Overcoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architectures

Abstract

We propose new and original mathematical connections between Hamilton–Jacobi (HJ) partial differential equations (PDEs) with initial data and neural network architectures. Specifically, we prove that some classes of neural networks correspond to representation formulas of HJ PDE solutions whose Hamiltonians and initial data are obtained from the parameters of the neural networks. These results do not rely on universal approximation properties of neural networks; rather, our results show that some classes of neural network architectures naturally encode the physics contained in some HJ PDEs. Our results naturally yield efficient neural network-based methods for evaluating solutions of some HJ PDEs in high dimension without using grids or numerical approximations. We also present some numerical results for solving some inverse problems involving HJ PDEs using our proposed architectures.

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Appendices

A Proofs of lemmas in Section 3.1

A.1 Proof of Lemma 3.1

Proof of (i): The convex and lower semicontinuous function \(J^*\) satisfies Eq. (12) by [68, Prop. X.3.4.1]. It is also finite and continuous over its polytopal domain \({\mathrm {dom}~}J^* = {\mathrm {conv}~}{\left( \{\textit{\textbf{p}}_i\}_{i=1}^{m}\right) }\) [133, Thms. 10.2 and 20.5], and moreover, the subdifferential \(\partial J^*(\textit{\textbf{p}})\) is non-empty by [133, Thm. 23.10].

Proof of (ii): First, suppose the vector \((\alpha _1, \dots , \alpha _m)\in \mathbb {R}^m\) satisfies the constraints (a)–(c). Since \(\textit{\textbf{x}}\in \partial J^*(\textit{\textbf{p}})\), there holds \(J^*(\textit{\textbf{p}}) = \langle \textit{\textbf{p}},\textit{\textbf{x}}\rangle - J(\textit{\textbf{x}})\) [68, Cor. X.1.4.4], and using the definition of the set \(I_{\textit{\textbf{x}}}\) (11) and constraints (a)–(c) we deduce that

$$\begin{aligned} \begin{aligned} J^*(\textit{\textbf{p}})&= \langle \textit{\textbf{p}},\textit{\textbf{x}}\rangle - J(\textit{\textbf{x}}) = \langle \textit{\textbf{p}}, \textit{\textbf{x}}\rangle - \sum _{i\in I_{\textit{\textbf{x}}}} \alpha _i J(\textit{\textbf{x}})\\&= \langle \textit{\textbf{p}}, \textit{\textbf{x}}\rangle - \sum _{i\in I_{\textit{\textbf{x}}}} \alpha _i (\langle \textit{\textbf{p}}_i, \textit{\textbf{x}}\rangle - \gamma _i)\\&= \left\langle \textit{\textbf{p}}- \sum _{i\in I_{\textit{\textbf{x}}}} \alpha _i\textit{\textbf{p}}_i, \textit{\textbf{x}}\right\rangle + \sum _{i\in I_{\textit{\textbf{x}}}} \alpha _i \gamma _i = \sum _{i=1}^m \alpha _i\gamma _i. \end{aligned} \end{aligned}$$

Therefore, \((\alpha _1,\dots , \alpha _m)\) is a minimizer in Eq. (12). Second, let \((\alpha _1,\dots , \alpha _m)\) be a minimizer in Eq. (12). Then, (a)–(b) follow directly from the constraints in Eq. (12). A similar argument as above yields

$$\begin{aligned} \begin{aligned} J(\textit{\textbf{x}})&= \langle \textit{\textbf{p}}, \textit{\textbf{x}}\rangle - J^*(\textit{\textbf{p}}) = \left\langle \sum _{i=1}^m \alpha _i\textit{\textbf{p}}_i, \textit{\textbf{x}}\right\rangle - \sum _{i=1}^m \alpha _i\gamma _i = \sum _{i=1}^m \alpha _i\left( \langle \textit{\textbf{p}}_i, \textit{\textbf{x}}\rangle - \gamma _i\right) . \end{aligned} \end{aligned}$$

But \(J(\textit{\textbf{x}})= \max _{i\in \{1,\dots ,m\}}\{\left\langle \textit{\textbf{p}}_{i},\textit{\textbf{x}}\right\rangle -\gamma _{i}\}\) by definition, and so there holds \(\alpha _i = 0\) whenever \(J(\textit{\textbf{x}})\ne \langle \textit{\textbf{p}}_i, \textit{\textbf{x}}\rangle - \gamma _i\). In other words, \(\alpha _i=0\) whenever \(i\not \in I_{\textit{\textbf{x}}}\).

