Editing Lagrange multiplier (section)

==Applications==

===Control theory===
In [[optimal control]] theory, the Lagrange multipliers are interpreted as [[costate]] variables, and Lagrange multipliers are reformulated as the minimization of the [[Hamiltonian (control theory)|Hamiltonian]], in [[Pontryagin's maximum principle]].

===Nonlinear programming===
The Lagrange multiplier method has several generalizations. In [[nonlinear programming]] there are several multiplier rules, e.g. the Carathéodory–John Multiplier Rule and the Convex Multiplier Rule, for inequality constraints.<ref>{{Cite journal |last=Pourciau |first=Bruce H. |date=1980 |title=Modern multiplier rules |journal=[[American Mathematical Monthly]] |volume=87 |issue=6 |pages=433–452 |doi=10.2307/2320250 |jstor=2320250 |url=http://www.maa.org/programs/maa-awards/writing-awards/modern-multiplier-rules}}</ref>

===Economics===

In many models in [[mathematical economics]] such as [[general equilibrium model]]s, consumer behavior is implemented as [[utility maximization problem|utility maximization]] and firm behavior as [[profit maximization]], both entities being subject to constraints such as [[budget constraint]]s and [[production function|production constraints]]. The usual way to determine an optimal solution is achieved by maximizing some function, where the constraints are enforced using Lagrangian multipliers.<ref>{{cite book |first1=M. I. |last1=Kamien |author-link=Morton Kamien |first2=N. L. |last2=Schwartz |author-link2=Nancy Schwartz |title=Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management |location=New York |publisher=Elsevier |edition=Second |year=1991 |isbn=0-444-01609-0 |url=https://books.google.com/books?id=0IoGUn8wjDQC }}</ref><ref>{{cite journal |last1=Glötzl |first1=Erhard |last2=Glötzl |first2=Florentin |last3=Richters |first3=Oliver |title=From constrained optimization to constrained dynamics: extending analogies between economics and mechanics |journal=Journal of Economic Interaction and Coordination |volume=14 |pages=623–642 |date=2019 |issue=3 |doi=10.1007/s11403-019-00252-7|hdl=10419/171974 |hdl-access=free }}</ref><ref>{{cite journal |first1=John V. |last1=Baxley |first2=John C. |last2=Moorhouse |title=Lagrange Multiplier Problems in Economics |journal=The American Mathematical Monthly |volume=91 |issue = 7 |date=1984 |pages=404–412|doi=10.1080/00029890.1984.11971446 }}.</ref><ref>{{cite journal |first1=Jitka  |last1=Janová |date=2011 |title=Applications of a constrained mechanics methodology in economics |journal=European Journal of Physics |volume=32 |issue=6 |pages=1443–1463 |doi=10.1088/0143-0807/32/6/001|arxiv=1106.3455 |bibcode=2011EJPh...32.1443J }}</ref>

===Power systems===
Methods based on Lagrange multipliers have applications in [[power systems]], e.g. in distributed-energy-resources (DER) placement and load shedding.<ref>
{{cite conference
 | last1 = Gautam | first1 = Mukesh
 | last2 = Bhusal | first2 = Narayan
 | last3 = Benidris | first3 = Mohammed
 | year = 2020
 | title = A sensitivity-based approach to adaptive under-frequency load shedding
 | conference = 2020 IEEE Texas Power and Energy Conference (TPEC)
 | publisher = [[Institute of Electronic and Electrical Engineers]]
 | pages=1–5
 | doi=10.1109/TPEC48276.2020.9042569
}}
</ref>

===Safe Reinforcement Learning===
The method of Lagrange multipliers applies to [[constrained Markov decision processes]].<ref>
{{cite book
 | last1 = Altman | first1 = Eitan
 | year = 2021
 | title = Constrained Markov Decision Processes
 | publisher = [[Routledge]]
}}
</ref> 
It naturally produces gradient-based primal-dual algorithms in safe reinforcement learning.<ref>
{{cite conference
 | last1 = Ding | first1 =Dongsheng
 | last2 = Zhang | first2 = Kaiqing
 | last3 = Jovanovic | first3 = Mihailo
 | last4 = Basar | first4 = Tamer
 | year = 2020
 | title = Natural policy gradient primal-dual method for constrained Markov decision processes
 | conference = Advances in Neural Information Processing Systems
}}
</ref>

===[[Normalized solution (mathematics)|Normalized solutions]]===
Considering the PDE problems with constraints, i.e., the study of the properties of the normalized solutions, Lagrange multipliers play an important role.