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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 $${\mathcal V}$$ parallel to the Clarke subdifferential at some point. Relative to its orthogonal subspace $${\mathcal U}={\mathcal V}^\perp$$ the function appears to be smooth, and it is actually possible to find smooth trajectories tangent to $${\mathcal U}$$ 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 $${\mathcal U}$$. Explicit expressions for the first and second order derivatives are given. Connections between second order epi-derivatives and $${\mathcal U}$$-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:
 90C31 Sensitivity, stability, parametric optimization 49J52 Nonsmooth analysis 90C46 Optimality conditions and duality in mathematical programming
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