Proof of (iii): Let \((\beta _1,\dots , \beta _m) \in \varLambda _m\) satisfy \(\sum _{i=1}^m\beta _i \textit{\textbf{p}}_i = \textit{\textbf{p}}_k\). By assumption (A2), we have \(\gamma _k=g(\textit{\textbf{p}}_k)\) with g convex, and hence, Jensen’s inequality yields

$$\begin{aligned} \sum _{i=1}^m\delta _{ik}\gamma _i = \gamma _k = g(\textit{\textbf{p}}_k) = g\left( \sum _{i=1}^m \beta _i \textit{\textbf{p}}_i\right) \leqslant \sum _{i=1}^m \beta _i g(\textit{\textbf{p}}_i) = \sum _{i=1}^m \beta _i \gamma _i. \end{aligned}$$

Therefore, the vector \((\delta _{1k},\dots ,\delta _{mk})\) is a minimizer in Eq. (12) at the point \(\textit{\textbf{p}}_k\), and \(J^*(\textit{\textbf{p}}_k) = \gamma _k\) follows.

A.2 Proof of Lemma 3.2

Proof of (i): Let \(\textit{\textbf{p}}\in {\mathrm {dom}~}J^*\). The set \(\mathcal {A}(\textit{\textbf{p}}) \subseteq \varLambda _m\) is non-empty and bounded by Lemma 3.1(i), and it is closed since \(\mathcal {A}(\textit{\textbf{p}})\) is the solution set to the linear programming problem (12). Hence, \(\mathcal {A}(\textit{\textbf{p}})\) is compact. As a result, we immediately have that \(H(\textit{\textbf{p}})<+\infty \). Moreover, for each \((\alpha _1,\dots ,\alpha _m)\in \mathcal {A}(\textit{\textbf{p}})\) there holds

$$\begin{aligned} -\infty< \min _{i=\{1,\dots , m\}} \theta _i \leqslant \sum _{i=1}^m \alpha _i \theta _i \leqslant \max _{i=\{1,\dots , m\}} \theta _i < +\infty \end{aligned}$$

from which we conclude that H is a bounded function on \({\mathrm {dom}~}J^*\). Since the target function in the minimization problem (14) is continuous, existence of a minimizer follows by compactness of \(\mathcal {A}(\textit{\textbf{p}})\).

Proof of (ii): We have already shown in the proof of (i) that the restriction of H to \({\mathrm {dom}~}J^*\) is bounded, and so it remains to prove its continuity. For any \(\textit{\textbf{p}}\in {\mathrm {dom}~}J^*\), we have that \((\alpha _1,\dots ,\alpha _m)\in \mathcal {A}(\textit{\textbf{p}})\) if and only if \((\alpha _1,\dots , \alpha _m)\in \varLambda _m\), \(\sum _{i=1}^m \alpha _i\textit{\textbf{p}}_i = \textit{\textbf{p}}\), and \(\sum _{i=1}^m \alpha _i\gamma _i = J^*(\textit{\textbf{p}})\). As a result, we have

$$\begin{aligned} H(\textit{\textbf{p}}) = \min \left\{ \sum _{i=1}^m\alpha _i \theta _i:\ (\alpha _1,\dots ,\alpha _m)\in \varLambda _m, \ \sum _{i=1}^m \alpha _i\textit{\textbf{p}}_i = \textit{\textbf{p}}, \ \sum _{i=1}^m \alpha _i\gamma _i = J^*(\textit{\textbf{p}}) \right\} . \end{aligned}$$
(32)

Define the function \(h:\ \mathbb {R}^{n+1}\rightarrow \mathbb {R}\cup \{+\infty \}\) by

$$\begin{aligned} \begin{aligned} h(\textit{\textbf{p}},r) {:}{=}\min \left\{ \sum _{i=1}^m\alpha _i \theta _i: (\alpha _1,\dots ,\alpha _m)\in \varLambda _m, \ \sum _{i=1}^m \alpha _i\textit{\textbf{p}}_i = \textit{\textbf{p}}, \ \sum _{i=1}^m \alpha _i\gamma _i = r \right\} , \end{aligned} \end{aligned}$$
(33)

for any \(\textit{\textbf{p}}\in \mathbb {R}^n\) and \(r\in \mathbb {R}\). Using the same argument as in the proof of Lemma 3.1(i), we conclude that h is a convex lower semicontinuous function, and in fact continuous over its domain \({\mathrm {dom}~}h = {\mathrm {conv}~}{\{(\textit{\textbf{p}}_i, \gamma _i)\}_{i=1}^m}\). Comparing Eq. (32) and the definition of h in (33), we deduce that \(H(\textit{\textbf{p}}) = h(\textit{\textbf{p}}, J^*(\textit{\textbf{p}}))\) for any \(\textit{\textbf{p}}\in {\mathrm {dom}~}J^*\). Continuity of H in \({\mathrm {dom}~}J^*\) then follows from the continuity of h and \(J^*\) in their own domains.

