Capra-Convexity, Convex Factorization and Variational Formulations for the ℓ₀ Pseudonorm

Abstract

The so-called ℓ₀ pseudonorm, or cardinality function, counts the number of nonzero components of a vector. In this paper, we analyze the ℓ₀ pseudonorm by means of so-called Capra (constant along primal rays) conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the ℓ_p-norms, but for the extreme ones). We obtain three main results. First, we show that the ℓ₀ pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex function. Second, we deduce an unexpected consequence, that we call convex factorization: the ℓ₀ pseudonorm coincides, on the unit sphere of the source norm, with a proper convex lower semicontinuous function. Third, we establish a variational formulation for the ℓ₀ pseudonorm by means of generalized top-k dual norms and k-support dual norms (that we formally introduce).

Appendices

Appendix A: Properties of relevant norms for the ℓ ₀ pseudonorm

We provide background on properties of norms that prove relevant for the ℓ₀ pseudonorm. In Section A.1, we review notions related to dual norms. We establish properties of orthant-monotonic and orthant-strictly monotonic norms in Appendix A.2 and of coordinate-k and dual coordinate-k norms in Appendix A.3.

1.1 A.1 Dual Norm, ∥ | ⋅ ∥ |-Duality, Normal Cone

For any norm ∥ | ⋅ ∥ | on $\mathbb {R}^{d}$, we recall that the following expression

$$ \|\!|\! y \!\|\!|_{\star} = \sup\limits_{\|\!|\! x \!\|\!| \leq 1 } \langle{x, y}\rangle , \forall y \in \mathbb{R}^{d} $$

(27)

defines a norm on $\mathbb {R}^{d}$, called the dual norm ∥ | ⋅ ∥ |_⋆ (in [20, Section 15], this operation is widened to a polarity operation between closed gauges). By definition of the dual norm in (27), we have the inequality

$$ \langle{x, y}\rangle \leq \|\!|\! x \!\|\!| \times \|\!|\! y \!\|\!| , \forall \left( {x,y}\right) \in \mathbb{R}^{d} \times \mathbb{R}^{d} . $$

(28a)

We are interested in the case where this inequality is an equality. One says that $y \in \mathbb {R}^{d}$ is ∥ | ⋅ ∥ |-dual to $x \in \mathbb {R}^{d}$, denoted by y ∥_{∥ | ⋅ ∥ |}x, if equality holds in inequality (28a), that is,

$$ y \parallel_{\|\!|\!\cdot\!\|\!|} x \iff \langle{x, y}\rangle = \|\!|\! x \!\|\!| \times \|\!|\! y \!\|\!| . $$

(28b)

The terminology ∥ | ⋅ ∥ |-dual comes from [11, page 2] (see also the vocable of dual vector pair in [7, Equation (1.11)] and of dual vectors in [8, p. 283], whereas it is refered as polar alignment in [6]). It will be convenient to express this notion of ∥ | ⋅ ∥ |-duality in terms of geometric objects of convex analysis. For this purpose, we recall that the normal cone N_C(x) to the (nonempty) closed convex subset ${C} \subset \mathbb {R}^{d}$ at x ∈C is the closed convex cone defined by [10, p.136]

$$ N_{C}(x) = \left\{ y \in \mathbb{R}^{d} \mid \langle{y}, x^{\prime}-x \rangle \leq 0 , \forall x^{\prime} \in C \right\} . $$

(29)

Now, an easy computation shows that the notion of ∥ | ⋅ ∥ |-duality can be rewritten in terms of normal cone $N_{\mathbb {B} }$ as follows:

$$ y \parallel_{\|\!|\! \cdot \!\|\!|} x \iff y \in N_{\mathbb{B} }\left( { \frac{ x }{\|\!|\! x \!\|\!| } }\right) , \forall \left( {x,y}\right) \in \left( { \mathbb{R}^{d}\setminus\{0\} }\right) \times \mathbb{R}^{d} . $$

(30)

1.2 A.2 Properties of Orthant-Strictly Monotonic Norms

We provide useful properties of orthant-monotonic and orthant-strictly monotonic norms (see Definition 4). We recall that $x_{K} \in {\mathcal {R}}_{K}$ denotes the vector which coincides with x, except for the components outside of K that vanish, and that the subspace ${\mathcal {R}}_{K}$ of $\mathbb {R}^{d}$ has been defined in (7).

