Modified SSOR Modelling for Linear Complementarity Problems

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1 Turish Joural of Aalysis ad Number Theory, 204, Vol. 2, No. 2, Available olie at Sciece ad Educatio Publishig DOI:0.269/tjat Modified SSOR Modellig for Liear Complemetarity Problems M.T. Yahyapour, S.A. Edalatpaah * Departmet of Mathematics, Ramsar Brach, Islamic Azad Uiversity, Ramsar, Ira *Correspodig author: saedalatpaah@gmail.com Received March 07, 204; Revised April 0, 204; Accepted April 7, 204 Abstract I this paper, we preset a efficiet umerical method for solvig the liear complemetarity problems based o precoditioig strategy. Furthermore, the covergece properties of the proposed method have bee aalyzed ad compared with symmetric successive over-relaxatio (SSOR method. Fially, some umerical experimets are illustrated to show the efficiecy of the proposed method. Keywords: liear complemetarity problem, precoditioig, iterative methods, SSOR method, compariso theorems, H-matrix Cite This Article: M.T. Yahyapour, ad S.A. Edalatpaah, Modified SSOR Modellig for Liear Complemetarity Problems. Turish Joural of Aalysis ad Number Theory, vol. 2, o. 2 (204: doi: 0.269/tjat Itroductio For a give real vector M R abbreviated as (, q R ad a give matrix the liear complemetarity problem LCP M q, cosists i fidig vectors z R such that T w = Mz + q, z 0, w 0, T z w= 0. (. Where, z deotes the traspose of the vector z. May problems i operatios research, maagemet sciece, various scietific computig ad egieerig areas ca lead to the solutio of a LCP of the form (.. For more details (see, [,2,3] ad the refereces therei. The early motivatio for studyig the LCP (M, q was because the KKT optimality coditios for liear ad quadratic programs costitute a LCP of the form (.. However, the study of LCP really came ito promiece sice the 960s ad i several decades, may methods for solvig the LCP (M, q have bee itroduced. Most of these methods origiate from those for the system of the liear equatios where these methods may be classified ito two pricipal ids, direct ad iterative methods (see, [,2,3]. Recetly, various authors have suggested differet model i the frame of the iterative methods for the above metioed problem. For example, Yua ad Sog i [4], based o the models i [5], proposed a class of modified accelerated over-relaxatio (MAOR methods to solve LCP( M, q. Furthermore, whe the system matrix M is a H-matrix they proposed some sufficiet coditios for covergece of the MAOR ad modified successive over-relaxatio (MSOR methods. Uder certai coditios, Li ad Dai [6], Saberi Najafi ad Edalatpaah [7] ad Dehgha ad Hajaria [8] also studied geeralized accelerated over-relaxatio (GAOR ad symmetric successive over-relaxatio (SSOR methods for solvig LCP( M, q, respectively. The case that M is No-Hermitia, Saberi Najafi ad Edalatpaah [8,9], proposed some ew iterative methods for solvig this class of LCP (to see that other iterative methods for LCP( M, q see [-7] ad the refereces therei. I this paper, we will propose a modificatio of SSOR method for LCP( M, q. To accomplish of this purpose, SSOR method is coupled with the precoditioig strategy. We also show that our method for solvig LCP is superior to the basic SSOR method. Numerical experimets show that the ew method is feasible ad efficiet for solvig large sparse liear complemetarity problems. 2. Prerequisite We begi with some basic otatio ad prelimiary results which we refer to later. Defiitio 2. [8,9]. (a The matrix A = ( a ij is oegative (positive if i, j a ij 0 (a ij > 0. I this case we write A 0(A>0.Similarly, for -dimesioal vectors x, by idetifyig them with matrices, we ca also defie x 0 (x>0.

