1 Definition
1.1 Basic epigraph of a function
Let \(f : X \to \mathbb{R}\cup\{+\infty\}\) be a real-valued (possibly extended) function on a set \(X\). The epigraph of \(f\) is the subset of the product space \(X\times \mathbb{R}\) defined by \[ \operatorname{epi}(f)=\{(x,t)\in X\times\mathbb{R}: t\ge f(x)\}. \] Geometrically, for each \(x\), the epigraph contains all points with vertical coordinate \(t\) at or above the height given by \(f(x)\).
1.2 Epigraph in extended-real-valued settings
In optimization and variational analysis, it is common to allow \(f(x)\) to take the extended values \(+\infty\) (and sometimes \(-\infty\), though this is less typical for epigraph methods). If \(f(x)=+\infty\), then the condition \(t\ge f(x)\) forces no finite \(t\) to qualify, so \((x,t)\) is excluded from \(\operatorname{epi}(f)\) for any real \(t\). This convention makes epigraphs suitable for representing constraints via “infinite penalties,” where infeasible points are ruled out.
When \(f(x)\in\mathbb{R}\), the epigraph contains the vertical ray \(\{(x,t): t\ge f(x)\}\). Thus the epigraph combines both finite and “forbidden” regions of the graph into one set in \(X\times\mathbb{R}\).
1.3 Subgraph versus epigraph
The subgraph of \(f\) is the set \[ \operatorname{sub}(f)=\{(x,t)\in X\times\mathbb{R}: t\le f(x)\}, \] which lies on or below the graph. Epigraphs and subgraphs are dual in the sense that many properties (such as convexity and semicontinuity) translate into corresponding geometric properties of these sets, often with inequalities reversed. For instance, convexity corresponds to convexity of the epigraph, while concavity corresponds to convexity of the hypograph (the dual construction to the epigraph).
2 Geometric interpretation
2.1 Points above the graph
For any \(x\in X\), the epigraph contains exactly those points whose second coordinate \(t\) is not less than the function value. In a plot, one can imagine drawing the graph of \(f\) and shading the region above it; the shaded region, expressed as a set of pairs, is the epigraph.
This viewpoint is especially useful because many analytic properties become statements about shapes of sets: the epigraph is a geometric proxy for the behavior of \(f\).
2.2 Closed and unbounded shapes
The epigraph can be bounded or unbounded depending on \(f\). Even when \(f\) is bounded below, the epigraph is generally unbounded upward because it contains all sufficiently large \(t\). In addition, whether the boundary is included depends on the limiting behavior of \(f\). For many functions, the epigraph is not automatically closed; it may fail to contain boundary points that correspond to “limiting values” of \(f\) not captured by direct evaluation.
Because of this, closedness of \(\operatorname{epi}(f)\) is closely tied to lower semicontinuity of \(f\) (discussed later).
2.3 Visual examples in one dimension
In one dimension, let \(X=\mathbb{R}\) and consider a few standard functions:
- Constant function \(f(x)=c\): the epigraph is the half-plane \(\{(x,t): t\ge c\}\), a region with a flat boundary line.
| - Absolute value \(f(x)= | x | \): the boundary of the epigraph forms a “V”-shaped curve, and the epigraph fills everything above that curve. |
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- Discontinuous function: if \(f\) jumps upward at some point, the epigraph boundary reflects those jump values; if \(f\) jumps downward, the epigraph may contain boundary points not aligned with naive expectations, depending on the notion of closure.
Such examples illustrate that the epigraph encodes both local values and global arrangement of the function’s heights.
3 Fundamental properties
3.1 Relationship to the function domain
The epigraph’s horizontal “shadow” on \(X\) corresponds to where \(f\) is finite. Specifically, if \(f(x)=+\infty\), then no pair \((x,t)\) with real \(t\) lies in the epigraph; hence the projection of \(\operatorname{epi}(f)\) onto \(X\) is essentially the effective domain \(\{x\in X: f(x)<+\infty\}\).
