The authors consider the problem of minimizing the function \(g\colon B_1\to\mathbf R\) over the set defined by the constraint \(-G(x)\in K\), where \(G\colon B_1\to B_2\), \(B_1\) and \(B_2\) are reflexive Banach spaces and \(K\) is a nonempty closed convex cone in \(B_2\). The methods the authors propose continue an idea originating in R. T. Rockafellar’s augmented Lagrangian method [SIAM J. Control 12, 268–285 (1974; Zbl 0257.90046)] which consists of applying a proximal point procedure to a dual form of the problem in order to generate sequences whose pairs via the Karush-Kuhn-Tucker conditions approximate solutions of the original problem. The authors further develop this idea in a direction initiated by D. Butnariu and A. N. Iusem [Totally convex functions for fixed points computation and infinite dimensional optimization. Applied Optimization. 40. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0960.90092)] and expanded by A. Iusem and R. Gárciga Otero [Numer. Funct. Anal. Optimization 22, No. 5–6, 609–640 (2001; Zbl 1018.90067)]. The new convergence results the authors prove are made possible by the so-called “inexact proximal point procedures” described in the paper under review.