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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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