Hypergraphical/Epigraphical Sets in Conic Duality

• Hypergraphical and epigraphical sets are convex-geometric constructs designed to capture constraint perturbations and duality in infinite-dimensional conic programming.
• They unify graphical and epigraphical representations to generalize Farkas-type alternatives and ensure strong duality through convex separation principles.
• Their analysis aids in duality gap evaluation, closure properties, and verifying algebraic conditions essential for robust infinite-dimensional conic linear programs.

Hypergraphical and epigraphical sets are advanced convex-geometric constructions that systematize both feasible-region/cost-pair representations and duality phenomena in infinite-dimensional conic linear programming. Closely tied to graphical/epigraphical objects in variational analysis and convex optimization, they encode, in a single convex set, all possible combinations of constraint perturbations and value functions, and play a pivotal role in the duality theory, separation theorems, and generalizations of the Farkas lemma. These sets are also fundamental in recent work on algebraic duality gaps, Kretschmer-type closedness, and perturbed conic LP (Khanh et al., 15 Jan 2026).

1. Algebraic and Functional-Analytic Setup

The construction operates in the context of dual pairs of real vector spaces $(X, Y)$ and $(Z, W)$, equipped with the canonical bilinear pairings $\langle x, y\rangle$, $\langle z, w\rangle$. Primal and dual feasible sets are generated from closed convex cones $P \subset X$ and $Q \subset Z$, and their respective algebraic dual cones, $P^* \subset Y$ and $Q^* \subset W$. The primal conic linear program is to minimize a linear form subject to affine and conic constraints,

$$(\mathcal P)\quad\min\{\langle x, c\rangle :\;Ax - b\in Q,\;x\in P\},$$

while the algebraic dual is

$$(\mathcal D)\quad\max\{\langle b,w\rangle :\;-A^*w + c\in P^*,\;w\in Q^*\},$$

where $A:X\to Z$ is a linear operator and $A^*$ its adjoint. The feasible sets are $F(\mathcal P)=\{x\in P:Ax-b\in Q\}$ and $F(\mathcal D)=\{w\in Q^*:-A^*w+c\in P^*\}$. Weak duality holds: $$\operatorname{val}(\mathcal P)\ge\operatorname{val}(\mathcal D)$$ (Khanh et al., 15 Jan 2026).

2. Definitions: Hypergraphical and Epigraphical Sets

The hypergraphical set ("graph-type" set, Editor's term) of the primal, $\mathcal H\subset Z\times\mathbb{R}$, encodes all attainable pairs of constraint slack and cost value: $$\mathcal H := \bigcup_{x\in P}\left((Ax - b - Q)\times[\langle x,c\rangle,+\infty)\right) = \left\{(z, r): \exists x\in P,\; Ax-b-z\in Q,\; r\ge\langle x,c\rangle\right\},$$ which is explicitly convex and nonempty. The optimal value is read off as

$$\operatorname{val}(\mathcal P)= \inf\{r:(0_Z, r)\in\mathcal H\}.$$

The epigraphical set ("epi-type" set, Editor's term) of the dual, $\mathcal N \subset Z\times\mathbb{R}$, is the epigraph of the perturbed dual value function,

$$v_{\mathcal D}(z) = \sup\{\langle b+z, w\rangle : w \in F(\mathcal D)\},$$

so

$$\mathcal N = \left\{(z, r): r\ge v_{\mathcal D}(z)\right\} = \bigcup_{z'}\{z'\}\times[v_{\mathcal D}(z'),+\infty).$$

Again, $\mathcal N$ is convex and $$\operatorname{val}(\mathcal D)=\sup\{r:(0_Z, r)\in \mathcal N\}$$ (Khanh et al., 15 Jan 2026).

Dually, in $Y\times \mathbb{R}$, the set $\mathcal K$ and the hypographical set $\mathcal M$ are defined analogously for the dual problem.

3. Convex Separation and Farkas-Type Theorems

A core property of these sets is the characterization of strong duality and feasibility via convex separation. The separation theorem [Theorem 5.3] states:

• For $(z, r)\notin\mathcal N$, there exists $(w, \alpha)\in W\times \mathbb{R},\ \alpha>0$ such that

$$\langle z', w\rangle + r' \alpha \le \langle z, w\rangle + r\alpha$$

for all $(z', r')\in\mathcal H$, with strict inequality at $(z, r)$.

