On the solution of the monotone and nonmonotone Linear Complementarity Problem by an infeasible interior-point algorithm

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1 On the solution of the monotone and nonmonotone Linear Complementarity Problem by an infeasible interior-point algorithm J. Júdice L. Fernandes A. Lima Abstract The use of an Infeasible Interior-Point (IIP) algorithm [14] is investigated for the solution of the Linear Complementarity Problem (LCP). Some monotone an nonmonotone LCPs from different sources are solved by two versions of the IIP algorithm, which differ on the line-search technique that computes the stepsize. The first version, denoted by SIIP, employs the simple maximum ratio technique commonly used in IIP methods for linear programming. On the other hand the second variant GIIP incorporates a more sophisticated Armijo-type line-search technique, that ensures global convergence for the procedure under some hypothesis. The computational experiments indicate that both the variants process efficiently monotone LCPs and LCPs with P -matrices. On the contrary, the algorithms face many difficulties for solving some LCPs with P 0 -matrices and LCPs associated with nonzero sum bimatrix games. However, the algorithm GIIP has succeeded in a large number of these problems than the method SIIP. A third version that employs an iterative solver for finding the search direction is also investigated and seems to be quite efficient for the solution of the monotone LCP that arises in a spatial equilibrium networ structured model. Keywords: Linear Complementarity Problem, Interior-Point algorithms, large-scale problems. 1 Introduction The Linear Complementarity Problem (LCP) consists of finding vectors z IR n and w IR n such that w = q + Mz, z 0, w 0, z T w = 0 (1) where q is an n-vector and M is a square matrix of order n. This problem has become one of the most important areas of optimization. Several applications of the LCP have appeared in many areas of science, economics and engineering and a large number of algorithms has been designed for its solution. We refer [3, 20] for two excellent boos on this subject. It is well-nown [3] that the LCP is equivalent to the following Affine Variational Inequality (AVI): Support of this wor has been provided by the Instituto de Telecomunicações. Departamento de Matemática da Universidade de Coimbra, Coimbra, Portugal. Escola Superior de Tecnologia de Tomar, Tomar, Portugal. Faculdade de Ciências da Universidade do Porto, Porto, Portugal. 1

2 Find z 0 such that (q + M z) T (z z) 0 for all z 0 (2) Due to this equivalence, the LCP has been classified as Monotone or Nonmonotone depending on the matrix M to be Positive Semi-Definite (P SD) or not. We recall that a matrix M is P SD if x T Mx 0 for all x IR n. This class contains the Positive Definite (P D) matrices, that is, those which satisfy x T Mx > 0 for all x IR n {0}. A monotone LCP can be solved in polynomial-time [15], while a nonmonotone LCP is NPhard [20]. In the last category of LCPs, one should distinguish the cases of M being a matrix P or P 0, where M is P (P 0 ) if and only if all its principal minors are positive (nonnegative). It is interesting to add that a LCP with a matrix P 0 is NP-hard [15] while the complexity of a LCP with a matrix P is still an open question. The following diagram shows the relationships among these four classes of matrices P D P (3) P SD P 0 where represents strict inclusion. Furthermore P = P D and P SD = P 0 for symmetric matrices [20]. Interior-Point algorithms have become one of the most interesting approaches for solving the monotone LCP. As the proper name indicates, these methods wor on the interior of the nonnegative orthant IR 2n +, that is, z > 0 and w > 0 are forced for each iterate (z, w ). Each iteration consists of a damped Newton step for the system of nonlinear equations { Mz w + q = 0 (4) ZW e = µ e where Z = diag(z 1,..., z n ), W = diag(w 1,..., w n ) and µ is the so-called central parameter that goes to zero as the algorithm proceeds. A stepsize is appropriately chosen so that the positivity of the variables z i and w i is maintained during the whole procedure. Global convergence of the algorithm is ensured by a criterious choice of the central parameter µ and of the stepsize that is used in these damped Newton iterations. There are two main categories of interior-point (IP) methods, namely the Feasible (FIP) and Infeasible (IIP) algorithms. In [15], Kojima et al have discussed an algorithm that unifies all the existing FIP methods. They have been able to establish global convergence for their unified method when the LCP is monotone or M is a P -matrix. The infeasible version of this algorithm seems to wor quite efficiently for monotone LCPs when the stepsize is computed by the simple maximum ratio technique commonly employed in linear programming [10]. Recently, Simantirai and Shanno [23], have discussed the use of an Armijo-type line search technique for the computation of the stepsize. They have been able to establish global convergence for the resulting method in the sense introduced by Kojima et al [14] for linear programming. They have also tested this algorithm in a number of monotone and 2

