Epigraph (mathematics) explained

f : X \to [ -inf ty, inf ty] valued in the extended real numbers [ -inf ty, inf ty] = \Reals \cup \{\pm inf ty\} is the set $$\backslash operatorname\; f\; =\; \backslash$$consisting of all points in the Cartesian product X x \Reals lying on or above the function's graph. Similarly, the strict epigraph \operatorname{epi} _{S} f is the set of points in X x \Reals lying strictly above its graph.

Importantly, unlike the graph of

f, the epigraph consists of points in X x \Reals (this is true of the graph only when f is real-valued). If the function takes \pm inf ty as a value then \operatorname{graph} f will be a subset of its epigraph \operatorname{epi} f. For example, if f\left(x _{0\right)} = inf ty then the point \left(x _{0,} f\left(x _{0\right)\right)} = \left(x _{0,} inf ty\right) will belong to \operatorname{graph} f but not to \operatorname{epi} f. These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa.

The study of continuous real-valued functions in real analysis has traditionally been closely associated with the study of their graphs, which are sets that provide geometric information (and intuition) about these functions. Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions valued in

[ -inf ty, inf ty] instead of continuous functions valued in a vector space (such as \Reals or \Reals ^{2} ). This is because in general, for such functions, geometric intuition is more readily obtained from a function's epigraph than from its graph. Similarly to how graphs are used in real analysis, the epigraph can often be used to give geometrical interpretations of a convex function's properties, to help formulate or prove hypotheses, or to aid in constructing counterexamples.

Definition

The definition of the epigraph was inspired by that of the graph of a function, where the of

f : X \to Y is defined to be the set $$\backslash operatorname\; f\; :=\; \backslash .$$

The or of a function

f : X \to [ -inf ty, inf ty] valued in the extended real numbers [ -inf ty, inf ty] = \Reals \cup \{\pm inf ty\} is the set

The is the epigraph with the graph removed:where all sets being unioned in the last line are pairwise disjoint, and some may be empty.

Relationships with other sets

Despite the fact that

f might take one (or both) of \pm inf ty as a value (in which case its graph would be a subset of X x \Reals ), the epigraph of f is nevertheless defined to be a subset of X x \Reals rather than of X x [ -inf ty, inf ty]. This is intentional because when X is a vector space then so is X x \Reals but X x [ -inf ty, inf ty] is a vector space (since the extended real number line [ -inf ty, inf ty] is not a vector space). This deficiency in X x [ -inf ty, inf ty] remains even if instead of being a vector space, X is merely a non-empty subset of some vector space. The epigraph being a subset of a vector space allows for tools related to real analysis and functional analysis (and other fields) to be more readily applied.

The domain (rather than the codomain) of the function is not particularly important for this definition; it can be any linear space or even an arbitrary set instead of

\Reals ^{n} .

The strict epigraph

\operatorname{epi} _{S} f and the graph \operatorname{graph} f are always disjoint.

The epigraph of a function

f : X \to [ -inf ty, inf ty] is related to its graph and strict epigraph by $$\backslash ,\backslash operatorname\; f\; \backslash ,\backslash subseteq\backslash ,\; \backslash operatorname\_S\; f\; \backslash ,\backslash cup\backslash ,\; \backslash operatorname\; f$$where set equality holds if and only if f is real-valued. However, $$\backslash operatorname\; f\; =\; \backslash left[\backslash operatorname\{epi\}\_S\; f\; \backslash ,\backslash cup\backslash ,\; \backslash operatorname\{graph\}\; f\backslash right]\; \backslash ,\backslash cap\backslash ,\; [X\; \backslash times\; \backslash Reals]$$ always holds.

Reconstructing functions from epigraphs

The epigraph is empty if and only if the function is identically equal to infinity.

Just as any function can be reconstructed from its graph, so too can any extended real-valued function

f on X be reconstructed from its epigraph E : = \operatorname{epi} f (even when f takes on \pm inf ty as a value). Given x \in X, the value f(x) can be reconstructed from the intersection E \cap (\{x\} x \Reals) of E with the "vertical line" \{x\} x \Reals passing through x as follows:

• case 1:
  E \cap (\{x\} x \Reals) = \varnothing if and only if f(x) = inf ty,
• case 2: E \cap (\{x\} x \Reals) = \{x\} x \Reals if and only if f(x) = -inf ty,
• case 3: otherwise, E \cap (\{x\} x \Reals) is necessarily of the form \{x\} x [f(x), inf ty), from which the value of f(x) can be obtained by taking the infimum of the interval.

The above observations can be combined to give a single formula for

f(x) in terms of E : = \operatorname{epi} f. Specifically, for any x \in X, $$f(x)\; =\; \backslash inf\_\; \backslash$$where by definition, inf \varnothing : = inf ty. This same formula can also be used to reconstruct f from its strict epigraph E : = \operatorname{epi} _{S} f.

Relationships between properties of functions and their epigraphs

g : \Reals ^{n} \to \Reals is a halfspace in \Reals ^{n+1} .

A function is lower semicontinuous if and only if its epigraph is closed.

References

• • Rockafellar, Ralph Tyrell (1996), Convex Analysis, Princeton University Press, Princeton, NJ. .