The paper contains an excellent overview about convergence theory for constrained optimization methods and gives a unifying framework of convergence analysis. The investigations are based on the diagonalized multiplier method (DMM) defined by the third author [J. Optim. Theory Appl. 22, 135-195 (1977; Zbl 0336.65034)]. The analysis can also be applied to other classes of methods, for instance to sequential quadratic programming methods. After some theoretical preliminaries in section 2 the third section gives the unified convergence theory of DMM. In the next section these results are applied to special multiplier updates. In this way previous convergence results can be improved. The last section is devoted to conditions which ensure the q-superlinear convergence. The obtained results are strongly related to well known characterizations of \textit{P. T. Boggs}, \textit{J. W. Tolle} and \textit{P. Wang} [SIAM J. Control Optimization 20, 161-171 (1982; Zbl 0494.65036)] but the applied assumptions are less restrictive.