An Algorithmic Characterization of P-matricity II: Adjustments, Refinements, and Validation

The paper "An algorithmic characterization of P-matricity " (SIAM Journal on Matrix Analysis and Applications, 34:3 (2013) 904–916, by the same authors as here) implicitly assumes that the iterates generated by the Newton-min algorithm for solving a linear complementarity problem of dimension n, which reads 0 ⩽ x ⊥ (M x + q) ⩾ 0, are uniquely determined by some index subsets of [[1, n]]. Even if this is satisfied for a subset of vectors q that is dense in R^n , this assumption is improper, in particular in the statements where the vector q is not subject to restrictions. The goal of the present contribution is to show that, despite this blunder, the main result of that paper is preserved. This one claims that a nondegenerate matrix M is a P-matrix if and only if the Newton-min algorithm does not cycle between two distinct points, whatever is q. The proof is hardly more complex, requiring only some additional refinements.

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