Remark 2.6. Observe that ${\mathcal{F}}(\rho |\mu)=\infty$ whenever $\rho \not \ll \mu$ and ${\mathcal{F}}(\rho |\nu) = \infty$ whenever $\rho \not \ll \nu$ . This guarantees that ${\pi _1}_{\#} \gamma \ll \mu$ and ${\pi _2}_{\#} \gamma \ll \nu$ for all feasible $\gamma$ , where feasible means that $\mathcal{E}(\gamma) \lt \infty$ . Thus, when $\mu \ll \mathcal{L}$ and $\nu$ is discrete, as in the semi-discrete setting (which will be discussed in the following section), then the first and second marginal of any feasible $\gamma$ will share these properties.

Remark 2.7. For simplicity we assume that the same marginal discrepancy is applied to both marginals in (2.5), but of course in some cases it may be more appropriate to consider two different discrepancies. All results in this article generalize to this case in a canonical way.

Proof. (i) Since $F$ is proper, we can find $s \in (0,\infty)$ with $F(s)\lt \infty$ . Then for all $z \in {\mathbb{R}}$ , $F^\ast (z) = \sup _{x \geq 0} (z \cdot x -F(x)) \geq z \cdot s -F(s) \gt -\infty$ .

(ii) Let $z_1 \leq z_2$ . Then $F^\ast (z_2) = \sup _{x \geq 0} (z_2 \cdot x - F(x)) \geq \sup _{x \geq 0} (z_1 \cdot x - F(x)) = F^\ast (z_1)$ .

(iii) Let $z \leq 0$ . Since $F \geq 0$ , then $F^\ast (z) = \sup _{x \geq 0} (z \cdot x - F(x)) \leq \sup _{x \geq 0} z \cdot x = 0$ .

(iv) Let $z \in (0,\infty)$ . Since $F \geq 0$ , $F^\ast (z)=\infty$ is only possible if any maximizing sequence $x_1,x_2,\ldots$ for $F^\ast (z) = \sup _{x \geq 0} (z \cdot x - F(x))$ diverges to $\infty$ . However, $\lim _{n \to \infty } (z \cdot x_n - F(x_n)) = \lim _{x \to \infty } x \big (z-\frac {F(x)}{x}\big) = -\infty$ since $\lim _{s \to \infty } \frac {F(s)}{s} = \infty$ . So $F^\ast (z)\lt \infty$ .

(v) (i), (iv) and (iii) imply ${\mathrm{dom}}(F^\ast)={\mathbb{R}}$ . By convexity, $F^\ast$ is therefore continuous.

(vi) This is a special case of a classical result in convex analysis, which can be found, for instance, in [Reference Rockafellar59, Thm. 26.3].

(vii) Let $z \in {\mathbb{R}}$ . Then $F^\ast (z) = \sup _{x \geq 0} (z \cdot x - F(x)) \geq -F(0)$ .

(viii) Let $z_1,z_2,\ldots$ and $u_1,u_2,\ldots$ be sequences with $z_n \to -\infty$ as $n \to \infty$ and $u_n \in \partial F^\ast (z_n)$ . By monotonicity of $F^\ast$ , (ii), we have $u_n \geq 0$ . By (iii) and convexity one finds for any $a \geq 0$ with $F(a)\lt \infty$ that $0 \geq F^\ast (0) \geq F^\ast (z_n) + u_n \cdot (0-z_n) \geq a \cdot z_n -F(a) + u_n \cdot |z_n|$ , which implies that $u_n \leq F(a)/|z_n|+a$ . This implies that $\limsup _n u_n \leq a$ . Sending now $a \to 0$ yields the claim.

Remark 2.11 (Feasibility for finite $F(0)$ ). Note that for $F(0)\lt \infty$ the trivial transport plan $\gamma =0$ leads to a finite cost in (2.5) so that $W(\mu ,\nu)\lt \infty$ for all $\mu ,\nu \in {\mathcal M_+}(\Omega)$ .

Remark 2.15 (Existence of optimal weights). Maximizers for (2.10) do not always exist, even when $W_{\textrm {OT}}(\mu ,\nu)\lt \infty$ . A simple sufficient condition for existence is that $c$ is bounded from above on $\Omega \times \Omega$ . More details can be found, for instance, in [Reference Villani69 , Thm. 5.10].

Proof. Fix $i \neq j$ and $w \in {\mathbb{R}}^M$ and recall that ${c}(x,y)=\ell (d(x,y))$ . We have

\begin{equation*} {C}_i(w)\cap {C}_j(w)=\bigcup _{n\in \mathbb{N}}A_n \quad \text{for } A_n=\{x\in \Omega \,|\,{c}(x,x_i)-w_i={c}(x,x_j)-w_j,\,{c}(x,x_i)\leq n\}\,, \end{equation*}

and we will show that the $d$ -dimensional Hausdorff measure of each $A_n$ is zero, ${\mathcal H}^d(A_n)=0$ , which implies $|A_n|=0$ and thus also $|{C}_i(w) \cap {C}_j(w)| = 0$ . Indeed, abbreviating $f=d(\cdot ,x_i)$ and noting that the Jacobian of $f$ equals $1$ almost everywhere, the coarea formula [Reference Ambrosio, Fusco and Pallara1, Thm. 2.93] yields

\begin{equation*} {\mathcal H}^d(A_n) =\int _{A_n}1\,{\textrm {d}}{\mathcal H}^d =\int _{\mathbb{R}}{\mathcal H}^{d-1}(A_n\cap f^{-1}(t))\,{\textrm {d}} t =\int _0^{\ell ^{-1}(n)}{\mathcal H}^{d-1}(A_n\cap f^{-1}(t))\,{\textrm {d}} t\,. \end{equation*}

Now, for $t \in [0,\ell ^{-1}(n)]$ ,

\begin{equation*} A_n\cap f^{-1}(t)=\{x\in \Omega \,|\,d(x,x_i)=t\text{ and }d(x,x_j)\in \ell ^{-1}(\ell (d(x,x_i))+w_j-w_i)\}, \end{equation*}

where $\ell ^{-1}(\ell (d(x,x_i))+w_j-w_i)$ is either empty or single-valued due to the strict monotonicity of $\ell$ . Hence, $A_n\cap f^{-1}(t)$ is contained in the intersection of two non-concentric $(d-1)$ -dimensional spheres and thus is ${\mathcal H}^{d-1}$ -negligible.

