K-convex function

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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 .


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.


K-convex functions are used, for example, in the formulation of conical programs or generalizations of the Lagrange duality .


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 .


  • Stephen Boyd, Lieven Vandenberghe: Convex Optimization . Cambridge University Press, Cambridge, New York, Melbourne 2004, ISBN 978-0-521-83378-3 ( online ).