A numerical imlementation of a redictor-corrector algorithm for sufcient linear comlementarity roblem BENTERKI DJAMEL University Ferhat Abbas of Setif-1 Faculty of science Laboratory of fundamental and numerical Mathematics ALGERIA dj benterki@yahoo.fr BOULOUDENINE NADJIBA University Ferhat Abbas of Setif-1 Faculty of science Deartment of Mathematics ALGERIA nadjiba010@hotmail.fr Abstract: In this aer, we imlement a version of a Mizuno-Todd-Ye redictor-corrector interior oint algorithm for the P*k)-matrix linear comlementarity roblem LCP) resented by T. Illes and M. Nagy. We make some numerical develoments to calculate the directions and the ste length in the redictor case. Our results shows that the roosed algorithm achieve high, accuracy when it alies to monotone linear comlementarity roblem. Key Words: Linear comlementarity roblem, P*k)-matrix, Predictor-corrector methods., 1 Introduction We consider the following linear comlementarity roblem LCP ) : nd vectors x; y R n which veries : 8 < LCP ) : Mx + y x; y 0 xy = 0: = q Where M R nn and q R n. Comlementarity roblems are known in 1940, they have many alications in mathematical rogramming and equilibrium roblems. Indeed, it is known that by exloiting the rst-order otimality conditions of the otimization roblem, any differentiable convex quadratic rogram can be formulated into a monotone LCP ), and vice versa [16]. Variational inequality roblems are widely used in the study of equilibriums in economics, transortation lanning, and game theory. They also have a close connection to the LCP )s. The reader can refer to [] for the basic theory. Although it's earlier aearance, it was not until 1960 that researchers interested in these issues. This certainly comes to the lack of numerical tools for solving such roblem. Since then, several algorithms are roosed in the literature, esecially after the aearance of the famous method of Karmarkar in 1984. The interior oint methods are more suitable for solving the LCP ) roblem, in articular, the central ath methods have become the methods of choice thanks to their good roerties rimal-dual methods, olynomial convergence). Many alternatives are available in the literature, the work of Kojima, Mizuno and Yoshise [9], Achache [1] are cited to solve a linear comlementarity roblem LCP ) monotone M is ositive semidenite ). Desite the remarkable success of these methods, the central ath methods suffer esecially from the initial oint. Much more, in general, they don't have a feasible initial oint. Worse, even if this is known, the convergence of the central ath methods is not guaranteed unless this oint is close to the central ath. To address this issue, different rocedures are roosed. This is the method of weight with central ath [3, 6, 14], which serves to return the iterated neighboring to the central ath. Moreover, these methods are not achievable the algorithm starts from any oint x > 0 ) and rove theoretically converge. Unfortunately, these statements are not erformed numerically. Moreover, such methods did not matter thereafter. Currently, the redictor-corrector methods are imortant to answer this question. The rst version of the redictor-corrector algorithm was initiated by Sonnevend, Stor and Zhao [14] for solving a linear rogramming roblems. This algorithm needs more corrector stes after each redictor ste in order to return to the aroriate neighborhood of the central ath. Mizuno, Todd and Ye [11] have roosed a redictor-corrector algorithm for solving a linear rograming in a olynomial time which each redictor ste is followed by a single corrector ste. This algo- ISBN: 978-1-61804-75-0 387