Proof of Theorem 2.14. By Kantorovich duality [Reference Villani69, Thm. 5.10] we can write

(2.12) \begin{align} W_{\textrm {OT}}(\mu ,\nu) = \sup \left \{ \int _\Omega \phi \,{\textrm {d}} \mu + \int _\Omega \psi \,{\textrm {d}} \nu \,\Bigg |\, \phi \in L^1(\mu),\, \psi \in L^1(\nu), \right . \nonumber \\[5pt] \left . \vphantom {\int _\Omega } \,\phi (x) + \psi (y) \leq c(x,y) \,\forall \,(x,y) \in \Omega \times \Omega \right \}\,. \end{align}

Since $\nu$ is discrete, $L^1(\nu)$ is isomorphic to ${\mathbb{R}}^M$ under the isomorphism $I:L^1(\nu)\to {\mathbb{R}}^M$ , $\psi \mapsto (\psi (x_1),\ldots ,\psi (x_M))$ . The above dual problem thus becomes

\begin{align*} W_{\textrm {OT}}(\mu ,\nu) = \sup \left \{ \int _\Omega \phi \,{\textrm {d}} \mu + \sum _{i=1}^M w_i\,m_i \,\middle |\, \phi \in L^1(\mu),\, w\in {\mathbb{R}}^M, \right . \\[5pt] \left . \vphantom {\int _\Omega } \, \phi (x) + w_i \leq c(x,x_i) \,\forall \,x\in \Omega ,i\in \{1,\ldots ,M\}\right \}\,. \end{align*}

Next, for fixed $w$ , one can explicitly maximize over $\phi$ , which corresponds to pointwise maximization subject to the constraint. We denote the maximizer by $\phi _{w}$ to emphasize the dependency on $w$ ,

(2.13) \begin{align} \phi _{w}(x) = \min \big \{ c(x,x_i)-w_i\,|\,i =1,\ldots ,M \big \}\,. \end{align}

Since $\infty \gt W_{\textrm {OT}}(\mu ,\nu)-\sum _{i=1}^Mw_i\,m_i\geq \int _\Omega \phi _w\,{\textrm {d}}\mu$ and $\phi _w$ is bounded from below (as $c$ is so in our setting) one must have $\phi _{w} \in L^1(\mu)$ for all $w \in {\mathbb{R}}^M$ , and we find

(2.14) \begin{align} W_{\textrm {OT}}(\mu ,\nu) = \sup \{ {\mathcal{E}}_{\textrm {SD}}(w)\,|\,w \in {\mathbb{R}}^M\} \quad \textrm {with} \quad {\mathcal{E}}_{\textrm {SD}}(w) = \int _\Omega \phi _{w}(x)\,{\textrm {d}} \mu (x) + \sum _{i=1}^M w_i\,m_i\,. \end{align}

Since $\phi _w\in L^1(\mu)$ for any $w\in {\mathbb{R}}^M$ , the residual set $R$ must be $\mu$ -negligible; likewise, the intersection of generalized Laguerre cells is $\mu$ -negligible by Lemma2.17. Consequently,

\begin{equation*} {\mathcal{E}}_{\textrm {SD}}(w) = \sum _{i=1}^M\int _{{C}_i(w)} \phi _{w}(x)\,{\textrm {d}} \mu (x) + \sum _{i=1}^M w_i\,m_i = \sum _{i=1}^M\int _{{C}_i(w)} [{c}(x,x_i)-w_i]\,{\textrm {d}} \mu (x) + \sum _{i=1}^M w_i\,m_i\,, \end{equation*}

which leads to the desired result.

Proof of Theorem 2.16. The condition $\gamma \in \Gamma (\mu ,\nu)$ implies that $\gamma$ can be written as $\gamma =\sum _{i=1}^M \gamma _i \otimes \delta _{x_i}$ where $\gamma _i \in {\mathcal M_+}(\Omega)$ , $\gamma _i(A) \;:\!=\; \gamma (A \times \{ x_i\})$ . Observe that $\sum _{i=1}^M \gamma _i = \mu$ and $\gamma _i(\Omega)=m_i$ . We obtain

(2.15) \begin{align} W_{\textrm {OT}}(\mu ,\nu) \leq \int _{\Omega \times \Omega } c\,{\textrm {d}}\gamma = \sum _{i=1}^M \int _\Omega c(x,x_i)\,{\textrm {d}}\gamma _i(x)\,, \end{align}

where the inequality is an equality if and only if $\gamma$ is optimal. Let $w \in {\mathbb{R}}^M$ . From (2.14) with $\phi _{w}$ given by (2.13), we find

(2.16) \begin{align} W_{\textrm {OT}}(\mu ,\nu) \geq \int _\Omega \phi _{w}(x)\,{\textrm {d}} \mu (x) + \sum _{i=1}^M w_i\,m_i = \sum _{i=1}^M \int _\Omega \left [\phi _{w}(x)+w_i\right ]\,{\textrm {d}}\gamma _i(x)\,, \end{align}

where the inequality is an equality if and only if $w$ is optimal. Subtracting (2.16) from (2.15) yields

(2.17) \begin{align} 0 \leq \sum _{i=1}^M \int _\Omega \left [c(x,x_i)-w_i-\phi _{w}(x)\right ]\,{\textrm {d}}\gamma _i(x) \end{align}

with equality if and only if $\gamma$ and $w$ are optimal. By definition of $\phi _{w}$ , the integrand in each term of the sum is nonnegative and strictly positive for $x \notin {C}_i(w)$ . Therefore (2.17) is an equality if and only if $\gamma _i$ is concentrated on ${C}_i(w)$ for all $i \in \{1,\ldots ,M\}$ . Combining absolute continuity with respect to the Lebesgue measure of $\mu$ and $\gamma _i$ and Lemma2.17 implies that the unique choice is $\gamma _i = \mu {{\LARGE \llcorner }}{C_i(w)}$ . Due to the second marginal constraint, this implies $\mu (C_i(w))=\gamma _i(\Omega)=m_i$ .