This formulation matches the algebraic generalizations of the Farkas lemma: for fixed $z$, equality of the slices $$\mathcal H\cap(\{z\}\times\mathbb{R}) = \mathcal N\cap(\{z\}\times\mathbb{R})$$ is shown to be equivalent to the precise Farkas alternatives (Theorem 4.1):

• (a) All dual feasible points $w$ satisfy $\langle b+z, w\rangle \le r$,
• (b) There exists $x\in P$ with $Ax-b-z\in Q$ and $\langle x, c\rangle \le r$ (Khanh et al., 15 Jan 2026).

4. Strong Duality, Zero-Gap, and Closedness

The geometry of hypergraphical and epigraphical sets yields a pointwise and global strong duality:

• For each $z\in Z$, equality of

$$\operatorname{val}(\mathcal P_z) = \inf\{r: (z, r)\in\mathcal H\} = \inf\{r: (z, r)\in\mathcal N\} = \operatorname{val}(\mathcal D_z)$$

occurs if and only if $$\mathcal H\cap(\{z\}\times\mathbb{R}) = \mathcal N\cap(\{z\}\times\mathbb{R})\neq\emptyset$$ (Theorem 5.7).

Global zero duality gap holds if and only if $\mathcal H = \mathcal N$. Kretschmer's classical topological criterion (Theorem 6.1) states that, in any locally convex topology consistent with the dual pair, the closure of $\mathcal H$ equals $\mathcal N$, so $\mathcal H=\mathcal N$ (i.e., no gap) if and only if $\mathcal H$ is topologically closed (Khanh et al., 15 Jan 2026).

5. Algebraic Sufficient Conditions and Examples

Algebraic sufficient conditions for $\mathcal H=\mathcal N$ involve "core" points in the dual cone $P^*$. If $P^*$ contains an algebraic core point of the form $-A^*\bar w + c$ and $P^{**}=P$, then algebraic separation ensures equality (Theorem 7.3). An explicit infinite-dimensional example (the "Gale" problem) demonstrates that $\mathcal H$ may be strictly smaller than its closure $\mathcal N$; only upon closure does the duality gap disappear (Khanh et al., 15 Jan 2026).

6. Context: Connections to Variational and Convex Analysis

While the terminology of hypergraphical/epigraphical sets in (Khanh et al., 15 Jan 2026) arises from infinite-dimensional duality, related convex-geometric constructions are central in several adjacent areas. In variational analysis, the epigraph $$\operatorname{epi} f := \{(x, \alpha): \alpha \ge f(x)\}$$ is key for defining Painlevé–Kuratowski epi-convergence, which underpins the stability theory of convex minimization and the Attouch theorem (Daniilidis et al., 2023). Epigraphical projections appear as practical algorithms in convex optimization and are linked to proximal point frameworks and variational-analytic properties (including SC$^1$ Newton methods for calculating projections onto $\operatorname{epi} f$ (Friedlander et al., 2021)).

The notion of set- or hypergraph-based representations also appears in combinatorics for encoding hypergraphs via families of sets, but these do not incorporate the cost-function/conic structure essential to the algebraic duality context (Basu et al., 2024).

7. Significance and Unified Perspective

The hypergraphical set $\mathcal H$ and the epigraphical set $\mathcal N$ provide a uniquely comprehensive convex-geometric framework unifying:

• Farkas-type alternatives (by slicing at fixed $z$),
• Strong duality and "zero duality gap" results (by global set equality),
• Closedness criteria (algebraic and topological),
• The existence of multipliers and validity of separation theorems.

All standard phenomena in conic linear programming—including dual attainment, dual gaps, and the structural role of feasible perturbations—are encoded via the inclusion, equality, and closure properties of these sets (Khanh et al., 15 Jan 2026). This perspective generalizes classical results and provides a toolkit for both algebraic and topological analysis of duality phenomena in infinite-dimensional settings.