2 48 Turish Joural of Aalysis ad Number Theory (b The matrix A = ( a ij is called a Z-matrix if for ay i ja, ij 0. (c The square matrix A = ( a ij is called M-matrix if A= α I B ; B 0 ad α > ρ( B ; ( we deote the spectral radius of B by ρ (B. (d Z-matrix is M-matrix, if A is osigular, ad if A 0. (e For ay matrix A = ( a ij A = ( mij R is defied by: the compariso matrix mii = aii, mij = aij, i j, i, j. (f The Matrix A = ( a ij is a H-matrix if ad oly if A is a M-matrix. Defiitio 2.2 [5]. For x R, vector x + is defied such that (x + j = max{0,x j }, j =,2,...,. The, for ay xy, R, the followig facts hold:. (x + y + x + + y +, 2. x + - y + (x - y +, 3. x = x + + (-x +, 4. x y implies x + y +. Defiitio 2.3 [8,9]. Let A be a real matrix. The splittig A=M- N is called (a Coverget if ρ ( M N <. (b Regular if M 0 ad N 0. (c Wea regular if M 0 ad M N 0. (d M-splittig if M is M-matrix ad N 0. Clearly, a M-splittig is regular ad a regular splittig is wea regular. Lemma 2. [8]. Let A be a Z-matrix. The A is M- matrix if ad oly if there is a positive vector x such that Ax >0. Lemma 2.2 [8]. Let A =M N be a M-splittig of A. The ρ( M N < if ad oly if A is M-matrix. Lemma 2.3 [20]. Let A, B be Z-matrices ad A is a M- matrix, if A B the B is a M-matrix too. Lemma2.4 [9]. If A 0, the ( A has a oegative real eigevalue equal to its spectral radius, (2 To ρ ( A > 0, there correspods a eigevector x 0, (3 ρ ( A does ot decrease whe ay etry of A is icreased. Lemma2.5 [8]. Let T 0. If there exist x > 0 ad a scalar α > 0 such that i Tx α x, the ρ( T α. Moreover, if Tx < α x, the ρ( T < α. ii Tx α x, the ρ( T α. Moreover, if Tx > α x, the ρ( T > α. Lemma 2.6 [5]. LCP( M, q ca be equivaletly trasformed to a fixed-poit system of equatios z = ( z α E( Mz + q +, (2. where α is some positive costat ad E is a diagoal matrix with positive diagoal elemets. Lemma 2.7 [5]. Let M R be a H-matrix with positive diagoal elemets. The the LCP( M, q has a * uique solutio z R. Let the matrix M be split as; M = D L U, (2.2 where D is diagoal, -L ad -U are strictly lower ad upper triagular matrices of M, respectively. The by choosig of α E = D ad i view of Lemma 2.6 we have, z ( z D ( Mz q +. = + (2.3 So, i order to solve LCP( M, q method is defied i [8] as follows;,ssor iterative + + wlz + ( w(2 w M wl z z = ( z D +. Also they proposed followig model; + + wuz + ( w(2 w M wu z z = ( z D +. where 0 < w < 2. However, because of similarity we igore this case. The operator f : R R that f( z Let (2.4 (2.5, is defied such = ξ, where ξ is the fixed poit of the system; wlξ + ( w(2 w M wl z ξ = ( z D +. (2.6 Q = I wd L, R = I D ( w(2 w M wl.(2.7 The i ext lemma we have the covergece theorem, proposed i [8] for the SSOR method. Lemma 2.8 [8]. Let M R be a H-matrix with positive diagoal elemets ad 0 < w <2. The, for ay 0 iitial vector z R, the iterative sequece { z } geerated by the SSOR method coverges to the uique solutio z* of the LCP (M, q ad ρ( Q R < 3. Precoditioig SSOR for LCP The developmet of efficiet ad authetic precoditioed iterative methods is the ey for the successful applicatio of scietific computatio to the solutio of may large scale Problems. Sice, the covergece rate of iterative methods depeds o spectral properties of the coefficiet matrix, i precoditioig schemes attempt to trasform the origial system ito aother oe that has the same solutio but more desirable properties for iterative solutio. I this sectio, SSOR method for LCP ad the effect of precoditioig for this method is coupled. I literature, various authors have suggested differet models of (I+S-type precoditioer for liear systems A=I-L-U; where I is the idetity matrix ad L,U are strictly lower ad strictly upper triagular matrices of A, respectively; (see, [2-28] ad the refereces therei. These precoditioers have reasoable

