Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix.

Ibtihel Ben Gharbia (1) , Jean Charles Gilbert (1)

1 ESTIME - Parameter estimation and modeling in heterogeneous media

The plain Newton-min algorithm for solving the linear complementarity problem (LCP for short) 0≤ x \perp Mx+q ≥0 can be viewed as a nonsmooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x,Mx+q)=0 , which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm is known to converge in at most n iterations. We show in this note that this result no longer holds when M is a P-matrix of order n ≥3, since then the algorithm may cycle. P-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally, convergence occurs for a P-matrix of order 1 or 2.

Mots clés: linear complementarity problem; Newton's method; nonconvergence; nonsmooth function; P-matrix.; P-matrix

Domaines: Optimisation et contrôle [math.OC]

Ibtihel Ben Gharbia, Jean Charles Gilbert. Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix.. [Research Report] RR-7160, 2009. ⟨inria-00442293v1⟩