Proposition 5

Let ∥ | ⋅ ∥ | be an orthant-monotonic norm on $\mathbb {R}^{d}$. Then, the dual norm ∥ | ⋅ ∥ |_⋆ is orthant-monotonic, and the norm ∥ | ⋅ ∥ | is increasing with the coordinate subspaces, in the sense that, for any $x \in \mathbb {R}^{d}$ and any J ⊂K ⊂⟦1,d⟧, we have ∥ | x_J ∥ |≤∥ | u_K ∥ |.

Proof

Let ∥ | ⋅ ∥ | be an orthant-monotonic norm on $\mathbb {R}^{d}$. Then, by [7, Theorem 2.23], the dual norm ∥ | ⋅ ∥ |_⋆ is also orthant-monotonic and, by [11, Proposition 2.4], we have that ∥ | u ∥ |≤∥ | u + v ∥ |, for any subset J ⊂⟦1,d⟧ and for any vectors $u \in {\mathcal {R}}_{J}$ and $v \in {\mathcal {R}}_{-J}$ (following notation from game theory, we have denoted by −J the complementary subset of J ⊂⟦1,d⟧, that is, J ∪(−J) = ⟦1,d⟧ and J ∩(−J) = ∅). We consider $x \in \mathbb {R}^{d}$ and J ⊂K ⊂⟦1,d⟧. By setting $u=x_{J} \in {\mathcal {R}}_{J}$ and v = x_K −x_J, we get that $v \in {\mathcal {R}}_{-J}$, hence that ∥ | x_J ∥ |≤∥ | u_K ∥ |. □

Proposition 6

Let ∥ | ⋅ ∥ | be an orthant-strictly monotonic norm on $\mathbb {R}^{d}$. Then

• (a) the norm ∥ | ⋅ ∥ | is strictly increasing with the coordinate subspaces in the sense that, for any $x \in \mathbb {R}^{d}$ and any $J \subsetneq K \subset {\llbracket 1,d \rrbracket }$, we have x_J≠x_K ⇒∥ | x_J ∥ |< ∥ | u_K ∥ |.

• (b) for any vector $u \in \mathbb {R}^{d}\setminus \{0\}$, there exists a vector $v \in \mathbb {R}^{d}\setminus \{0\}$ such that supp(v) = supp(u), that u ∘ v ≥ 0, and that v is ∥ | ⋅ ∥ |-dual to u, that is, 〈u,v〉= ∥ | u ∥ |×∥ | v ∥ |_⋆.

Proof

• (a) Let $x \in \mathbb {R}^{d}$ and $J \subsetneq K \subset {\llbracket 1,d \rrbracket }$ be such that x_J≠x_K. We will show that ∥ | u_K ∥ |> ∥ | x_J ∥ |.

For this purpose, we set u = x_J and v = x_K −x_J. Thus, we get that $u \in {\mathcal {R}}_{K}$ and $v \in {\mathcal {R}}_{-K}\setminus \{0\}$ (since $J \subsetneq K$ and x_J≠x_K), that is, u = u_K and v = v_−K≠ 0. We are going to show that ∥ | u + v ∥ |> ∥ | u ∥ |. On the one hand, by definition of the module of a vector, we easily see that |w|= |w_K|+ |w_−K|, for any vector $w \in \mathbb {R}^{d}$. Thus, we have ${|u+v|} = {|\left ({u+v}_{K} \right )|} + {| \left ({u+v}_{-K} \right )|} = {| u_{K}+v_{K} |} + {| u_{-K}+v_{-K} |} = {| u_{K}+0 |} + {| 0+v_{-K} |} = {| u_{K} |} + {|v_{-K} |} > {| u_{K} |} ={| u |}$ since |v_−K|> 0 as v = v_−K≠ 0, and since u = u_K. On the other hand, we easily get that $\left ({u+v}\right )~\circ ~u = \left ({ \left ({u+v}\right )_{K}~\circ ~u_{K} }\right ) + \left ({ \left ({u+v}\right )_{-K}~\circ ~u_{-K} }\right ) = \left ({ u_{K}~\circ ~u_{K} }\right ) + \left ({ v_{-K}~\circ ~u_{-K} }\right ) = \left ({ u_{K}~\circ ~u_{K} }\right )$, because u_−K = 0. Therefore, we get that $\left ({u+v}\right )~\circ ~u = \left ({ u_{K}~\circ ~u_{K} }\right ) \geq 0$.

