and Ix I = u + ~p Ev. Thus, using the change of variables, we have shown that the complementarity problem can be formulated as a fixed point problem of solving:

x=F(x)=

(12)

(4).

}

q

I)-112I--)tpE(M-L) I

where x = u -- ½PEv and the pair (u, v) is the solution of

where p ~>0, E E R nxn is a positive diagonal matrix. It is obvious that u ) 0 and v/> 0. Using (7) and the technique of Van Bokhoven,s one can prove that the linear complementarity problem (4) is equivalent to finding x E R n such that

Ixl+x

(11)

x Ix,, - x l

Xn+l --x,, = (1--)t) and v = ( p E ) - l ( l x l - x )

Ixn--Xn-11

and

1-- )t 2 +

(Ix,,l+x,,-Ix,,-xl-x,,-O

I-

)(Ixnl+xn-lx,,-11-x,,-a)

+ -)t peL (x,, - x , , _ 0 - ~X pEL (~,,+, --x,,) 2 Taking absolute value of both sides and using the inequalities,

la--b and


I l a l - - b ll <~ l a -- b l,

foralla, b E R n for all a, b E R n

we obtain . ]Xn+ 1 - - X n

pEM

< (1--)t)IXn--Xn_11 + )t l----~-- I lXn--Xn-a I

+

)tpE + )toe .... IL [Ix,,-Xn-xl IL IIx,,-l-Xn I 2

An iterative scheme for complementarity problems: M. Aslam Noor and S. Zarae Algorithm 3.5 Given xo E R n, compute

XoE (M-L) l x.-x._~ I 2

<<.I I - ' - - + XpE

X.+l = (1 - - ~k) [xnl+Xn - - t - ~ 2

I L II X.÷l - - x . I

2

n =0, 1,2,...

Hence (21--XpE IL I) IXn+l--xn 1<<.I21--XpE(M--L ) I x Ixn--xn-ll Since L is strictly upper or lower triangular matrix, then

(1--X(p/2)EIL I) is an M-matrix. Recall that a square matrix with non-positive off-diagonal elements and with a non-negative inverse is called an M-matrix, see ref. 10 for various characterisation of a M-matrix. Thus (I-Ot/2)pE IL I)-1 exists and is non-negative. Hence we obtain:

1)-11 2 / x Ixn--Xn-l[

IXn+l -- Xn I <~ (21 -- hoE tL

[ [ X n l + X n P T {[Xn~Xn}] 2 2

;

(16)

where p > 0 is a constant and ~ > 0 is the relaxation parameter. To prove the convergence of the approximate solution obtained from (16), to the exact solution u of (14), we need the following concepts:

Definition 3.6 A non-linear operator T:Rn~R" is said to be: (i) Strongly monotone, if there exists a constant ot > 0 such that:

(Tu-- Tv, u-- v)>>-allu--vll 2, forallu, v E R n

;kpE(M -- L ) I

the required (11). Similarly, we can prove (12). We now state the following principal convergence result for the iterative scheme (10).

(ii) Lipschitz continuous, if there exists a constant/3 > 0 such that:

II T u - T v l l < < - ~ l l u - v l l ,

forallu, v E R n

In particular e, ~<¢]. Now we state and prove the main result of this section.

Theorem 3.2 Suppose that:

Theorem 3. 7

o(G) < 1

Let T be a strongly monotone and Lipschitz continuous operator with monotonicity constant ~ > 0 and Lipschitz continuity constant /~>0 respectively. If {Xn+l)is the sequence generated by algorithm 3.5, then

where G = (21 - XpE IL

I)-112/- XpE(M -- L) I

(13)

with o denoting the spectral radius and ~ > 0 is the relaxation parameter. Then for any initial vector Xo, the sequence {Xn) obtained from (10) converges to a solution of (4). Its proof is similar to that of theorem 4.1 of Ahn.6 (See also Pang ~x and Noor and Zarae12). Note that since an Hmatrix with positive diagnosis is a P-matrix,1~ there exists a unique solution.

Xn+ 1 -*" X

for 4~

O
Remark 3.3 Note that for )t = 2, the results of theorem 3.1 are exactly the same as proved by Gana 7 for the algorithm of Van Bokhoven. Furthermore, it is noted that results hold for all values of?, > 0, whereas the results of Gana 7 are only true for 0
in R n

1

(1 - V / 1 - o l p

+~)

where x is the solution of (14).

