In this paper we develop a unified theory for establishing the local and q-superlinear convergence of the secant methods from the convex class that 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 from the convex class for the 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.