Epigraph (mathematics)

In mathematics , the epigraph of a real-valued function denotes the set of all points that lie on or above its graph . ${\displaystyle f\colon X\to \mathbb {R} }$

${\displaystyle \mathrm {epi} \,f:=\left\{(x,\mu )\in X\times \mathbb {R} \,:\,f(x)\leq \mu \right\}\subseteq X\times \mathbb {R} }$

If the image space of the function is provided with a generalized inequality , then the epigraph is defined as ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \preccurlyeq _{K}}$

${\displaystyle \mathrm {epi} \,f:=\left\{(x,\mu )\in X\times \mathbb {R} ^{n}\,:\,f(x)\preccurlyeq _{K}\mu \right\}\subseteq X\times \mathbb {R} ^{n}}$.

properties

The epigraph of a convex function is a convex set

Let be a normalized vector space. The following applies to functions : ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$

• ${\displaystyle f}$is convex if and only if the epigraph of forms a convex set . ${\displaystyle f}$
• ${\displaystyle f}$is semi-continuous from below if and only if the epigraph of forms a closed set . ${\displaystyle f}$
• ${\displaystyle f}$is exactly then slightly lower continuous when the epigraph of a weakly sequentially closed set is. ${\displaystyle f}$
• If an affine-linear function, then its epigraph defines a half-space in . ${\displaystyle f}$ ${\displaystyle X}$

If the image space of the function is , then it is K-convex if and only if the epigraph is convex. ${\displaystyle \mathbb {R} ^{n}}$

See also

• Hypograph

literature

• Ralph Tyrell Rockafellar: Convex Analysis. Princeton University Press, Princeton 1997, ISBN 0-691-01586-4
• Johannes Jahn: Introduction to the Theory of Nonlinear Optimization . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2007, ISBN 978-3-540-49378-5 .