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 . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{m}}$

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 ${\displaystyle K\subset \mathbb {R} ^{m}}$ ${\displaystyle \preccurlyeq _{K}}$ ${\displaystyle \prec _{K}}$ ${\displaystyle D}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle f\colon D\to \mathbb {R} ^{m}}$

is called K-convex on the set if and only if ${\displaystyle D}$

${\displaystyle f(\lambda x+(1-\lambda )y)\preccurlyeq _{K}\lambda f(x)+(1-\lambda )f(y)}$

applies to everyone and everyone . The function is called strictly K-convex on the set if ${\displaystyle \lambda \in [0,1]}$ ${\displaystyle x,y\in D}$ ${\displaystyle f}$ ${\displaystyle D}$

${\displaystyle f(\lambda x+(1-\lambda )y)\prec _{K}\lambda f(x)+(1-\lambda )f(y)}$

applies to everyone and everyone in . ${\displaystyle \lambda \in (0,1)}$ ${\displaystyle x\neq y}$ ${\displaystyle D}$

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. ${\displaystyle m=1}$ ${\displaystyle K=\{x\geq 0\}}$
• 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. ${\displaystyle K=\{x\leq 0\}}$
• The cone is the crowd

${\displaystyle K=\{x\in \mathbb {R} ^{m}\,|\,x_{i}\geq 0\,{\text{ für alle }}\,i=1,\dots ,n\}}$, 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. ${\displaystyle K}$ ${\displaystyle K^{*}}$ ${\displaystyle v}$ ${\displaystyle 0\preccurlyeq _{K^{*}}v}$ ${\displaystyle v^{T}f}$

For differentiable functions

If there is a differentiable function , then it is K-convex if and only if ${\displaystyle f}$

${\displaystyle f(x)+Df(x)(y-x)\preccurlyeq _{K}f(y)}$for everyone . Here is the Jacobian matrix . ${\displaystyle x,y}$ ${\displaystyle D}$

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. ${\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle h\colon \mathbb {R} ^{m}\to \mathbb {R} }$ ${\displaystyle {\tilde {h}}}$ ${\displaystyle h(g(x))}$

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. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle S^{n}}$ ${\displaystyle \succcurlyeq _{S_{+}^{n}}}$ ${\displaystyle g(x)=z^{T}f(x)z}$ ${\displaystyle z\in \mathbb {R} ^{n}}$ ${\displaystyle f(x)}$

For example, the function defined by is matrix-convex because convex is the norm because of convexity. ${\displaystyle f\colon \mathbb {R} ^{n\times m}\to S^{n}}$ ${\displaystyle f(X)=X\cdot X^{T}}$ ${\displaystyle z^{T}XXz=\Vert X^{T}z\Vert _{2}^{2}}$

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 ${\displaystyle f\colon V_{1}\mapsto V_{2}}$ ${\displaystyle V_{2}}$ ${\displaystyle K}$

${\displaystyle \lambda f(x)+(1-\lambda )f(y)-f(\lambda x+(1-\lambda )y)\in K}$

for all and put out of the convex set . Then the map is called a convex map again . ${\displaystyle \lambda \in [0,1]}$ ${\displaystyle x,y}$ ${\displaystyle M}$ ${\displaystyle f}$

literature

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