[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86545-en":3,"doc-seo-86545-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86545,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","An augmented Lagrangian algorithm for constrained nonlinear least squares","An augmented Lagrangian algorithm addresses nonlinear least-squares problems with mixed nonlinear and linear constraints. Nonlinear constraints are incorporated by reformulating the objective into an augmented Lagrangian function, while linear constraints are treated directly. Each iteration reduces to solving a linearly constrained subproblem using a gradient projection technique, together with a structured approximation of the augmented Lagrangian Hessian. The method guarantees global convergence and is validated through numerical experiments.","An augmented Lagrangian algorithm for constrained nonlinear least-squares  \narXiv :2607 . 11239v1 [math .OC] 13 Jul 2026  \nPierre Borie ∗ Fabian Bastin ∗  \n[pierre. borie@umontreal. ca](pierre. borie@umontreal. ca) [bastin@iro. umontreal. ca](bastin@iro. umontreal. ca)  \nSt´ephane Dellacherie †‡  \n[dellacherie. stephane@hydroquebec. com](dellacherie. stephane@hydroquebec. com)  \nJuly 14, 2026  \nAbstract  \nWe present an algorithm for solving nonlinear least-squares problems subject to a mix of nonlinear and linear constraints. The nonlinear constraints are handled by reformulating the objective as the augmented Lagrangian function while linear constraints are handled directly. Each iteration consists of approximately solving a linearly constrained problem by means of a gradient projection technique. Our approach also involves a structured approximation of the augmented Lagrangian Hessian. We show global convergence of the method and assess the performance through numerical experiments.  \nKeywords: constrained nonlinear least-squares, augmented Lagrangian method, structured quasi-Newton update.  \n1 Introduction  \nWe consider the constrained nonlinear least-squares (NLS) problem  \nmin  1 ∥r(x)∥2  \nx∈Rn 2  \n(1.1)  \ns.t. c (x) = 0 x ∈ Ω ,  \nwhere ∥ · ∥ denotes the ℓ2 norm. The residuals r : Rn → Rnr and constraints c: Rn → Rnc  \nfunctions are assumed to be twice continuously differentiable with respective Jacobians J and C. We may also write the least-squares objective as a general function f with gradient  ∇f (x) = J (x)⊤ r (x) .  \n∗ Universit´e de Montr´eal, Department of Computer Science and Operations Research, Montreal, QC, Canada †Hydro-Qu´ebec, Montreal, QC, Canada  \n‡Universit´e du Qu´ebec `a Montr´eal, Montreal, QC, Canada  \nThe set Ω consists of linear constraints and is defined as  \nΩ = {x ∈ Rn | Ax = b, l ≤ x ≤ u} , (1.2)  \nwith A ∈ Rm ×n , m \u003C n, b ∈ Rm , and l, u ∈ Rn. Without loss of generality, some components of the latter two vectors can be set to ±∞ for unbounded parameters. For convenience, we also assume that the linear equalities do not involve single-variable constraints. Formulation (1.1) is also suitable for problems with nonlinear inequality constraints of the form g (x) ≥ 0. The latter are transformed into equality constraints by adding non-negative slack variables, which gives the new constraints  \ng (x) − ν = 0, ν ≥ 0.  \nThe lower and upper bounds associated with these slack variables are thus 0 and ∞ respectively. Linear inequality constraints can be converted into equalities similarly.  \nThe Lagrangian for problem (1.1), with respect to the nonlinear constraints, is defined by  \nL (x,λ) = f(x) + ⟨λ, c(x)⟩ ,  \nwhere λ ∈ Rnc is the vector of Lagrange multipliers and ⟨· , ·⟩ is the standard inner product. An algorithm for solving problem (1.1) aims to find a first-order critical point (x ∗ ,λ∗ ) that  \nsatisfies the necessary condition  \nx∗ ∈ Ω and c (x∗ ) = 0  \nPΩ [x∗ − ∇xL (x∗ ,λ∗ )] = x∗ ,  \nwhere PΩ [·] is the projector operator onto Ω, defined, for any vector x, as  \nPΩ [x] = argmin ∥v − x∥2 .  \nv∈Ω  \nThe NLS structure arises in data fitting, parameter estimation, and scientific computing, and its efficient resolution has attracted a sustained research effort [13, 25 , 48] . A distinctive feature of NLS problems is that the Hessian of the objective decomposes into a first-order part J (x)⊤ J (x), where J (x) is the Jacobian of r, and a second-order part involving the individual Hessians of the residuals weighted by their values. Exploiting this decomposition is the central challenge in designing efficient algorithms. The Gauss–Newton (GN) method retains only the first-order term, which is effective when residuals are small at the solution but can lead to poor convergence otherwise. The Levenberg–Marquardt (LM) method, initiated by Levenberg [38] and Marquardt [44] and further studied, for instance, in [7, 9], regularizes the GN matrix by a scalar multiple of the identity, providing robustness at the","cbCaikbR4qZVHXiy","https://ap.wps.com/l/cbCaikbR4qZVHXiy","pdf",711122,3,1,37,"English","en",105,"# Introduction\n## Constrained nonlinear least-squares formulation\n## Lagrangian and optimality conditions\n## Related structured nonlinear least-squares methods\n## Constrained extensions and SQP context","[{\"question\":\"How are nonlinear constraints and linear constraints handled in the proposed method?\",\"answer\":\"Nonlinear constraints are handled by reformulating the objective into an augmented Lagrangian function, while linear constraints are handled directly in the iteration subproblems.\"},{\"question\":\"What does each iteration approximately solve?\",\"answer\":\"Each iteration approximately solves a linearly constrained problem using a gradient projection technique, paired with a structured approximation of the augmented Lagrangian Hessian.\"},{\"question\":\"What convergence and evaluation results are reported?\",\"answer\":\"The document states global convergence of the method and assesses performance through numerical experiments.\"}]",1784212539,93,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"an-augmented-lagrangian-algorithm-for-constrained-nonlinear-least-squares","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/an-augmented-lagrangian-algorithm-for-constrained-nonlinear-least-squares/86545/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How are nonlinear constraints and linear constraints handled in the proposed method?","Question",{"text":75,"@type":76},"Nonlinear constraints are handled by reformulating the objective into an augmented Lagrangian function, while linear constraints are handled directly in the iteration subproblems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does each iteration approximately solve?",{"text":80,"@type":76},"Each iteration approximately solves a linearly constrained problem using a gradient projection technique, paired with a structured approximation of the augmented Lagrangian Hessian.",{"name":82,"@type":73,"acceptedAnswer":83},"What convergence and evaluation results are reported?",{"text":84,"@type":76},"The document states global convergence of the method and assesses performance through numerical 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