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${\cal V}{\cal U}$-smoothness and proximal point results for some nonconvex functions. (English) Zbl 1097.90059
Summary: This article is concerned with a function $f$ having a primal-dual gradient structure at a point $\overline{x}$ which satisfies a property called strong transversality. The structure is related to $\cal {VU}$-space decomposition, depending on the subdifferential of $f$ at $\overline{x}$. It is shown that there exists a $C^2$ primal track leading to $\overline{x}$ and a space decomposition mapping that is $C^1$. As a result, there exists a second-order expansion of f on the primal track, an associated subdifferential that is $C^1$ in a certain sense, and a corresponding dual track. For $\overline{x}$ a minimizer, conditions on $f$ are given to ensure that for any point near $\overline{x}$ its corresponding proximal point is on the primal track.

90C31Sensitivity, stability, parametric optimization
49J52Nonsmooth analysis (other weak concepts of optimality)
65K10Optimization techniques (numerical methods)
47N10Applications of operator theory in optimization, convex analysis, programming, economics
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