---
title: Fenchel Conjugate for Set-Valued Mappings
url: https://www.emergentmind.com/topics/fenchel-conjugate-for-set-valued-mappings
type: topic
---

# Fenchel Conjugate for Set-Valued Mappings

Searching arXiv for relevant papers on Fenchel conjugates of set-valued mappings and duality.
Fenchel conjugation for set-valued mappings extends classical Legendre–Fenchel duality from scalar functions to mappings whose values are sets, and, in related vector-optimization formulations, to vector-valued mappings whose conjugates are intrinsically set-valued. In the contemporary literature, at least three closely related lines of development coexist: scalarization-based conjugates on lattices of upper sets, graph-based conjugates defined as support functions of multifunction graphs, and direct order-theoretic constructions for vector-valued mappings using weak supremum and weak infimum. These approaches share the duality intuition of encoding a primal mapping by supporting dual objects, but they differ in codomain, order structure, biconjugation, and calculus rules [1011.5860] [2305.17612] [2105.13299].

## 1. Order structures and image spaces

The subject is formulated on ordered topological vector spaces. A closed convex cone induces a preorder on the codomain, and the choice of image space determines the algebra available for conjugation. In the upper-set framework, one works with families such as
\[
F(Z,C)=\{A\subset Z: A=\operatorname{cl}(A+C)\},
\]
\[
P_A(Z,C)=\{A\subset Z: A=A+C\},
\]
or
\[
G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},
\]
typically ordered by reverse inclusion. These spaces are complete lattices, and their lattice operations make Minkowski addition, infima, suprema, and residuation compatible with convex-analytic constructions [1112.1315] [1011.5860] [2306.15906].

A second line of work treats a multifunction \(F:X\rightrightarrows Y\) through its graph
\[
\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},
\]
so that conjugation becomes the support function of \(\operatorname{gph}F\). This approach is geometric and is formulated in real locally convex Hausdorff topological vector spaces, in both finite and infinite dimensions [2305.17612].

A third line, developed for vector optimization problems, starts from proper mappings \(F:X\to Y\cup\{+\infty_Y\}\) and defines a conjugate that is itself set-valued through weak supremum in the ordered space \((Y^\bullet,<_K)\). There the basic dual objects live in \(L(X,Y)\times \mathcal P_p(Y)\), not only in \(X^*\times Y^*\) [2105.13299].

| Framework | Conjugate object | Characteristic feature |
|---|---|---|
| Upper-set/lattice | set-valued half-space map | scalarization and reverse-inclusion order |
| Graph-support | extended real-valued function | support function of \(\operatorname{gph}F\) |
| Vector weak-order | set-valued map on \(L(X,Y)\) | \(WSup/WInf\), extended epigraphs |

These frameworks are not merely notational variants. They encode different dual viewpoints: one represents sets by half-spaces, another represents graphs by support functions, and another represents vector orders directly.

## 2. Principal definitions of the conjugate

For a multifunction \(F:X\rightrightarrows Y\), Nam, Sandine, Thieu, and Yen define the Fenchel conjugate by
\[
F^*(x^*,y^*)=\sup\{\langle x^*,x\rangle+\langle y^*,y\rangle\mid (x,y)\in \operatorname{gph}F\},
\]
equivalently,
\[
F^*=\sigma_{\operatorname{gph}F}.
\]
In this formulation the conjugate is an extended real-valued convex function on \(X^*\times Y^*\), proper and lower semicontinuous whenever \(\operatorname{gph}F\neq\varnothing\) [2305.17612].

In the upper-set approach, the conjugate remains set-valued. For a map \(F:X\to P_A(Z,C)\), scalarizations are defined by
\[
\varphi_{z^*}(x)=\inf\{-\langle z^*,z\rangle\mid z\in F(x)\},\qquad z^*\in C^-,
\]
and the Legendre–Fenchel conjugate is assembled as a family of half-spaces. In one standard formulation,
\[
F^*(x^*,z^*,r)=H_{\varphi_{z^*}^*(x^*,r)}(z^*)
=\{z\in Z\mid \varphi_{z^*}^*(x^*)-r\le -\langle z^*,z\rangle\},
\]
so the set-valued conjugate is identified with the scalar conjugates of the scalarizations [1011.5860].

