Journal
SIAM JOURNAL ON OPTIMIZATION
Volume 13, Issue 3, Pages 702-725Publisher
SIAM PUBLICATIONS
DOI: 10.1137/S1052623401387623
Keywords
active set; nonsmooth analysis; subdifferential; generalized gradient; sensitivity; U-Lagrangian; eigenvalue optimization; spectral abscissa; identifiable surface
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Nonsmoothness pervades optimization, but the way it typically arises is highly structured. Nonsmooth behavior of an objective function is usually associated, locally, with an active manifold : on this manifold the function is smooth, whereas in normal directions it is vee-shaped. Active set ideas in optimization depend heavily on this structure. Important examples of such functions include the point wise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix. Among possible foundations for practical nonsmooth optimization, this broad class of partly smooth functions seems a promising candidate, enjoying a powerful calculus and sensitivity theory. In particular, we show under a natural regularity condition that critical points of partly smooth functions are stable: small perturbations to the function cause small movements of the critical point on the active manifold.
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