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DOI: 10.23952/cot.2027.13
Received January 15, 2026; Accepted March 10, 2026; Published online October 2, 2026
Abstract. In the mid-1970s, in his seminal article “Monotone Operators and the Proximal Point Algorithm,” Rockafellar developed a convergence analysis for a generalized proximal-point algorithm, which is related to maximal monotone operators. That article thus built a general framework for convergence analysis for methods of Lagrange multipliers, since they can be viewed as proximal-point algorithms. This paper is concerned with two well-known methods of Lagrange multipliers. One method for solving optimization problems with equality constraints is an augmented Lagrangian (AL) method introduced by Hestenes and Powell in the late 1960s. Another Lagrange multiplier method for optimization problems with inequality constraints is based on nonlinear rescaling of the functions that define the constraints with properly selected transformations. This approach was introduced by Polyak in the 1980s. Rescaling the constraints, however, leads to a dynamic rescaling of the corresponding monotone operators, which, in turn, leads to the need for an extension of Rockafellar’s methodology to include such a case. This manuscript reviews the relation among the two Lagrange multiplier methods, proximal-point algorithms, and monotone operators. It also outlines the basic steps of convergence analysis of the augmented Lagrangian and nonlinear rescaling methods as proximal-point methods, discussing similarities and differences between them. Thus the manuscript pays tribute to Rockafellar’s methodology that made such an analysis possible. It also provides some numerical results demonstrating the value of proximal-point methods.
How to Cite this Article:
I. Griva, Convergence analysis of two methods of Lagrange multipliers using Rockafellar’s theory on monotone operators and proximal-point algorithms, Commun. Optim. Theory 2027 (2027) 13.