K-monotonic function

A K-monotonic function is a generalization of a real monotonic function to functions that map from to . The order on the real numbers is generalized to a partial order using a real cone . K-monotonic functions can be understood as a special case of a monotonic mapping . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$

definition

Given a function with and a real cone im as well as the generalized inequality defined by it and the strict generalized inequality . Then the function is called ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle D\subset \mathbb {R} ^{n}}$ ${\displaystyle K}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \preccurlyeq _{K}}$ ${\displaystyle \prec _{K}}$

• K-monotonically increasing or K-monotonically increasing , if that is true for all with . ${\displaystyle x,y\in D}$ ${\displaystyle x\preccurlyeq _{K}y}$ ${\displaystyle f(x)\leq f(y)}$
• K-monotonically decreasing , if that is true for all with . ${\displaystyle x,y\in D}$ ${\displaystyle x\preccurlyeq _{K}y}$ ${\displaystyle f(x)\geq f(y)}$
• strictly K-monotonically increasing or strictly K-monotonically increasing , if that is true for all with . ${\displaystyle x,y\in D,\,x\neq y}$ ${\displaystyle x\preccurlyeq _{K}y}$ ${\displaystyle f(x)<f(y)}$
• strictly K-monotonically decreasing if that is true for all with . ${\displaystyle x,y\in D,\,x\neq y}$ ${\displaystyle x\preccurlyeq _{K}y}$ ${\displaystyle f(x)>f(y)}$
• strictly K-monotonically if it is either strictly K-monotonically increasing (strictly K-monotonically increasing) or strictly K-monotonically decreasing .
• K-monotonic if it is either K-monotonically increasing (K-monotonically increasing) or K-monotonically decreasing .

Examples

• Every monotonically increasing function is K-monotonically increasing with respect to the cone . ${\displaystyle K=\mathbb {R} _{+}=[0,\infty )}$
• Every monotonically decreasing function is K-monotonically increasing with respect to the cone . The specification of the cone is therefore essential to prevent mix-ups. ${\displaystyle K=\mathbb {R} _{-}=(-\infty ,0]}$
• If the functions are increasing monotonically, the function is ${\displaystyle f_{i}(x_{i})}$

${\displaystyle f(x)=f_{1}(x_{1})+\dots +f_{n}(x_{n})}$

K-monotonically increasing with respect to the positive orthant . This follows directly from the monotony of the . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f_{i}}$

properties

Let be differentiable and a convex set as well as the dual cone of the cone . Then: ${\displaystyle h:\mathbb {R} ^{n}\supset D\to R}$ ${\displaystyle D}$ ${\displaystyle K^{D}}$ ${\displaystyle K}$

• ${\displaystyle h}$is K-monotonically increasing to if and only for all . ${\displaystyle D}$ ${\displaystyle \nabla h(x)\succcurlyeq _{K^{D}}0}$ ${\displaystyle x\in D}$
• ${\displaystyle h}$is K-monotonically decreasing to if and only for all . ${\displaystyle D}$ ${\displaystyle \nabla h(x)\preccurlyeq _{K^{D}}0}$ ${\displaystyle x\in D}$
• If applies to all , then is strictly K-monotonically growing on . ${\displaystyle \nabla h(x)\succ _{K^{D}}0}$ ${\displaystyle x\in D}$ ${\displaystyle h}$ ${\displaystyle D}$
• If applies to all , then is strictly K-monotonically falling on . ${\displaystyle \nabla h(x)\prec _{K^{D}}0}$ ${\displaystyle x\in D}$ ${\displaystyle h}$ ${\displaystyle D}$

Matrix monotonic functions

If one chooses the vector space instead of the (the vector space of all real symmetric matrices), the corresponding functions are called matrix-monotonic functions . The cone of the semidefinite matrices is chosen as the cone , which is equivalent to using the Loewner partial order . The naming follows the scheme above. The determinant is strictly matrix-monotonically growing on the cone of positively definite matrices. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle S^{n}}$ ${\displaystyle h\colon S^{n}\to \mathbb {R} }$ ${\displaystyle S_{+}^{n}}$ ${\displaystyle \det \colon S^{n}\to \mathbb {R} }$ ${\displaystyle S_{++}^{n}}$

use

K-monotonic functions are used in the theory of convex functions. For example, the concatenation of a K-monotonically growing convex function and a K-convex function is convex again.

literature

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