KeywordsLCP Splitting method SOR method
Splitting methods were originally proposed as a generalization of the classical SOR method for solving a system of linear equations [8,25], and in the late 1970s they were extended to the linear complementarity problem (LCP; cf. Linear complementarity problem) [1,2, Chap. 5], [10,13,18]. These methods are iterative and are best suited for problems in which exploitation of sparsity is important, such as large sparse linear programs and the discretization of certain elliptic boundary value problems with obstacle.
[12, Thm. 3.2]. (Earlier results of this kind that further assumed M is positive semidefinite or nondegenerate can be found in [2 Chap. 5], [5,11,19,20] and references therein.) For the SOR method, corresponding to B given by (3) with ω ∊ (0, 2), it can be verified that (B, C) is a regular Q-splitting provided M has positive diagonal entries. The proof of the above convergence result uses two key facts about the LCP, namely, that f(x) assumes only a finite number of values on the solution set and that the distance to the solution set from any point x near the solution set is in the order of the‘residual’ at x, defined to be the difference in the two sides of (1). In addition, the function f(x) can be used in a line-search strategy to accelerate convergence of the splitting methods [2, Sec. 5.5].
B − C is positive definite and for every x there exists a solution x′ to (2)
[2, Thm. 5.6.1], [24, Cor. 5.3]. One choice of B that satisfies the above assumption is
B − M is symmetric positive definite
In summary, building on the early work of Hildreth and H.B. Keller and others, splitting methods have been well developed in the last twenty years to solve the LCP (1) when the matrix M is either symmetric or positive semidefinite. Computationally, these methods are best suited when M is symmetric, possibly having some sparsity structure (e. g., M = AA ⊺ with A sparse), and the function (4) is used in a line-search strategy to accelerate convergence. Extensions of these methods to problems where the box l ≤ x ≤ u is replaced by a general polyhedral set, including as special cases the extended linear / quadratic programming problem of R.T. Rockafellar and R.J-B. Wets and the quadratic program formulation of the LCP with row sufficient matrix , have also been studied [2, Sec. 5.5], [6,12,22,23]. Inexact computation of x′ is discussed in [2, Sec. 5.7], [9,12,15]. Acceleration of the methods in the case where M is not symmetric remains an open issue. In fact, if M is not symmetric nor positive semidefinite, convergence of the splitting methods is known only for the case where M is an H- matrix with positive diagonal entries and B is likewise, with the comparison matrix of B having a contractive property [2, p. 418], [18,19]. Thus, even if M is a P-matrix , it is not known whether the splitting methods converge for some practical choice of B.