Generalized convexity

The generalized convexity ( English generalized convexity is) a generalization of the ordinary of convexity for functions and levels that are particularly in the treatment of non-convex optimization problems proves to be useful.

Φ convexity

Given is a set and the set of all mappings from to ${\displaystyle X\neq \emptyset }$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$

${\displaystyle F(X)=\{\varphi \mid \varphi \colon X\to \mathbb {R} \}}$

A set is called a reference system for if and only if: ${\displaystyle \Phi \subseteq F(X)}$ ${\displaystyle X}$

1. ${\displaystyle \forall \varphi \in \Phi ,\lambda \geq 0:\lambda \varphi \in \Phi }$
2. ${\displaystyle \forall \varphi \in \Phi ,c\in \mathbb {R} :\varphi +c\in \Phi }$

Φ-convex function

An (extended) real-valued function is called -convex if and only if there is a set such that ${\displaystyle f\colon X\mapsto {\overline {\mathbb {R} }}}$ ${\displaystyle \Phi }$ ${\displaystyle \Phi _{0}\subset \Phi }$

${\displaystyle f(x)=\sup _{\varphi \in \Phi _{0}}\varphi (x)}$

applies.

Φ-convex set

A set is called -convex if and only if there is a set and for each one exists such that ${\displaystyle A\subseteq X}$ ${\displaystyle \Phi }$ ${\displaystyle \Phi _{0}\subseteq \Phi }$ ${\displaystyle \varphi \in \Phi _{0}}$ ${\displaystyle a_{\varphi }}$

${\displaystyle A=\bigcap _{\varphi \in \Phi _{0}}\{x\in X:\varphi (x)\leq a_{\phi }\}}$

Examples

• For example, if one takes the affine functions as the reference system , then the convexity coincides with the usual convexity. ${\displaystyle \Phi =\{\varphi \,|\,\varphi (x)=\langle v,x\rangle +c,v\in \mathbb {R} ^{n},c\in \mathbb {R} \}}$ ${\displaystyle \Phi }$

• The Lipschitz continuous functions are convex to the reference system of the peak functions . ${\displaystyle \Phi =\{\varphi \,|\,\varphi (x)=-k\cdot d(x,x_{0})+c,x_{0}\in X,c\in \mathbb {R} \}}$ ${\displaystyle \Phi }$

See also

• Convex function
• Convex set
• Convex optimization

literature

• Szymon Dolecki, Stanisl Aw Kurcyusz: On -Convexity in Extremal Problems ${\displaystyle \Phi }$ . In: SIAM Journal on Control and Optimization . 16, No. 2, 1978, pp. 277-300. doi : 10.1137 / 0316018 .
• Pallaschke, D., Rolewicz, S .: Foundations of Mathematical Optimization: Convex Analysis Without Linearity . Kluwer Academic Publishers, 1997, ISBN 0-7923-4424-3 .