Local convergence of secant methods for nonlinear constrained optimization

Abstract

In this paper a new class of algorithms is proposed for solving nonlinear equality constrained problems. The Hessian of the Lagrangian is approximated using the DFP or the BFGS secant updates. When the Hessian is only positive definite in a subspace of $R^n $ one shows that the algorithms generate a sequence $\{ x_k \} $ converging 2-step q-superlinearly. Furthermore, if one extra evaluation of the constraints is carried out at each iteration the convergence is q-superlinear. The algorithms, however, require one extra gradient evaluation over the standard Successive Quadratic Programming algorithm.