Iterative methods for linear complementarity problems with upperbounds on primary variables

This paper is concerned with iterative methods for the linear complementarity problem (LCP) of findingx and y in Rn such thatc+Dx+y≥0, b-x≥0, andxT(c+Dx+y)=yT(b-x)=0 whenb>0. This is the LCP (M, q) withM=(l 0D t), which is in turn equivalent to a linear variational inequality over a rectangle. This type of problem arises, for example, from quadratic programming (QP) problems with upper and lower bounds ifD is symmetric, and from multicommodity market equilibrium problems with institutional price controls imposed ifD is not necessarily symmetric.Iterative methods for the LCP (M, q) with symmetricM are well developed and studied using QP applications. For nonsymmetric cases withM being either an H-matrix with positive diagonals, or a Z-matrix, there exists a robust iterative method with guaranteed convergence.This paper extends this algorithm so that the LCP (M, q) withM=(l 0D t) which is neither symmetric, nor an H-matrix with positive diagonals, nor a Z-matrix, can be processed when onlyD notM satisfies such properties. The case whereD is nonsymmetric is explicitly discussed.

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