More precisely, if one considers the set-valued projection \[ \pi_X(\operatorname{epi}(f))=\{x\in X:\exists t\in\mathbb{R}\text{ with }(x,t)\in \operatorname{epi}(f)\}, \] then \(\pi_X(\operatorname{epi}(f))\) equals the points where \(f(x)\) is not \(+\infty\).
3.2 Closure properties
3.2.1 Closed epigraphs
Whether \(\operatorname{epi}(f)\) is closed depends on how \(f\) behaves under limits. For functions on topological spaces, a closed epigraph means that whenever \((x_n,t_n)\in \operatorname{epi}(f)\) converges to \((x,t)\), the limit point must still satisfy \(t\ge f(x)\). This requirement rules out certain kinds of downward discontinuities in the limit and precisely captures a standard semicontinuity notion.
3.2.2 Epigraph closure and lower semicontinuity
Let \(X\) have a topology, and \(f : X\to \mathbb{R}\cup\{+\infty\}\) be extended-real-valued. A central result in convex and variational analysis states:
- \(f\) is lower semicontinuous if and only if its epigraph is closed in \(X\times\mathbb{R}\).
Intuitively, lower semicontinuity means the function cannot “fall below” its limit infimum. In epigraph terms, the region above the graph cannot lose boundary points when taking limits.
3.3 Convexity properties
3.3.1 Convex functions and convex epigraphs
A fundamental characterization is:
- \(f\) is convex if and only if \(\operatorname{epi}(f)\) is a convex set in \(X\times\mathbb{R}\).
Concretely, convexity of the epigraph means that for any \((x_1,t_1)\) and \((x_2,t_2)\) in the epigraph, the entire segment joining them is also contained in the epigraph. Since the epigraph inequality is \(t\ge f(x)\), this geometric condition yields the standard inequality for convex functions.
3.3.2 Strict and non-strict convexity
Convexity of the epigraph corresponds to non-strict convexity of \(f\). Stronger notions correspond to finer geometric structure. For example, if \(f\) is strictly convex (in an appropriate sense), then the boundary of the epigraph has properties that prevent boundary segments from lying entirely on the graph except in degenerate cases. However, strict convexity is typically expressed directly via the functional inequality rather than purely by convexity of sets, because strictness concerns uniqueness or curvature effects rather than mere inclusion of segments.
4 Epigraphs in convex analysis
4.1 Characterization of convex functions
Convex analysis uses epigraphs as a primary object because convexity becomes a statement about geometry. When \(\operatorname{epi}(f)\) is convex, one can apply many tools from convex geometry, including separation theorems and support function arguments, to obtain properties of \(f\) such as existence of minimizers under suitable compactness and coercivity conditions.
This approach also extends smoothly to functions that are not everywhere finite, because the epigraph remains well-defined when the function takes \(+\infty\).
4.2 Supporting hyperplanes
If \(\operatorname{epi}(f)\) is convex and closed, then at boundary points one can often describe supporting hyperplanes. A supporting hyperplane at a boundary point \((\bar x,\bar t)\) provides a linear inequality that the epigraph satisfies, which translates into a subgradient inequality for the function at \(\bar x\). In this way, geometric separation becomes an analytic subdifferential characterization.
4.3 Epigraphical minimization
“Epigraphical” viewpoints simplify certain minimization problems. Minimizing \(f(x)\) over \(x\) is equivalent to locating the lowest vertical position where the epigraph intersects a horizontal line or, equivalently, identifying the point(s) in \(\operatorname{epi}(f)\) with minimal \(t\). This perspective is useful in reformulations where one introduces an auxiliary variable representing the objective value.
4.4 Indicator functions and epigraphs
For a set \(C\subseteq X\), the indicator function \(\delta_C\) is defined by \[ \delta_C(x)= \begin{cases} 0, & x\in C,\\ +\infty, & x\notin C. \end{cases} \] Its epigraph is particularly simple: \[ \operatorname{epi}(\delta_C)=\{(x,t): x\in C,\ t\ge 0\}. \] Indicator functions therefore encode constraints into convex-analytic structures. When combined with other functions (e.g., via addition), they restrict the effective domain while preserving the epigraph-based framework.