The order-theoretic extension of Hamel’s framework replaces ordinary difference by residuation. For \(g:X\to\mathcal G_A(Z)\), one introduces conaffine maps
\[
S^A(\xi,r,z^*)(x)=\{z\in Z:\xi_r(x)\le -z^*(z)\},
\]
and defines
\[
g^*(\xi,r,z^*)=\sup_{x\in X}\big[S^A(\xi,r,z^*)(x)-g(x)\big].
\]
Its scalarization formula is explicit:
\[
g^*(\xi,r,z^*)=\{z\in Z:(\varphi_{g,z^*})^*(\xi,r)\le -z^*(z)\}.
\]
This construction is designed to treat proper and improper cases uniformly [1011.3179].

In the vector-valued formulation, the conjugate of \(F:X\to Y^\bullet\) is
\[
F^*(L)=WSup\{L(x)-F(x):x\in X\},\qquad L\in L(X,Y),
\]
with \(K\)-epigraph
\[
\operatorname{epi}_K F^*=\{(L,y)\in L(X,Y)\times Y: y\in F^*(L)+K\}.
\]
A basic characterization is
\[
(L,y)\in \operatorname{epi}_K F^*
\iff
F(x)-L(x)+y\notin -\operatorname{int}K,\quad \forall x\in X.
\]
Here the conjugate is set-valued because \(F^*(L)\) is a subset of \(Y^\bullet\), not a single point [2105.13299].

## 3. Scalarization, representation, and Fenchel–Young inequalities

Scalarization is the main bridge between set-valued and scalar convex analysis. In the upper-set framework, the family of scalarizations completely characterizes the original map. For convex and closed \(F\),
\[
F(x)=\bigcap_{z^*\in C^-\setminus\{0\}}\{z\in Z:\varphi_{z^*}(x)\le -\langle z^*,z\rangle\},
\]
and more generally the same intersection reconstructs \(\operatorname{cl}\operatorname{co}F\). The same principle underlies the description of the conjugate by half-spaces and the representation of convex closed set-valued maps as pointwise suprema of conaffine minorants [1011.5860] [1011.3179].

Heyde and Schrage work in \(F(Z,C)\), the family of upper closed subsets of a preordered topological vector space, and use scalarizations
\[
\varphi_{(f,z^*)}(x)=\inf_{z\in f(x)}(-z^*(z)).
\]
They show that a convex-valued map can be reconstructed from these scalarizations by intersecting the corresponding supporting half-spaces, and that continuity properties of the set-valued map can be transferred to semicontinuity properties of the scalarizations. Their fundamental duality theorem uses the weakest regularity among the continuity notions considered: upper semicontinuity at \(0\) of all scalarizations of \(y\mapsto f(x_0,y)\) [1112.1315].

In the graph-support framework, scalarization is replaced by direct graph support, but the classical Fenchel–Young pattern survives. For every \((x,y)\in\operatorname{gph}F\),
\[
\langle x^*,x\rangle\le \langle y^*,y\rangle + F^*(x^*,-y^*).
\]
If \(F\) is convex, equality holds if and only if
\[
x^*\in D^*F(x,y)(y^*),
\]
so equality is characterized by a coderivative condition. For proper, closed, convex \(F\), the subdifferential of \(F^*\) is linked to coderivatives by
\[
(x,y)\in \partial F^*(x^*,y^*)
\iff
x^*\in D^*F(x,y)(-y^*).
\]
This places multifunction conjugacy directly inside convex generalized differentiation [2305.17612].

A recurrent point of confusion is the role of scalarization. In the upper-set literature, scalarization is the primary representation device. In the vector weak-order literature, by contrast, the general vector case explicitly avoids scalarization and works directly with \(Y\)-ordered structures, weak suprema, and positive operators \(L_+(S,K)\) [2105.13299].