3 nonmonotone LCPs. The results seem to indicate that this global version GIIP is always more efficient than the simple version SIIP of the infeasible interior-point method that simply uses the stepsize computed from the maximum ratio criterion mentioned before. In this paper we have studied the efficiency of these two versions of the IIP method for monotone and nonmonotone LCPs. In the first experience we have solved some monotone and nonmonotone LCPs with matrices P and P 0. The numerical results show that the methods perform in a similar way for these type of problems and there is no clear winner between them. We have also studied these two versions of the infeasible interior-point method for the solution of some NP-hard nonmonotone LCPs with a P 0 matrix and LCPs associated with two formulations of bimatrix games. The experiments have shown that both the versions face many difficulties for processing these LCPs. However, the version GIIP has succeeded in a larger number of instances than the variant SIIP. This seems to indicate that the former version should be probably a better starting point for the possible design of an interior-point method capable of dealing with nonmonotone LCPs. As it is usual in interior-point methods, both the versions of the algorithm have been implemented using a direct solver for finding the search direction in each iteration. However, some large LCPs with a networ structure [1, 11, 21] cannot be processed with this type of implementation and require an iterative solver for that propose. The use of this iterative solver needs a special stopping criterion that leads to a small number of iterations for this method. This had lead to the design of a truncated infeasible interior-point method TIIP [22] which consider in each iteration a residual vector r in (4) to get an equation of the form Mz w + q = r. The norm of this residual vector can be controlled in such a way that the resulting procedure possesses global convergence in a sense stated in Kojima et al [14]. Some computational experience presented in this paper and elsewhere [22] shows that this TIIP algorithm is quite appropriate for the solution of these large monotone LCPs with a networ structure. The organization of this paper is as follows. In section 2 the (truncated) infeasible interiorpoint method is introduced, together with its variants. The section 3 reports the experiments performed with these variants for the solution of the monotone and nonmonotone LCPs mentioned before. Finally some concluding remars are presented at the end of the paper. 2 An Infeasible Interior-Point Algorithm In this section we discuss a general interior-point algorithm that can be seen as an extension of the method described in [14] for the LCP. See also [23, 25] for other papers on the same ideas. As stated in [22] we further introduce a residual vector r in each iteration, that is quite appropiate when the search direction is found approximately by an iterative solver. Consider the LCP (1) and let N be the set of the points (z, w) defined by the following equations z > 0, w > 0 z i w i γ zt w n i = 1,..., n (5) z T w γ w q Mz or w q Mz ɛ 2 where 0 < γ < 1, γ > 0 are given real numbers and ɛ 2 is a small positive tolerance. Furthermore let 0 β 0 < β 1 < β 2 < β 3 < 1 be given real numbers. Then the steps of the algorithm can be stated as follows 3