5 Lower semicontinuity and continuity
5.1 Epigraph characterization of lower semicontinuity
For topological spaces, the closedness of \(\operatorname{epi}(f)\) is equivalent to lower semicontinuity of \(f\). In practice, this characterization allows one to verify semicontinuity by studying the limiting behavior of sets rather than directly checking pointwise limit conditions.
Because lower semicontinuity is often a prerequisite for existence of minimizers in variational problems, the epigraph provides a geometric route to establishing hypotheses for optimization.
5.2 Continuity points and local behavior
Even when \(f\) is not globally continuous, one can analyze continuity points using epigraph geometry. Lower semicontinuity concerns behavior from above in the sense that the function cannot drop below its limit infimum. If one also has upper semicontinuity, then continuity follows, and the epigraph and hypograph become simultaneously well-behaved (closedness in appropriate senses for both constructions).
Local regularity can sometimes be read off from the local shape of the epigraph boundary: smoothness or sharp corners in the boundary correspond to particular local forms of \(f\).
5.3 Proper functions
An extended-real-valued function \(f\) is typically called proper if it never takes the value \(-\infty\) and is finite at least at one point, i.e., \(f(x)>-\infty\) for all \(x\) and \(\operatorname{dom}(f)=\{x: f(x)<+\infty\}\) is nonempty. Properness ensures that the epigraph is nonempty and meaningful for optimization. Many convex-analytic results assume properness to avoid degenerate cases where the epigraph might be empty or ill-suited for separation arguments.
6 Applications in optimization
6.1 Reformulation of optimization problems
Consider minimizing \(f(x)\) over \(x\in X\). Introduce an auxiliary variable \(t\) and consider the problem of minimizing \(t\) subject to the constraint \(t\ge f(x)\). This converts an objective involving \(f\) into a feasibility/constraint structure determined by the epigraph: \[ \min\ t \quad \text{subject to}\quad (x,t)\in \operatorname{epi}(f). \] The equivalence holds because for any fixed \(x\), the smallest feasible \(t\) is \(f(x)\).
6.2 Epigraphical constraints
In constrained optimization, epigraph constraints appear when one needs to enforce inequalities of the form \[ t \ge f(x) \] rather than directly working with the function value. This formulation can be advantageous because it allows one to treat nonlinear expressions through geometric or convex set constraints, sometimes making the problem amenable to standard convex programming tools.
6.3 Auxiliary variables and epigraph form
Many classical reformulations use epigraphs to linearize or convexify certain structures by introducing auxiliary variables. Examples include handling norms, log-sum-exp functions, or other expressions where the epigraph can be represented using convex constraints (possibly after additional auxiliary variables).
The general principle is: replace complicated objective terms with constraints describing an epigraph, then optimize the new variable representing the objective level.
6.4 Convex programming
For convex optimization problems, epigraph representations play a key role. When the epigraph of the objective is convex and closed and the feasible set is convex, standard duality and optimality conditions can often be applied. In practical modeling, solvers frequently implement epigraph-based transformations to express objectives through allowed convex constraints.
As a result, epigraphs serve as the bridge between mathematical definitions of convex functions and the constraint systems used in optimization algorithms.
7 Related constructions
7.1 Hypograph
The hypograph of \(f\) is the set \[ \operatorname{hypo}(f)=\{(x,t): t\le f(x)\}. \] While the epigraph corresponds to lower bounds on \(f\), the hypograph corresponds to upper bounds. Convexity of the hypograph characterizes concavity of \(f\), mirroring the epigraph characterization of convexity.
7.2 Level sets
The level set of \(f\) at height \(\alpha\) is \[ \{x\in X: f(x)\le \alpha\}. \] These sets are closely related to the epigraph: the epigraph cut by the horizontal line \(t=\alpha\) projects onto (or corresponds to) the sublevel set. Specifically, points \((x,t)\in\operatorname{epi}(f)\) with \(t=\alpha\) satisfy \(f(x)\le \alpha\) at the boundary, linking level-set geometry with epigraph geometry.
