R. W. Cottle, J.-S. Pang and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:, 9, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....LCP of the form (1.1) where M is a positive semi definite matrix. However, there are applications for which M is not positive semidefinite. The most general results obtained so far assume that M belongs to the class SU of sufficient matrices introduced in 1989 by Cottle, Pang and Venkateswaran [4]. A matrix M is column sufficient if for all x 2 R I [x] i [Mx] i 0; i = 1; n ) x] i [Mx] i = 0; i = 1; n where components of an n vector u are denoted by [u] i ; i = 1; n, and row sufficient if M is column sufficient. M is sufficient if it is both row and column ....

R. W. Cottle, J.-S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:231--249, 1989.

....s satisfying (1) or decides that no such vectors exist. The known methods for dealing with the LCP s only are guaranteed to solve the problem if the matrix M belongs to a certain class. We briefly recall two examples of this. The class of sufficient matrices (SU) was introduced by Cottle et al. [2]. A matrix M 2 IR n Thetan is column sufficient if for all x 2 IR n , x i (Mx) i 0; 8i ) x i (Mx) i = 0; 8i; and row sufficient if M T is column sufficient. The matrix M is sufficient if it is both row and column sufficient. In that case the LCP is called a sufficient LCP. Cottle showed ....

R. W. Cottle, J.-S. Pang and V. Venkateswaran. Sufficient Matrices and the Linear Complementarity Problem. Linear Algebra and Its Applications, 114/115:231-249, 1989.

....j (Mx) j 0 for some j 2 f1; 2; Delta Delta Delta ; ng: 15) A matrix M is called row sufficient if its transpose is column sufficient. A matrix M is called sufficient if it is both row and column sufficient. The notion of the sufficiency was first introduced by Cottle, Pang and Venkateswaran [19]. 11 11 The class of sufficient matrices is identical to the class of P matrices introduced by Kojima e.al. 49] This result was proved by Valiaho [74] 28 For a given n Theta n matrix M and an n vector b, we consider the linear system V (M; b) fx 2 R E : I ; GammaM; Gammab]x = 0g, ....

R.W. Cottle, J.-S. Pang and V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra and Its Applications 114/115 (1987) 235--249.

....it is a P ( matrix for some nonnegative , i.e. P = 0 P ( One easily verifies that M is a P (0) matrix if and only if M is positive semidefinite. Furthermore, if M is P ( for some 0 then M is P ( for all . The class of sufficient matrices (SU) was introduced by Cottle et al. [1]. A matrix M 2 n Thetan is column sufficient if for all x 2 n , x i (Mx) i 0; 8i 2 I ) x i (Mx) i = 0; 8i 2 I; and row sufficient if M T is column sufficient. The matrix M is sufficient if it is both row and column sufficient. Recently, Valiaho [14] proved that P = SU . Now we turn ....

....x i 0 for some x 2 Gamma g; N : fi 2 I : s i (x) 0 for some x 2 Gamma g; T : fi 2 I : x i = s i (x) 0 for all x 2 Gamma g: Then the index sets B; N and T form the so called optimal partition of I (see Lemma 3. 1 in [4] It is known that the solution set Gamma is convex [1]. If x 2 Gamma and xB 0; s N (x) 0, then we call x a maximally complementarity solution of LCP. The first condition number for LCP we use here is defined by [4] oe x LCP : min i2B max x2 Gamma fx i g; oe s LCP : min i2N max x2 Gamma fs i (x)g oe LCP = minfoe x LCP ; oe ....

Cottle, R.W., J.S. Pang and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:231-249, 1989.

....vector x (i.e. if x i (M T x) i 0 for all i, then x i (M T x) i = 0 for every i) ffl column sufficient if XMx 0 implies XMx = 0 for every vector x. ffl sufficient if it is both row and column sufficient. The concept of (column, row) sufficient matrices was introduced by Cottle et al. [6, 7]. This definition of (column, row) sufficient matrices formalizes the sign (non) reversibility property of matrices. Observe that for a column sufficient matrix the vectors x and Mx cannot have opposite sign for each coordinate. 26 Exercise 1.19 Let us consider the following three matrices: M ....

R. W. Cottle, J.-S. Pang and V. Venkateswaran. Sufficient Matrices and the Linear Complementarity Problem. Linear Algebra and Its Applications, 114/115:231-249, 1989.

....both zero in all solutions of the (LCP ) Complementary solutions with the maximal number of nonzero coordinates will be referred to as maximally complementary solutions. The existence of maximally complementary solutions follows from the convexity of the solution set, proved by Cottle et al. in [4]. Kojima et al. 18] under some additional assumptions showed that solutions on the central path converge to a maximally complementary solution of (LCP ) All known algorithms for solving (LCP ) need some assumption on the matrix M . So do Interior Point Methods (IPMs) as well. IPMs for solving ....

