Journal of Optimization Theory and Applications
, Volume 12, Issue 6, pp 555562
The multiplier method of Hestenes and Powell applied to convex programming
 R. T. RockafellarAffiliated withDepartment of Mathematics, University of Washington
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For nonlinear programming problems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagrangian, and a certain simple rule is used for updating the Lagrange multipliers after each cycle. Powell has essentially shown that the rate of convergence is linear if one starts with a sufficiently high penalty factor and sufficiently near to a local solution satisfying the usual secondorder sufficient conditions for optimality. This paper furnishes the corresponding method for inequalityconstrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement that an exact minimum be calculated at each cycle. Furthermore, the Lagrange multipliers are shown to converge, even though the optimal multipliers may not be unique.
 Title
 The multiplier method of Hestenes and Powell applied to convex programming
 Journal

Journal of Optimization Theory and Applications
Volume 12, Issue 6 , pp 555562
 Cover Date
 197312
 DOI
 10.1007/BF00934777
 Print ISSN
 00223239
 Online ISSN
 15732878
 Publisher
 Kluwer Academic PublishersPlenum Publishers
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 Authors

 R. T. Rockafellar ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Washington, Seattle, Washington