dorsal/arxiv
View SchemaOn directional second-order tangent sets of analytic sets and applications in optimization
| Authors | Le Cong Trinh |
|---|---|
| Categories | |
| ArXiv ID | 2601.09991vv1 |
| URL | https://arxiv.org/abs/2601.09991 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
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 second-order tangent set $T^2_{0,u}X$, defined via second-order expansions of analytic arcs, with the algebraic second-order tangent set $T^{2,a}_{0,u}X$, defined by initial forms of the defining equations. We prove the general inclusion $T^2_{0,u}X\subseteq T^{2,a}_{0,u}X$ and construct explicit real and complex analytic examples showing that the inclusion is strict. We introduce a second-jet formulation along fixed tangent directions and show that $T^2_{0,u}X=T^{2,a}_{0,u}X$ if and only if the natural second-jet map from analytic arcs in $X$ to jets on the tangent cone $C_0X$ is surjective. This surjectivity is established for 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 analytic sets.
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"abstract": "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 second-order tangent set $T^2_{0,u}X$, defined via second-order expansions of analytic arcs, with the algebraic second-order tangent set $T^{2,a}_{0,u}X$, defined by initial forms of the defining equations. We prove the general inclusion $T^2_{0,u}X\\subseteq T^{2,a}_{0,u}X$ and construct explicit real and complex analytic examples showing that the inclusion is strict.\n We introduce a second-jet formulation along fixed tangent directions and show that $T^2_{0,u}X=T^{2,a}_{0,u}X$ if and only if the natural second-jet map from analytic arcs in $X$ to jets on the tangent cone $C_0X$ is surjective. This surjectivity is established for 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 analytic sets.",
"arxiv_id": "2601.09991",
"authors": [
"Le Cong Trinh"
],
"categories": [
"math.AG",
"math.CV",
"math.OC"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "On directional second-order tangent sets of analytic sets and applications in optimization",
"url": "https://arxiv.org/abs/2601.09991",
"version": "v1"
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