On directional second-order tangent sets of analytic sets and applications in optimization
In this paper we study directional second-order tangent sets of real and complex analytic sets. For an analytic set $X\subseteq \mathbb K^n$ and a nonzero tangent direction $u\in T_0X$, we compare the geometric directional second-order tangent set $T^2_{0,u}X$, defined through second-order expansions of analytic curves in $X$, with the algebraic directional second-order tangent set $T^{2,a}_{0,u}X$, defined by the initial forms of the equations of $X$. We first prove the general inclusion $T^2_{0,u}X\subseteq T^{2,a}_{0,u}X$ and exhibit explicit real and complex analytic examples showing that this inclusion can be strict. These examples show that algebraically admissible second-order coefficients need not be geometrically realizable by analytic curves in $X$. To address this gap, we reformulate the equality $T^2_{0,u}X=T^{2,a}_{0,u}X$ as a realizability problem: the two sets coincide whenever every algebraically admissible second-order coefficient is realized by an analytic curve in $X$ with prescribed first two terms. We establish this realizability property for several important classes of analytic sets, including smooth analytic germs, homogeneous analytic cones, hypersurfaces with nondegenerate tangent directions, and nondegenerate analytic complete intersections. As an application, we derive second-order necessary and sufficient optimality conditions for $C^2$ optimization problems on closed sets. In the analytic setting, whenever the above equality holds, the geometric directional second-order tangent sets appearing in these conditions may be replaced by their algebraic counterparts, so that the second-order tests become explicitly computable from the defining equations of the feasible set.