Active sets, nonsmoothness, and sensitivity. (English) Zbl 1055.90072
The author introduces the notions of partly smooth functions and partly smooth sets together with illustrative examples including the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix. He gives some properties of such functions and sets in terms of subderivatives and subgradients due to R. T. Rockafellar and R. J.-B. Wets [Variational analysis (Grundlehren der Mathematischen Wissenschaften 317, Springer, Berlin) (1998; Zbl 0888.49001)]. He shows invariant properties of the class of partly smooth functions by proving a variety of calculus rules. He also shows under a suitable regularity condition that critical points of partly smooth functions are stable on active manifolds, and develops the connection of partly smoothness with the \({\mathcal U}\)-Lagragian theory for convex functions.
Reviewer: Do Van Luu (Hanoi)
MSC:
90C31 | Sensitivity, stability, parametric optimization |
49K40 | Sensitivity, stability, well-posedness |
49J52 | Nonsmooth analysis |