Linearized tangential cone

A linearized tangential cone is a term from nonlinear optimization . It represents a simplification of a tangential cone and is mostly used to derive optimality criteria or regularity conditions like the Abadie CQ . The linearized tangential cone is always a superset of the tangential cone.

definition

Given a non-empty set , which is described by the inequalities and the equations . Then for a point is called the amount ${\displaystyle X\subset \mathbb {R} ^{n}}$ ${\displaystyle k}$ ${\displaystyle g_{i}(x)\leq 0}$ ${\displaystyle l}$ ${\displaystyle h_{j}(x)=0}$ ${\displaystyle x\in X}$

${\displaystyle {\mathcal {T}}_{\text{lin}}(x)=\{d\in \mathbb {R} ^{n}\,|\,\nabla h_{j}(x)^{T}d=0,\nabla g_{i}(x)^{T}d\leq 0{\text{ falls }}g_{i}(x)=0\}}$

the linearized tangential cone at the point . ${\displaystyle x}$

example

If one considers the implicit function the unit circle as an example , then is ${\displaystyle h(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}-1=0}$

${\displaystyle \nabla h(x)^{T}=(2x_{1},2x_{2})}$

So at the point is the linearized tangential cone ${\displaystyle (1,0)}$

${\displaystyle {\mathcal {T}}_{\text{lin}}(1,0)=\{0\}\times \mathbb {R} }$.

If the function had been defined as an inequality and not an equation, it would be

${\displaystyle {\mathcal {T}}_{\text{lin}}(1,0)=(-\infty ,0]\times \mathbb {R} }$.

literature

• C. Geiger, C. Kanzow: Theory and numerics of restricted optimization tasks . Springer, 2002. ISBN 3-540-42790-2 . http://books.google.de/books?id=spmzFyso_b8C