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$𝒱𝒰$-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 $\mathrm{𝒱𝒰}$-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 (other weak concepts of optimality) 65K10 Optimization techniques (numerical methods) 47N10 Applications of operator theory in optimization, convex analysis, programming, economics