Proof of (iii): Let \(k\in \{1,\dots ,m\}\). On the one hand, Lemma 3.1(iii) implies \((\delta _{1k},\dots , \delta _{mk})\in \mathcal {A}(\textit{\textbf{p}}_k)\), so that

$$\begin{aligned} H(\textit{\textbf{p}}_k)\leqslant \sum _{i=1}^m \delta _{ik}\theta _i = \theta _k. \end{aligned}$$
(34)

On the other hand, let \((\alpha _1,\dots , \alpha _m)\in \mathcal {A}(\textit{\textbf{p}}_k)\) be a vector different from \((\delta _{k1},\dots , \delta _{km})\). Then, \((\alpha _1,\dots , \alpha _m) \in \varLambda _m\) satisfies \(\sum _{i=1}^m \alpha _i\textit{\textbf{p}}_i = \textit{\textbf{p}}\), \(\sum _{i=1}^m \alpha _i\gamma _i = J^*(\textit{\textbf{p}})\), and \(\alpha _k<1\). Define \((\beta _1,\dots , \beta _m) \in \varLambda _m\) by

A straightforward computation using the properties of \((\alpha _1,\dots , \alpha _m)\), Lemma 3.1(iii), and the definition of \((\beta _1,\dots , \beta _m)\) yields

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{}(\beta _1,\dots , \beta _m)\in \varLambda _m \quad \text { with }\beta _k = 0,\\ &{}\sum _{i\ne k}\beta _i \textit{\textbf{p}}_i = \sum _{i\ne k}\frac{\alpha _i \textit{\textbf{p}}_i}{1-\alpha _k} = \frac{\textit{\textbf{p}}_k - \alpha _k\textit{\textbf{p}}_k}{1-\alpha _k} = \textit{\textbf{p}}_k,\\ &{}\sum _{i\ne k}\beta _i \gamma _i = \sum _{i\ne k}\frac{\alpha _i \gamma _i}{1-\alpha _k} = \frac{J^*(\textit{\textbf{p}}_k) - \alpha _k\gamma _k}{1-\alpha _k}= \frac{\gamma _k - \alpha _k\gamma _k}{1-\alpha _k} =\gamma _k. \end{aligned} \end{array}\right. } \end{aligned}$$

In other words, Eq. (9) holds at index k, which, by assumption (A3), implies that \(\sum _{i\ne k}\beta _i \theta _i > \theta _k\). As a result, we have

$$\begin{aligned} \sum _{i=1}^m\alpha _i \theta _i = \alpha _k\theta _k + (1-\alpha _k)\sum _{i\ne k}\beta _i \theta _i > \alpha _k\theta _k + (1-\alpha _k)\theta _k = \theta _k = \sum _{i=1}^m \delta _{ik}\theta _i. \end{aligned}$$

Taken together with Eq. (34), we conclude that \((\delta _{1k},\dots , \delta _{mk})\) is the unique minimizer in (14), and hence, we obtain \(H(\textit{\textbf{p}}_k) = \theta _k\).

B Proof of Theorem 3.1

To prove this theorem, we will use three lemmas whose statements and proofs are given in Sect. B.1, B.2, and B.3, respectively. The proof of Theorem 3.1 is given in Sect. B.4.

B.1 Statement and proof of Lemma B.1

Lemma B.1

Suppose the parameters \(\{(\textit{\textbf{p}}_i,\theta _i,\gamma _i)\}_{i=1}^{m}\subset \mathbb {R}^n \times \mathbb {R}\times \mathbb {R}\) satisfy assumptions (A1)-(A3). Let J and H be the functions defined in Eqs. (10) and (14), respectively. Let \(\tilde{H}:\mathbb {R}^n\rightarrow \mathbb {R}\) be a continuous function satisfying \(\tilde{H}(\textit{\textbf{p}}_i) = H(\textit{\textbf{p}}_i)\) for each \(i\in \{1,\dots ,m\}\) and \(\tilde{H}(\textit{\textbf{p}})\geqslant H(\textit{\textbf{p}})\) for all \(\textit{\textbf{p}}\in {\mathrm {dom}~}J^*\). Then, the neural network f defined in Eq. (8) satisfies

$$\begin{aligned} f(\textit{\textbf{x}},t) {:}{=}\max _{i\in \{1,\dots ,m\}}\{\left\langle \textit{\textbf{p}}_{i},\textit{\textbf{x}}\right\rangle -t\theta _{i}-\gamma _{i}\} = \sup _{\textit{\textbf{p}}\in {\mathrm {dom}~}J^*}\left\{ \langle \textit{\textbf{p}},\textit{\textbf{x}}\rangle - t\tilde{H}(\textit{\textbf{p}}) - J^*(\textit{\textbf{p}})\right\} . \end{aligned}$$
(35)

Proof

Let \(\textit{\textbf{x}}\in \mathbb {R}^n\) and \(t\geqslant 0\). Since \(\tilde{H}(\textit{\textbf{p}})\geqslant H(\textit{\textbf{p}})\) for every \(\textit{\textbf{p}}\in {\mathrm {dom}~}J^*\), we get

$$\begin{aligned} \langle \textit{\textbf{p}},\textit{\textbf{x}}\rangle - t\tilde{H}(\textit{\textbf{p}}) - J^*(\textit{\textbf{p}}) \leqslant \langle \textit{\textbf{p}},\textit{\textbf{x}}\rangle - tH(\textit{\textbf{p}}) - J^*(\textit{\textbf{p}}). \end{aligned}$$
(36)

