Optimality Conditions And Duality Theory for Minimizing Sums of the Largest Eige

Cover Optimality Conditions And Duality Theory for Minimizing Sums of the Largest Eige
Optimality Conditions And Duality Theory for Minimizing Sums of the Largest Eige
Michael L Overton
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The dual matrix U provides the information which leads to the generation of a descent direction, just as negative Lagrange multipliers provide similar information in constrained optimization. The distinction between the cases ^ 1 is as follows: if ^ 1 one eigenvalue is separated from the others by an increase, again reducing the approximate mul- tiplicity but increasing the number of larger eigenvalues (to first order). In either case the theorem guarantees an overall reduction in /„.
20 Case
... 2 applies generically ii m + 1 = t{t + l)/2, i. E. The generic dimension of the manifold defined by (3. 53) consists of a single point. It also applies if x minimizes /k on this manifold; see Overton (1988) for a further explanation.
Case 3. Neither of Cases 1 and 2 apply.


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