Fenchel Conjugate for Set-Valued Mappings

• Fenchel conjugation for set-valued mappings is an extension of classical convex duality that encodes multifunctions via supporting dual objects through scalarization and order methods.
• Methodologies include scalarization-based conjugates, graph-support functions, and residuated order-theoretic constructions, each offering distinct dual representations in optimization.
• Calculus rules and biconjugation vary by framework, with applications in vector optimization, control, and PDEs highlighting the approach’s practical significance.

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 (Schrage, 2010, Nam et al., 2023, Dinh et al., 2021).

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 (Heyde et al., 2011, Schrage, 2010, Ararat, 2023).

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 (Nam et al., 2023).

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 $X^*\times Y^*$ (Dinh et al., 2021).

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 $P_A(Z,C)=\{A\subset Z: A=A+C\},$0
Vector weak-order	set-valued map on $P_A(Z,C)=\{A\subset Z: A=A+C\},$1	$P_A(Z,C)=\{A\subset Z: A=A+C\},$2, 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 $P_A(Z,C)=\{A\subset Z: A=A+C\},$3, Nam, Sandine, Thieu, and Yen define the Fenchel conjugate by

$P_A(Z,C)=\{A\subset Z: A=A+C\},$4

equivalently,

$P_A(Z,C)=\{A\subset Z: A=A+C\},$5

In this formulation the conjugate is an extended real-valued convex function on $P_A(Z,C)=\{A\subset Z: A=A+C\},$6, proper and lower semicontinuous whenever $P_A(Z,C)=\{A\subset Z: A=A+C\},$7 (Nam et al., 2023).

In the upper-set approach, the conjugate remains set-valued. For a map $P_A(Z,C)=\{A\subset Z: A=A+C\},$8, scalarizations are defined by

$P_A(Z,C)=\{A\subset Z: A=A+C\},$9

and the Legendre–Fenchel conjugate is assembled as a family of half-spaces. In one standard formulation,

$$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$0

so the set-valued conjugate is identified with the scalar conjugates of the scalarizations (Schrage, 2010).

The order-theoretic extension of Hamel’s framework replaces ordinary difference by residuation. For $$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$1, one introduces conaffine maps

$$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$2

and defines

$$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$3

Its scalarization formula is explicit: $$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$4 This construction is designed to treat proper and improper cases uniformly (Hamel et al., 2010).

In the vector-valued formulation, the conjugate of $$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$5 is

$$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$6

with $$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$7-epigraph

$$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$8

A basic characterization is

$$G(Z,C)=\{A\subset Z: A=\operatorname{cl}\operatorname{conv}(A+C)\},$$9

Here the conjugate is set-valued because $F:X\rightrightarrows Y$0 is a subset of $F:X\rightrightarrows Y$1, not a single point (Dinh et al., 2021).

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:X\rightrightarrows Y$2,

$F:X\rightrightarrows Y$3

and more generally the same intersection reconstructs $F:X\rightrightarrows Y$4. 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 (Schrage, 2010, Hamel et al., 2010).

Heyde and Schrage work in $F:X\rightrightarrows Y$5, the family of upper closed subsets of a preordered topological vector space, and use scalarizations

$F:X\rightrightarrows Y$6

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 $F:X\rightrightarrows Y$7 of all scalarizations of $F:X\rightrightarrows Y$8 (Heyde et al., 2011).

In the graph-support framework, scalarization is replaced by direct graph support, but the classical Fenchel–Young pattern survives. For every $F:X\rightrightarrows Y$9,

$$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$0

If $$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$1 is convex, equality holds if and only if

$$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$2

so equality is characterized by a coderivative condition. For proper, closed, convex $$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$3, the subdifferential of $$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$4 is linked to coderivatives by

$$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$5

This places multifunction conjugacy directly inside convex generalized differentiation (Nam et al., 2023).

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 $$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$6-ordered structures, weak suprema, and positive operators $$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$7 (Dinh et al., 2021).

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 $$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$8,

$$\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\},$$9

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{gph}F$0, quasi-relative interior, strong quasi-relative interior, closedness of graphs, and polyhedral convexity (Nam et al., 2023).

A recent extension studies set-valued convex compositions in the lattice $\operatorname{gph}F$1. If $\operatorname{gph}F$2, then the scalarization satisfies

$\operatorname{gph}F$3

Under a weak$\operatorname{gph}F$4-compact generator assumption for the dual cone of the intermediate order, together with unboundedness or strict monotonicity assumptions, the conjugate obeys the chain rule

$\operatorname{gph}F$5

and this yields an exact dual representation of $\operatorname{gph}F$6 when the composition is proper and scalarly closed (Ararat, 2023).

