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1.
We propose a new smoothing Newton method for solving the P
0-matrix linear complementarity problem (P
0-LCP) based on CHKS smoothing function. Our algorithm solves only one linear system of equations and performs only one line search per iteration. It is shown to converge to a P
0-LCP solution globally linearly and locally quadratically without the strict complementarity assumption at the solution. To the best of author's knowledge, this is the first one-step smoothing Newton method to possess both global linear and local quadratic convergence. Preliminary numerical results indicate that the proposed algorithm is promising. 相似文献
2.
Jianming Miao 《Mathematical Programming》1993,61(1-3):351-356
We consider the linear complementarity problem (LCP),w=Az + q, w0,z0,w
T
z=0, when all the off-diagonal entries ofA are nonpositive (the class of Z-matrices), all the proper principal minors ofA are positive and the determinant ofA is negative (the class of almost P-matrices). We shall call this the class of F-matrices. We show that ifA is a Z-matrix, thenA is an F-matrix if and only if LCP(q, A) has exactly two solutions for anyq0,q0, and has at most two solutions for any otherq.
Research supported by AFOSR-89-0512. 相似文献
3.
4.
The class of realn × n matricesM, known asK-matrices, for which the linear complementarity problemw – Mz = q, w 0, z 0, w
T
z =0 has a solution wheneverw – Mz =q, w 0, z 0 has a solution is characterized for dimensionsn <4. The characterization is finite and practical. Several necessary conditions, sufficient conditions, and counterexamples pertaining toK-matrices are also given. A finite characterization of completelyK-matrices (K-matrices all of whose principal submatrices are alsoK-matrices) is proved for dimensions <4.Partially supported by NSF Grant MCS-8207217.Research partially supported by NSF Grant No. ECS-8401081. 相似文献
5.
Jiri Rohn 《Mathematical Programming》1990,46(1-3):255-256
We give a short proof of the finiteness of Murty's principal pivoting algorithm for solving the linear complementarity problemy = Mz + q, y
T
z = 0,y 0,z 0 withP-matrixM. 相似文献
6.
In this paper we consider not necessarily symmetric co-positive as well as semi-monotoneQ-matrices and give a set of sufficient conditions for such matrices to beR
0-matrices. We give several examples to show the sharpness of our results. Construction of these examples is based on the following elementary proposition: IfA is a square matrix of ordern whose first two rows are identical and bothA
11 andA
22 areQ-matrices whereA
ii
stands for the principal submatrix ofA obtained by deleting rowi and columni fromA, thenA is aQ-matrix.Dedicated to our colleague and friend B. Ramachandran on his sixtieth birthday.Corresponding author. 相似文献
7.
Given a square matrixM of ordern and a vectorq
n
, the linear complementarity problem is the problem of either finding aw
n
and az
n
such thatw–Mz=q,w0,z0 andw
T
z=0 or showing that no such (w, z) exists. This problem is denoted asLCP(q, M). We say that a solution (w, z) toLCP(q, M) is degenerate if the number of positive coordinates in (w, z) is less thann. As in linear programming, degeneracy may cause cycling in an adjacent vertex following methods like Lemke's algorithm. Moreover, ifLCP(0,M) has a nontrivial solution, a condition related to degeneracy, then unless certain other conditions are satisfied, the algorithm may not be able to decide about the solvability of the givenLCP(q, M). In this paper we review the literature on the implications of degeneracy to the linear complementarity theory. 相似文献
8.
Jiri Rohn 《Mathematical Programming》1993,60(1-3):229-231
We give a characterization of unique solvability of an infinite family of linear complementarity problems of a special form by means of a finite subset of this family. 相似文献
9.
The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP 总被引:2,自引:0,他引:2
** Email: zhenghaihuang{at}yahoo.com.cn; huangzhenghai{at}hotmail.com In this paper, we propose a non-interior continuation algorithmfor solving the P0-matrix linear complementarity problem (LCP),which is conceptually simpler than most existing non-interiorcontinuation algorithms in the sense that the proposed algorithmonly needs to solve at most one linear system of equations ateach iteration. We show that the proposed algorithm is globallyconvergent under a common assumption. In particular, we showthat the proposed algorithm is globally linearly and locallyquadratically convergent under some assumptions which are weakerthan those required in many existing non-interior continuationalgorithms. It should be pointed out that the assumptions usedin our analysis of both global linear and local quadratic convergencedo not imply the uniqueness of the solution to the LCP concerned.To the best of our knowledge, such a convergence result hasnot been reported in the literature. 相似文献
10.
