Convex figure

In mathematics, a convex mapping is a generalization of a convex function to general ordered vector spaces . It contains several different classes of convex functions as special cases.

definition

Two real vector spaces as well as a convex set and an order cone are given . Then a map is called convex on the set if and only if ${\displaystyle V_{1},V_{2}}$ ${\displaystyle M\subset V_{1}}$ ${\displaystyle K}$ ${\displaystyle V_{2}}$ ${\displaystyle f\colon M\to V_{2}}$ ${\displaystyle M}$

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

is for everyone and . ${\displaystyle x,y\in M}$ ${\displaystyle \lambda \in [0,1]}$

Examples

• Every convex function is a convex map with respect to the cone of order . ${\displaystyle f\colon V\supset M\to \mathbb {R} }$ ${\displaystyle K=[0,\infty )}$
• Each concave function is a convex map with respect to the cone of order . ${\displaystyle f\colon V\supset M\to \mathbb {R} }$ ${\displaystyle K=(-\infty ,0]}$
• Every K-convex function is a convex map with respect to the cone of order , which in this case is even a real cone . ${\displaystyle f\colon \mathbb {R} ^{m}\supset M\to \mathbb {R} ^{n}}$ ${\displaystyle K}$
• Every matrix-convex function is a convex map. denotes the vector space of the symmetric real matrices. The order cone is the semidefinite cone , the corresponding order the Loewner partial order . ${\displaystyle f\colon V\supset M\to S^{n}}$ ${\displaystyle S^{n}}$
• Every linear map is a convex map. Its ever ${\displaystyle l}$

${\displaystyle \lambda l(x)+(1-\lambda )l(y)-l(\lambda x+(1-\lambda )y)=0}$.

However, since an order cone always contains zero, every linear mapping is convex.

properties

• Sub-level sets of a convex map, i.e. sets of the form

${\displaystyle {\mathcal {L}}_{f}(c):=\{x\in M\mid c-f(x)\in K\}}$

are convex. This follows from the convexity of the cone of order .

• If the order cone is acute and both the map and the map are convex, then it is linear. The additional requirement for the order cone cannot be dispensed with, since only this guarantees the necessary antisymmetry of the order relation. ${\displaystyle l}$ ${\displaystyle -l}$ ${\displaystyle l}$

use

Apart from the various applications of the special cases of convex mapping listed above, convex maps are used, for example, in convex optimization in infinite-dimensional spaces to model restriction sets. Due to the convexity of the sub-level sets, these restriction sets are convex and thus guarantee, in the case of convex target functions, that every local optimum is a global optimum.

generalization

The almost convex functions are a generalization of a convex map . The only requirement for them is that a certain amount above their graph is convex. Every convex map is almost convex.

literature

• Johannes Jahn: Introduction to the Theory of Nonlinear Optimization . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2007, ISBN 978-3-540-49378-5 .