rithm has oened the way for other more comlicated alication to solve linear comlementarity roblems. Indeed Anstricher and Ye [17] extended this result to the linear comlementarity roblem with a ositive semidenite matrix with the same iteration comlexity. Potra have roosed a Mizuno-Todd-Ye algorithm in a larger neighborhood of the central ath [1]. In 1991, Kojima and al have ublished their book on interior oint method for LCP )s [8]. Since then the quality of a variant of an interior oint algorithm is measured by the fact whether it can be generalized to the P k)-matrix linear comlementarity roblem or not. Several variants of Muzino -Todd-Ye tye redictor-corrector interior oint algorithm are known in the literature. In 007, Illes and Nagy [5] have roosed a redictor-corrector algorithm tye Mizuno - Todd -Ye for sufcient linear comlementarity roblems. Our work focuses on the comrehensive theoretical and numerical study of this algorithm [5], where we have made theoretical and numerical develoments in order to make the algorithm The work is divided into three sections organized as follows : In the rst section, we mention some matrix classes denitions, and some results. The second section contains a summary of Illes and Nagy's [5] algorithm, followed by the descrition of the algorithm itself, and the last section resent our numerical exeriments of the algorithm and a conclusion. We denote by: e = 1; 1; :::; 1) R n : xy = x 1 y 1 ; x y ; :::; x n y n ) R n Hadamard roduct of vectors x and y. x = x 1 ; x ; :::; x n ) R n. x y = x 1 y 1 ; x y ; :::; xn y n ) R n : X = diagx 1 ; x ; ::; x n ): F = x; y) R n + : y = Mx + q ; the feasible set of the LCP ) roblem. F + = x; y) R n ++ : y = Mx + q ; the strict feasible set of the LCP ) roblem. F = x; y) R n + : y = Mx + q; xy = 0, the solution set of the LCP ) roblem. T = fx; y) F + : xy = e; > 0g ; the central ath. The P k)-matrices and their roerties Denition 1 [8] Let k R ++. A matrix M R nn is called P k)-matrix if for all x R n we have: 1 + 4k) X x i Mx) i + X x i Mx) i 0; ii + x) ii where I + x) = f1 i n : x i Mx) i > 0 g and I x) = f1 i n : x i Mx) i < 0 g : - If k = 0; then M is called a semidenite matrix M is a P 0)-matrix). - Lets k 1 and k tow ositif real numbers. If k 1 k then P k 1 ) P k ): Denition A matrix M R nn is called a P - matrix if it is a P k)-matrix for some k > 0, i.e., P = [P k) k0 Denition 3 A matrix M R nn is called a column sufcient matrix if: 8x R n ; x i Mx) i 0 =) x i Mx) i = 0; 8i = 1; :::; n: Denition 4 A matrix M R nn is called row sufcient if M T is a column sufcient matrix. Denition 5 A matrix M R nn is called sufcient if it is both row and column sufcient. P matrices are sufcient. Kojima and al [8] roved that a P k) matrix is column sufcient and Guu and Cottle [4] roved that it is row sufcient, too. Valiaho [15] roved that each sufcient matrix is a P matrix, so the class of P -matrices is equal to the class of sufcient matrices. Proosition 6 [8] If M R nn is a P k)-matrix then M 0 M I = Y X is a nonsingular matrix for any ositive diagonal matrices X; Y R nn. Corollary 7 Let M R nn be a P k) matrix, x; y R n + then for all a R n the system x) M4x + 4y = 0 Y 4x + X4y = e xy has a unique solution 4x; 4y): ISBN: 978-1-61804-75-0 388
Theorem 8 [8] Let a linear comlementarity roblem with a P k)-matrix M be given. Then the following statements are equivalent: 1 F + 6=?. 8w R n +; 9!x; y) F + : xy = w: 3 8 > 0; 9!x; y) F + : xy = e: Throughout the aer we make the following assumtions: 1. F + 6=?:. An initial oint x 0 ; y 0 ) F + exists. 3. M is a P k)-matrix. 3 Descrition of the method After the relaxation of the LCP ) roblem, we obtain the following central ath roblem: 8 < Mx + y = q LCP ) x; y > 0 : xy = e When! 0; the sequence x); y)) solutions of the central ath LCP ) roblem aroach the solution x; y) of the LCP ) roblem. By using the Newton method to solve the LCP ) roblem, we obtain the following system: M4x + 4y = 0 1) Y 4x + X4y = e xy Let the roximity measure be x; y; ) = v) = v 1 v ; q where v = xy. For the oint x; y) of the central ath, the roximity must be zero, i.e., xy = e for some > 0. The roximity measure aroaches the innity if some coordinates of x or y tends to 0 or innity. Let 0 < < 0 be the suitable roximity arameters dening the neighborhood of the central ath. In redictor ste, from a given initial oint x 0 ; y 0 ; 0 ) F + and satises x 0 ; y 0 ; 0 ) <, we solve the Newton system with = 0 and we denote the solution by 4x; 4y). The new oint will be: x = x 0 +4x; y = y 0 +4y; = 1 ) 0 : Where 0; 1] is the maximal real number for which x ; y ; ) F + and x ; y ; ) 0. In the corrector ste, we return from the 0 neighborhood to the neighborhood of the central ath. Starting from the oint x ; y ; ); we solve the Newton system and we get 4ex; 4ey). Let the new