3 Turish Joural of Aalysis ad Number Theory 49 effectiveess ad low costructio cost. For example, Milaszewicz [2] preseted the precoditioer of (I+ S- type, where the elemets of the first colum below the diagoal of A elimiate. Usui et al.[22] cosidered the alterative precoditioer, with the followig form P = I + L. I the preset sectio, we will propose precoditioig methods for solvig liear complemetarity problem. Geerally, we rewrite M to (I+SM. So, let M i (2.2 be osigular. The precoditioig i M is; M = ( I + S M = D L U + SD SL SU = D L U. (3. Where DLU,, are diagoal, strictly lower ad strictly upper triagular parts of M, respectively. Ad, q= ( I+ Sq. (3.2 We cosider Usui et al s precoditioer [22] as (I+S. Therefore we have, Where, P0 ( I K0 ( I S. = + = + ( m m m3 m32 0 S =. m m m m2 m, 0 m m22 m, Note. We ca also cosider other (I+S-type precoditioers but here, we use Usui et al s precoditioer, sice covergece rate via usig this precoditioer slightly is better tha others; see, [24]. Now, we propose ew precoditioers for M. Our precoditioers for LCP are; { P = ( I + K = ( I + S 2 I M ( I + S, (3.4 P2 = ( I + K2 = ( I + S 3 I M( I + S(3 I M( I + S. Ad for i = 0,, 2, we have, { M = ( I + Ki M, q = ( I + Ki q. Thus the precoditioed SSOR method for LCP is: + + wlz + ( w(2 w M wl z z = ( z D +. } } (3.5 (3.6 Lemma 3.. Let M be a H-matrix. The the precoditioed M = ( I + Ki M also is H-matrix. Proof. Let M be a H-matrix. The <M> is M-matrix ad by Lemma 2., x> 0 ST. < M > x> 0. Sice < M >= ( I + Ki < M >, The, < M > x = ( I + Ki < M > x > 0. Therefore < M > is M-matrix ad the proof is completed. Theorem 3.2. Let M R with positive diagoal elemets be a H-matrix ad M = ( I + Ki M is precoditioed form of M with precoditioers (3.3-(3.5. The if Q = I wd L & R = I D ( w(2 w M wl we have, ρ ( Q R ρ( Q R <. Proof. By Lemma 3. M is a H-matrix. Hece < M >= Q R is M-matrix ad by Lemma 2.2 ρ( Q R <. Sice ( < M > < Q, by Lemma 2.3 Q is M-matrix. As same demostratio Q is M-matrix too. Thus, Q 0, Q R 0, Q 0, Q R 0. The by Lemma 2.4 (Perro-Frobeius Theorem, there exist a positive vector x such that, Therefore, ( Q Rx = ρ( Q Rx. ( Q Rx =< M> x= QI ( Q Rx ρ( Q R = Rx 0. ρ( Q R Furthermore, for ay P i (i=0,, 2 we have; < M >= ( I + K < M >= ( I + K ( D L U = ( D L U. i = D L U + K D K L K U i i i D+ L+ U D = D D D L = L + L L U= U+ U + SU SD Thus, Q Q ad i view of the fact that both QQ, are M-matrices, we have; Therefore, Q ( I + Ki Q Q. 0 [ Q ( I+ Ki Q ]( Q Rx = ( I Q R x ( I Q R x = Q Rx Q Rx = ρ( Q R x Q Rx. Ad by Lemma 2.5 we have; ρ ( Q R ρ( Q R. i

4 50 Turish Joural of Aalysis ad Number Theory Therefore by Lemma 2.8, the proof is completed. Now, we show that i LCP, the covergece rate of precoditioed SSOR methods is faster tha of the SSOR method. Theorem 3.3. Let M R with positive diagoal elemets be a H-matrix, 0 <w < 2. Also M = PM is precoditioed form of M with precoditioers (3.3- (3.5.The covergece rate of precoditioed SSOR methods is faster tha of the SSOR method. Proof. Let, iterative sequece {z i } i=0,,, geerated by (3.6. From the assumptio that M is a H-matrix, it follows, by Lemma 3. M is a H-matrix ad therefore by Lemma 2.7, the vector sequece{z i } is uiquely defied ad the LCP(M, q has a uique solutio z R. Similar to (2.6,we defie the operator v: R R, such that vz ( = ξ, where ξ is the fixed poit of the followig system, Let wlξ + ( w(2 w M wl z ξ = ( z D +. (3.7 wlψ + ( w(2 w M wl x ψ = vx ( = ( x D.