4 Step 0: Let (z 0, w 0 ) N and = 0. Step 1: If (z ) T w ɛ 1 and w q Mz ɛ 2 (ɛ 1 ɛ 2 ) stop: (z, w ) is a solution of the LCP. ( ) (z Step 2: Search Direction - Let µ = σ ) T w u n, β 0 σ < β 1 and compute v by ( ) ( ) ( ) ( ) M I u Mz v = q + w r + (6) Z W e µ e W Z where and r β 0 Mz q + w (7) Z = diag(z 1,..., z n), W = diag(w 1,..., w n). Step 3: Stepsize (i) Let ᾱ be the maximum α > 0 such that (z, w ) + α(u, v ) N (8) (z + αu ) T (w + αv ) (1 α(1 β 2 ))[(z ) T w ] for all α [0, ᾱ ] (9) (ii) If ᾱ ɛ 2 stop. Otherwise let α be such that 0 < α ᾱ and (z + α u ) T (w + α v ) (1 ᾱ (1 β 3 ))(z ) T w (10) Step 4: Set = + 1 and go to Step 1. z +1 = z + α u w +1 = w + α v (11) Next we discuss some further issues of the algorithm. In step 2, the matrix Z is diagonal with positive diagonal elements. So its inverse Z 1 exists and is easily computable. The system (6) can be rewritten as follows v = w + µ Z 1 e Z 1 W u (M + Z 1 W )u = Mz q + µ Z 1 e + r (12) Hence the algorithm wors if the matrix M + Z 1 W is nonsingular in each iteration. The following result holds concerning the nonsingularity of this matrix. Theorem 1 [15] - M is a matrix P 0 if and only if M + Z 1 W is nonsingular for all z > 0 and w > 0. It is not difficult to extend the theory presented in Kojima et al [14] to show that all the iterates belong to the set N, provided (z 0, w 0 ) N and M + Z 1 W is nonsingular for each (z, w ) N. This means that the algorithm may wor for the nonmonotone LCP whose matrix is not P 0. Furthermore the method always stops in one of the following terminations: 4

5 (i) in Step 1 with a solution of the LCP (ii) in Step 3 with a failure. The drawbac of this type of methodology concerns the possibility of occurence of this failure. As we see later, the termination can occur when the LCP has no solution, but the method may terminate with such form in cases where the LCP has a solution. Actually, this type of difficulty is shared by all the direct and iterative methods for the LCP [3, 20]. So, as these other linear complementarity procedures, this algorithm cannot process the LCP in all cases. It is important at this stage to distinguish three categories of the interior-point algorithm: (i) The Feasible Interior-Point (FIP) algorithm in which for all iterations. r = 0, w = q + Mz (ii) The Infeasible Interior-Point (IIP) method in which r = 0 for all iterations and w q + Mz in general. (iii) The Truncated Interior-Point (TIP) algorithm, where r 0 in general and satisfies the inequality (7). It is important to add that for the first category of method the termination in Step 1 always occurs when M is a matrix P SD or P [15]. As stated before, all those forms of the interior-point algorithm possess global convergence to one of the terminations mentioned in the Steps 1 and 3, provided the initial point belongs to the neighborhood N of the central path defined by (5). The computation of the stepsize α is done according to the formulas (8) - (10) of Step 3. In practice it is not easy to find a value α satisfying these conditions.a quite relaxed criterion for the choice of α has been employed in practice and simply consists of finding the largest positive value such that the variables zi and wi are positive in each iteration. The most common choice for α taes the following form α = max{α > 0 : z + αu > 0, w + αv > 0} This choice for α does not guarantee global convergence, but has shown to perform quite well in practice for linear programming [19] and monotone LCPs [10]. Recently Simantirai and Shanno [23] have adapted an idea of El-Bary et al [8] and have proposed a line-search that guarantees global convergence for the infeasible interior-point method and seems to wor well in practice. This line-search is an Armijo type tecnhique for the merit function Φ(z, w) = [ ZW e 2 + w Mz q 2] 1 2 where e IR n is a vector of ones and denotes the l 2 norm. This function is a quite natural choice, since the algorithm pursues a point (z, w ) that is a solution of the following system of nonlinear equations { w Mz q = 0 ZW e = 0 (13) 5