....if it is a P ( matrix for some nonnegative : P = 0 P ( One easily verifies that M is a P (0) matrix if and only if M is positive semidefinite. Furthermore, if M is P ( for some 0 then M is P ( for all . The class of sufficient matrices (SU) was introduced by Cottle et al. [4]. A matrix M 2 IR n Thetan is column sufficient if for all x 2 IR n , X(Mx) 0 ) X(Mx) 0 and row sufficient if M T is column sufficient. The matrix M is sufficient if it is both row and column sufficient. Recently, Valiaho [29] proved that P = SU . The sets of feasible and positive ....

[Article contains additional citation context not shown here]

R. W. Cottle, J.-S. Pang and V. Venkateswaran. Sufficient Matrices and the Linear Complementarity Problem. Linear Algebra and Its Applications, 114/115:231-249, 1989.

....both zero in all solutions of the (LCP ) Complementary solutions with the maximal number of nonzero coordinates will be referred to as maximally complementary solutions. The existence of maximally complementary solutions follows from the convexity of the solution set, proved by Cottle et al. in [4]. Kojima et al. 17] under some additional assumptions showed that solutions on the central path converge to a maximally complementary solution of (LCP ) All known algorithms for solving (LCP ) need some assumption on the matrix M . So do Interior Point Methods (IPMs) as well. IPMs for solving ....

....if it is a P ( matrix for some nonnegative : P = 0 P ( One easily verifies that M is a P (0) matrix if and only if M is positive semidefinite. Furthermore, if M is P ( for some 0 then M is P ( for all . The class of sufficient matrices (SU ) was introduced by Cottle et al. [4]. A matrix M 2 IR n Thetan is column sufficient if for all x 2 IR n , X(Mx) 0 ) X(Mx) 0 and row sufficient if M T is column sufficient. The matrix M is sufficient if it is both row and column sufficient. Recently, Valiaho [27] proved that P = SU . The sets of feasible and positive ....

[Article contains additional citation context not shown here]

R. W. Cottle, J.-S. Pang and V. Venkateswaran. Sufficient Matrices and the Linear Complementarity Problem. Linear Algebra and Its Applications, 114/115:231-249, 1989.

....as a linear optimization problem. Thus it goes without surprise that the least index criss cross method is generalized to this class of optimization problems as well [17] Linear Complementarity Problems. The largest solvable class of LCPs is the class of LCPs with a sufficient matrix [6, 5]. The LCP least index criss cross method is a proper generalization of the LO criss cross method. When the LCP arises from a LO problem, the LO criss cross method is obtained. Convex quadratic optimization problems give an LCP with a bisymmetric coefficient matrix. Because a bisymmetric matrix is ....

Cottle, R.W., Pang, J.-S., and Venkateswaran, V.: `Sufficient matrices and the linear complementarity problem', Linear Algebra and Its Applications 114/115 (1987), 235--249.

....x i (Mx) i X i2I Gamma x i (Mx) i 0 where I = fi : x i (Mx) i 0g and I Gamma = fi : x i (Mx) i 0g: Clearly, for linear map f(x) Mx q; f is a P map if and only if M is a P matrix. Valiaho [40] showed that the class of P matrices coincides with the class of sufficient matrices [8, 9]. A new equivalent definition of the P matrix is given in [46] The next concept is a generalization of the quasi monotone function and the P map. Definition 2.3. 46] A function f : R n R n is said to be a quasi P map if there exists a constant 0 such that the following implication ....

.... 0: Particularly, a matrix M 2 R n Thetan is a P matrix if and only if there is a constant 0 such that (1 ) max 1in x i (Mx) i min 1in x i (Mx) i 0: This is an equivalent definition of the concept of a P matrix (sufficient matrix) introduced by Kojima et al. 26] and Cottle et al. [9]. The following result follows immediately from Theorem 4.3. Corollary 4.6. Let f be a continuous P 0 and P( ff; fi) map. If NCP is strictly feasible, then the central path exists and any slice of it is bounded. It is worth noting that each P map is a P 0 and a P( ff; fi) function. The ....

R. W. Cottle, J. S. Pang and V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra Appl., 114/115 (1989), pp. 231-249.