(3.8 ] + By subtractig (3.7 & (3.8, we get; wl( ξ ψ ξ ψ (( z x D +, + ( w(2 w M wl( z x wl( ψ ξ ψ ξ (( x z D +. + ( w(2 w M wl( x z Therefore, by above relatios we have; = ( + + ( + Q Rz x. (3.9 ξ ψ ξ ψ ψ ξ Thus from the defiitio of the precoditioed SSOR method ad (3.9 we ca write + * * * z z = f( z f( z Q R z z. Hece, the iterative sequece {z }, =0,,, coverges to z * if ρ( Q R ad sice by Theorem 3.2, ρ( Q R ρ( Q R we coclude that for solvig LCP, the precoditioed SSOR iterative methods is better tha of the SSOR method from poit of view of the covergece speed ad the proof is completed. 4. Numerical Results I this sectio, we give examples to illustrate the results obtaied i previous Sectios. These examples computed with MATLAB7 o a persoal computer Petium 4. Example 4.. Cosider LCP (M, q with followig system N N M R N ad q R N N M = G I I + I F I + I I F R, 3 T N q = (,,, ( R, N N where I R ad deotes the Kroecer product. Also G ad F are tridiagoal Matrices give by; 2+ 2h 2 2h G = tridiagoal[ (,, ( ], h 2 h F = tridiagoal[ (,0, ( ], & h = / ; N =. Evidetly, M is a H-matrix with positive diagoal elemets. The LCP (M, q has a uique solutio. The, we solved the 3 3 H-matrix yielded by the iterative methods, ad Precoditioed forms. I this experimet, we choose Usui et al. s model ad our models (P,P 2 as precoditioers. The iitial 0 approximatio of z is z = (,,, T R N ad as a stoppig criterio we choose; 6 mi( Mz + q, z 0 6 mi( Mz + q, z 0 I Table, with several values we report the CPU time (CPU ad umber of iteratios (Iter for the correspodig SSOR ad precoditioed SSOR methods (whe N=000. Here, the precoditioed SSOR method with Usui et al. s precoditioer is deoted by PREC(USUI, while PREC(P, PREC(P 2 correspods to our precoditioers (P i ; i=,2. Table. The results of example 4.2 for SSOR Method SSOR PREC(USUI PREC(P PREC(P2 w Iter CPU Iter CPU Iter CPU Iter CPU

5 Turish Joural of Aalysis ad Number Theory 5 From the table, we ca see that the precoditioed iterative methods are superior to the basic iterative methods ad our precoditioers are better tha Usui et al. s precoditioer. Example 4.2. Cosider LCP (M, q Where 0 if i = j 4 if i = j if i = j + 2 M =,.5 if i = j if i = j 0 else T q = (,,, R 0 T z = (0.5,.05,,0.5 R It is easy to see that, M is a H-matrix with positive diagoal elemets. The spectral radius of the correspodig iterative matrix with differet parameters w ad above precoditioers(p 0,P,P 2 for N=500 i semilogarithmic scale is give i the Figure. From the followig figures, we ca see that the precoditioed SSOR methods are superior to the classic SSOR. Figure. Spectral radius of iterative methods of example4.2 with some values of w, =500 Furthermore, I Table 2, we report the CPU time of iterative methods for differet values of (whe w=. Table 2. CPU time SSOR Usui et al. s model (I+K (I+K Coclusio I this paper, we have proposed the precoditioed SSOR method for liear complemetarity problem ad aalyzed the covergece for these methods uder certai coditios. We have also studied how the iterative method for LCP is affected if the system is precoditioed by our models. Acowledgemets The authors would lie to tha Ramsar Brach of Islamic Azad Uiversity for the fiacial support of this research, which is based o a research project cotract. Refereces [] Murty, K.G. (988. Liear Complemetarity, Liear ad Noliear Programmig. Helderma Verlag: Berli. [2] Cottle, R.W., Pag, J.S.