6 As stated in [5] the merit function (13) is usualy employed for this type of nonlinear system. The procedure starts by finding the maximum value ᾱ such that (z +αu, w +αv ) N for all 0 α ᾱ. This is done according to the formulas (17) - (19) in [23]. Next the stepsize α is computed by α = ᾱ 2 t where t is the smallest nonnegative integer satisfying Φ(z + α u, w + α v ) Φ(z, w ) + α [ ] 10 4 Φ(z, w ) T u v. with Φ(z, w ) the gradient of the merit function. As is shown in Simantirai and Shanno [23] this line-search technique is sufficient to guarantee global convergence for the infeasible interior-point method when r = 0 in all iterations. To show that the truncated infeasible interior-point method mantains global convergence it is necessary to prove that the search direction computed by this procedure is a descent direction for the merit function (13). To do this we need the following result. Theorem 2 If Φ(z, w ) denotes the gradient of the function Φ at (z, w ) and (u, v ) is the search direction computed by the IIP algorithm, then [ ] Φ(z, w ) T u Φ(z, w )(1 σ ) v Proof: Consider the system associated to the iteration [ ] [ w Mz q r F (z, w) = = ZW e µ e ] If F (z, w ) denotes the jacobian of this system, then by computations similar to those presented in [23] we have Φ(z, w ) T [ u But and v ] = = Furthermore σ β 0 and (7) imply [ [ 1 F (z, w ) F (z, w ) T F (z, w )F (z, w ) 1 F (z, w r ) + µ e ( [ 1 F (z, w F (z, w ) 2 + F (z, w ) T ) r µ e = F (z, w ) + ( Mz + w q) T r + µ (z ) T w F (z, w ) F (z, w ) = Φ(z, w ) µ (z ) T w σ Z W 2 r Mz + w q β 0 σ 6 ]) ]]

7 Hence by the Cauchy-Schwarz inequality we have ( Mz + w q) T r Mz + w q r σ Mz + w q 2 (14) Therefore Φ(z, w ) T [ u v ] Φ(z, w ) + σ [ ] 2 Φ(z, w ) Φ(z, w ) It follows from the theorem that one of the two possible cases occur: = Φ(z, w )(1 σ ) (i) Φ(z, w ) = 0 and the algorithm stops with a solution of the LCP. [ ] u (ii) Φ(z, w ) T < 0 and (u, v ) is a descent direction. v Hence the theory presented in [23] can now be used to achieve exactly the same conclusion stated in this paper. Therefore the TIIP algorithm possesses global convergence if the Armijotype line-search described before is used throughout the procedure. In the next section we describe some experiments made with the three variants of the interior-point method incorporating the two line-searches discussed in this section. As stated in the first section, we denote by SIIP and GIIP the two versions of these methods that use the simple line-search and the Armijo type technique respectively. The so-called Feasible and Infeasible interior-point versions employ a direct solver for the solution of the system (12) that computes the search direction. Furthermore an iterative solver is used in the truncated variant of the interior-point algorithm. The choice of the direct or iterative solvers depends on the symmetry and sparsity of the matrix M of the LCP. 3 Computational Experience In this section we describe some computational experience on a SUN SPARCstation 10 (48Mhz, 64Mb RAM) with the variants of the interior-point algorithm discussed in the previous section. Three sets of experiments have been performed, namely the solution of some LCPs with P and P 0 matrices, LCPs associated with nonzero sum bimatrix games and LCPs corresponding to a networ spatial equilibrium model. 3.1 LCPs with matrices P and P 0 In this first experience we have started by considering a number of symmetric P and P 0 matrices from the literature. Since symmetric matrices P (P 0 ) are P D(P SD) then the corresponding LCPs are monotone. These matrices are presented below. (i) Pentadiagonal P D matrices, denoted by P DL, that have been introduced in [18]. The nonzero elements are given by m ii = 6 m i,i 1 = m i,i+1 = 4 m i,i 2 = m i,i+2 = 1 (15) 7