7.3 Subdifferentials
For a convex function \(f\), subdifferentials describe how the function can be supported by affine functions. In epigraph terms, a subgradient at \(\bar x\) corresponds to a supporting hyperplane to \(\operatorname{epi}(f)\) at some boundary point \((\bar x, f(\bar x))\). This relationship makes epigraph geometry a tool for understanding subgradients and for deriving optimality conditions.
7.4 Minkowski sums and transformations
Epigraphs behave well under certain operations that correspond to set operations. For example, affine transformations of the variables induce corresponding transformations of the epigraph. Under some structural assumptions, sums or infimal convolutions of functions translate into operations on their epigraphs or related sets. Such transformations are part of the broader toolkit of convex analysis, where geometric constructions often simplify algebraic expressions.
8 Examples
8.1 Linear functions
Let \(f(x)=a^\top x+b\) on \(X=\mathbb{R}^n\). Then \[ \operatorname{epi}(f)=\{(x,t): t\ge a^\top x+b\}. \] This is a half-space in \(\mathbb{R}^n\times\mathbb{R}\), bounded below by an affine hyperplane. As a consequence, linear functions have convex (indeed, flat-faced) epigraphs.
8.2 Quadratic functions
| For \(f(x)=\|x\|^2\) on \(X=\mathbb{R}^n\), |
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\[
| \operatorname{epi}(f)=\{(x,t): t\ge \|x\|^2\}. |
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\]
| The boundary \(t=\|x\|^2\) forms a paraboloid; the epigraph is the region above it. Since \(f\) is convex and continuous, the epigraph is both convex and closed. |
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8.3 Absolute value and piecewise-linear functions
| In one dimension, \(f(x)= | x | \) gives |
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\[
| \operatorname{epi}(f)=\{(x,t): t\ge | x | \}. |
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\] The boundary consists of two line segments meeting at the “corner” \((0,0)\). For piecewise-linear convex functions, the epigraph is a polyhedron: it is formed by finitely many half-spaces corresponding to the linear pieces. This structure is central in modeling and computational geometry.
8.4 Functions with discontinuities
Consider \(f(x)=0\) for \(x\ne 0\) and \(f(0)=1\). Then for \(x\ne 0\), the epigraph contains all \((x,t)\) with \(t\ge 0\); at \(x=0\) it contains only \((0,t)\) with \(t\ge 1\). The boundary near \((0,0)\) reveals a failure of lower semicontinuity. Indeed, points \((x,0)\) with \(x\ne 0\) approach \((0,0)\), but \((0,0)\) is not in the epigraph, so \(\operatorname{epi}(f)\) is not closed.
This illustrates how discontinuities show up as gaps or missing boundary points in the epigraph.
9 Generalizations
9.1 Multivariable functions
For \(f:X\to\mathbb{R}\cup\{+\infty\}\) with \(X\subseteq \mathbb{R}^n\), the definition remains: \[ \operatorname{epi}(f)=\{(x,t)\in X\times\mathbb{R}: t\ge f(x)\}. \] The epigraph then lives in \(\mathbb{R}^{n+1}\), where convexity and closedness become geometric properties in higher dimensions. The characterization of convexity via convexity of the epigraph still holds.
9.2 Functions with infinite values
Allowing \(f\) to take \(+\infty\) enables one to encode constraints and exclude regions of the domain. The epigraph method handles this naturally because points with \(f(x)=+\infty\) contribute no real vertical coordinates. This is particularly useful for penalty and barrier interpretations, as well as for modeling feasible sets through indicator functions.
In convex optimization, properness and closedness hypotheses ensure that the epigraph has enough structure to apply duality and separation results.
9.3 Epigraphs in topology and functional analysis
In more abstract settings, such as when \(X\) is a topological vector space, the epigraph is still defined as a subset of \(X\times\mathbb{R}\) using the inequality \(t\ge f(x)\). Continuity and semicontinuity are then discussed in terms of topological properties of this set.
In functional analysis and variational problems, one often studies functionals on infinite-dimensional spaces where direct pointwise analysis is difficult. Epigraphical methods remain valuable because they convert functional properties into geometric/topological properties of sets in product spaces, enabling the use of separation theorems and compactness arguments.