From |u + v|> |u| and $\left ({u+v}\right )~\circ ~u \geq 0$, we deduce that ∥ | u + v ∥ |> ∥ | u ∥ | by Definition 4 as the norm ∥ | ⋅ ∥ | is orthant-strictly monotonic. Since u = x_J and v = x_K −x_J, we conclude that ∥ | u_K ∥ |> ∥ | x_J ∥ |.

• (b) Let $u \in \mathbb {R}^{d}\setminus \{0\}$ be given and let us put K = supp(u)≠∅. As the norm ∥ | ⋅ ∥ | is orthant-strictly monotonic, it is orthant-monotonic; hence, by [11, Proposition 2.4], there exists a vector $v \in \mathbb {R}^{d}\setminus \{0\}$ such that supp(v) ⊂supp(u), that u ∘ v ≥ 0 and that v is ∥ | ⋅ ∥ |-dual to u, as in (28b), that is, 〈u,v〉= ∥ | u ∥ |×∥ | v ∥ |_⋆. Thus J = supp(v) ⊂K = supp(u). We will now show that $J \subsetneq K$ is impossible, hence that J = K, thus proving that Item (b) holds true with the above vector v.

Writing that 〈u,v〉= ∥ | u ∥ |×∥ | v ∥ |_⋆ (using that u = u_K and v = v_K = v_J), we obtain

$$ \|\!|\! u \!\|\!| \times \|\!|\! v \!\|\!|_{\star} = \langle{u, v}\rangle = \langle{u_{K}, v}\rangle = \langle{u_{K}, v_{K}}\rangle = \langle{u_{K}, v_{J}}\rangle = \langle{u_{J}, v_{J}}\rangle = \langle{u_{J}, v}\rangle , $$

by obvious properties of the scalar product 〈⋅,⋅〉. As a consequence, we get that $\{u_{K},u_{J} \} \subset \arg \max \limits _{\|\!|\! x \!\|\!| \leq \|\!|\! u \!\|\!| } \langle {x, v}\rangle$, by definition (27) of ∥ | v ∥ |_⋆, because ∥ | u ∥ |= ∥ | u_K ∥ |≥∥ | u_J ∥ |, by Proposition 5 since J ⊂K and the norm ∥ | ⋅ ∥ | is orthant-monotonic. But any solution in $\arg \max \limits _{\|\!|\! x \!\|\!| \leq \|\!|\! u \!\|\!| } \langle {x, v}\rangle$ belongs to the frontier of the ball of radius ∥ | u ∥ |, hence has exactly norm ∥ | u ∥ |. Thus, we deduce that ∥ | u ∥ |= ∥ | u_K ∥ |= ∥ | u_J ∥ |. If we had $J = \text {supp}({v}) \subsetneq K = \text {supp}({u})$, we would have u_J≠u_K, hence ∥ | u_K ∥ |> ∥ | u_J ∥ | by Item (a); this would be in contradiction with ∥ | u_K ∥ |= ∥ | u_J ∥ |. Therefore, J = supp(v) = K = supp(u).

This ends the proof. □

1.3 A.3 Properties of Coordinate-k and Dual Coordinate-k Norms, and of Generalized top-k and k-Support Dual Norms

We establish useful properties of coordinate-k and dual coordinate-k norms (Definition 2), and of generalized top-k and k-support dual norms (Definition 3).

Proposition 7

Let ∥ | ⋅ ∥ | be a source norm on $\mathbb {R}^{d}$.

Coordinate-k norms are greater than k-support dual norms, that is,

$$ \|\!|\! x \!\|\!|^{\mathcal{R}}_{(k)} \geq \|\!|\! x \!\|\!|^{\star \text{sn}}_{\star, (k)} , \forall x \in \mathbb{R}^{d} , \forall k\in{\llbracket 1,d \rrbracket} , $$

(31a)

whereas dual coordinate-k norms are lower than generalized top-k dual norms, that is,

$$ \|\!|\! y \!\|\!|^{\mathcal{R}}_{(k), \star} \leq \|\!|\! y \!\|\!|^{\text{tn}}_{\star, (k)} , \forall y \in \mathbb{R}^{d} , \forall k\in{\llbracket 1,d \rrbracket} . $$

(31b)