oint be x c = x + 4ex; y c = y + 4ey; c = : The oint x c ; y c ; c ) will be in the neighborhood again, so we can continue the iteration from this oint as an initial oint x 0 ; y 0 ) while the duality ga is not small enough. Algorithm Inut An accuracy arameter " > 0; Proximity arameters ; 0 ; 0 < < 0 ; An initial oint x 0 ; y 0 ; 0 ); such that x 0 ; y 0 ; 0 ) ; Begin x = x 0 ; y = y 0 ; = 0 ; While x T y " do Predictor ste Determine 4x; 4y) by solving 1) with = 0 and let x = x 0 +4x; y = y 0 +4y; = 1 ); Determine : max f > 0 : x ; y ; ) F + and x ; y ; ) 0 g Corrector ste Determine 4ex; 4ey) by solving 1) with x = x ; y = y ; = and let x c = x +4ex; y c = y +4ey; c = ; x = x c ; y = y c ; = c ; End while End. 4 The algorithm iteration In this section, we describe the comutation of the redictor and corrector directions, and the ste-length = us it was q resented in q [5]: Let v = xy ; d = x y ; d 1 = e d ; d x = d 1 4x = v4x x ; d y = d 1 4y = v4y y. After rescaling the Newton-system we have Mdx + d y = 0 d x + d y = v 1 v; where M = DMD; and D = diagd i ): ISBN: 978-1-61804-75-0 389
4.1 The redictor ste Starting from a oint x; y) F + where x; y; ) <, we solve the Newton system with arameter = 0; that means to solve the following system: Mdx + d y = 0 d x + d y = v; which has the following solution: d x = I + M) 1 v; and d y = MI + M) 1 v: We choose the ste-length so that the following two conditions are satised: 1. x ; y ) F +, i.e., x y > 0,. x ; y ; ) < 0, where x = x + 4x; y = y + 4y; and = 1 ); i.e., the ositivity condition is satised at the new oint and we do not deviate from the central ath too much. We have: and x y = 1 )v + d x d y ; v ) = x y 1 ) = v + 1 d xd y : For the rst condition, x y > 0 imlies v + 1 d xd y > 0: Here we have tow cases: Case 1: If d x d y ) i 0; then obviously v ) i > 0 is true for all 0:1): Case : If d x d y ) i < 0; then the condition 1 < vi d xd y) i must hold for the ste-length. Let v i b' = min : d x d y ) i < 0 d x d y ) i and ' = 1 ; then ' [0; b') is the necessary condition so that x ; y ) F + is satised. To examine the second condition about choosing the ste-length, we comute the roximity measure of the vector v so we have: v ) = v ) 1 v = v) + e T e v + 'd x d y e v + 'd xd y ): Let f') : [0; b')! R, therefore f') = v ) v) = e T e v + 'd x d y e v + 'd xd y ): The function f is strictly convex on the interval [0; b') and veries, f0) = 0 and lim '!b' f') = +1: From the roerties of function f it follows that the equation f') = 0 v) has a unique solution ' on interval [0; b'): We can ut ' = ) 1 ; ) then the ste length can be comuted as = ' + ' ) + 4' : Remark 9 To comute the ste length ), we can use a numerical method such as for examle Newton method. Proosition 10 [5] is the maximal feasible ste length such that v ) 0 and x ; y ) F + : 4. The corrector ste In this ste, our aim is to return from the 0 - neighborhood of the central ath. To obtain the corrector directions g 4x; f 4y); we solve the following Newton-system: M f d x + e d y = 0 fd x + e d y = v ) 1 v where fd x = v g 4x x ; e dy = v f 4y y ; e d = Let r x y ; f M = e DM e D: x c = x + g 4x; y c = y + f 4y; c = = 1 ): The new oint x c ; y c ; c ) must be feasible witch means the following tow conditions are satised: 1. x c ; y c ) F +,. x c ; y c ; c ) <. ISBN: 978-1-61804-75-0 390
For that, we will choose a suitable value of and 0 where the full Newton ste is feasible. r q Proosition 11 [5]If < 1 + 1 + 16, 1+4k) 1+4k 4 + then there exists a 0 that satises 0 < and the full Newton ste is feasible at corrector ste. 4.3 Iteration comlexity Theorem 1 [5] Let the linear comlementarity roblem for any P k)-matrix M be given, where k 0 and let 0 = 1. Then the Mizuno-Todd-Ye algorithm generates an x; y; ) oint satisfying x T y < " in at most [ Where: n 1 log 4n+ 0 4" ] iterations. 1 = 0 0 + 1 0 ) 0 = m)1 + 4k) ) = min 0 3 ; h) 8m) ; r m) = 1 + + + 4 4 h) = 8 m ) 1 5 11m ) ) Corollary 13 [5] Let the linear comlementarity roblem for any P k)-matrix M be given, where k 0 and let 0 = 1; = 1 1+4k and 0 = 1+4k. Then the Mizuno-Todd-Ye algorithm generates an x; y; ) oint satisfying x T y < " in at most O1 + k) 3 n log n " ) iterations. 