Epigraph

analysis optimisation convex-analysis

Definition

Epigraph

Let $f:X\to \mathbb{R}$ be a function with domain $X\subseteq {\mathbb{R}}^d$. The epigraph of $f$ is the set of all points lying on or above the graph of $f$:

$$\mathrm{epi}(f)=\{(x,t)\in X\times \mathbb{R}\mid t\ge f(x)\}.$$

The variable $x$ chooses a point in the domain, while $t$ is a height. The inequality $t\ge f(x)$ keeps every height above the function value and discards every height below it.

Interpretation

The epigraph turns a function into a geometric set. Instead of only looking at the graph, it fills everything above the graph.

This is useful because many properties of functions become properties of sets. In particular, a function is convex exactly when its epigraph is a convex set.

Optimisation Meaning

Minimising $f$ can be written as finding the lowest height $t$ such that $(x,t)$ lies in the epigraph:

$$\underset{x\in X}{\min }f(x)⟺\underset{(x,t)\in \mathrm{epi}(f)}{\min }t.$$

The function becomes a feasible region, and optimisation becomes the search for its lowest point.