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# A remark on multiplier methods for nonlinear programming

- M. Cirinà
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## Abstract

This paper is concerned with certain aspects of multiplier methods where the solution of a constrained minimization problem is obtained by means of a sequence of unconstrained minimizations of an augmented Lagrangian L(x,y,r), followed each by an iteration on the La — grange multiplier vector y. In spite of the growing recognition that multiplier methods are among the most effective constrained minimization methods, the value to be given to the penalty parameter r does not seem yet to have received enough attention. This paper — related to work done recently by Bertsekas and Polyak — contains a result in such direction, namely the following one: if G and Q are given matrices and Q is positive definite on the kernel of G, then it is produced r* such that for all r > r*, Q + r G^{T}G is positive definite on the whole space. Also we prove a lemma — related to a known one — about Hilbert space operators with uniformly bounded inverses, that may be useful in extending the result above to more general situations. To test the computational value of the estimate r* arrived at here, a computer program is being tested and some numerical results are reported.

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- Title
- A remark on multiplier methods for nonlinear programming
- Book Title
- Optimization Techniques Modeling and Optimization in the Service of Man Part 2
- Book Subtitle
- Proceedings, 7th IFIP Conference Nice, September 8–12, 1975
- Pages
- pp 283-292
- Copyright
- 1976
- DOI
- 10.1007/3-540-07623-9_293
- Print ISBN
- 978-3-540-07623-0
- Online ISBN
- 978-3-540-38150-1
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- 41
- Series ISSN
- 0302-9743
- Publisher
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag
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