By Glashoff K., Gustafson S.A.

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2 Basic Newton-type Methods Let a function G : Ω ⊆ IRn → IRn be given. 1) when G is nonsmooth is motivated by the classical Newton algorithm for a continuously diﬀerentiable G. The latter algorithm is the prototype of many local, fast algorithms for solving smooth equations. Such algorithms have excellent convergence rates in a neighborhood of a zero of G, but may fail to converge if the starting point is far from the desired zero. The key idea in a general Newton-type method is to replace the function G by an approximation depending on the current iterate, resulting in an approximated problem that can be solved more easily.

Therefore F is a global homeomorphism from IR2 onto itself. Obviously all the matrices B i (Ai )−1 belong to ∂F (0). Since B 1 ( A1 )−1 = I2 and B 4 ( A4 )−1 = −I2 , we deduce that ∂F (0) contains the zero matrix. Consequently ∂F (0) is not nonsingular. 2 for necessary and suﬃcient conditions for a locally Lipschitz continuous function to be a locally Lipschitz homeomorphism. 18. 2 Basic Newton-type Methods Let a function G : Ω ⊆ IRn → IRn be given. 1) when G is nonsmooth is motivated by the classical Newton algorithm for a continuously diﬀerentiable G.

We often refer to the Clarke generalized Jacobian simply as the generalized Jacobian of G. When m = 1 there is a (traditional) notational problem, in that the notion of the (generalized) gradient is not consistent with that of the (generalized) Jacobian because of a transposition operation. Hopefully this won’t cause any confusion. We can illustrate these deﬁnitions with the simple function |x|. This function is globally Lipschitz continuous with a Lipschitz constant L = 1, and it is continuously diﬀerentiable everywhere except at the origin.