Augmented Lagrangian methods for infeasible convex optimization problems and diverging proximal-point algorithms
It addresses convergence issues in ALMs for infeasible convex optimization, which is incremental as it builds on classical relationships with proximal-point algorithms.
This work investigates the convergence of augmented Lagrangian methods (ALMs) for convex optimization problems that may be infeasible, showing that iterates converge to solutions of the 'closest feasible problem' under mild assumptions, with results ranging from basic convergence to precise rates.
This work investigates the convergence behavior of augmented Lagrangian methods (ALMs) when applied to convex optimization problems that may be infeasible. ALMs are a popular class of algorithms for solving constrained optimization problems. We demonstrate that, under mild assumptions, the sequences of iterates generated by ALMs converge to solutions of the ``closest feasible problem''. We establish progressively stronger convergence results, ranging from basic sequence convergence to more precise convergence rates, under a hierarchy of assumptions. This study leverages the classical relationship between ALMs and the proximal-point algorithm applied to the dual problem. A key technical contribution is a set of concise results on the behavior of the proximal-point algorithm when applied to functions that may lack minimizers. These results pertain to its convergence in terms of its subgradients and of the values of the convex conjugate.