If the source norm norm ∥ | ⋅ ∥ | is orthant-monotonic, then equalities hold true, that is,

$$ \|\!|\!\cdot\!\|\!| \text{is orthant-monotonic} \Rightarrow \forall k\in{\llbracket 1,d \rrbracket} \quad \begin{cases} \|\!|\! \cdot \!\|\!|^{\mathcal{R}}_{(k)} \! &= \|\!|\! \cdot \!\|\!|^{\star \text{sn}}_{\star, (k)} , \\ \|\!|\! \cdot \!\|\!|^{\mathcal{R}}_{(k), \star} \! &= \|\!|\! \cdot \!\|\!|^{\text{tn}}_{\star, (k)} . \end{cases} $$

(32)

Proof

It is known that, for any nonempty subset K ⊂⟦1,d⟧, we have the inequality ∥ | ⋅ ∥ |_K,⋆ ≤∥ | ⋅ ∥ |_⋆,K (see [11, Proposition 2.2]). From the definition (10) of the generalized top-k dual norm, and the definition (8) of the dual coordinate-k norm, we get that $\|\!|\! y \!\|\!|^{\mathcal {R}}_{(k), \star } = \sup _{{|K|} \leq k} \|\!|\! y_{K} \!\|\!|_{K,\star } \leq \sup _{{|K|} \leq k} \|\!|\! y_{K} \!\|\!|_{\star ,K} = \|\!|\! y \!\|\!|^{\text {tn}}_{\star , (k)}$, hence we obtain (31b). By taking the dual norms, we get (31a).

The norms for which the equality ∥ | ⋅ ∥ |_K,⋆ = ∥ | ⋅ ∥ |_⋆,K holds true for all nonempty subsets K ⊂⟦1,d⟧, are the orthant-monotonic norms ([7, Characterization 2.26], [11, Theorem 3.2]). Therefore, if the norm ∥ | ⋅ ∥ | is orthant-monotonic, from the definition (10) of the generalized top-k dual norm, we get that the inequality (31b) becomes an equality. Then, the inequality (31a) also becomes an equality by taking the dual norm as in (27). Thus, we have obtained (32).

This ends the proof. □

Proposition 8

Let ∥ | ⋅ ∥ | be a source norm on $\mathbb {R}^{d}$. Let $y\in \mathbb {R}^{d}$ and l ∈⟦1,d⟧. If the dual norm ∥ | ⋅ ∥ |_⋆ is orthant-strictly monotonic, we have that

$$ \ell_0({y}) = l \implies \begin{cases} \|\!|\! y \!\|\!|^{\mathcal{R}}_{(1), \star} < {\cdots} < \|\!|\! y \!\|\!|^{\mathcal{R}}_{(l-1), \star} < \|\!|\! y \!\|\!|^{\mathcal{R}}_{(l), \star} = {\cdots} = \|\!|\! y \!\|\!|^{\mathcal{R}}_{(d), \star} =\|\!|\! y \!\|\!|_{\star} , \\ \|\!|\! y \!\|\!|^{\text{tn}}_{\star,(1)} < {\cdots} < \|\!|\! y \!\|\!|^{\text{tn}}_{\star,(l-1)} < \|\!|\! y \!\|\!|^{\text{tn}}_{\star,(l)} = {\cdots} = \|\!|\! y \!\|\!|^{\text{tn}}_{\star,(d)} =\|\!|\! y \!\|\!|_{\star} . \end{cases} $$

(33)

Proof

We consider $y\in \mathbb {R}^{d}$. We put L = supp(y) and we suppose that ℓ₀(y) = |L|= l.

Since the norm ∥ | ⋅ ∥ |_⋆ is orthant-strictly monotonic, it is orthant-monotonic and so is ∥ | ⋅ ∥ | by Proposition 5. By (32) in Proposition 7, we get that $\|\!|\! \cdot \!\|\!|^{\mathcal {R}}_{(j)} = \|\!|\! \cdot \!\|\!|^{\star \text {sn}}_{\star , (j)}$ and $\|\!|\! \cdot \!\|\!|^{\mathcal {R}}_{(j), \star } = \|\!|\! \cdot \!\|\!|^{\text {tn}}_{\star , (j)}$, for j ∈⟦1,d⟧ (with the convention that these are the null seminorms in the case j = 0). Therefore, we can translate all the results, obtained in [4], with coordinate-k and dual coordinate-k norms, into results regarding generalized top-k and k-support dual norms. As an application, by [4, Equation (18)], we get, from ℓ₀(y) = l, that