Let \((\alpha _1,\dots , \alpha _m)\) be a minimizer in (14). By Eqs. (12), (13), and (14), we have

$$\begin{aligned} \textit{\textbf{p}}= \sum _{i=1}^{m}\alpha _i\textit{\textbf{p}}_i,\quad H(\textit{\textbf{p}}) = \sum _{i=1}^{m}\alpha _i\theta _i, \quad \text { and } \quad J^*(\textit{\textbf{p}})=\sum _{i=1}^{m}\alpha _i\gamma _i. \end{aligned}$$
(37)

Combining Eqs. (36), (37), and (8), we get

$$\begin{aligned} \begin{aligned} \langle \textit{\textbf{p}},\textit{\textbf{x}}\rangle - t\tilde{H}(\textit{\textbf{p}}) - J^*(\textit{\textbf{p}})&\leqslant \sum _{i=1}^m \alpha _i (\langle \textit{\textbf{p}}_i, \textit{\textbf{x}}\rangle -t \theta _i - \gamma _i)\\&\leqslant \max _{i\in \{1,\dots ,m\}} \{\langle \textit{\textbf{p}}_i, \textit{\textbf{x}}\rangle -t \theta _i - \gamma _i\} = f(\textit{\textbf{x}},t), \end{aligned} \end{aligned}$$

where the second inequality follows from the constraint \((\alpha _1,\dots ,\alpha _m)\in \varLambda _m\). Since \(\textit{\textbf{p}}\in {\mathrm {dom}~}J^*\) is arbitrary, we obtain

$$\begin{aligned} \sup _{\textit{\textbf{p}}\in {\mathrm {dom}~}J^*}\left\{ \langle \textit{\textbf{p}},\textit{\textbf{x}}\rangle - t\tilde{H}(\textit{\textbf{p}}) - J^*(\textit{\textbf{p}})\right\} \leqslant f(\textit{\textbf{x}},t). \end{aligned}$$
(38)

Now, by Lemmas 3.1(iii), 3.2(iii), and the assumptions on \(\tilde{H}\), we have

$$\begin{aligned} \tilde{H}(\textit{\textbf{p}}_k)= H(\textit{\textbf{p}}_k) = \theta _k \quad \text { and } \quad J^*(\textit{\textbf{p}}_k) = \gamma _k, \end{aligned}$$

for each \(k\in \{1,\dots , m\}\). A straightforward computation yields

$$\begin{aligned} \begin{aligned} f(\textit{\textbf{x}},t)&= \max _{i\in \{1,\dots ,m\}} \{\langle \textit{\textbf{p}}_i, \textit{\textbf{x}}\rangle -t \theta _i - \gamma _i\} \\&= \max _{i\in \{1,\dots ,m\}} \left\{ \langle \textit{\textbf{p}}_i, \textit{\textbf{x}}\rangle -t \tilde{H}(\textit{\textbf{p}}_i) - J^*(\textit{\textbf{p}}_i)\right\} \\&\leqslant \sup _{\textit{\textbf{p}}\in {\mathrm {dom}~}J^*} \left\{ \langle \textit{\textbf{p}}, \textit{\textbf{x}}\rangle -t \tilde{H}(\textit{\textbf{p}}) - J^*(\textit{\textbf{p}})\right\} , \end{aligned} \end{aligned}$$
(39)

where the inequality holds since \(\textit{\textbf{p}}_i\in {\mathrm {dom}~}J^*\) for every \(i\in \{1,\dots ,m\}\). The conclusion then follows from Eqs. (38) and (39). \(\square \)

B.2 Statement and proof of Lemma B.2

Lemma B.2

Suppose the parameters \(\{(\textit{\textbf{p}}_i,\theta _i,\gamma _i)\}_{i=1}^{m}\subset \mathbb {R}^n \times \mathbb {R}\times \mathbb {R}\) satisfy assumptions (A1)-(A3). For every \(k\in \{1,\dots , m\}\), there exist \(\textit{\textbf{x}}\in \mathbb {R}^n\) and \(t>0\) such that \(f(\cdot , t)\) is differentiable at \(\textit{\textbf{x}}\) and \(\nabla _{\textit{\textbf{x}}} f(\textit{\textbf{x}},t) = \textit{\textbf{p}}_k\).

