Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets
نویسنده
چکیده
We present a generalized Newton method and a quasiNewton method for solving H(x) := F(nc(x))+x-nc(x) = 0, when C is a polyhedral set. For both the Newton and quasi-Newton methods considered here, the subproblem to be solved is a linear system of equations per iteration. The other characteristics of the quasi-Newton method include: (i) a g-superlinear convergence theorem is established without assuming the existence of H' at a solution x* of H(x) = 0; (ii) only one approximate matrix is needed; (iii) the linear independence condition is not assumed; (iv) Q-superlinear convergence is established on the original variable x; and (v) from the QR-factorization of the kth iterative matrix, we need at most O((1 +2\Lk\ +2\Lk\)n2) arithmetic operations to get the QR-factorization of the (k+ l)th iterative matrix.
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