8 (ii) P D matrices that arise on the solution of the Laplace equation by finite differences [2]. We denote by LAPL this type of matrices and a description of their elements appears in [10]. (iii) P D matrices of the form M = LL T, where L is a unit lower triangular matrix with all off-diagonal elements equal to 2 [9]. This type of matrix is denoted by F AT. (iv) The well-nown Hilbert matrices [20] whose elements are given by m ij = 1 i + j 1 These matrices are P SD and are denoted by HILB. (v) The so-called Pascal P SD matrices [20] that are denoted by P ASC. Their elements satisfy (i + j 2)! θ m ij = (i 1)!(j 1)! where! denotes factorial and θ is a small positive real number (we have set θ = 10 7 ). We have also considered in this first experience two well-nown matrices P that are stated next. (vi) The matrices L referred in (iii), which are P but not P D and are denoted by MU. (vii) P D matrices introduced by Chandrasearan, Pang and Stone [20], whose elements are given by m ii = 1 m ij = 2 if j > i and i + j is odd m ij = 1 if j > i and i + j is even (16) m ij = 1 if j < i and i + j is odd m ij = 2 if j < i and i + j is even These matrices are denoted by P ANG. Finally we have constructed some matrices of the form DAE where A is a matrix of the categories mentioned before and D and E are diagonal matrices whose diagonal elements d ii and e ii are randomly generated numbers belonging to the intervals (0, 0.1) and (0, 1) respectively. We denote by P SNAME the matrices of this form, where NAME represents one of the classes of the matrices stated before. All these matrices have been used for constructing the LCPs of our first experience. According to the suggestions of their authors, the right-hand side vectors for the LCPs with the matrices F AT, MU and P ANG are given by q = e, where e is a vector of ones. Furthermore all the right-hand side vectors q of the remaining LCPs have been generated by the technique discussed in [10], with scal = 0, F = n 2 and F d = 0. The symmetric monotone LCPs have been solved by the two variants SIIP and GIIP of the infeasible interior-point method, that have been implemented according to the lines stated in [10]. The LCPs with nonsymmetric matrices have been solved by a similar implementation in 8

9 which the subroutine MA27 [6] has been replaced by MA28 [6]. We note that this procedure is certainly not the best way of implementing the interior-point methods in the latter case. However, this code is suitable for comparing the efficiency of the two variants of the interiorpoint algorithm. These two versions SIIP and GIIP require some parameters that have been set according to the recommendations presented in [10] and [23] respectively. Our experiences have shown that σ = 1 n is usually preferable for the LCPs of smaller dimension. We have used this value for all the problems but LAP L and P SLAP L, for which the choice presented in [23] has been followed. As suggested in [10], we have used the initial point as follows zi 0 = λ, i = 1,..., n wi 0 = q + Mz, i = 1,..., n (17) where λ is a small number belonging to the interval [1, 5]. Finally the stopping criteria for both the variants of the infeasible interior-point use the tolerances ɛ 1 = ɛ 2 = Test Problem n SIIP GIIP PDL LAPL HILB PASC FAT MUC PANG PSDPL PSLAPL PSHILB PSPASC PSPANG Table 1 - Number of iterations for LCPs with matrices P and P 0 The results displayed in Table 1 indicate that both the versions SIIP and GIIP of the infeasible interior-point algorithm are quite efficient for solving all the LCPs. We note that the matrices P SHILB and P SP ASC are P 0 but are not P. Actually, they belong to the socalled class of sufficient matrices introduced in [4]. It is interesting to note that the Feasible IP 9

10 algorithms possess global convergence for LCPs with matrices of this class, as these matrices coincide with the matrices P introduced in [15]. See [24] for a proof of this result. In the second set of experiments we have considered some matrices P 0 that are neither P nor even sufficient. We start with the LCP defined by q = , M = It is easy to see that the famous Leme s algorithm [16] cannot process this LCP. Furthermore the LCP is feasible and possesses an infinity number of solutions of the form and z = (α, 0, 0, 0), α > 0 z = (1, 0, β, 0), β > 0 Both the variants of the infeasible interior-point methods have attained a solution of the first form in 4 iterations. It is well-nown that if M is a matrix P 0 then one of the following three possible cases must occur: (i) The LCP has a solution. (ii) The LCP is infeasible, that is, its linear constraints are incompatible. (ii) The LCP is feasible but has no solution. In our next experiment we have considered some LCPs satisfying the last two cases. In all the problems the infeasible interior-point method has terminated with a failure. After all these successful experiments we have tested both the variants of the infeasible interior-point algorithm on some NP-hard LCPs with matrices P 0 that have been introduced in [15]. Both the variants of the method have stopped with a failure in Step 3 despite the existence of a solution for these LCPs. So these experiments seem to indicate that both the variants are efficient for solving monotone LCPs and LCPs with P and sufficient matrices. Furthermore there is no clear winner between the two versions. The algorithm can also solve more difficult LCPs with matrices P 0 for which other traditional methods are unable to process. However, the two versions do not seem to be able to solve NP-hard LCPs with matrices P Nonzero Sum Bimatrix Games It is well-nown that any nonzero sum bimatrix game can be reduced into a nonmonotone LCP. There are two possible formulations of the bimatrix games in terms of a LCP. In the most common formulation (FORM1) the right-hand side vector q and the matrix M tae the following forms [ ] [ ] e m 0 A q = e r, M = B 0 10