Proof

Since f is the supremum of a finite number of affine functions by definition (8), it is finite-valued and convex for \(t\geqslant 0\). As a result, \(\nabla _{\textit{\textbf{x}}} f(\textit{\textbf{x}},t) = \textit{\textbf{p}}_k\) is equivalent to \(\partial (f(\cdot , t))(\textit{\textbf{x}}) = \{\textit{\textbf{p}}_k\}\), and so it suffices to prove that \(\partial (f(\cdot , t))(\textit{\textbf{x}}) = \{\textit{\textbf{p}}_k\}\) for some \(\textit{\textbf{x}}\in \mathbb {R}^n\) and \(t>0\). To simplify the notation, we use \(\partial _{\textit{\textbf{x}}} f(\textit{\textbf{x}},t)\) to denote the subdifferential of \(f(\cdot , t)\) at \(\textit{\textbf{x}}\).

By [67, Thm. VI.4.4.2], the subdifferential of \(f(\cdot , t)\) at \(\textit{\textbf{x}}\) is the convex hull of the \(\textit{\textbf{p}}_i\)’s whose indices i’s are maximizers in (8), that is,

$$\begin{aligned} \partial _{\textit{\textbf{x}}} f(\textit{\textbf{x}},t) = {{\mathrm {co}}~}\{\textit{\textbf{p}}_i: i \text { is a maximizer in (8)}\}. \end{aligned}$$

It suffices then to prove the existence of \(\textit{\textbf{x}}\in \mathbb {R}^n\) and \(t>0\) such that

$$\begin{aligned} \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}\rangle -t\theta _k - \gamma _k > \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -t\theta _i - \gamma _i \quad \text { for every } i\ne k. \end{aligned}$$
(40)

First, consider the case when there exists \(\textit{\textbf{x}}\in \mathbb {R}^n\) such that \(\langle \textit{\textbf{p}}_k, \textit{\textbf{x}}\rangle -\gamma _k > \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -\gamma _i\) for every \(i\ne k\). In that case, by continuity, there exists small \(t>0\) such that \(\langle \textit{\textbf{p}}_k,\textit{\textbf{x}}\rangle -t\theta _k - \gamma _k > \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -t\theta _i - \gamma _i\) for every \(i\ne k\) and so (40) holds.

Now, consider the case when there does not exist \(\textit{\textbf{x}}\in \mathbb {R}^n\) such that \(\langle \textit{\textbf{p}}_k, \textit{\textbf{x}}\rangle -\gamma _k > \max _{i\ne k}\{\langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -\gamma _i\}\). In other words, we assume

$$\begin{aligned} J(\textit{\textbf{x}}) = \max _{i\ne k} \{\langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle - \gamma _i\}\quad \text { for every }\textit{\textbf{x}}\in \mathbb {R}^n. \end{aligned}$$
(41)

We apply Lemma 3.1(i) to the formula above and obtain

$$\begin{aligned} J^*(\textit{\textbf{p}}_k) = \min \left\{ \sum _{i=1}^{m}\alpha _{i}\gamma _{i}: (\alpha _{1},\dots ,\alpha _{m})\in \varLambda _m,\ \sum _{i=1}^{m}\alpha _{i}\textit{\textbf{p}}_{i}=\textit{\textbf{p}}_k,\ \alpha _k = 0\right\} . \end{aligned}$$
(42)

Let \(\textit{\textbf{x}}_0\in \partial J^*(\textit{\textbf{p}}_k)\). Denote by \(I_{\textit{\textbf{x}}_0}\) the set of maximizers in Eq. (41) at the point \(\textit{\textbf{x}}_0\), i.e.,

$$\begin{aligned} I_{\textit{\textbf{x}}_0}:= {\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{i\ne k}} \{\langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle - \gamma _i\}. \end{aligned}$$
(43)

Note that we have \(k\not \in I_{\textit{\textbf{x}}_0}\) by definition of \(I_{\textit{\textbf{x}}_0}\). Define a function \(h:\mathbb {R}^n\rightarrow \mathbb {R}\cup \{+\infty \}\) by

$$\begin{aligned} h(\textit{\textbf{p}}) {:}{=}{\left\{ \begin{array}{ll} \theta _i &{}\quad \text {if }\;\;\textit{\textbf{p}}= \textit{\textbf{p}}_i \text { and }i\in I_{\textit{\textbf{x}}_0},\\ +\infty &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$
(44)

Denote the convex lower semicontinuous envelope of h by \(\overline{{\mathrm {co}}}~h\). Since \(\textit{\textbf{x}}_0\in \partial J^*(\textit{\textbf{p}}_k)\), we can use [67, Thm. VI.4.4.2] and the definition of \(I_{\textit{\textbf{x}}_0}\) and h in Eqs. (43) and (44) to deduce

$$\begin{aligned} \textit{\textbf{p}}_k \in \partial J(\textit{\textbf{x}}_0) = {{\mathrm {co}}~}\{\textit{\textbf{p}}_i: i\in I_{\textit{\textbf{x}}_0}\} = {\mathrm {dom}~}\overline{{\mathrm {co}}}~h. \end{aligned}$$
(45)