Proof. In analogy to the Kantorovich duality (2.12) for the classical optimal transport problem (2.3), we make use of a corresponding duality result for the unbalanced transport problem (2.6),

\begin{align*} W(\mu ,\nu) = \sup \left \{ -\int _\Omega F^\ast (\!-\!\phi (x))\,{\textrm {d}}\mu (x) -\int _\Omega F^\ast (\!-\!\psi (x))\,{\textrm {d}}\nu (x) \,\Bigg |\, \phi , \psi \in \mathcal{C}(\Omega), \right . \\[5pt] \left . \vphantom {\int _\Omega } \phi (x)+\psi (y) \leq c(x,y) \,\forall \,(x,y) \in \Omega \times \Omega \right \}. \end{align*}

This follows from [Reference Liero, Mielke and Savaré45, Thm. 4.11 and Cor. 4.12], where the former establishes the duality formula with $\phi$ and $\psi$ ranging over all lower semi-continuous simple functions and the latter allows us to use continuous functions instead, exploiting the fact that $F^\ast$ is continuous on $\mathbb{R}$ by Lemma2.10 (v). Analogously to the proof of Theorem2.14 (dual tessellation formulation), we now parameterize the function $\psi$ on the set $\{x_i\}_{i=1}^M$ by a vector $w \in {\mathbb{R}}^M$ , $w_i=\psi (x_i)$ , and obtain

(3.2) \begin{align} W(\mu ,\nu) = \sup \left \{ -\int _\Omega F^\ast (\!-\!\phi (x))\,{\textrm {d}}\mu (x) -\sum _{i=1}^Mm_iF^\ast (\!-\!w_i) \,\Bigg |\, \phi \in \mathcal{C}(\Omega), w\in {\mathbb{R}}^M,\right . \nonumber \\[5pt] \left . \vphantom {\sum _{i=1}^M} \phi (x)+w_i \leq c(x,x_i) \,\forall \,x \in \Omega , i\in \{1,\ldots ,M\} \right \}. \end{align}

Next, given $w\in {\mathbb{R}}^M$ we would like to optimize for $\phi$ as we did in (2.13). Note though that $\phi _w=\infty$ on the residual set $R$ , which in unbalanced transport may be nonnegligible despite finite $W(\mu ,\nu)$ . For this reason we argue by truncation: For given $w \in {\mathbb{R}}^M$ and $n \in \mathbb{N}$ , the function $\phi =\phi _{w,n}$ with

\begin{align*} \phi _{w,n}\;:\;\Omega \to {\mathbb{R}},\quad \phi _{w,n}(x) = \min \{n, \min \{c(x,x_i)-w_i\,|\,i \in \{1,\ldots ,M\} \} \} \end{align*}

lies in $\mathcal{C}(\Omega)$ and is feasible in (3.2). Moreover, for fixed $w$ , the sequence $(\phi _{w,n})_{n \in \mathbb{N}}$ is a maximizing sequence for the maximization over $\phi$ , and it converges pointwise monotonically to the function $\phi _{w}$ defined in (2.13). By Lemma2.10 (ii) and (v), $z \mapsto -F^\ast (\!-\!z)$ is real-valued, continuous, and increasing. Since $\phi _{w,n}(x) \geq -\max _i w_i$ , $-F^\ast (\!-\!\phi _{w,n}(x))$ is uniformly bounded from below with respect to $n$ and $x$ . Consequently, also $\phi _w$ is bounded from below. Therefore, the monotone convergence theorem implies that

\begin{align*} \lim _{n \to \infty } -\int _\Omega F^\ast (\!-\!\phi _{w,n}(x))\,{\textrm {d}}\mu (x) = -\int _\Omega F^\ast (\!-\!\phi _{w}(x))\,{\textrm {d}}\mu (x), \end{align*}

where by convention $F^\ast (\!-\!\infty)=\lim _{z\to -\infty }F^\ast (z)=-F(0)$ (see Lemma2.10). With this, (3.2) finally becomes

(3.3) \begin{align} W(\mu ,\nu) = \sup \left \{ -\int _\Omega F^\ast (\!-\!\phi _{w}(x))\,{\textrm {d}}\mu (x)-\sum _{i=1}^M m_iF^\ast (\!-\!w_i) \,\Bigg |\, w \in {\mathbb{R}}^M \right \}\,. \end{align}

Note that this objective is $\gt -\infty$ for all $w \in {\mathbb{R}}^M$ by the lower bound on $\phi _w$ and the monotonicity and real-valuedness of $F^*$ . Now we decompose the integration domain $\Omega$ into $\{{C}_i(w)\}_{i=1}^M$ and $R$ (using once more $\mu \ll \mathcal{L}$ and Lemma2.17; it is only here that we use the radiality of the cost function $c$ so that the cells $C_i(w)$ are well defined with negligible overlap). For $x \in {C}_i(w)$ one finds $\phi _{w}(x)=c(x,x_i)-w_i$ , while for $x \in R$ one obtains $\phi _{w}(x)=\infty$ and therefore $F^\ast (\!-\!\phi _{w}(x))=-F(0)$ . This leads to expression (3.1a) and also implies that ${\mathcal{G}}(w)\gt -\infty$ .

For fixed $x \in \Omega$ the map $w \mapsto \phi _{w}(x)$ is concave (since it is a minimum over affine functions). Moreover, the map $z \mapsto -F^\ast (\!-\!z)$ is concave and increasing (cf. Lemma2.10 (ii)). Therefore, the objective function in (3.3) and consequently $\mathcal{G}$ are concave functions of $w$ .

