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For convex smooth optimization, first and second order necessary and sufficient conditions are well known. Does such standard second order necessary and sufficient conditions exist for convex nonsmooth optimization. For first order, we have the well known condition that zero must belong to the subgradient set of the function. In that sense what is a second order analogue of Hessian being positive semidefinite in convex nonsmooth settings. What will differ in constrained and unconstrained settings.

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