$$ \|\!|\! y \!\|\!|^{\text{tn}}_{\star, (1)} \leq {\cdots} \leq \|\!|\! y \!\|\!|^{\text{tn}}_{\star, (j)} \leq \|\!|\! y \!\|\!|^{\text{tn}}_{\star, (j+1)} \leq {\cdots} \leq \|\!|\! y \!\|\!|^{\text{tn}}_{\star, (d)} =\|\!|\! y \!\|\!|_{\star} , \forall y \in \mathbb{R}^{d} . $$

(34)

We now prove (33) in two steps.

We first show that $\|\!|\! y \!\|\!|^{\text {tn}}_{\star , (l)} = {\cdots } = \|\!|\! y \!\|\!|^{\text {tn}}_{\star , (d)} =\|\!|\! y \!\|\!|_{\star }$ (the right hand side of (33)). Since y = y_L, by definition of the set L = supp(y), we have that $\|\!|\! y \!\|\!|_{\star } = \|\!|\! y_{L} \!\|\!|_{\star } \leq \sup _{{|K|} \leq l} \|\!|\! y_{K} \!\|\!|_{\star } = \|\!|\!y\!\|\!|^{\text {tn}}_{\star , (l)}$ by the very definition (10) of the generalized top-l dual norm $\|\!|\! \cdot \!\|\!|^{\text {tn}}_{\star , (l)}$. By (34), we conclude that $\|\!|\!y\!\|\!|^{\text {tn}}_{\star , (l)} = {\cdots } = \|\!|\!y\!\|\!|^{\text {tn}}_{\star , (d)} =\|\!|\! y \!\|\!|_{\star }$.

Second, we show that $\|\!|\!y\!\|\!|^{\text {tn}}_{\star , (1)} < {\cdots } < \|\!|\!y\!\|\!|^{\text {tn}}_{\star , (l-1)} < \|\!|\!y\!\|\!|^{\text {tn}}_{\star , (l)}$ (the left hand side of (33)). There is nothing to show for l = 0. Now, for l ≥ 1 and for any k ∈⟦0,1 − 1⟧, we have

$$ \begin{array}{@{}rcl@{}} \|\!|\! y \!\|\!|^{\text{tn}}_{\star, (k)} \!&=& \sup\limits_{{|K|} \leq k} \|\!|\! y_{K} \!\|\!|_{\star} \qquad\quad (\text{by definition (10) of the generalized top- \textit{k} dual norm})\\ \!&=& \sup\limits_{{|K|} \leq k} \|\!|\! y_{K \cap L} \!\|\!|_{\star} \qquad\quad(\text{because} y_{L}=y \text{by definition of the set} L=\text{supp}({y})) \\ \!&=& \sup\limits_{{|K^{\prime}|} \leq k, K^{\prime} \subset L} \|\!|\! y_{K^{\prime}} \!\|\!|_{\star} \qquad\qquad\qquad\qquad\qquad\qquad\quad(\text{by setting} K^{\prime}=K \cap L )\\ \!&=& \sup\limits_{{|K|} \leq k, K \subset L} \|\!|\! y_{K} \!\|\!|_{\star} \qquad\qquad\quad\quad(\text{the same but with notation \textit{K} instead of} K^{\prime})\\ \!&=& \sup\limits_{{|K|} \leq k, K \subsetneq L } \|\!|\! y_{K} \!\|\!|_{\star} \quad(\text{because} {|K|} \leq k \leq l-1 < l ={|L|} \text{implies that} K \neq L )\\ \!&<& \underset{\underset{K \subsetneq L}{{|K|} \leq k, j \in L\setminus K}}{\sup} \|\!|\! y_{K \cup \{{j}\}} \!\|\!|_{\star} \end{array} $$

because the set L ∖K is nonempty (having cardinality |L|−|K|= l −|K|≥k + 1 −|K|≥ 1), and because, since the norm ∥ | ⋅ ∥ |_⋆ is orthant-strictly monotonic, using Item 6 in Proposition 6, we obtain that ∥ | y_K ∥ |_⋆ < ∥ | y_K∪{j} ∥ |_⋆ as y_K≠y_K∪{j} for at least one j ∈L ∖K since L = supp(y)

