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Primal-dual gradient structured functions: second-order results; links to epi-derivatives and partly smooth functions. (English) Zbl 1036.90067
The paper studies second order expansions for the recently introduced class of nonsmooth functions with primal-dual gradient structure. For this class of lower semi-continuous and not necessarily convex functions it is possible to explicitly give a basis for the subspace 𝒱 parallel to the Clarke subdifferential at some point. Relative to its orthogonal subspace 𝒰=𝒱 the function appears to be smooth, and it is actually possible to find smooth trajectories tangent to 𝒰 along which the function is C 2 . Along with this smooth restriction a smooth multiplier function can be defined. Having these two smooth objects at hand, a C 2 Lagrangian is defined which leads to a second order expansion of the nonsmooth function along the subspace 𝒰. Explicit expressions for the first and second order derivatives are given. Connections between second order epi-derivatives and 𝒰-Hessians are made, and expressions for manifold restricted Hessians are given for partly smooth functions. A number of illuminating examples accompany the results of the article.
MSC:
90C31Sensitivity, stability, parametric optimization
49J52Nonsmooth analysis (other weak concepts of optimality)
90C46Optimality conditions, duality