&Stoe, R.E. (992. The Liear Complemetarity Problem. Academic Press: New Yor. [3] Bazaraa, M.S., Sherali, H.D.& Shetty CM. (2006. Noliear programmig, Theory ad algorithms. Third editio. Hoboe, NJ: Wiley-Itersciece. [4] Yua, D., Sog, Y.Z. (2003. Modified AOR methods for liear complemetarity problem. Appl. Math. Comput. 40, [5] Bai, Z.Z., Evas, D.J. (997. Matrix multisplittig relaxatio methods for liear complemetarity Problems. It. J. Comput. Math. 63, [6] Li, Y., Dai, P. (2007. Geeralized AOR methods for liear complemetarity problem. Appl. Math. Comput. 88, 7-8. [7] Saberi Najafi, H., Edalatpaah, S. A. (203. O the Covergece Regios of Geeralized AOR Methods for Liear Complemetarity Problems, J. Optim. Theory. Appl. 56, [8] Dehgha, M., Hajaria, M. (2009. Covergece of SSOR methods for liear complemetarity problems, Operatios Research Letters. 37, [9] Saberi Najafi, H., Edalatpaah, S. A. (202. A id of symmetrical iterative methods to solve special class of LCP (M,q, Iteratioal Joural of Applied Mathematics ad Applicatios, 4 ( [0] Saberi Najafi, H., Edalatpaah, S. A. (203. SOR-lie methods for o-hermitia positive defiite liear complemetarity problems, Advaced Modelig ad Optimizatio, 5(3, [] Cvetovic Lj., Rapajic S. (2007. How to improve MAOR method covergece area for liear complemetarity problems. Appl. Math. Comput. 62, [2] Dog J.-L., Jiag M.-Q. (2009. A modified modulus method for symmetric positive-defiite liear complemetarity problems. Numer. Liear Algebra Appl. 6, [3] Bai Z.-Z. (200. Modulus-based matrix splittig iteratio methods for liear complemetarity problems. Numer. Liear Algebra Appl. 7, [4] Zhag L.-L. (20. Two-step modulus based matrix splittig iteratio method for liear complemetarity problems. Numer. Algor. 57, [5] Cvetovic Lj., Kostic V. (203. A ote o the covergece of the MSMAOR method for liear complemetarity problems. Numer. Liear Algebra Appl. [6] Saberi Najafi H., Edalatpaah S. A. (203. Modificatio of iterative methods for solvig liear complemetarity problems. Egrg. Comput. 30, [7] Saberi Najafi H., Edalatpaah S. A.(203. Iterative methods with aalytical precoditioig techique to liear complemetarity problems. RAIRO-Oper. Res. 47, [8] Berma, A., Plemmos, R.J. (979. Noegative Matrices i the Mathematical Scieces. Academic Press: New Yor. [9] Varga, R.S. (2000. Matrix Iterative Aalysis, secod ed., Berli: Spriger. [20] Frommer, A., Szyld, D.B. (992. H-splittig ad two-stage iterative methods. Numer. Math. 63,

6 52 Turish Joural of Aalysis ad Number Theory [2] Milaszewicz, J.P. (987. Improvig Jacobi ad Gauss Seidel iteratios. Liear Algebra Appl 93, [22] Usui, M., Nii, H. & Koho, T. (994. Adaptive Gauss Seidel method for liear systems. Iter. J. Computer Math. 5, [23] Hirao, H., Nii, H. (20. Applicatio of a precoditioig iterative method to the computatio of fluid flow. Numer.Fuct.Aal.Ad Optimiz. 22, [24] Li, J.C., Li., W. (2008. The Optimal Precoditioer of Strictly Diagoally Domiat Z-matrix. Acta Mathematicae Applicatae Siica, Eglish Series. 24, [25] Saberi Najafi, H., Edalatpaah, S. A. (203. Compariso aalysis for improvig precoditioed SOR-type iterative method, Numerical Aalysis ad Applicatios. 6, [26] Saberi Najafi, H., Edalatpaah, S. A. (203. A collectio of ew precoditioers for solvig liear systems, Sci. Res. Essays. 8(3, [27] Saberi Najafi, H., Edalatpaah, S. A, Gravvais, G.A. (204. A efficiet method for computig the iverse of arrowhead matrices. Applied Mathematics Letters, 33, -5. [28] Saberi Najafi, H., Edalatpaah, S. A, Refahi Sheihai, A. H. (204. Covergece Aalysis of Modified Iterative Methods to Solve Liear Systems, Mediterraea Joural of Mathematics.