Hence, the point \(\textit{\textbf{p}}_k\) is in the domain of the polytopal convex function \(\overline{{\mathrm {co}}}~h\). Then, [133, Thm. 23.10] implies \(\partial (\overline{{\mathrm {co}}}~h) (\textit{\textbf{p}}_k)\ne \emptyset \). Let \(\textit{\textbf{v}}_0 \in \partial (\overline{{\mathrm {co}}}~h)(\textit{\textbf{p}}_k)\) and \(\textit{\textbf{x}}=\textit{\textbf{x}}_0+t\textit{\textbf{v}}_0\). It remains to choose suitable positive t such that (40) holds. Letting \(\textit{\textbf{x}}=\textit{\textbf{x}}_0+t\textit{\textbf{v}}_0\) in (40) yields

$$\begin{aligned} \begin{aligned}&\langle \textit{\textbf{p}}_k,\textit{\textbf{x}}\rangle -t\theta _k - \gamma _k - \left( \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -t\theta _i - \gamma _i\right) \\&\quad = \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}_0+t\textit{\textbf{v}}_0\rangle - t\theta _k -\gamma _k - (\langle \textit{\textbf{p}}_i,\textit{\textbf{x}}_0+t\textit{\textbf{v}}_0\rangle -t\theta _i- \gamma _i)\\&\quad = \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}_0\rangle -\gamma _k - (\langle \textit{\textbf{p}}_i,\textit{\textbf{x}}_0\rangle - \gamma _i) + t(\theta _i-\theta _k - \langle \textit{\textbf{p}}_i-\textit{\textbf{p}}_k, \textit{\textbf{v}}_0\rangle ). \end{aligned} \end{aligned}$$
(46)

Now, we consider two situations, the first when \(i\not \in I_{\textit{\textbf{x}}_0} \cup \{k\}\) and the second when \(i\in I_{\textit{\textbf{x}}_0}\). It suffices to prove (40) hold in each case for small enough positive t.

If \(i\not \in I_{\textit{\textbf{x}}_0}\cup \{k\}\), then i is not a maximizer in Eq. (41) at the point \(\textit{\textbf{x}}_0\). By (45), \(\textit{\textbf{p}}_k\) is a convex combination of the set \(\{\textit{\textbf{p}}_i: i\in I_{\textit{\textbf{x}}_0}\}\). In other words, there exists \((c_1,\dots ,c_m)\in \varLambda _m\) such that \(\sum _{j=1}^mc_j\textit{\textbf{p}}_j=\textit{\textbf{p}}_k\) and \(c_j=0\) whenever \(j\not \in I_{\textit{\textbf{x}}_0}\). Taken together with assumption (A2) and Eqs. (10), (41), (43), we have

$$\begin{aligned} \begin{aligned} J(\textit{\textbf{x}}_0)&\geqslant \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}_0\rangle - \gamma _k = \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}_0\rangle - g(\textit{\textbf{p}}_k) = \left\langle \sum _{j\in I_{\textit{\textbf{x}}_0}} c_j\textit{\textbf{p}}_j,\textit{\textbf{x}}_0\right\rangle - g\left( \sum _{j\in I_{\textit{\textbf{x}}_0}} c_j\textit{\textbf{p}}_j\right) \\&\geqslant \sum _{j\in I_{\textit{\textbf{x}}_0}} c_j(\langle \textit{\textbf{p}}_j,\textit{\textbf{x}}_0\rangle - g(\textit{\textbf{p}}_j)) = \sum _{j\in I_{\textit{\textbf{x}}_0}} c_j J(\textit{\textbf{x}}_0) = J(\textit{\textbf{x}}_0). \end{aligned} \end{aligned}$$

Thus, the inequalities become equalities in the equation above. As a result, we have

$$\begin{aligned} \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}_0\rangle - \gamma _k = J(\textit{\textbf{x}}_0) > \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}_0\rangle - \gamma _i, \end{aligned}$$

where the inequality holds because \(i\not \in I_{\textit{\textbf{x}}_0}\cup \{k\}\) by assumption. This inequality implies that the constant \(\langle \textit{\textbf{p}}_k,\textit{\textbf{x}}_0\rangle -\gamma _k - (\langle \textit{\textbf{p}}_i,\textit{\textbf{x}}_0\rangle - \gamma _i)\) is positive, and taken together with (46), we conclude that the inequality in (40) holds for \(i\not \in I_{\textit{\textbf{x}}_0}\cup \{k\}\) when t is small enough.