Remark 3.2 (Finiteness of $\mathcal{G}$ ). ${\mathcal{G}}\not \equiv \infty$ if and only if $\mathcal{G}$ is finite everywhere. Indeed, the last summand in (3.1a) is independent of $w$ , and $-\sum _{i=1}^MF^\ast (\!-\!w_i)\,m_i$ is finite by Lemma 2.10 (v). Finally, if $\tilde {\mathcal{G}}(w)\lt \infty$ for some $w\in {\mathbb{R}}^M$ , where $\tilde {\mathcal{G}}(w)=-\sum _{i=1}^M\int _{{C}_i(w)}F^\ast \big (\!-\!c(x,x_i)+w_i\big)\,{\textrm {d}}\mu (x)$ , then for $g_i(x)\in \partial F^\ast (\!-\!c(x,x_i)+w_i)$ and $\tilde g_i\in \partial F^\ast (w_i)$ we have

\begin{align*} \tilde {\mathcal{G}}(\tilde w) & \leq -\sum _{i=1}^M\int _{{C}_i(\tilde w)}F^\ast \big (\!-\!c(x,x_i)+w_i\big)+g_i(x)(\tilde w_i-w_i)\,{\textrm {d}}\mu (x)\\ & \leq \tilde {\mathcal{G}}(w)+\sum _{i=1}^M|\tilde w_i-w_i|\int _{{C}_i(w)}\tilde g_i\,{\textrm {d}}\mu (x) \lt \infty \end{align*}

for all $\tilde w\in {\mathbb{R}}^M$ by the convexity and monotonicity of $F^\ast$ from Lemma 2.10 (i).

${\mathcal{G}}\not \equiv \infty$ is for instance ensured by $F(0)\lt \infty$ or by boundedness of $\ell$ (which implies $R=\emptyset$ ). Indeed, the former implies boundedness of $F^\ast$ from below by Lemma 2.10 (vii), while the latter implies uniform boundedness of $-c(x,x_i)+w_i$ from below so that in either case the integrand of $\tilde {\mathcal{G}}$ is uniformly bounded from below.

Proof. Let $\gamma \in {\mathcal M_+}(\Omega \times \Omega)$ be such that ${\mathcal{E}}(\gamma)$ in (2.5) is finite. This implies that $\gamma$ can be written as $\gamma =\sum _{i=1}^M \gamma _i \otimes \delta _{x_i}$ for $\gamma _i \in {\mathcal M_+}(\Omega)$ , $\sum _{i=1}^M \gamma _i = {\pi _1}_{\#} \gamma = \rho \ll \mu$ and $\rho (R)=0$ . (Note that the same holds true if (3.4) is assumed instead of ${\mathcal{E}}(\gamma)\lt \infty$ .) We obtain

\begin{align*} {\mathcal{E}}(\gamma) &= \int _{\Omega \times \Omega } c\,{\textrm {d}}\gamma + {\mathcal{F}}(\rho |\mu) + {\mathcal{F}}({\pi _2}_{\#} \gamma |\nu) \\[5pt] &= \sum _{i=1}^M \int _{\Omega \setminus R} {c}(x,x_i)\,{\textrm {d}}\gamma _i(x) + \int _{\Omega \setminus R} F\bigg (\frac {{\textrm {d}} \rho }{{\textrm {d}} \mu }(x)\bigg)\,{\textrm {d}}\mu (x) + F(0) \cdot \mu (R) + \sum _{i=1}^M F\bigg (\frac {\gamma _i(\Omega)}{m_i}\bigg) \cdot m_i \end{align*}

so that the duality gap between the primal and dual formulations (2.6) and (3.1) reads

\begin{align*} {\mathcal{E}}(\gamma)-{\mathcal{G}}(w)&= \sum _{i=1}^M \int _{\Omega \setminus R} {c}(x,x_i)\,{\textrm {d}}\gamma _i(x) + \int _{\Omega \setminus R} \left [ F\bigg (\frac {{\textrm {d}} \rho }{{\textrm {d}} \mu }(x)\bigg)+F^\ast (\!-\!\phi _{w}(x)) \right ] {\textrm {d}}\mu (x) \\[5pt] &\quad + \sum _{i=1}^M \left (F\bigg (\frac {\gamma _i(\Omega)}{m_i}\bigg) +F^\ast (\!-\!w_i) \right) \cdot m_i\,. \end{align*}

Using the Fenchel–Young inequality, which states that $F(s)+F^\ast (z) \geq s \cdot z$ with equality if and only if $z \in \partial F(s)$ or equivalently $s \in \partial F^\ast (z)$ [Reference Bauschke and Combettes6, Prop. 13.13 and Thm. 16.23], we obtain the lower bound

\begin{align*} {\mathcal{E}}(\gamma)-{\mathcal{G}}(w)&\geq \sum _{i=1}^M \int _{\Omega \setminus R} {c}(x,x_i)\,{\textrm {d}}\gamma _i(x) - \int _{\Omega \setminus R} \phi _{w}(x) \, {\textrm {d}}\rho (x) -\sum _{i=1}^M w_i \cdot \gamma _i(\Omega)\\[5pt] &= \sum _{i=1}^M \int _{\Omega \setminus R} \left [ {c}(x,x_i)-w_i-\phi _{w}(x) \right ] {\textrm {d}}\gamma _i(x)\geq 0\,, \end{align*}

where the first inequality is an equality if and only if $\frac {{\textrm {d}} \rho }{{\textrm {d}} \mu }(x) \in \partial F^\ast (\!-\!\phi _{w}(x))$ for $\mu$ -almost every $x \in \Omega \setminus R$ and $\frac {\gamma _i(\Omega)}{m_i} \in \partial F^\ast (\!-\!w_i)$ for $i=1,\ldots ,M$ , and where the second inequality is an equality if and only if $\operatorname{spt}\gamma _i\subset {C}_i(w)$ and thus $\gamma _i=\rho {{\LARGE \llcorner }}{C}_i(w)$ for $i=1,\ldots ,M$ . As a consequence, we have ${\mathcal{E}}(\gamma)-{\mathcal{G}}(w)=0$ if and only if (3.4) holds.