$$ \begin{array}{@{}rcl@{}} & \leq& \sup\limits_{{|J|} \leq k+1, J \subset L } \|\!|\! y_{J} \!\|\!|_{\star} \\ &&\qquad\qquad(\text{as all the subsets} K^{\prime}=K \cup \{j\} \text{are such that} K^{\prime} \subset L \text{and} {|K^{\prime}|}=k+1 )\\ & \leq& \|\!|\! y \!\|\!|^{\text{tn}}_{\star, (k+1)} \end{array} $$

by definition (10) of the generalized top-(k + 1) dual norm (in fact the last inequality is easily shown to be an equality as y_L = y). Thus, for any k ∈⟦0,1 − 1⟧, we have established that $\|\!|\! y \!\|\!|^{\text {tn}}_{\star , (k)} < \|\!|\!y\!\|\!|^{\text {tn}}_{\star , (k+1)}$.

This ends the proof. □

Appendix B: Proposition 9

We reproduce here [4, Proposition 4.5] in order to simplify the reading of the proof of Proposition 3.

Proposition 9

([4, Proposition 4.5]) Let ∥ | ⋅ ∥ | be a norm on $\mathbb {R}^{d}$, with associated sequence $\left \{\|\!|\! \cdot \!\|\!|^{\mathcal {R}}_{(j)}\right \}_{j\in {\llbracket 1,d \rrbracket }}$ of coordinate-k norms and sequence $\left \{\|\!|\! \cdot \!\|\!|^{\mathcal {R}}_{(j), \star }\right \}_{j\in {\llbracket 1,d \rrbracket }}$ of dual coordinate-k norms, as in Definition 2, and with associated Capra coupling ¢ in (4).

• 1. For any function $\varphi : {\llbracket 0,d \rrbracket } \to {\overline {\mathbb R}}$, we have

  $$ ({ \varphi \circ \ell_0})^{\cent{\cent^{\prime}}}({x}) = \left( \left( { \varphi \circ \ell_0}\right){\cent} \right)^{\star \prime} \left( { \frac{x}{\|\!|\! x \!\|\!|} }\right) , \forall x \in \mathbb{R}^{d}\backslash\{0\} , $$

  (35a)

  where the closed convex function $\left (\left ({\varphi \circ \ell _{0}}\right )^{\cent } \right )^{\star \prime }$ has the following expression as a Fenchel conjugate

  $$ \left( \left( { \varphi \circ \ell_0}\right){\cent} \right)^{\star \prime} = \left( { \sup\limits_{J \in{\llbracket 1,d \rrbracket}} \left[{ \|\!|\! \cdot \!\|\!|^{\mathcal{R}}_{(j), \star} -\varphi(j) }\right] }\right)^{\star \prime} , $$

  (35b)

  and also has the following four expressions as a Fenchel biconjugate

  $$ = \left( \inf_{J \in{\llbracket 1,d \rrbracket}} \left[ \delta_{\mathbb{B}^{\mathcal {R}}_{(j)} } \dotplus \varphi(j) \right] \right)^{\star\star \prime} , $$

  (35c)

  hence the function $\left (\left ({ \varphi \circ \ell _0}\right ){\cent } \right )^{\star \prime }$ is the largest closed convex function below the integer valued function $\inf _{J \in {\llbracket 1,d \rrbracket }} \left [ \delta _{\mathbb {B}^{\mathcal {R}}_{(j)} } \dotplus \varphi (j) \right ]$, such that $x \in \mathbb {B}^{\mathcal {R}}_{(j)} \backslash \mathbb {B}^{\mathcal {R}}_{(j-1)} \mapsto \varphi (j)$ for l ∈⟦1,d⟧, and $x \in \mathbb {B}^{\mathcal {R}}_{(0)} = \{0\} \mapsto \varphi ({0})$, the function being infinite outside $\mathbb {B}^{\mathcal {R}}_{(d)}= \mathbb {B}$, that is, with the convention that $\mathbb {B}^{\mathcal {R}}_{(0)}=\{0\}$ and that $\inf \emptyset = +\infty$

  $$ \begin{array}{@{}rcl@{}} &=& \left( x \mapsto \inf \left\{ \varphi(j) \right\} { x \in \mathbb{B}^{\mathcal {R}}_{(j)} , j \in {\llbracket 0,d \rrbracket} } \right)^{\star\star \prime} , \end{array} $$

