Abstract.
In this paper we present a nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n×n symmetric matrices. We show how a simple bundle method, the approximate eigenvalue method can be used to globalize the second-order method developed by M.L. Overton in the eighties and recently revisited in the framework of the ?-Lagrangian theory. With no additional assumption, the resulting algorithm generates a minimizing sequence. A geometrical and constructive proof is given. To prove that quadratic convergence is achieved asymptotically, some strict complementarity and non-degeneracy assumptions are needed. We also introduce new variants of bundle methods for semidefinite programming.
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Received: February 9, 1998 / Accepted: May 2, 2000¶Published online September 20, 2000
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Oustry, F. A second-order bundle method to minimize the maximum eigenvalue function. Math. Program. 89, 1–33 (2000). https://doi.org/10.1007/PL00011388
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DOI: https://doi.org/10.1007/PL00011388