K-convex function
A K-convex function is a generalization of the notion of convexity of a function to real-vector-valued functions . For this, the strict order is weakened and it is worked with partial orders , the so-called generalized inequalities .
definition
Given is a closed, pointed and convex cone with a non-empty interior and / or the partial order or strict partial order induced by this cone . Furthermore, let us be a convex subset of the . The function
is called K-convex on the set if and only if
applies to everyone and everyone . The function is called strictly K-convex on the set if
applies to everyone and everyone in .
Examples and characteristics
- If one sets that the function is real-valued and chooses the set as the cone , then the K-convex functions are exactly the convex functions. This is because the order induced by the cone is the ordinary order on the real numbers.
- On the other hand, if you choose the set as the cone , then the K-convex functions are exactly the concave functions, since the cone reverses the order on the real numbers.
- The cone is the crowd
- , the induced general inequality is less than or equal to the component wise . The K-convex functions are then the functions whose components are all convex.
- Affine functions are always K-convex, regardless of the cone used. This follows directly from the linearity of the function and the reflexivity of the generalized inequality.
- The sub-level set of a K-convex function is a convex set.
- A function is K-convex if and only if its epigraph is a convex set. The epigraph is defined in this case by means of the generalized inequality and not with the conventional less or equal .
Alternative characterizations
About duality
The K-convexity of a function can also be well described by means of the partial order induced by the dual cone . A function is then exactly (strictly) K-convex if for each different from the zero vector Vector with valid that is (strictly) is convex in the conventional sense.
For differentiable functions
If there is a differentiable function , then it is K-convex if and only if
- for everyone . Here is the Jacobian matrix .
Concatenations of K-convex functions
The compositions of K-convex functions are again convex under certain circumstances.
- If K-convex and convex and if the extended function K-monotonically increasing , then is convex. In particular, the two cones that define the K-convexity and the K-monotony must match.
Matrix convex functions
If one considers mappings from into the space of the symmetrical real matrices , provided with the Loewner half-order , then the corresponding K-convex functions are also called matrix-convex functions. An equivalent characterization of matrix convexity is that the function is convex for all if and only if matrix is convex.
For example, the function defined by is matrix-convex because convex is the norm because of convexity.
use
K-convex functions are used, for example, in the formulation of conical programs or generalizations of the Lagrange duality .
Generalizations
In some cases, mappings between two real vector spaces are also considered and only provided with an order cone , not with a generalized inequality. The requirement is attached to the figure
for all and put out of the convex set . Then the map is called a convex map again .
literature
- Stephen Boyd, Lieven Vandenberghe: Convex Optimization . Cambridge University Press, Cambridge, New York, Melbourne 2004, ISBN 978-0-521-83378-3 ( online ).