Now let $W(\mu ,\nu)\lt \infty$ and $\gamma$ and $w$ be optimal in (2.6) and (3.1) so that $W(\mu ,\nu)={\mathcal{E}}(\gamma)={\mathcal{G}}(w)\lt \infty$ . Then necessarily ${\mathcal{E}}(\gamma)-{\mathcal{G}}(w)=0$ and so (3.4) holds. Conversely, if (3.4) holds, then if ${\mathcal{E}}(\gamma)\lt \infty$ or ${\mathcal{G}}(w)\lt \infty$ (so that the difference ${\mathcal{E}}(\gamma)-{\mathcal{G}}(w)$ is well defined), the above argument shows that ${\mathcal{E}}(\gamma)-{\mathcal{G}}(w)=0$ , which due to ${\mathcal{E}}(\gamma)\geq W(\mu ,\nu)\geq {\mathcal{G}}(w)$ implies $W(\mu ,\nu)={\mathcal{E}}(\gamma)={\mathcal{G}}(w)$ and thus the optimality of $\gamma$ and $w$ . If on the other hand ${\mathcal{E}}(\gamma)={\mathcal{G}}(w)=\infty$ , then $W(\mu ,\nu)=\infty$ so that $\gamma$ and $w$ are trivially optimal.

Proof. We first show that (3.4b) fully specifies $\rho$ . Let $S$ be the set where $\partial F^\ast$ is not a singleton. By convexity, $S$ is countable. In analogy to Lemma2.17, for any $s \in S$ , the set $\{ x \in {\mathbb{R}}\,|\,-{c}(x,x_i)+w_i=s\}$ is Lebesgue-negligible. Since $S$ is countable, the set $\{ x \in {\mathbb{R}}\,|\,-{c}(x,x_i)+w_i \in S \}$ is Lebesgue-negligible and thus also $\mu$ -negligible. Consequently, $\frac {{\textrm {d}} \rho }{{\textrm {d}} \mu }$ is uniquely defined by (3.4b) on $\Omega$ up to a $\mu$ -negligible set.

For $W(\mu ,\nu)\lt \infty$ , conditions (3.4) are necessary and must therefore be satisfied by any minimizer $\gamma$ (which exists by Theorem2.5). Therefore, as $\rho$ is uniquely determined by (3.4b), so is $\gamma$ by (3.4a). Optimality of $\gamma$ and $w$ ensures that (3.4c) also holds.

Remark 3.5 (Balanced transport). For classical optimal transport with $F=\iota _{\{1\}}$ (such as the Wasserstein-2 distance from Example 2.9 (a)) one obtains $-F^\ast (\!-\!z)=z$ . Then (3.1) becomes (2.10) (and finiteness of $W_{\textrm {OT}}(\mu ,\nu)$ implies that $\mu (R)=0$ ). Furthermore, with $\partial F^\ast (z)=1$ for all $z$ , equation (3.4b) implies $\rho =\mu {{\LARGE \llcorner }}{(\Omega \setminus R)}=\mu$ . Then (3.4a) and (3.4c) become (2.11).

Proof. For any $w \in {\mathbb{R}}^M$ and $\rho \in {\mathcal M_+}(\Omega)$ with $\rho {{\LARGE \llcorner }}{R}=0$ , the objective function in (3.5) is equal to ${\mathcal{E}}(\gamma)$ for $\gamma =\sum _{i=1}^M \rho {{\LARGE \llcorner }}{{C}_i(w)} \otimes \delta _{x_i}$ . Therefore, minimizing (3.5) corresponds to minimizing $\mathcal{E}$ over a particular subset of ${\mathcal M_+}(\Omega \times \Omega)$ , which implies that the right-hand side of (3.5) is no smaller than $W(\mu ,\nu)$ . Now, if $\gamma$ and $w$ are a pair of optimizers for (2.6) and (3.1), then by (3.4), the objective function in (3.5) for $w$ and $\rho ={\pi _1}_{\#} \gamma$ becomes ${\mathcal{E}}(\gamma)=W(\mu ,\nu)$ so that the right-hand side of (3.5) actually equals $W(\mu ,\nu)$ and $w$ and $\rho$ are minimizers of (3.5).

Conversely, if $w$ and $\rho$ minimize (3.5), the induced $\gamma$ must minimize $\mathcal{E}$ .

Remark 3.7 (Optimality of dual variable). The converse conclusion that optimal $w$ in (3.5) are optimal in (3.1) is in general not true. Indeed, (3.5) only depends on $w$ via the cells $\{{C}_i(w)\}_{i=1}^M$ and therefore is invariant with respect to adding the same constant to all components of $w$ , which does not change the cells. In the balanced case, when $F^*(z)=z$ , then (3.1) is also invariant under such transformations (as long as $\mu (\Omega)=\sum _{i=1}^M m_i$ ). However, for general $F$ , $F^*$ is nonlinear, and the objective function of (3.1) is no longer invariant.

Similarly, when the support of the optimal $\rho$ in (3.5) is bounded away from the boundary of some ${C}_i(w)$ (see Fig. 2, right), then slightly changing the corresponding $w_i$ will not affect the value of (3.5), whereas (3.1) will usually not exhibit this invariance.

Figure 2. Semi-discrete transport between the Lebesgue measure on $\Omega =[0,L]^2$ , $L=5$ and a discrete measure with $M=4$ Dirac masses of locations $(x_1,x_2,x_3,x_4)= L \cdot ((0.375,0.375),(0.75,0.35),$ $(0.65,0.75),(0.25,0.8))$ and weights $(m_1,m_2,m_3,m_4)=|\Omega |\cdot (0.38,0.29,0.19,0.14)$ . Top row: optimal cells $\{{C}_i(w)\}_{i=1}^M$ ; the residual set $R$ is represented by white; location of the discrete points $(x_1,\ldots ,x_M)$ is indicated with red dots. Bottom row: optimal marginal $\rho$ (identical colour scale in all figures; regions with $\frac {{\mathrm {d}} \rho }{{\mathrm {d}} \mu }(x)=0$ are white for emphasis) and boundaries of cells $\{{C}_i(w)\}_{i=1}^M$ (red) are shown for models (a)–(b) from Examples2.9 and 3.14. Figure (e) shows the same model as (d), only with ${c}(x,y)=[d(x,y)/2]^2$ instead of ${c}(x,y)=d(x,y)^2$ ; on some cells $\operatorname{spt} \rho$ is now strictly bounded away from the boundaries of ${C}_i(w)$ .