If \(i\in I_{\textit{\textbf{x}}_0}\), then both i and k are maximizers in Eq. (10) at \(\textit{\textbf{x}}_0\), and hence, we have

$$\begin{aligned} \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}_0\rangle - \gamma _k = J(\textit{\textbf{x}}_0) = \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}_0\rangle - \gamma _i. \end{aligned}$$
(47)

Together with Eq. (46) and the definition of h in Eq. (44), we obtain

$$\begin{aligned} \begin{aligned} \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}\rangle -t\theta _k - \gamma _k - \left( \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -t\theta _i - \gamma _i\right)&= 0 + t(h(\textit{\textbf{p}}_i)-\theta _k - \langle \textit{\textbf{p}}_i-\textit{\textbf{p}}_k, \textit{\textbf{v}}_0\rangle )\\&\geqslant t(\overline{{\mathrm {co}}}~h(\textit{\textbf{p}}_i)-\theta _k - \langle \textit{\textbf{p}}_i-\textit{\textbf{p}}_k, \textit{\textbf{v}}_0\rangle ). \end{aligned} \end{aligned}$$
(48)

In addition, since \(\textit{\textbf{v}}_0\in \partial (\overline{{\mathrm {co}}}~h)(\textit{\textbf{p}}_k)\), we have

$$\begin{aligned} \overline{{\mathrm {co}}}~h(\textit{\textbf{p}}_i)\geqslant \overline{{\mathrm {co}}}~h(\textit{\textbf{p}}_k) +\langle \textit{\textbf{p}}_i-\textit{\textbf{p}}_k,\textit{\textbf{v}}_0\rangle . \end{aligned}$$
(49)

Combining Eqs. (48) and (49), we obtain

$$\begin{aligned} \langle \textit{\textbf{p}}_k,\textit{\textbf{x}}\rangle -t\theta _k - \gamma _k - \left( \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -t\theta _i - \gamma _i\right) \geqslant t(\overline{{\mathrm {co}}}~h(\textit{\textbf{p}}_k)-\theta _k). \end{aligned}$$
(50)

To prove the result, it suffices to show \(\overline{{\mathrm {co}}}~h(\textit{\textbf{p}}_k)>\theta _k\). As \(\textit{\textbf{p}}_k \in \overline{{\mathrm {co}}}~h\) (as shown before in Eq. (45)), then according to [68, Prop. X.1.5.4] we have

$$\begin{aligned} \overline{{\mathrm {co}}}~h(\textit{\textbf{p}}_k) = \sum _{j\in I_{\textit{\textbf{x}}_0}} \alpha _j h(\textit{\textbf{p}}_j) = \sum _{j\in I_{\textit{\textbf{x}}_0}} \alpha _j \theta _j, \end{aligned}$$
(51)

for some \((\alpha _1,\dots ,\alpha _m)\in \varLambda _m\) satisfying \(\textit{\textbf{p}}_k = \sum _{j=1}^m \alpha _j \textit{\textbf{p}}_j\) and \(\alpha _j=0\) whenever \(j\not \in I_{\textit{\textbf{x}}_0}\). Then, by Lemma 3.1(ii) \((\alpha _1,\dots ,\alpha _m)\) is a minimizer in Eq. (42), that is,

$$\begin{aligned} \gamma _k = J^*(\textit{\textbf{p}}_k) = \sum _{j=1}^m \alpha _j \gamma _j = \sum _{j\in I_{\textit{\textbf{x}}_0}} \alpha _i \gamma _i = \sum _{i\ne k}\alpha _i \gamma _i. \end{aligned}$$

Hence, Eq. (9) holds for the index k. By assumption (A3), we have \(\theta _k < \sum _{j\ne k}\alpha _j \theta _j\). Taken together with the fact that \(\alpha _j=0\) whenever \(j\not \in I_{\textit{\textbf{x}}_0}\) and Eq. (51), we find

$$\begin{aligned} \theta _k < \sum _{j\ne k}\alpha _j \theta _j = \sum _{j\in I_{\textit{\textbf{x}}_0}} \alpha _j \theta _j= \overline{{\mathrm {co}}}~h(\textit{\textbf{p}}_k). \end{aligned}$$
(52)

Hence, the right-hand side of Eq. (50) is strictly positive, and we conclude that \(\langle \textit{\textbf{p}}_k,\textit{\textbf{x}}\rangle -t\theta _k - \gamma _k > \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -t\theta _i - \gamma _i\) for \(t>0\) if \(i\in I_{\textit{\textbf{x}}_0}\).

Therefore, in this case, when \(t>0\) is small enough and \(\textit{\textbf{x}}\) is chosen as above, we have \(\langle \textit{\textbf{p}}_k,\textit{\textbf{x}}\rangle -t\theta _k - \gamma _k > \langle \textit{\textbf{p}}_i,\textit{\textbf{x}}\rangle -t\theta _i - \gamma _i\) for every \(i\ne k\), and the proof is complete. \(\square \)

