In this note, we develop some new properties of a fundamental quantity associated with a P-matrix introduced by Mathias and Pang[1]. Also, based on extensions of such a quantity, we obtain global error bounds for the vertical and horizontal linear complementarity problems.

In this paper, we study the equivalence characterizations of several modified fixedpoint equations to variational inequalities (VI). Based on these equations, we give some applications in constructing iterative methods for the solution of the VI. Especially, we show global convergence, the sublinear convergence, and the finite termination of a new iterative algorithm under certain conditions.

We propose a noninterior continuation method for the monotone linear complementarity problem (LCP) by modifying the Burke-Xu framework of the noninterior predictor-corrector path-following method (Refs. 1-2). The new method solves one system of linear equations and carries out only one line search at each iteration. It is shown to converge to the LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution. Our analysis of the continuation method is based on a broader class of the smooth functions introduced by Chen and Mangasarian (Ref. 3).

In this paper we develop the convergence theory of a general class of projection and contraction algorithms (PC method), where an extended stepsize rule is used, for solving variational inequality (VI) problems. It is shown that, by defining a scaled projection residue, the PC method forces the sequence of the residues to zero. It is also shown that, by defining a projected function, the PC method forces the sequence of projected functions to zero. A consequence of this result is that if the PC method converges to a nondegenerate solution of the VI problem, then after a finite number of iterations, the optimal face is identified. Finally, we study local convergence behavior of the extragradient algorithm for solving the KKT system of the inequality constrained VI problem.

For the extended linear complementarity problem over an affine subspace, we first study some characterizations of (strong) column/row monotonicity and (strong) R0-property. We then establish global s-type error bound for this problem with the column monotonicity or R0-property, especially for the one with the nondegeneracy and column monotonicity, and give several equivalent formulations of such error bound without the square root term for monotone affine variational inequality. Finally, we use this error bound to derive some properties of the iterative sequence produced by smoothing methods for solving such a problem under suitable assumptions.