Biconjugation is similarly framework-dependent. In the graph-support theory,

$\operatorname{gph}F$7

so the biconjugate is the indicator of the closed convex hull of the graph. In the upper-set theory of scalar representation and conjugation,

$\operatorname{gph}F$8

and if $\operatorname{gph}F$9 is convex and closed then $F:X\to Y\cup\{+\infty_Y\}$0. In the residuated framework for $F:X\to Y\cup\{+\infty_Y\}$1, the Fenchel–Moreau type theorem states that closed convex $F:X\to Y\cup\{+\infty_Y\}$2 satisfies $F:X\to Y\cup\{+\infty_Y\}$3 (Nam et al., 2023, Schrage, 2010, Hamel et al., 2010).

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:X\to Y\cup\{+\infty_Y\}$4-valued framework of Heyde and Schrage, for a convex $F:X\to Y\cup\{+\infty_Y\}$5 the marginal map

$F:X\to Y\cup\{+\infty_Y\}$6

satisfies weak duality

$F:X\to Y\cup\{+\infty_Y\}$7

and, under the scalarization regularity condition at $F:X\to Y\cup\{+\infty_Y\}$8, the fundamental duality theorem gives

$F:X\to Y\cup\{+\infty_Y\}$9

Thus the primal upper set is reconstructed as an intersection of set-valued conjugates indexed by dual variables (Heyde et al., 2011).

For vector optimization, the paper on epigraphs of conjugate mappings studies

$(Y^\bullet,<_K)$0

with $(Y^\bullet,<_K)$1, $(Y^\bullet,<_K)$2, and feasible set $(Y^\bullet,<_K)$3. The key composite object is $(Y^\bullet,<_K)$4. Using extended epigraphs

$(Y^\bullet,<_K)$5

the $(Y^\bullet,<_K)$6-sum $(Y^\bullet,<_K)$7, and the boxplus operation on extended epigraphs, the paper derives new representations of $(Y^\bullet,<_K)$8 through simpler conjugates such as $(Y^\bullet,<_K)$9, $L(X,Y)\times \mathcal P_p(Y)$0, and $L(X,Y)\times \mathcal P_p(Y)$1 (Dinh et al., 2021).

Under convexity of $L(X,Y)\times \mathcal P_p(Y)$2 and $L(X,Y)\times \mathcal P_p(Y)$3, convexity of $L(X,Y)\times \mathcal P_p(Y)$4, and regularity conditions $L(X,Y)\times \mathcal P_p(Y)$5, $L(X,Y)\times \mathcal P_p(Y)$6, and $L(X,Y)\times \mathcal P_p(Y)$7, these representations become exact: $L(X,Y)\times \mathcal P_p(Y)$8 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 $L(X,Y)\times \mathcal P_p(Y)$9, $X^*\times Y^*$0, and $X^*\times Y^*$1. In the special case $X^*\times Y^*$2, these duals reduce to the usual scalar Lagrange and Fenchel–Lagrange dual problems (Dinh et al., 2021).

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 $X^*\times Y^*$3, $X^*\times Y^*$4-e-convexity means that the $X^*\times Y^*$5-epigraph is evenly convex, that is, an intersection of open half-spaces. The associated $X^*\times Y^*$6-conjugate is defined with a domain-filtered coupling depending on $X^*\times Y^*$7, and the resulting biconjugation theorem states

$X^*\times Y^*$8

Moreover, $X^*\times Y^*$9-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 (Fajardo, 10 Jan 2025).

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 $P_A(Z,C)=\{A\subset Z: A=A+C\},$00,

$P_A(Z,C)=\{A\subset Z: A=A+C\},$01

and for $P_A(Z,C)=\{A\subset Z: A=A+C\},$02 one obtains the explicit scalarized formula

$P_A(Z,C)=\{A\subset Z: A=A+C\},$03

This converts the set-valued Hamiltonian into a family of half-spaces parameterized by $P_A(Z,C)=\{A\subset Z: A=A+C\},$04, and the characteristic system is then driven by the scalar conjugates $P_A(Z,C)=\{A\subset Z: A=A+C\},$05 (Visetti, 2022).

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 (Nam et al., 2023, Schrage, 2010, Dinh et al., 2021). Second, exact calculus and exact biconjugation are qualification-sensitive; without convexity or without the stated interior, continuity, $P_A(Z,C)=\{A\subset Z: A=A+C\},$06, $P_A(Z,C)=\{A\subset Z: A=A+C\},$07, Slater, or polyhedral assumptions, the theory typically yields only inequalities rather than identities (Nam et al., 2023, Dinh et al., 2021). 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 (Dinh et al., 2021).

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.