A remark on multiplier methods for nonlinear programming

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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 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.