Superlinearly convergent approximate Newton methods for LC1 optimization problems

In the literature, the proof of superlinear convergence of approximate Newton or SQP methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions. Sometimes, the second-order derivatives of those functions are required to be Lipschitzian. In this paper, we present approximate Newton or SQP methods for solving nonlinear programming problems whose objective and constraint functions have locally Lipschitzian derivatives, and establish Q-superlinear convergence of these methods under the assumption that these derivatives are semismooth. This assumption is weaker than the second-order differentiability. The extended linear-quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives.