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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 ${\cal V}$ parallel to the Clarke subdifferential at some point. Relative to its orthogonal subspace ${\cal U}={\cal V}^\perp$ the function appears to be smooth, and it is actually possible to find smooth trajectories tangent to ${\cal U}$ 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 ${\cal U}$. Explicit expressions for the first and second order derivatives are given. Connections between second order epi-derivatives and ${\cal U}$-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.

90C31Sensitivity, stability, parametric optimization
49J52Nonsmooth analysis (other weak concepts of optimality)
90C46Optimality conditions, duality
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