Lagrange Multipliers and Optimality

1993/06/01 by R. T. Rockafellar, R. Tyrrell Rockafellar · 586 citations
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Applied mathematics #Constraint algorithm #Geometry #Lagrange multiplier #Mathematical optimization #Mathematics #Matrix Theory and Algorithms #Minification #Multiplier (economics) #Optimization and Variational Analysis #Tangent

paper · doi:10.1137/1035044

published in SIAM Review 35(2), 183-238 (Society for Industrial and Applied Mathematics)

openalex publication_date 1993/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of one-sided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the game-theoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black-and-white constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a free-standing exposition of basic nonsmooth analysis as motivated by and applied to this subject.

Citations

• Multiplier and gradient methods
• Zur Theorie der Gesellschaftsspiele

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