Finally, if $c(x,x_i)$ becomes infinite for sufficiently small $d(x,x_i)$ , then there exists an isolated cell ${C}_i(w)$ that is strictly bounded away from any other cell (see Fig. 4, right). In that case, none of the cells $\{{C}_j(w)\}_{j=1}^M$ depend on $w_i$ , and so neither does (3.5). However, the objective function of (3.1) in general still depends on $w_i$ via $F^\ast$ .

Remark 3.8 (Primal tessellation formulation for classical optimal transport). For classical optimal transport with $F=\iota _{\{1\}}$ , the term ${\mathcal{F}}(\rho |\mu)$ in (3.5) is finite (and zero) if and only if $\rho =\mu$ . Likewise, $\sum _{i=1}^M F\big (\frac {\rho ({C}_i(w))}{m_i}\big) \cdot m_i$ is finite (and zero) if and only if $\rho ({C}_i(w))=m_i$ . These are the optimality conditions given in Theorem 2.16. Thus, the objective function in (3.5) is finite only where it is optimal, making it somewhat pathological.

Proof. This is similar to the proof of Theorem 2.27 in [Reference Folland26], which however only covers the case where $M=1$ and $f(x,\cdot)$ is differentiable for all $x \in \Omega$ . We show that the directional derivative of $\mathcal H$ in an arbitrary direction $\hat u\in {\mathbb{R}}^M$ exists and is of the desired form. Indeed, let $L\gt 0$ be the Lipschitz constant of $f$ in its second argument. By assumption, there exists $S\subset \Omega$ Lebesgue-negligible such that $f(x,\cdot)$ is differentiable at $u$ for all $x\in \Omega \setminus S$ . Now for $t\neq 0$ ,

\begin{equation*} \frac {\mathcal H(u+t\hat u)-\mathcal H(u)}t =\int _{\Omega \setminus S}\frac {f(x,u+t\hat u)-f(x,u)}t\,{\textrm {d}}\mu (x)\,. \end{equation*}

Since the integrand is bounded in absolute value by $L\|\hat u\|$ and since it converges pointwise to $\frac {\partial f}{\partial u}(x,u)\cdot \hat u$ as $t\to 0$ , by the Dominated Convergence Theorem, we have

\begin{equation*} \lim _{t\to 0}\frac {\mathcal H(u+t\hat u)-\mathcal H(u)}t =\int _{\Omega \setminus S}\frac {\partial f}{\partial \tilde u}(x,u)\cdot \hat u\,{\textrm {d}}\mu (x) =\int _{\Omega }\frac {\partial f}{\partial \tilde u}(x,u) \,{\textrm {d}}\mu (x)\cdot \hat u\,. \end{equation*}

The arbitrariness of $\hat u$ , the linearity of the directional derivative, and the Lipschitz continuity of $\mathcal H$ imply that $\mathcal{H}$ is differentiable and has the desired form.

Proof. Define

\begin{equation*} f(x,w)=\min \{-F^\ast (\!-\!{c}(x,x_j)+w_j)\,|\,j=1,\ldots ,M\}. \end{equation*}

Since $F^\ast$ is increasing by Lemma2.10 (ii), then $f(x,w) = -F^\ast (\!-\!{c}(x,x_i)+w_i)$ for $x \in {C}_i(w)$ . If $\ell$ is bounded, the residual set $R$ is empty; otherwise, we have $f(x,w)=-F^\ast (\!-\!\infty)=F(0)$ for $x \in R$ . Therefore

\begin{equation*} {\mathcal{G}}(w)=\int _\Omega f(x,w)\,{\textrm {d}}\mu (x)-\sum _{i=1}^MF^\ast (\!-\!w_i)\cdot m_i. \end{equation*}

By Remark3.2, ${\mathcal{G}}(w)$ is finite for all $w\in {\mathbb{R}}^M$ . Thus, there exists $\tilde \Omega \subset \Omega$ with $\mu (\Omega \setminus \tilde \Omega)=0$ such that $f(x,w)$ is finite for all $x\in \tilde \Omega$ and $w\in {\mathbb{R}}^M$ . Now consider the function $\Omega \times {\mathbb{R}} \ni (x,v) \mapsto f_i(x,v)=-F^\ast (\!-\!{c}(x,x_i)+v)$ , where $i \in \{1 , \ldots ,M\}$ . Furthermore, by Lemma2.10 (vi) the strict convexity of $F$ implies continuous differentiability of its conjugate $F^\ast$ so that $f_i(x,\cdot)$ is differentiable for any $x\in \Omega$ with $f_i(x,\cdot)$ finite. Moreover, since $F^\ast$ is convex and increasing, ${\partial f_i}/{\partial v}$ is nonpositive and decreasing so that $f_i(x,\cdot)$ (if finite) is Lipschitz on $(\!-\!\infty ,\omega ]$ for any $\omega \in {\mathbb{R}}$ with Lipschitz constant $L\leq -\frac {\partial f_i}{\partial v}(x,\omega)\leq (F^\ast)'(\omega)$ . Consequently, $(\!-\!\infty ,\omega ]^M \ni \hat {w} \mapsto f(x,\hat {w})$ is Lipschitz with constant $\sqrt M(F^\ast)'(\omega)$ for all $x\in \tilde \Omega$ and differentiable in $\hat {w}$ for all $x\in \tilde \Omega \setminus S$ , where $S=\bigcup _{i=1}^M\partial {C}_i(w)$ is Lebesgue-negligible and thus also $\mu$ -negligible. Thus, by the previous Lemma, $\mathcal{G}$ is differentiable with

\begin{equation*} \frac {\partial {\mathcal{G}}(w)}{\partial w_i} = (F^\ast)'(\!-\!w_i)\cdot m_i + \int _{\Omega } \frac {\partial f}{\partial w_i}(x,w)\,{\textrm {d}}\mu (x)\,, \end{equation*}

where $\frac {\partial f}{\partial w_i}(x,w)=-(F^\ast)'(\!-\!c(x,x_i)+w_i)$ for $\mu$ -almost all $x\in {C}_i(w)$ and $\frac {\partial f}{\partial w_i}(x,w)=0$ for $\mu$ -almost all $x\notin {C}_i(w)$ .

