Abadie Constraint Qualification

The Abadie Constraint Qualification (or Abadie CQ ) is an important prerequisite for the necessary optimality criteria to apply in non-linear optimization . The Abadie CQ is a condition on the regularity of a permissible point. If the Abadie CQ is fulfilled at one point and this point is a local minimum , then the Karush-Kuhn-Tucker conditions are also fulfilled at this point. ${\ displaystyle {\ tilde {x}}}$

It is named after the French mathematician Jean Abadie .

definition

There is an optimization problem in the form

{\ displaystyle \ min _ {x \ in X} f (x)}

,

in which

{\ displaystyle X = \ {x \ in \ mathbb {R} ^ {n} \, | \, g (x) \ leq 0, h (x) = 0 \}}

and all functions should be continuously differentiable . Then a feasible point of the restricted optimization problem fulfills the Abadie CQ if the tangential cone at the point coincides with the linearized tangential cone at the point . ${\ displaystyle x \ in X}$ ${\ displaystyle x}$ ${\ displaystyle x}$

example

As an example, consider the functions . The inequalities describe a set of restrictions and are all continuously differentiable. We now investigate whether the Abadie-CQ is fulfilled. It is then ${\ displaystyle g_ {1} (x_ {1}, x_ {2}) = - x_ {2}, \, g_ {2} (x_ {1}, x_ {2}) = x_ {1} ^ {3 } + x_ {2}}$ ${\ displaystyle g_ {i} (x) \ leq 0}$ ${\ displaystyle (0,0)}$

{\ displaystyle \ nabla g_ {1} (x) = (0, -1) ^ {T}, \, \ nabla g_ {2} (x) = (3x_ {1} ^ {2}, 1) ^ { T}}

.

In point both inequalities are active. According to the definition, the second component of the linearized tangential cone must then always be 0. The first component is arbitrary, since it disappears for both gradients at the examined point. So is . ${\ displaystyle (0,0)}$ ${\ displaystyle {\ mathcal {T}} _ {\ text {lin}} (0,0) = \ mathbb {R} \ times \ {0 \}}$

The tangential cone is only the ray and thus a real subset of the linearized tangential cone. Thus the Abadie QC is not fulfilled. ${\ displaystyle (- \ infty, 0] \ times \ {0 \}}$

Comparison with other constraint qualifications

In comparison with the other constraint qualifications, the Abadie CQ is very general, but difficult to use in practice due to the tangential cone. Therefore, one usually uses a different constraint qualification such as the MFCQ or the LICQ . If these are given, the Abadie CQ also applies. However, the reverse is not true. A constraint qualification that is weaker than the Abadie CQ is the Guinard CQ. The implications apply

{\ displaystyle {\ text {LICQ}} \ Rightarrow {\ text {MFCQ}} \ Rightarrow {\ text {Abadie CQ}} \ Rightarrow {\ text {Guinard CQ}}}

.

Furthermore, in the case of convex problems, the Slater condition implies the Abadie CQ, but the converse does not apply here either.

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

Individual evidence

↑ J. Abadie: On the Kuhn-Tucker Theorem, In: J. Abadie (Ed.), Nonlinear Programming, North-Holland, 1967, pp. 21-36

[1] J. Abadie: On the Kuhn-Tucker Theorem, In: J. Abadie (Ed.), Nonlinear Programming, North-Holland, 1967, pp. 21-36