  (35d)
  $$ \begin{array}{@{}rcl@{}} &=& \left( \inf_{J \in{\llbracket 1,d \rrbracket}} \left[\delta_{\mathbb{S}^{\mathcal {R}}_{(j)} } \dotplus \varphi(j)\right] \right)^{\star\star \prime} , \end{array} $$

  (35e)

  hence the function $\left (\left ({ \varphi \circ \ell _0}\right ){\cent } \right )^{\star \prime }$ is the largest closed convex function below the integer valued function $\inf _{J \in {\llbracket 1,d \rrbracket }} \left [ \delta _{\mathbb {S}^{\mathcal {R}}_{(j)} } \dotplus \varphi (j)\right ]$, that is, with the convention that $\mathbb {S}^{\mathcal {R}}_{(0)}=\{0\}$ and that $\inf \emptyset = +\infty$

  $$ =\left( x \mapsto \inf \left\{ \varphi(j) \mid x \in \mathbb{S}^{\mathcal {R}}_{(j)} , \j \in {\llbracket 0,d \rrbracket} \right\} \right) . $$

  (35f)

• 2. For any function $\varphi : {\llbracket 0,d \rrbracket } \to \mathbb {R}$, that is, with finite values, the function $\left (\left ({ \varphi \circ \ell _0}\right ){\cent } \right )^{\star \prime }$ is proper convex lsc and has the following variational expression (where Δ_d+ 1 denotes the simplex of $\mathbb {R}^{d+1}$)

  $$ \left( \left( { \varphi \circ \ell_0}\right){\cent} \right)^{\star \prime} ({x}) = \underset{\underset{x \in \sum\limits_{j=1 }^{d } \lambda_j \mathbb{B}^{\mathcal {R}}_{(j)}}{ \left( {\lambda_{0},\lambda_{1},\ldots,\lambda_{d}}\right) \in {\Delta}_{d+1}}}{\min} {\sum}_{j=0}^{d } \lambda_j \varphi(j) , \forall x \in \mathbb{R}^{d} . $$

  (35g)

• 3. For any function $\varphi : {\llbracket 0,d \rrbracket } \to \mathbb {R}_{+}$, that is, with nonnegative finite values, and such that φ(0) = 0, the function $\left (\left ({ \varphi \circ \ell _0}\right ){\cent } \right )^{\star \prime }$ is proper convex lsc and has the following two variational expressions (notice that, in (35g), the sum starts from j = 0, whereas in (35h) and in (35i), the sum starts from j = 1)

  $$ \begin{array}{@{}rcl@{}} \left( \left( { \varphi \circ \ell_0}\right){\cent} \right)^{\star \prime} ({x}) &=& \underset{\underset{x \in \sum\limits_{j=1 }^{d } \lambda_j \mathbb{S}^{\mathcal {R}}_{(j)}}{ \left( {\lambda_{0},\lambda_{1},\ldots,\lambda_{d}}\right) \in {\Delta}_{d+1}} }{\min} \sum\limits_{j=1 }^{d } \lambda_j \varphi(j) , \forall x \in \mathbb{R}^{d} , \end{array} $$

  (35h)
  $$ \begin{array}{@{}rcl@{}} &=& \underset{ \begin{array}{ccc} z^{(1)} \in \mathbb{R}^{d}, \ldots, z^{(d)} \in \mathbb{R}^{d}\\ \sum\limits_{j=1 }^{d } \|\!|\! z^{(j)} \!\|\!|j \leq 1\\ \sum\limits_{j=1 }^{d } z^{(j)} = x \end{array} }{\min} \sum\limits_{j=1 }^{d } \varphi(j) \|\!|\! z^{(j)} \!\|\!|^{\mathcal{R}}_{(j)} , \forall x \in \mathbb{R}^{d} , \end{array} $$

  (35i)

  and the function $({ \varphi \circ \ell _0})^{\cent {\cent ^{\prime }}}$ has the following variational expression

  $$ ({ \varphi \circ \ell_0})^{\cent{\cent^{\prime}}}({x}) = \frac{ 1 }{ \|\!|\! x \!\|\!| } \underset{ \begin{array}{ccc} z^{(1)} \in \mathbb{R}^{d}, \ldots, z^{(d)} \in \mathbb{R}^{d}\\ \sum\limits_{j=1 }^{d } \|\!|\! z^{j} \!\|\!|j \leq \|\!|\! x \!\|\!|\\ \sum\limits_{j=1 }^{d } z^{(j)} = x \end{array} }{\min} \sum\limits_{j=1 }^{d } \|\!|\! z^{j} \!\|\!|j \varphi(j) , \forall x \in \mathbb{R}^{d}\backslash\{0\} . $$