Proof." Since the solution x of (14) is also a solution of (15) and conversely, so from (16) and (15), we have:

Xn+x-x = (1-~,) }(IXn+x l + xn) +x.

3.4. Non-linear complementarity problem It can be shown by using (7), that the non-linear complementarity problem of finding u ER n such that

u>>-O, v= Tu>~O, (u,v)=O

,x.+x__ 2

P T(L~ 2

)

/ xl+x\

(14)

where T is a non4inear mapping from R n into itself, is equivalent to finding x E R n such that:

x-

k

Hence

Ilxn+l-x I1"<<(1- k ) I I x , - x II +~11 . x n l + x ~ l x I - x

(15)

This equivalence enables ut to suggest the following algorithm for finding the approximate solution of (14).

I+x,~

(17)

Engineering Analysis, 1986, Vol. 3, No. 4 223

An iterative scheme for complementarity problems: M. Aslam Noor and S. Zarae Now by using the strongly monotonicity and Lipschitz continuity of T, we obtain

I

u>~O, Mu+q>~O, (u,Mu + q ) = 0 with M=

=1x" +x"

x "l'

[i011 4

--3

--

[i]

and q = --

This problem is only solvable by the modified algorithm 2.3, but not by Mangasarian's algorithm 2.1 and Van Bokhoven's algorithm 2.2.

,x?x) O2p= ~< 1-ctp +

4

Ilxn-X I?

(18)

Example 4.2 Consider the linear complementarity problem of the form: u>~O, Mu+q>~O, ( u , M u + q ) = O with

by definition 3.6. Thus from (17) and (18), we obtain:

Ilxn+l--xll<~

-~'p+'--~)'~llxn-xll

1--X+X

= 0 II x .

-

x

where 1

for

4a 0 < p < - - p2 and

1 X ~< 1 -- V/ 1 - ~ o + - - /32p~ 4

Since 0 < 1, so by the Banach-Picard theorem, we see that the approximate solution Xn÷1 obtained from (1 6)converges to x, the exact solution of (1 4), which is the requked result.

4. EXAMPLES We now give some examples for which the linear complementarity problem can be solved by the modified algorithm 2.3, but not by Mangasarian's algorithm 2.1 oe Van Bokhoven's algorithm 2.2. Example 4.1 Consider the linear complementarity problem of the form:

224

12 --21],

and q = [ - - 1 4 ]

This problem is unsolvable by Van Bokhoven's algorithm 2.2, but can be solved by Mangasarian's algorithm 2.1 and modified algorithm 2.3.

II

0 = [1 --X + X (V/1 -o~p + ~ - ~ ) ] <

M=[

Engineering Analysis, 1986, Vol. 3, No. 4

REFERENCES 1 Cottle, R. W. Complementarity and variational problems, Symposia Math. 1976, 19, 177 2 Lin, Y. and Cryer, C. W. An alternating direction implicit algorithm for the solution of linear complementarity problems arising free boundary problems, Appl. Math. Opt. 1985, 13, 1 3 Mangasarian, O. L. Solution of symmetric linear complementarity problems by iterative methods, J. Opt. Theor. Appl. 1977, 22,465 4 Aganagic, M. Newton's method for linear complementarity problems, Math. Prog. 1984, 28,349 5 Van Bokhoven, W. M. G. A class of linear complementarity problems is solvable in polynomial time, Tech. Report, Department of Electrical Engineering, Technical University, Eindhoven, Holland, 1980 6 Ahn, B. H. Solution of nonsymmetrie linear complementarity problems by iterative methods, J. Opt. Theor. Appl. 1981, 33, 175 7 Gana, A. Studies in the complementarity problem, PhD Thesis, University of Michigan, USA, 1982 8 Cryer, C. W. The solution of a quadratic programming problem using systemic over relaxation, SIAMJ. Control 1971, 9, 385 9 Aim, B. H. Iterative methods for linear complementarity problems with upper bounds on primary variables, Math. Prog. 1983, 26,295 10 Plemmons, R. J. M-matrix eharacterisation, I: Nonsingular M-matrices, Linear AppL 1977, 18,175 11 Pang,J. S. On the convergenceof a basic iterative method for the implicit complementarity problem, J. Opt. Theor. Appl. 1982, 37,149 12 Aflam Noor, M. and Zarae, S. Linear quasi-complementarity problem, UtilitasMath. 1985, 27,249