In this chapter, we present local analysis of Newton-type algorithms for variational problems, starting with the fundamental Josephy–Newton method for generalized equations. This method is an important extension of classical Newtonian techniques to more general variational problems. For example, as a specific application, the Josephy–Newton method provides a convenient tool for analyzing the sequential quadratic programming algorithm for optimization. We also discuss semismooth Newton methods for complementarity problems, and active-set methods with identifications based on error bounds.
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