11 where m and r are positive integer numbers, e t IR t is a vector of ones and A IR mxr, B IR rxm are positive matrices [17]. On the other hand in the second formulation (FORM2) e r is replaced by e r and B is a negative matrix [7]. It is easy to find a feasible positive starting point (z 0, w 0 ) for both the variants of the interior-point method. Next we present a process for computing the initial point z 0 that has been used in our experiments. The vector w 0 is given by q + Mz 0. Let Then in FORM1 we set m b ij 1 i r j=1 1 = min r a ij 1 i m j=1 2 = min z 0 i = α 1 1 i = 1,..., m z 0 i = α 2 2 i = m + 1,..., m + r where α 1 = max{r max i,j On the other hand in FORM2 we set b ij, 1.5} z 0 i = 1 α 1 1 α 2 = max{m max i,j i = 1,..., m a ij, 1.5} where α 1 = { 1 r min b ij if 1.5 < α 1 < otherwise Furthermore, the components m + 1,..., m + r of z 0 are given as in FORM1. As before, we have tested both the variants SIIP and GIIP of the interior-point algorithm. The elements of the matrices A and B of the test problems are randomly generated in the intervals [a, b] and [c, d] respectively. The values of these numbers a, b, c and d and the values of m and r are displayed in Table 2 together with the results of the performances of the two variants on these problems. In case of failure the value of the complementarity gap z T w at that stage is included after the letter F. FORM m r a b c d n SIIP GIIP F(1E-4) F(1E-4) F(1E-5) F(1E-5) F(1E-3) F(1E-1) F(1E-5) E-5 4E-2-4E-2-5E-5 60 F(2E+0) 182 Table 2 - Nonzero Sum Bimatrix Games 11

12 The results displayed in Table 2 show that the variant GIIP is more recommendable for this type of LCPs. In fact this procedure has been successful in 8 of the 10 test problems. Furthermore the value of the complementarity gap was quite small in the two failures. On the other hand the SIIP has failed in 6 problems. So these results seem to indicate that there is probably room for the use of interior-point algorithms on some nonmonotone LCPs whose matrices are not P 0. However, there is much to be done for the design of an efficient algorithm of this type in this case. 3.3 A Spatial Equilibrium Model This model has been discussed in [1, 11, 21] and consists of r regions that ship m commodities (m 1) among them. Furthermore there is connection between each pair of regions and the supply/demand functions are affine. This model leads into a monotone LCP with matrix M = JBJ T, where B 11 B 12 B 1m B 21 B 22 B 2m B = IRmrxmr B m1 B m2 B mm is a (symmetric or unsymmetric) P D matrix such that each submatrix B ij is diagonal with positive elements. Furthermore J is a bloc diagonal matrix of the form G G J =... G where G IR rxr(r 1) is an incidence matrix. Therefore the dimension of the LCP is n = mr(r 1), which can be quite large even for small number of regions and commodities. For instance a model of 50 commodities and 100 regions leads into a LCP of dimension Since B is a P D matrix, then M = JBJ T is a P SD matrix and the LCP is monotone. So the infeasible interior-point is recommended for solving this type of LCP. However, the large dimension of the LCP and the structure of its matrix prohibits the use of a direct solver to find the search direction in each iteration. Therefore the truncated version of the interior-point is much more suited in this application. In [22], we have developed an efficient implementation of the Truncated IIP method for the solution of this LCP. One of most important features of this implementation lies on the fact that the search direction is computed by solving a linear system of order much smaller than n. To see this, we note that each iteration of the TIIP method requires the solution of a linear system with matrix M +E = JBJ T +E, where E is a diagonal matrix with positive diagonal elements. Since B is a P D matrix then B = LDU where L and U are unit lower and upper triangular matrices respectively. Furthermore no fill-in occurs during this factorization due to the special structure of B. Now suppose we wish to solve the system (M + E )y = b (18) 12