Remark 3.11 (Balanced transport). For classical optimal transport with $F=\iota _{\{1\}}$ as in Remark 3.5, Theorem 3.10 reduces to well-known results. In particular, (3.6) becomes

(3.7) \begin{align} \frac {\partial {\mathcal{G}}(w)}{\partial w_i} = m_i-\mu ({C}_i(w))\,. \end{align}

For more details we refer, for example, to [Reference Kitagawa, Mérigot and Thibert37 , Thm. 1.1] or [Reference Mérigot, Thibert, Bonito and Nochetto52 , Thm. 40]. For marginals $\mu =\tilde \mu \mathcal{L}$ with $\tilde \mu \in \mathcal{C}(\Omega)$ the Hessian

(3.8) \begin{align} \frac {\partial ^2 {\mathcal{G}}(w)}{\partial w_i \partial w_j} =- \frac {\partial \mu ({C}_i(w))}{\partial w_j} \end{align}

can also be computed explicitly in terms of face integrals (see, for instance, [Reference Kitagawa, Mérigot and Thibert37 , Thm. 1.3] and [Reference Mérigot, Thibert, Bonito and Nochetto52 , Thm. 45]). Therefore (2.10) lends itself to efficient numerical optimization [Reference Aurenhammer, Hoffmann and Aronov2, Reference Kitagawa, Mérigot and Thibert37, Reference Lévy44, Reference Mérigot50]. For special cost functions, most prominently for the squared Euclidean distance, the gradient (3.7) and Hessian (3.8) can be evaluated numerically efficiently and with high precision, allowing the application of Newton’s method [Reference Kitagawa, Mérigot and Thibert37].

The semi-discrete unbalanced problem (3.1) is more complicated due to the influence of the marginal fidelity $\mathcal{F}$ and since we are often interested in non-standard cost functions such as $c_{\textrm {HK}}$ . Generalizing the above methods for balanced transport to the unbalanced case is therefore beyond the scope of this article.

Remark 3.12 (Discretization). Problem (3.1) is already finite-dimensional. We must however evaluate the integrals over ${C}_i(w)$ . For classical optimal transport and special cost functions $c$ , these integrals can be evaluated essentially in closed form (see Remark 3.11 ). For simplicity, in this section, we approximate $(\Omega ,\mu)$ with Dirac masses on a fine Cartesian grid. The cells $\{{C}_i(w)\}_{i=1}^M$ are approximated using brute force by computing $c(x,x_i)-w_i$ for each point $x$ in the Cartesian grid for each $i \in \{1,\ldots ,M\}$ . Points $x$ on the common boundaries of several cells $\{{C}_i(w)\}_{i=1}^M$ are arbitrarily assigned to one of those cells. (An efficient GPU-implementation of this brute force method can be found in [Reference Buze, Feydy, Roper, Sedighiani and Bourne15].) Note that for the special cost $c(x,y)=|x-y|^2$ , the Laguerre diagram $\{{C}_i(w)\}_{i=1}^M$ can be computed exactly, up to machine precision, and much more efficiently using, e.g., the lifting method [Reference Aurenhammer, Klein and Lee3 , Sec. 6.2.2], which has complexity $\mathcal{O}(M \log M)$ in ${\mathbb{R}}^2$ and $\mathcal{O}(M^2)$ in ${\mathbb{R}}^3$ . Our discretization yields an approximation of ${\mathcal{G}}(w)$ from (3.1a) and of $\nabla {\mathcal{G}}(w)$ from (3.6), as required for the quasi-Newton method. In the numerical examples below, we use $\Omega =[0,L]^2$ for some $L\gt 0$ and approximate it by a regular Cartesian grid with $1000$ points along each dimension.

Remark 3.13 (Primal-dual gap). The sub-optimality of any vector $w \in {\mathbb{R}}^M$ for (3.1) can be bounded by the primal-dual gap between (3.1a) and the objective of (3.5). We avoid the remaining optimization over $\rho$ in (3.5) by generating a feasible candidate via (3.4b). Corollaries 3.4 and 3.6 guarantee that the primal-dual gap vanishes for optimal $w$ .

Remark 4.4. An intuitive strategy for proving Theorem 4.2 could be as follows. One starts from the primal tessellation formulation in Corollary 3.6 and in addition minimizes over masses $(m_1,\ldots ,m_M)$ and positions $(x_1,\ldots ,x_M)$ . By (4.4) we find that minimizing masses are given by $m_i=\rho ({C}_i(w))$ . Next, only the transport term depends on the weights $w$ , and since the cost $c$ is a strictly increasing function of distance, the term is minimized for $w=0$ , thus essentially reducing the generalized Laguerre cells ${C}_i(w)$ into (truncated) Voronoi cells. Finally, the remaining minimization over $\rho$ can be handled with arguments from convex analysis, similar to those of Theorem 3.3, thus arriving at (4.8). We give a shorter proof, using results from the dual tessellation formulation and its optimality conditions.

