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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 x ¯ which satisfies a property called strong transversality. The structure is related to 𝒱𝒰-space decomposition, depending on the subdifferential of f at x ¯. It is shown that there exists a C 2 primal track leading to 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 x ¯ a minimizer, conditions on f are given to ensure that for any point near x ¯ its corresponding proximal point is on the primal track.
MSC:
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