A remark on multiplier methods for nonlinear programming
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- Cirinà M. (1976) A remark on multiplier methods for nonlinear programming. In: Cea J. (eds) Optimization Techniques Modeling and Optimization in the Service of Man Part 2. Optimization Techniques 1975. Lecture Notes in Computer Science, vol 41. Springer, Berlin, Heidelberg
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 GTG 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.