....which is an ultimate goal. We hope that our existence results provide many researchers with a good incentive and some guidance to look for such a jewel. Finally let we make some remarks on possible extensions. It appears to be hard to extend our main result to the case of sufficient LCP s [3, 9, 8], although it is possible for the nondegenerate case [7] So the following problem remains unsolved: Open Problem 4.1 Let a sufficient n Theta n LCP be given. Without any nondegeneracy assumption prove that the length of the shortest admissible pivot sequences from any (not necessarily feasible) ....

R.W. Cottle, J.-S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:231--249, 1989.

....of the form (1.1) where M is a positive semi definite matrix. However, there are applications for which M is not positive semidefinite. The most general results obtained so far assume 2 that M belongs to the class SU of sufficient matrices introduced in 1989 by Cottle, Pang and Venkateswaran [4]. A matrix M is column sufficient if for all x 2 R I n [x] i [Mx] i 0; i = 1; n ) x] i [Mx] i = 0; i = 1; n and row sufficient if M T is column sufficient. M is sufficient if it is both row and column sufficient. Row sufficient matrices are linked to the existence of ....

R. W. Cottle, J.-S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:231--249, 1989.

.... 0 where I (x; y) fi : x i Gamma y i ) f i (x) Gamma f i (y) 0g and I Gamma (x; y) f1; 2; Delta Delta Delta ; ngnI : When f = Mx q; where M 2 R n Thetan and q 2 R n , it is easy to see that f is a P map if and only if M is a P matrix, i.e. the sufficient matrix, see [16, 4]. In what follows, we establish sufficient conditions for the strict feasibility of CPs with such functions as positively homogeneous, uniform) semimonotone and quasi monotone functions. We consider first the situation where the mapping G(x) f(x) Gammaf (0) is positively homogeneous on R n ....

R.W.Cottle, J.S.Pang and V.Venkateswaran, Sufficient Matrices and the Linear Complementarity Problems, Linear Algebra and its Applications, 114/115 (1989), 231-249.

.... semimonotone matrix, if 0 6= x 0 ) x i 0 and (Mx) i 0 for some i: b) 5] a copositive (strictly copositive) matrix if 0 6= x 0 ) x T (Mx) 0: c) 5] a P 0 (P) matrix if x 6= 0 ) max x i 6=0 x i (Mx) i 0( 0) d) 5] a R 0 matrix if the solution set SOL(M; 0) f0g: e) [20, 6] a P matrix if there exists a scalar 0 such that (1 ) X i2I x i (Mx) i X i2I Gamma x i (Mx) i 0 where I = fi : x i (Mx) i 0g and I Gamma = fi : x i (Mx) i 0g: f) 40] M 2 R n Thetan is said to be an exceptionally regular matrix if there exists no (x; 2 R n 1 ....

R.W. Cottle, J.S.Pang and V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra Appl., 114/115(1989), pp.231-249.

....in the introduction. M is row sufficient if its transpose is column sufficient and M is sufficient if it is both column and row sufficient. Clearly, P ae S c ae P c since the solutions sets are: a singleton, convex and connected, respectively. The following corollary, which also follows from [5] is now immediate. 9 Corollary 3. Suppose M 2 S c Q 0 . Then Lemke s algorithm terminates at a solution of LCP(q; M ) or determines that FEA(q; M ) Proof. Since M is column sufficient, SOL(q; M ) is convex, and is hence connected for all q. The corollary now follows from Theorem 2. It ....

R. W. Cottle, J. S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:231--249, 1989.

....which is an ultimate goal. We hope that our existence results provide many researchers with a good incentive and some guidance to look for such a jewel. Finally let we make some remarks on possible extensions. It appears to be hard to extend our main result to the case of sufficient LCP s [2,8,7], although it is possible for the nondegenerate case [6] So the following problem remains unsolved: Open problem Let a sufficient n Theta n LCP be given. Without any nondegeneracy assumption prove that the length of the shortest admissible pivot sequences from any (not necessarily feasible) ....

R.W. Cottle, J.-S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:231--249, 1989.

....method is also generalized to solve sufficient oriented matroid LCP s. To generalize the characterization theorems of this paper still remains a subject of further research. 3. Identification of Matrix Classes. The class of (column, row) sufficient matrices was introduced by Cottle et al. [6]. They showed that (column, row) sufficient matrices are common generalizations of P matrices (i.e. matrices with positive principal submatrices) and PSD matrices (positive semidefinite matrices) Later Cottle [4] generalized the principal pivoting method for row sufficient LCP s. Recently ....