  (36)

Appendix C: Background on the Fenchel conjugacy on $\mathbb {R}^{d}$

We review concepts and notations related to the Fenchel conjugacy (we refer the reader to [18]). For any function $h: \mathbb {R}^{d} \to {\overline {\mathbb R}}$, its epigraph is $\left \{(w,t) \in \mathbb {R}^{d}\times \mathbb {R}\mid h(w) \leq t\right \}$, its effective domain is dom$h = \left \{w\in \mathbb {R}^{d} \mid h(w)<+\infty \right \}$. A function $h : \mathbb {R}^{d} \to {\overline {\mathbb R}}$ is said to be convex if its epigraph is a convex set, proper if it never takes the value $-\infty$ and that domh≠∅, lower semi continuous (lsc) if its epigraph is closed, closed if it either lsc and nowhere having the value $-\infty$, or is the constant function $-\infty$ [18, p. 15]. Closed convex functions are the two constant functions $-\infty$ and $+\infty$ united with all proper convex lsc functions. In particular, any closed convex function that takes at least one finite value is necessarily proper convex lsc.

For any functions $f : \mathbb {R}^{d} \to {\overline {\mathbb R}}$ and $g : \mathbb {R}^{d} \to {\overline {\mathbb R}}$, we denote

$$ \begin{array}{@{}rcl@{}} {f}^{\star}({y}) &=& \sup\limits_{x \in \mathbb{R}^{d}} \left( { \langle{x, y}\rangle \substack{+}{.} \left( { -f({x}) }\right) }\right) , \forall y \in \mathbb{R}^{d} , \end{array} $$

(37a)

$$ \begin{array}{@{}rcl@{}} g^{\star \prime}({x}) &=& \sup\limits_{y \in \mathbb{Y} } \left( { \langle{x, y}\rangle \substack{+}{.} \left( { -g({y}) }\right) }\right) , \forall x \in \mathbb{R}^{d}, \end{array} $$

(37b)

$$ \begin{array}{@{}rcl@{}} {f}^{\star\star \prime}({x}) &=& \sup\limits_{y \in \mathbb{R}^{d}} \left( { \langle{x, y}\rangle \substack{+}{.} \left( -{f}^{\star}({y})\right) }\right) , \forall x \in \mathbb{R}^{d} . \end{array} $$

(37c)

In convex analysis, one does not use the notation $^{\star \prime }$ in (37b) and $^{\star \star \prime }$ in (37c), but simply ^⋆ and ^⋆⋆. We use $^{\star \prime }$ and $^{\star \star \prime }$ to be consistent with the notation (6b) for general conjugacies.

It is proved that the Fenchel conjugacy (indifferently f↦f^⋆ or $g \mapsto g^{\star \prime }$) induces a one-to-one correspondence between the closed convex functions on $\mathbb {R}^{d}$ and themselves [18, Theorem 5].

In [20, p. 214-215] (see also the historical note in [19, p. 343]), the notions of (Moreau) subgradient and of (Rockafellar) subdifferential are defined for a convex function. Following the definition of the subdifferential of a function with respect to a duality in [1], we define the (Rockafellar-Moreau) subdifferential∂f(x) of a function $f : \mathbb {R}^{d} \to {\overline {\mathbb R}}$ at $x \in \mathbb {R}^{d}$ by

$$ \partial{f}({x}) = \left\{ y \in \mathbb{R}^{d} \mid {f}^{\star}({y}) = \langle{x, y}\rangle \substack{+}{.} \left( { -f({x}) }\right) \right\} . $$

(38a)

When the function f is proper convex and x ∈domf, we recover the classic definition that

$$ \partial{f}({x}) = \left\{ y \in \mathbb{R}^{d} \mid \langle{x^{\prime}-x},{y}\rangle+f({x}) \leq f\left( {x^{\prime}}\right) , \forall x^{\prime} \in \text{dom}f \right\} . $$

(38b)