In this paper, we propose a new smooth function that possesses a property not satisfied by the existing smooth functions.
Based on this smooth function, we discuss the existence and continuity of the smoothing path for solving theP
0 function nonlinear complementarity problem ( NCP). Using the characteristics of the new smooth function, we investigate the
boundedness of the iteration sequence generated by the non-interior continuation methods for solving theP
0 function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that
the solution set of the NCP is nonempty and bounded is weaker than those required by a few existing continuation methods for
solving the NCP 相似文献
11.
The paper generalizes the Mangasarian–Ren (Ref. 1) error bounds forlinear complementarity problems (LCPs) to nonlinear complementarity problems(NCPs). This is done by extending the concept of R
0-matrixto several R
0-type functions, which include a subset ofmonotone functions as a special case. Both local and global error bounds areobtained for R
0-type NCPs and some monotone NCPs. 相似文献
12.
In this paper, we first investigate a two-parametric class of smoothing functions which contains the penalized smoothing Fischer-Burmeister function and the penalized smoothing CHKS function as special cases. Then we present a smoothing Newton method for the nonlinear complementarity problem based on the class of smoothing functions. Issues such as line search rule, boundedness of the level set, global and quadratic convergence are studied. In particular, we give a line search rule containing the common used Armijo-type line search rule as a special case. Also without requiring strict complementarity assumption at the P0-NCP solution or the nonemptyness and boundedness of the solution set, the proposed algorithm is proved to be globally convergent. Preliminary numerical results show the efficiency of the algorithm and provide efficient domains of the two parameters for the complementarity problems. 相似文献
13.
14.
We introduce the notion of weak dually residuated lattice ordered semi-groups (WDRL-semigroups) and investigate the relation
between R
0-algebras and WDRL-semigroups. We prove that the category of R
0-algebras is equivalent to the category of some bounded WDRL-semigroups. Moreover, the connection between WDRL-semigroups
and DRL-semigroups is studied. 相似文献
15.
16.
A simple and unified analysis is provided on the rate of local convergence for a class of high-order-infeasible-path-following algorithms for the P*-linear complementarity problem (P*-LCP). It is shown that the rate of local convergence of a -order algorithm with a centering step is + 1 if there is a strictly complementary solution and ( + 1)/2 otherwise. For the -order algorithm without the centering step the corresponding rates are and /2, respectively. The algorithm without a centering step does not follow the fixed traditional central path. Instead, at each iteration, it follows a new analytic path connecting the current iterate with an optimal solution to generate the next iterate. An advantage of this algorithm is that it does not restrict iterates in a sequence of contracting neighborhoods of the central path. 相似文献
17.
M. Seetharama Gowda 《Mathematical Programming》1990,49(1-3):139-141
Recently, Jeter and Pye gave an example to show that Pang's conjecture, thatL
1 Q
, is false. We show in this article that the above conjecture is true for symmetric matrices. Specifically, we show that a symmetric copositive matrix is inQ if and only if it is strictly copositive. 相似文献
18.
19.
Uwe Schäfer 《Operations Research Letters》2004,32(4):350-354
Concerning three subclasses of P-matrices the modulus algorithm and the projected successive overrelaxation (PSOR) method solving the linear complementarity problem are compared to each other with respect to convergence. It is shown that the modulus algorithm is convergent for all three subclasses whereas the convergence of the PSOR method is only guaranteed for two of them. 相似文献
20.
In this article, we present a weaker version of the class of generalized positive subdefinite matrices introduced by Crouzeix and Komlósi [J.P. Crouzeix and S. Komlósi, The Linear Complementarity Problem and the Class of Generalized Positive Subdefinite Matrices, Applied Optimization, Vol. 59, Kluwer, Dordrecht, 2001, pp. 45–63], which is new in the literature, and obtain some properties of weak generalized positive subdefinite (WGPSBD) matrices. We show that this weaker class of matrices is also captured by row-sufficient matrices introduced by Cottle et al. [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] and show that for WGPSBD matrices under appropriate assumptions, the solution set of a linear complementarity problem is the same as the set of Karush–Kuhn–Tucker-stationary points of the corresponding quadratic programming problem. This further extends the results obtained in an earlier paper by Neogy and Das [S.K. Neogy and A.K. Das, Some properties of generalized positive subdenite matrices, SIAM J. Matrix Anal. Appl. 27 (2006), pp. 988–995]. 相似文献