{"paper":{"title":"Smoothness of the Augmented Lagrangian Dual in Convex Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SY","eess.SY"],"primary_cat":"math.OC","authors_text":"Jingwang Li, Vincent Lau","submitted_at":"2025-05-03T14:04:04Z","abstract_excerpt":"This paper focuses on the general linearly constrained optimization problem: $\\min_{x \\in \\mathbb{R}^d} f(x) \\ \\text{s.t.} \\ Ax = b$, where $f: \\mathbb{R}^d \\rightarrow \\mathbb{R} \\cup \\{+\\infty\\}$ is a closed proper convex function, $A \\in \\mathbb{R}^{p \\times d}$, and $b \\in \\mathbb{R}^p$. We define the standard dual function $\\phi(\\lambda) = \\inf_x \\{f(x) + \\langle \\lambda, A x - b \\rangle\\}$, the augmented Lagrangian $\\mathcal{L}_{\\rho}(x, \\lambda) = f(x) + \\langle \\lambda, Ax - b \\rangle + \\frac{\\rho}{2}\\|Ax - b\\|^2$ ($\\rho > 0$), and the augmented Lagrangian dual function $\\phi_{\\rho}(\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01824","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.01824/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}