....it and guarantee its finiteness are presented in Section 4. The characterization of the class of sufficient matrices by the criss cross method is discussed in Section 5. 2 Basic properties of sufficient matrices The concept of (column, row) sufficient matrices was introduced by Cottle et al. [6]. For ease of understanding, the definition and basic properties of sufficient matrices are summarized here. The proofs and further details can be found in [6, 4, 7] Definition 1 A matrix M is called ffl row sufficient if XM T x 0 implies XM T x = 0 for every vector x (i.e. if x i (M T x) ....

[Article contains additional citation context not shown here]

Cottle, R.W., Pang, J.--S., and Venkateswaran, V., (1989), Sufficient Matrices and the Linear Complementarity Problem. Linear Algebra and Its Applications 114/115, 231--249.

....1 Introduction In this paper we consider a linear complementarity problem (LCP) of the form, s = Mx q; x 0; s 0; x T s = 0; 1:1) where q 2 R I n and M 2 R I n Thetan is a sufficient matrix. The class SU of sufficient matrices was introduced in 1989 by Cottle, Pang and Venkateswaran [1]. A matrix M is column sufficient if for all x 2 R I n [x] i [Mx] i 0; i = 1; n ) x] i [Mx] i = 0; i = 1; n and row sufficient if M T is column sufficient. M is sufficient if it is both row and column sufficient. Row sufficient matrices are linked to the existence of ....

R. W. Cottle, J.-S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:231--249, 1989.

....the LCP literature. For example, results concerning L and 18 L matrices, which are extensions of those for copositive plus matrices, can be found in [12] and [13] see also [37] Termination results on P 0 matrices, which are extensions of those for P matrices can be found in [1] and [9]. Matrix classes related to LCP are numerous ( see [37] and [8] Here is a brief survey of those that are closely related to this work. Most of these matrix classes are defined with respect to IR n . Later in this thesis we generalize some of these, e.g. copositive plus, L, P , column ....

....matrices known as L as demonstrated by the work of Eaves ( see [12] and [13] and our work earlier in this chapter. Another approach for generalizing the termination results for Lemke s pivotal method on P matrices is through the notion of P 0 matrices and the work of Cottle et.al. in [1] and [9]. In this section, we explore the possibility of generalizing the notion of P 0 for polyhedral convex sets. We begin with an analysis on the standard LCP. Our study focuses on geometric and topological properties of the sets K(M) and SOL(q; M) that are crucial in analyzing termination behavior of ....

R.W. Cottle, J.S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and its Applications, 114115: 231--249, 1989.

....P = 0 P ( i.e. M is a P matrix if M 2 P ( for some 0. Obviously, P (0) PSD (the class of positive semi definite matrices) From the recent paper of Valiaho [11] it follows that the class of P matrices coincides with the class of sufficient matrices introduced in Cottle et al. [2]. The algorithm proposed in [4] terminates in O ( 1) 2 nL) iterations either by finding a solution or by determining that the problem is not solvable. If the problem is feasible and the initial point is close to the feasible set, then only O ( 1) p nL) iterations are needed. The ....

R. W. Cottle, J.-S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, (114/115):231--249, 1989.

....class of positive semi definite matrices) It is easily seen that P (0) ae P ( for all 0. We also note that from a recent paper of Valiaho [12] it follows that the class P = S 0 P ( coincides with the class SU of sufficient matrices introduced in 1989 by Cottle, Pang and Venkateswaran [5]. If M 2 P ( then the SLCP (1.1) is called a P ( GammaSLCP. Similarly, the MLCP (1.3) is called a P ( GammaMLCP if M 2 P ( while the HLCP (1.3) is called a P ( GammaHLCP if M 1 x M 2 s = 0 = x T s Gamma4 X i2I x i s i : If the above property is satisfied we also say that M; ....

R. W. Cottle, J.-S. Pang, and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:231--249, 1989.

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R. W. Cottle, J.-S. Pang and V. Venkateswaran. Sufficient matrices and the linear complementarity problem. Linear Algebra and Its Applications, 114/115:, 9, 1989.

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Cottle, R.W., J.S. Pang, V. Venkateswaran (1989). Sufficient matrices and the linear complementarity problem. Linear Algebra Appl. 114/115 231-249.

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R.W. Cottle, J.-S. Pang and V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra and Its Applications 114/115 (1987) 235--249.

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COTTLE, R. W., PANG, J. S., and VENKATESWARAN, V., Sufficient Matrices and the Linear Complementarity Problems, Linear Algebra and its Applications, Vols. 114/115, pp. 231-249, 1989.

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Cottle, R.W., Pang, J.S., and Venkateswaran, V., Sufficient Matrices and the Linear Complementarity Problems, Linear Algebra and Its Applications, Vol.114/115, pp.231--249, 1989.

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