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On directional second-order tangent sets of analytic sets and applications in optimization

Le Cong Trinh

TL;DR

This work analyzes directional second-order tangent sets for analytic sets, introducing both geometric $T^2_{0,u}X$ (via analytic arcs) and algebraic $T^{2,a}_{0,u}X$ (via initial forms) and proving the general inclusion $T^2_{0,u}X \subseteq T^{2,a}_{0,u}X$ with explicit real and complex examples showing strictness. It then develops a second-jet framework with spaces $J^2(X)_{0,u}$ and $J^2(C_0X)_u$ connected by a map $\Phi_u$, establishing that equality $T^2_{0,u}X=T^{2,a}_{0,u}X$ holds under surjectivity of $\Phi_u$; surjectivity is proved for several important classes, including smooth germs, homogeneous cones, nondegenerate hypersurfaces, and nondegenerate complete intersections. The results permit algebraic verification of second-order conditions when jet-surjectivity holds, tying parabolic regularity to the exactness of second-order tangent models. As an application, the authors derive second-order necessary and sufficient optimality conditions for $C^2$ problems constrained to analytic sets, clarifying when geometric and algebraic tangent notions yield the same conclusions and highlighting the practical impact for constrained optimization on singular sets.

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.

On directional second-order tangent sets of analytic sets and applications in optimization

TL;DR

This work analyzes directional second-order tangent sets for analytic sets, introducing both geometric (via analytic arcs) and algebraic (via initial forms) and proving the general inclusion with explicit real and complex examples showing strictness. It then develops a second-jet framework with spaces and connected by a map , establishing that equality holds under surjectivity of ; surjectivity is proved for several important classes, including smooth germs, homogeneous cones, nondegenerate hypersurfaces, and nondegenerate complete intersections. The results permit algebraic verification of second-order conditions when jet-surjectivity holds, tying parabolic regularity to the exactness of second-order tangent models. As an application, the authors derive second-order necessary and sufficient optimality conditions for problems constrained to analytic sets, clarifying when geometric and algebraic tangent notions yield the same conclusions and highlighting the practical impact for constrained optimization on singular sets.

Abstract

In this paper we study directional second-order tangent sets of real and complex analytic sets. For an analytic set and a nonzero tangent direction , we compare the geometric second-order tangent set , defined via second-order expansions of analytic arcs, with the algebraic second-order tangent set , defined by initial forms of the defining equations. We prove the general inclusion 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 if and only if the natural second-jet map from analytic arcs in to jets on the tangent cone 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 optimization problems on analytic sets.
Paper Structure (15 sections, 13 theorems, 135 equations)

This paper contains 15 sections, 13 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Directional second-order tangent sets of analytic sets
  3. Geometric directional second-order tangent sets
  4. Algebraic directional second-order tangent sets of analytic sets
  5. Distinction of tangent sets
  6. Case 1: $u_{1}u_{2}>0$.
  7. Case 2: $u_{1}u_{2}<0$.
  8. Case 3: $u_{1}=0,\ u_{2}\neq 0$.
  9. Case 4: $u_{2}=0,\ u_{1}\neq 0$.
  10. Second jet spaces and directional second-order tangent sets under the surjectivity of second jet maps
  11. Second jets along a fixed direction
  12. Representation of algebraic second-order tangent sets via second jets
  13. Directional second-order tangent sets under surjectivity of second jet spaces
  14. Some classes of analytic sets whose second jet maps are surjective
  15. Applications of directional second-order tangent sets in optimization

Key Result

Proposition 2.4

Let $X, X_1, X_2$ be subsets of $\mathbb K^n$; $u \in \mathbb K^n$, and $p \in X \cap X_1 \cap X_2$. Then

Theorems & Definitions (32)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4: KTZ15
  • Remark 2.5
  • Proposition 2.6: Analytic-arc characterization of $T^2_{0,u}X$
  • proof
  • Definition 2.7
  • Remark 2.8
  • Proposition 2.9
  • ...and 22 more