A family of improved secant methods via nonmonotone curvilinear paths technique for equality constrained optimization

This paper presents a family of improved secant algorithms via two preconditional curvilinear paths, the preconditional modified gradient path and preconditional optimal path, for solving general nonlinear optimization problems with nonlinear equality constraints. We employ the stable Bunch–Parlett factorization method of symmetric matrices so that two preconditional curvilinear paths are very easily formed. The nonmonotone curvilinear search technique, by introducing a nonsmooth merit function and adopting a dogleg-typed movement, is used to speed up the convergence progress in the contours of objective function with large curvature. Global convergence of the proposed algorithms is obtained under some reasonable conditions. Furthermore, the dogleg-typed step overcomes the Maratos effect to bring the local superlinear convergence rate. The results of numerical experiments are reported to show the effectiveness of the proposed algorithms.