Geometric directional second-order tangent sets coincide with their algebraic counterparts for smooth analytic germs, homogeneous cones, nondegenerate hypersurfaces, and complete intersections, allowing algebraic replacement in second-order optimality conditions.
Lojasiewicz,Introduction to complex analytic geometry, Birkh¨ auser
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On directional second-order tangent sets of analytic sets and applications in optimization
Geometric directional second-order tangent sets coincide with their algebraic counterparts for smooth analytic germs, homogeneous cones, nondegenerate hypersurfaces, and complete intersections, allowing algebraic replacement in second-order optimality conditions.