Proof of Theorem 4.2. First we consider minimization over the masses $(m_i)_i$ for fixed positions $(x_i)_i$ . Let $\nu =\sum _{i=1}^Mm_i \cdot \delta _{x_i}$ be any admissible measure for (4.1). From (3.1b) we find $W(\mu ,\nu) \geq {\mathcal{G}}(0)$ for any positions $x_1,\ldots ,x_M$ and masses $m_1,\ldots ,m_M$ . Note that ${\mathcal{G}}(0)$ does not depend on $m_1,\ldots ,m_M$ since we assume $F^\ast (0)=0$ , (4.7). Consider now the case where ${\mathcal{G}}(0)\lt \infty$ (and hence $\mathcal{G}$ is finite for all values of $w$ , see Remark3.2). We now show $W(\mu ,\nu)={\mathcal{G}}(0)$ for a particular choice of $m_1,\ldots ,m_M$ , which therefore must be optimal (for given locations $x_1,\ldots ,x_M$ ). We first define $\rho$ via (3.4b) and then $\gamma$ via (3.4a) for $w=0$ ( $\rho$ and $\gamma$ are fully determined, see Corollary3.4). Furthermore, since $1 \in \partial F^\ast (0)$ by (4.7), equation (3.4c) is satisfied by the choice $m_i=\rho ({C}_i(0))$ . (Note that had we chosen $F(\hat z)=0$ instead of $F(1)=0$ , one would simply use $m_i=\rho ({C}_i(0))/\hat z$ so that (4.9) would change by the factor $\hat z$ .) By Theorem3.3 (optimality conditions), $\gamma$ and $w$ are optimizers of $\mathcal{E}$ and $\mathcal{G}$ for these mass coefficients, which implies that $W(\mu ,\nu)={\mathcal{G}}(0)$ . Using $F^\ast (0)=0$ from (4.7), we have

\begin{align*} \min _{(m_1,\ldots ,m_M)} W(\mu ,\nu) = {\mathcal{G}}(0) = -\sum _{i=1}^M \int _{{C}_i(0)} F^\ast \big (\!-\!c(x,x_i)\big)\,{\textrm {d}}\mu (x) + F(0) \cdot \mu (R)\,. \end{align*}

Since ${c}(x,y)$ is a strictly increasing function of the distance $d(x,y)$ , for $w=0$ we find ${C}_i(0) \subset V_i(x_1,\ldots ,x_M)$ . With the convention $-F^\ast (\!-\!\infty)=F(0)$ (cf. Lemma2.10), the term $F(0) \cdot \mu (R)$ becomes $\int _{R}-F^\ast (\!-\!\phi _0(x))\,{\textrm {d}} \mu (x)$ , where $\phi _0$ was defined in equation (2.13). Since $\mu \ll \mathcal{L}$ , integrating over $R$ and $\Omega \setminus R$ is equivalent to integrating over all Voronoi cells $\{V_i(x_1,\ldots ,x_M)\}_{i=1}^M$ , and we arrive at

(4.10) \begin{align} \min _{(m_1,\ldots ,m_M)} W(\mu ,\nu) = -\sum _{i=1}^M \int _{V_i(x_1,\ldots ,x_M)} F^\ast \big (\!-\!c(x,x_i)\big)\,{\textrm {d}}\mu (x)\,, \end{align}

which establishes equivalence between (4.1) and (4.8).

Finally, with $m_i=\rho ({C}_i(0))$ and $\rho$ given by (3.4b) one obtains (4.9), where the integral runs over ${C}_i(0)$ instead of $V_i(x_1,\ldots ,x_M)$ . If the minimum is finite, then either $\mu (R)=0$ or $F(0)$ is finite, which implies the convention $(F^\ast)'(\!-\!\infty)=0$ (cf. Lemma2.10 (viii)). In both cases we can extend the area of integration to $V_i(x_1,\ldots ,x_M)$ without changing its value. Equation (3.4a) implies that mass is only transported from each Voronoi cell $V_i(x_1,\ldots ,x_M) \supset {C}_i(0)$ to the corresponding point $x_i$ .

If on the other hand ${\mathcal{G}}(0)=\infty$ , then by (3.1b) for all $\nu$ concentrated on the positions $(x_i)_i$ one has $W(\mu ,\nu)=\infty$ . This establishes the equality of the minimal values in both the finite and infinite cases.

Now we consider minimization over the positions $(x_i)_i$ . We merely need to consider the case where the minimal value is finite, as otherwise any configuration is optimal. Let $((x_i^k)_{i=1}^M)_{k \in \mathbb{N}}$ be a minimizing sequence of points for the right-hand side of (4.10). After selection of a subsequence and relabelling the order of the points, there will be an integer $N \in \{0,\ldots ,M\}$ and points $(x_i)_{i=1}^N$ such that $\lim _k x_i^k=x_i$ for $i \in \{1,\ldots ,N\}$ and $\lim _k |x_i^k|=\infty$ for $i\gt N$ . Let $(\rho ^k)_{k \in \mathbb{N}}$ be the sequence induced via (3.4b) from $((x_i^k)_i)_k$ (for $w=0$ ). Using non-negativity of $c$ , convexity of $F^\ast$ and the fact that $1 \in \partial F^\ast (0)$ , we get that $\rho ^k \leq \mu$ for all $k$ and hence the sequence is tight, allowing extraction of a weak (i.e. in duality with bounded continuous functions) cluster point $\rho$ . Let $m_i^k=\rho ^k(V_i((x_j^k)_j))$ be the corresponding optimal masses, as above, and let $m_i=\rho (V_i((x_j)_{j=1}^N))$ be the masses induced by $\rho$ for $i \in \{1,\ldots ,N\}$ . Using weak convergence of $\rho ^k \to \rho$ and absolute continuity of $\rho$ one finds that $m^k_i \to m_i$ for $i \in \{1,\ldots ,N\}$ and $m_i^k \to 0$ for $i\gt N$ . One then has that $\nu ^k \;:\!=\; \sum _{i=1}^M m_i^k \cdot \delta _{x_i^k}$ converges weakly to $\nu \;:\!=\; \sum _{i=1}^N m_i \cdot \delta _{x_i}$ . By [Reference Liero, Mielke and Savaré45, Lemma3.9], $W$ is weakly lower-semicontinuous and therefore the points $(x_i)_{i={1}}^N$ are candidates for a minimizer in (4.10). However, if $N\lt M$ , they are too few. But adding arbitrary points on the right-hand side of (4.10) will not increase the objective, and thus any such extension must yield a minimizer.