13 Hence its solution satisfies y = (M + E ) 1 b By the Shermann-Morrison formula [5] we have y = E 1 b E 1 Hence y can be computed by solving the systems and setting J T U 1 (D 1 + UJE 1 J T L) 1 L 1 Jb Lα = Jb (19) (D 1 + UJE 1 J T L)β = α (20) Uθ = β (21) y = E 1 (b J T θ) Since the matrices L and U are triangular, then the major computational effort for solving the system (18) lies on the solution of (20). The dimension of this latter system is mr, which is much smaller than the dimension of the LCP. To have an idea of the gap we note that for m = 50 and r = 100 the order of the system (20) is 5000, while the dimension of the LCP is The system (20) is solved by an iterative solver, which depends on the fact of B being symmetric or not. In the first case the Preconditioned Conjugate Gradient (PCG) method should be employed. On the other hand a nonsymmetric iterative method such as the Preconditioned QMR algorithm is quite suitable for the second case. Both these iterative methods can efficiently do their job if a good preconditioner is available. An incomplete QR factorization preconditioner has shown to be recommendable in this instance and is fully described in [22]. Another important feature of the implementation is the use of a dynamic stopping criterion for the iterative solver. If we set the vector r of the truncated IP method as the residual of the iterative solver, then this latter procedure can stop even when the residual vector has a relatively large norm. In effect the theory of the truncated IP method only requires that the inequality (7) holds in each iteration and this can be satisfied with r large. Based on the interesting features mentioned before, the implementation of the truncated IIP method (version SIIP) has shown to be quite efficient for solving this type of LCP [22]. The next figures illustrate the performance of this method for the case of some spatial equilibrium models with a symmetric P D matrix B, 50 commodities and a different number of regions. In each figure the dimension of the LCP is in bracets on the right of the corresponding number of regions. The first and the second figures represent the number of iterations and the CPU time in seconds that the truncated IP method has required for solving all these problems. The third figure contains the average number of iterations that the PCG method requires during the application of the truncated IP algorithm. We also show the dimension of the system (20) to be solved in each iteration, according to the procedure explained above. The results displayed in the first two figures show that the truncated IP method is quite efficient for this type of LCPs. Furthermore the performance of the algorithm does not seem to be quite affected by an increase of the dimension of the LCPs. On the other hand the average number of iterations of the PCG method grows linearly with the order of the LCP and is allways smaller than 0.5p, where p is the order of the system to be solved. We recommend [22] for a full report of the performance of the truncated IP algorithm for this 13

