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Optim."],"published-print":{"date-parts":[[2026,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Penalty methods are a well-known class of algorithms for constrained optimization. They transform a constrained problem into a sequence of unconstrained penalized problems in the hope that approximate solutions of the latter converge to a solution of the former. If Lagrange multipliers exist, exact penalty methods ensure that the penalty parameter only need increase a finite number of times, but they are typically scorned in smooth optimization because the penalized problems are not smooth. This led researchers to consider the implementation of exact penalty methods inconvenient. Recent advances in proximal methods have led to increasingly efficient solvers for nonsmooth optimization. We study a general exact penalty algorithm and use it to show that the exact [Formula: see text]-penalty method for equality-constrained optimization can, in fact, be implemented efficiently by solving the penalized problem using a proximal-type algorithm. We study the convergence of our algorithm and establish a worst-case complexity bound of [Formula: see text] to bring a stationarity measure below [Formula: see text] under the Mangasarian\u2013Fromovitz constraint qualification and Lipschitz continuity of the objective gradient and constraint Jacobian. While the Lipschitz continuity of the objective gradient is not required for convergence in view of recent works, it is used in our analysis to derive the complexity bound. In a degenerate scenario where the penalty parameter grows unbounded, the complexity becomes [Formula: see text], which is worse than another bound found in the literature. We justify the difference by arguing that our feasibility measure is properly scaled. Finally, we report numerical experience on small-scale problems from a standard collection and compare our solver with an augmented-Lagrangian and an SQP method. Our preliminary implementation is superior to the augmented Lagrangian in terms of robustness and efficiency and is competitive with the SQP method in terms of robustness, though the latter retains a slight edge in terms of number of problem function evaluations.<\/jats:p>","DOI":"10.1137\/24m1705974","type":"journal-article","created":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T07:36:09Z","timestamp":1775633769000},"page":"626-650","source":"Crossref","is-referenced-by-count":0,"title":["Nonsmooth Exact Penalty Methods for Equality-Constrained Optimization: Complexity and Implementation"],"prefix":"10.1137","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6609-7330","authenticated-orcid":true,"given":"Youssef","family":"Diouane","sequence":"first","affiliation":[{"name":"GERAD and Department of Mathematics and Industrial Engineering, Polytechnique Montr\u00e9al, Montr\u00e9al H3C 3A7, QC, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-3158-7912","authenticated-orcid":true,"given":"Maxence","family":"Gollier","sequence":"additional","affiliation":[{"name":"GERAD and Department of Mathematics and Industrial Engineering, Polytechnique Montr\u00e9al, Montr\u00e9al H3C 3A7, QC, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8017-7687","authenticated-orcid":true,"given":"Dominique","family":"Orban","sequence":"additional","affiliation":[{"name":"GERAD and Department of Mathematics and Industrial Engineering, Polytechnique Montr\u00e9al, Montr\u00e9al H3C 3A7, QC, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,4,8]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/060654797"},{"key":"ref2","volume-title":"Cahier du GERAD G-2021-12-SM","author":"Aravkin A.","year":"2024"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/21M1409536"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s11081-015-9287-9"},{"key":"ref5","unstructured":"R. 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