Abstract:In this work we provide theoretical and practical results of the Safeguarded Augmented Lagrangian Method (SALM) for constrained composite optimization problems whose objective is the sum of a smooth and a nonsmooth function. We obtain global convergence results to an M-stationary point for SALM under the Polyak-Lojasiewicz constraint qualification (PLCQ). For this result the boundedness of the Lagrange multipliers is crucial, and this is shown under the assumption that the nonsmooth part of the objective function is locally Lipschitz continuous. A counterexample shows that one cannot expect to get bounded multipliers without such an assumption. The performance of the algorithm is evaluated numerically on a set of sparse portfolio optimization problems with two different regularization terms, one being Lipschitz and the other one being non-Lipschitz. The results are significantly better for the Lipschitz sparsity term, whereas the underlying method generates seemingly unbounded multipliers in many instances when using the non-Lipschitz sparsity function.
From: Christian Kanzow [view email]
[v1]
Wed, 24 Jun 2026 08:44:22 UTC (42 KB)
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