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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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