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$${\mathcal V}{\mathcal U}$$-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 $$\mathcal {VU}$$-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.

##### MSC:
 90C31 Sensitivity, stability, parametric optimization 49J52 Nonsmooth analysis 65K10 Numerical optimization and variational techniques 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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