In this thesis we develop a unified theory for establishing the local and q-superlinear convergence of secant methods which use updates from Broyden's convex class and have been modified to take advantage of the structure present in the Hessian in constructing approximate Hessians. As an application of this theory, we show the local and q-superlinear convergence of any structured secant method which use updates from the convex class for the equality-constrained optimization problem and the nonlinear least-squares problem. Particular cases of these methods are the SQP augmented scale BFGS and DFP secant methods for constrained optimization problems introduced by Tapia. Another particular case, for which local and q-superlinear convergence is proved for the first time here, is the Al-Baali and Fletcher modification of the structured BFGS secant method considered by Dennis, Gay and Welsch for the nonlinear least-squares problem and implemented in the current version of the NL2SOL code.