## 4. Calculus rules and biconjugation

A central issue is whether classical conjugate calculus survives for set-valued objects. In the graph-support framework, the answer is affirmative under explicit qualification conditions. For multifunctions \(F_1,F_2:\mathbb R^n\rightrightarrows \mathbb R^p\),
\[
(F_1+F_2)^*(u,v)\le
\inf\{F_1^*(u_1,v)+F_2^*(u_2,v):u_1+u_2=u\},
\]
with equality under relative interior or polyhedral assumptions; analogous exact formulas are proved for composition and intersection. In infinite-dimensional locally convex or Banach settings, the relevant qualifications use \(\operatorname{int}\), quasi-relative interior, strong quasi-relative interior, closedness of graphs, and polyhedral convexity [2305.17612].

A recent extension studies set-valued convex compositions in the lattice \(G(Z,C)\). If \(H=F\circ G\), then the scalarization satisfies
\[
\varphi_{H,z^*}(x)=\inf_{y\in G(x)}\varphi_{F,z^*}(y).
\]
Under a weak\(^*\)-compact generator assumption for the dual cone of the intermediate order, together with unboundedness or strict monotonicity assumptions, the conjugate obeys the chain rule
\[
(\varphi_{H,z^*})^*(x^*)
=
\inf_{y^*\in Y_G}\big[(\varphi_{G,y^*})^*(x^*)+(\varphi_{F,z^*})^*(y^*)\big],
\]
and this yields an exact dual representation of \(H\) when the composition is proper and scalarly closed [2306.15906].

Biconjugation is similarly framework-dependent. In the graph-support theory,
\[
F^{**}(x,y)=\delta_{\operatorname{cl}\operatorname{co}(\operatorname{gph}F)}(x,y),
\]
so the biconjugate is the indicator of the closed convex hull of the graph. In the upper-set theory of scalar representation and conjugation,
\[
F^{**}=\operatorname{cl}\operatorname{co}F,
\]
and if \(F\) is convex and closed then \(F=F^{**}\). In the residuated framework for \(\mathcal G_A(Z)\), the Fenchel–Moreau type theorem states that closed convex \(g\) satisfies \(g^{**}=g\) [2305.17612] [1011.5860] [1011.3179].

These results show that “the” biconjugate is not universal. What it returns depends on what has been conjugated: a graph, an upper-set-valued map, or an order-theoretic object.

## 5. Duality in set-valued and vector optimization

The duality role of set-valued conjugation is explicit in set-valued optimization. In the \(F(Z,C)\)-valued framework of Heyde and Schrage, for a convex \(f:X\times Y\to F(Z,C)\) the marginal map
\[
f_X(y)=\bigcup_{x\in X}f(x,y)
\]
satisfies weak duality
\[
f_X(0)\subset (-f^*)((0,y^*),z^*),
\]
and, under the scalarization regularity condition at \(0\), the fundamental duality theorem gives
\[
f_X(0)=\bigcap_{(y^*,z^*)\in Y^*\times(C^-\setminus\{0\})}(-f^*)((0,y^*),z^*).
\]
Thus the primal upper set is reconstructed as an intersection of set-valued conjugates indexed by dual variables [1112.1315].

For vector optimization, the paper on epigraphs of conjugate mappings studies
\[
({\rm VP})\qquad \operatorname{WInf}\{F(x):x\in C,\ G(x)\in -S\},
\]
with \(F:X\to Y\cup\{+\infty_Y\}\), \(G:X\to Z\cup\{+\infty_Z\}\), and feasible set \(A=C\cap G^{-1}(-S)\). The key composite object is \(F+I_A\). Using extended epigraphs
\[
\mathbb Epi\,F^*=\{(L,U)\in L(X,Y)\times\mathcal P_p(Y):F^*(L)\preceq_K U\},
\]
the \(WS\)-sum \(U\sqcup V=WSup(U+V)\), and the boxplus operation on extended epigraphs, the paper derives new representations of \(\operatorname{epi}(F+I_A)^*\) through simpler conjugates such as \(F^*\), \(I_C^*\), and \((T\circ G)^*\) [2105.13299].