14 type of monotone LCP when M is symmetric or unsymmetric. These results show the great efficiency of the interior-point in this instance. 4 Concluding Remars In this paper we have investigated the use of an infeasible interior-point algorithm for the solution of the Linear Complementarity Problem (LCP). The method has shown to be quite efficient for solving monotone LCPs and LCPs with P matrices and other subclasses of P 0 matrices. However, it has not been able to process some NP-hard LCPs with P 0 matrices. The algorithm also faces difficulties for solving LCPs associated with nonzero sum bimatrix games, despite being successful in many cases. A robust nonenumerative method for solving a large class of nonmonotone LCPs is still to be designed. We hope that this paper will increase the interest of the research community on the nonmonotone LCP and some of its generalizations that appear quite often in the solution of nonlinear programs [20] variational inequalities [3] and global optimization [12, 13]. Acnowledgment We are grateful to Luís Portugal for is suggestions on a first draft of this manuscript. References [1] R. Asmuth, B. Eaves, and E. Peterson. Computing economic equilibria on affine networs with Leme s algorithm. Mathematics of Operations Research, 4: , [2] R. L. Burden, J. D. Faires, and A. Reynolds. Numerical Analysis. Prindle, Weber and Schmidt, Boston, [3] R. Cottle, J. Pang, and R. Stone. The Linear Complementarity Problem. Academic Press, New Yor, [4] R. Cottle, J. Pang, and Venateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and its Applications, 114: , [5] J. Dennis Jr. and R. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, New Yor, [6] I. Duff, A. Erisman, and J. Reid. Direct Methods for Sparse Matrices. Clarendon Press, Oxford, [7] B. Eaves. The linear complementarity problem. Management Science, 17: , [8] A. El-Bary, R. Tapia, Y. Zhang, and T. Touchiya. On the formulation of the primal-dual interior-point method for nonlinear programming for linear complementarity problems. TR 92-40, Rice University, USA, (revised April). [9] Y. Fathi. Computational complexity of linear complementarity problems associated with positive definite matrices. Mathematical Programming, 17: ,

15 number of commodities: TN iteration (30.000) 50 ( ) 75 ( ) 100 ( ) 125 ( ) 2500 regions (variabl 2000 CPU time (sec (30.000) 50 ( ) 75 ( ) 100 ( ) 125 ( ) regions (variabl average PCG iterations per s dimension of the systems (30.000) 50 ( ) 75 ( ) 100 ( ) 125 ( ) regions (variabl 15

16 [10] L. Fernandes, J. Júdice, and J. Patricio. An investigation of interior-point and bloc pivoting algorithms for large-scale symmetric monotone linear complementarity problems. Technical report, Department of Mathematics, University of Coimbra, Coimbra, Portugal, To appear in Computational Optimization and Applications. [11] F. Guder, J. Morris, and S. Yoon. Parallel and serial successive overrelaxation for multicommodity spatial price equilibrium problems. Transportation Science, 26:48 58, [12] J. Judice and A. Faustino. A computational analysis of LCP methods for bilinear and concave quadratic programming. Computers and Operations Research, 18: , [13] J. Judice and A. Faustino. A sequential LCP algorithm for bilinear linear programming. Annals of Operations Research, 34:89 106, [14] M. Kojima, N. Meggido, and S. Mizuno. A primal-dual infeasible interior-point algorithm for linear programming. Mathematical Programming, 61: , [15] M. Kojima, N. Megiddo, T. Noma, and A. Yoshise. A Unified Approach to Interior-Point Algorithms for Linear Complementarity Problems. Lecture Notes in Computer Science 538. Springer-Verlag, Berlin, [16] C. Leme. On complementary pivot theory. In G. Dantzig and A. Veinott, editors, Mathematics of Decision Sciences, pages American Mathematical Society, Providence, [17] C. Leme and J. Howson Jr. Equilibrium points of bimatrix games. SIAM Journal of Applied Mathematics, 12: , [18] Y. Lin and J. Pang. Iterative methods for large convex quadratic programs: a survey. SIAM Journal on Control and Optimization, 25: , [19] L. Lustig, R. Marsten, and D. Shanno. Computational experience with a primal-dual method for linear programming. Linear Algebra and its Applications, 152: , [20] K. Murty. Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin, [21] J. Pang. A hybrid method for the solution of some multicommodity spatial equilibrium problems. Management Science, 27: , [22] L. Portugal, L.Fernandes, and J. Judice. A truncated Newton interior-point algorithm for the solution of a multicommodity spatial equilibrium model. Technical report, Department of Mathematics, University of Coimbra, Coimbra, Portugal, [23] E. Simantirai and D. Shanno. An infeasible interior-point method for linear complementarity problems. RRR 7-95, RUTCOR, New Brunswic, New Jersey, USA, [24] H. Valiaho. P -matrices are just sufficient. Technical report, Department of Mathematics, University of Helsini, Finland, [25] Y. Zhang. On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem. SIAM Jounal on Optimization, 4: ,