5 Numerical exeriments In this section, we resent some numerical exeriments for a sufcient linear comlementarity roblems. Unfortunately, because the comutation of the arameter k is difcult those exeriments are limited on the monotone linear comlementarity roblems where the matrix M is a ositive semidenite. Our imlementation was tested on a Pentium 3 and was rogrammed in Dev-Pascal on a number of roblems taken from the literature and reorted in [1, 7]. In all the following roblems, we take " = 10 6, = 1:, 0 = 1:76, and we denote by exn) the taille of different examles and Iter reresents the number of iterations necessary to obtain the otimal solution. Examle 14 with xed size n) [1, 7] examle exn) iter ex) 4 ex3) 4 ex4) 5 ex5) 4 ex10) 7 Examle 15 with variable size n) [7] 0 1 ::: 0 1 ::: 0 0 1 ::: M = : : : : : B : : : ::: : @ : : : : : 0 0 0 ::: 1 q = 1; :::; 1 ) T ; x 0 = 0:05; :::; 0:05; 1:05 ) T ; Conclusion x = 0 ; :::; 0 ; 1 ) T : Size n Iter 4 3 7 4 10 4 15 5 0 5 4 6 1 ; C A The theoretical study on redictor-corrector algorithms for sufcient linear comlementarity roblems has allowed us to imrove the behavior of this tye of algorithm acting on the best values of and 0 rediction and correction neighborhood), numerical imlementation conrmed that. Our imlementation suose treated sufcient linear comlementarity roblem, because the calculation of the arameter k is difcult. The imlementation was limited on monotone linear comlementarity roblem. Our testes roved that the redictor-corrector algorithm worked very satisfactorily. References [1] M. Achache, Comlexity analysis and numerical imlementation of a short-ste rimal-dual algorithm for linear comlementarity roblems, Alied M athematics and Comutation 16, 010,. 1889 1895. ISBN: 978-1-61804-75-0 391
[] R.W. Cottle, J.S. Pang, R.E. Stone, The linear comlementarity roblem, Academic P ress Inc, San Diego, CA, 199. [3] J. Ding and T. Y. Li, An algorithm based on weighted logarithmic barrier function for linear comlementarity roblem, T he Arabian Journal f or Science and Engineering 15, N4B 1990. [4] S. M. Guu, R. W. Cottle, On a subclass of P 0 linear algebra and its alications 3/4; 1995. 35 335. [5] T. Illés, M. Nagy, A Mizuno-Todd-Ye tye redictor-corrector algorithm for sufcient linear comlementarity roblems, Euroean Journal of Oerational Research 181, 007,. 1097-1111. [6] Z. Kebbiche, Etude et extension d'algorithmes de oints intérieurs our la rogrammation non linéaire, T hese de doctorat; U niversite F erhat Abbas de Setif; 007. [13] E. M. Simantiraki et D. F. Shanno, An infeasible-interior oint method for linear comlementarity roblems, SIAM: J: otim:.7, 1997,. 60 640: [14] Gy. Sonnevend, J. Stoer, G. Zhao, On the comlexity of following the central ath of linear rograms by linear extraolation, M ethods of Oerations Research; 63, 1989,. 19 31. [15] H. Valiaho, P k) matrices are just sufcient, Linear Algebra and its Alications; 39, 1996,. 103 108. [16] S. J. Wright, Primal-dual interior oint methods, SIAM:P hiladelhia 1997. [17] Y. Ye, K. Anstreicher, On quadratic and O nl) convergence of a redictor-corrector algorithm for LCP ), M athematical P rogramming 6, 1993,. 537 551. [7] R. Khebchache, Etude théorique et numérique de quelques méthodes de oints intérieurs de trajectoire avec oids our le comlémentarité linéaire monotone, T hese de magisterse; Universite F erhat Abbas de Setif 010. [8] M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A unied aroach to interior oint algorithms for linear comlementarity roblems, Lecture N otes in Comuter Science; vol: 538; Sringer V erlag; Berlin; Germany; 1991. [9] M. Kojima, S. Mizuno and A. Yoshise, A olynomial time algorithm for class of linear comlementarity roblems, M athematical P rogramming; 44, 1989,. 1-6: [10] J. Miao, A quadratically convergent Ok + 1) nl) -iteration algorithm for the P k) matrix linear comlementarity roblem, Mathematical Programming 69 1995). 355 368. [11] S. Mizuno, M. J. Todd, Y. Ye, On adative-ste rimal-dual interior-oint algorithms for linear rogramming, M athematics of Oerations Research; 18,4, 1993,. 964 981. [1] F. A. Potra, The Mizuno-Todd-Ye algorithm in a larger neighborhood of the central ath, Euroean Journal of Oerational Research; 143 00,. 57 67. ISBN: 978-1-61804-75-0 39