Under convexity of \(F\) and \(G\), convexity of \(C\), and regularity conditions \((C_1)\), \((C_2)\), and \((C_3)\), these representations become exact:
\[
\operatorname{epi}(F+I_A)^*=A_1,\quad
\operatorname{epi}(F+I_A)^*=A_2,\quad
\operatorname{epi}(F+I_A)^*=A_3,
\]
depending on how many regularity assumptions are imposed. The same epigraph identities are equivalent to three stable vector Farkas lemmas and to stable strong duality for a Lagrange dual problem and two Fenchel–Lagrange dual problems, denoted \((VD_1)\), \((VD_2)\), and \((VD_3)\). In the special case \(Y=\mathbb R\), these duals reduce to the usual scalar Lagrange and Fenchel–Lagrange dual problems [2105.13299].

The duality message is therefore twofold. In set-valued optimization, conjugacy provides intersection-type dual representations. In vector optimization, conjugacy of composite mappings controls Farkas certificates, perturbation stability, and strong duality.

## 6. Later variants, applications, and conceptual distinctions

A recent extension replaces closed convexity by even convexity. For a cone-ordered set-valued map \(f:X\to P(Z)\), \(K\)-e-convexity means that the \(K\)-epigraph is evenly convex, that is, an intersection of open half-spaces. The associated \(c\)-conjugate is defined with a domain-filtered coupling depending on \((x^*,y^*,z^*,\alpha)\in X^*\times X^*\times(K^*\setminus\{0\})\times\mathbb R\), and the resulting biconjugation theorem states
\[
f^{cc'}=K\text{-eco}\,f.
\]
Moreover, \(K\)-e-convex maps are exactly the pointwise suprema of their e-affine minorants. This is a genuine variant of Fenchel–Moreau theory, with the e-convex hull replacing the closed convex hull [2501.06079].

Set-valued conjugates also appear in PDE and control-related constructions. In Visetti’s set-valued Hamilton–Jacobi framework, the Hamiltonian is the Fenchel conjugate of a set-valued Lagrangian \(\bar L=L+C\),
\[
\bar L^*(p,\zeta)=\sup_{x\in\mathbb R^n}\big[S_{(p,\zeta)}(x)-_\zeta \bar L(x)\big],
\]
and for \(\zeta\in B^+(\hat z)\) one obtains the explicit scalarized formula
\[
\bar L^*(p,\zeta)=L_\zeta^*(p)\hat z + H^+(\zeta),
\qquad
\inf_{z\in \bar L^*(p,\zeta)}\zeta\cdot z=L_\zeta^*(p).
\]
This converts the set-valued Hamiltonian into a family of half-spaces parameterized by \(\zeta\), and the characteristic system is then driven by the scalar conjugates \(L_\zeta^*\) [2201.01231].

Several conceptual distinctions follow from the literature. First, there is no single universal notion of Fenchel conjugate for set-valued mappings: some definitions produce a real-valued support function of the graph, others a set-valued half-space map, and still others a set-valued weak-supremum map on operator spaces [2305.17612] [1011.5860] [2105.13299]. Second, exact calculus and exact biconjugation are qualification-sensitive; without convexity or without the stated interior, continuity, \(qri\), \(sqri\), Slater, or polyhedral assumptions, the theory typically yields only inequalities rather than identities [2305.17612] [2105.13299]. Third, scalarization is powerful but not exhaustive: direct ordered constructions based on weak supremum and extended epigraphs show that non-scalarized duality is possible in the vector case [2105.13299].

Taken together, these developments establish Fenchel conjugation for set-valued mappings as a family of rigorous duality formalisms rather than a single definition. The common core is representation by supporting dual objects; the major differences lie in whether one supports graphs, values, or ordered epigraphs, and in whether the relevant hull under biconjugation is a closed convex hull, a graph hull, or an evenly convex hull.

Source: https://www.emergentmind.com/topics/fenchel-conjugate-for-set-valued-mappings