B.3 Statement and proof of Lemma B.3

Lemma B.3

Suppose the parameters \(\{(\textit{\textbf{p}}_i,\theta _i,\gamma _i)\}_{i=1}^{m}\subset \mathbb {R}^n \times \mathbb {R}\times \mathbb {R}\) satisfy assumptions (A1)-(A3). Define a function \(F:\ \mathbb {R}^{n+1}\rightarrow \mathbb {R}\cup \{+\infty \}\) by

$$\begin{aligned} F(\textit{\textbf{p}},E^{-}) {:}{=}{\left\{ \begin{array}{ll} J^{*}(\textit{\textbf{p}}) &{}\quad \mathrm{if } \;\; E^{-}+H(\textit{\textbf{p}})\leqslant 0,\\ +\infty &{} \quad \mathrm{otherwise,} \end{array}\right. } \end{aligned}$$
(53)

for all \(\textit{\textbf{p}}\in \mathbb {R}^n\) and \(E^-\in \mathbb {R}\). Then, the convex envelope of F is given by

$$\begin{aligned} {{\mathrm {co}}~}F(\textit{\textbf{p}},E^-) = \inf _{(c_1,\dots , c_m)\in C(\textit{\textbf{p}},E^-) }\sum _{i=1}^{m}c_{i}\gamma _{i}, \end{aligned}$$
(54)

where the constraint set \(C(\textit{\textbf{p}},E^-)\) is defined by

$$\begin{aligned} C(\textit{\textbf{p}},E^-){:}{=}\left\{ (c_1, \dots , c_m)\in \varLambda _m:\sum _{i=1}^{m}c_{i}\textit{\textbf{p}}_{i}=\textit{\textbf{p}}, \ \sum _{i=1}^{m}c_{i}\theta _{i}\leqslant -E^- \right\} . \end{aligned}$$

Proof

First, we compute the convex hull of \({\mathrm {epi}~}F\), which we denote by \({{\mathrm {co}}~}({\mathrm {epi}~}F)\). Let \((\textit{\textbf{p}}, E^-,r)\in {{\mathrm {co}}~}({\mathrm {epi}~}F)\), where \(\textit{\textbf{p}}\in \mathbb {R}^n\) and \(E^-,r\in \mathbb {R}\). Then there exist \(k\in \mathbb {N}\), \((\beta _1, \dots , \beta _k)\in \varLambda _k\) and \((\textit{\textbf{q}}_i, E_i^-,r_i)\in {\mathrm {epi}~}F\) for each \(i\in \{1,\dots , k\}\) such that \((\textit{\textbf{p}}, E^-,r) = \sum _{i=1}^k \beta _i (\textit{\textbf{q}}_i, E_i^-,r_i)\). By definition of F in Eq. (53), \((\textit{\textbf{q}}_i, E_i^-,r_i)\in {\mathrm {epi}~}F\) holds if and only if \(\textit{\textbf{q}}_i\in {\mathrm {dom}~}J^*\), \(E_i^-+H(\textit{\textbf{q}}_i)\leqslant 0\) and \(r_i\geqslant J^*(\textit{\textbf{q}}_i)\). In conclusion, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} (\beta _1, \dots , \beta _k)\in \varLambda _k,\\ (\textit{\textbf{p}}, E^-,r) = \sum _{i=1}^k \beta _i (\textit{\textbf{q}}_i, E_i^-,r_i),\\ \textit{\textbf{q}}_1,\dots ,\textit{\textbf{q}}_k\in {\mathrm {dom}~}J^*,\\ E_i^-+H(\textit{\textbf{q}}_i)\leqslant 0\quad \text { for each }i\in \{1,\dots , k\},\\ r_i\geqslant J^*(\textit{\textbf{q}}_i) \quad \text { for each }i\in \{1,\dots , k\}. \end{array}\right. } \end{aligned}$$
(55)

For each i, since we have \(\textit{\textbf{q}}_i\in {\mathrm {dom}~}J^*\), by Lemma 3.2(i) the minimization problem in (14) evaluated at \(\textit{\textbf{q}}_i\) has at least one minimizer. Let \((\alpha _{i1}, \dots , \alpha _{im})\) be such a minimizer. Using Eqs. (12), (14), and \((\alpha _{i1}, \dots , \alpha _{im})\in \varLambda _m\), we have

$$\begin{aligned} \sum _{j=1}^m \alpha _{ij}(1, \textit{\textbf{p}}_j, \theta _j, \gamma _j) = (1,\textit{\textbf{q}}_i, H(\textit{\textbf{q}}_i), J^*(\textit{\textbf{q}}_i)). \end{aligned}$$
(56)

Define the real number \(c_j{:}{=}\sum _{i=1}^k \beta _i \alpha _{ij}\) for any \(j\in \{1,\dots ,m\}\). Combining Eqs. (55) and (56), we get that \(c_j\geqslant 0\) for any j and

$$\begin{aligned} \begin{aligned}&\sum _{j=1}^m c_j(1,\textit{\textbf{p}}_j, \theta _j, \gamma _j) =\sum _{j=1}^m \sum _{i=1}^k \beta _i \alpha _{ij}(1,\textit{\textbf{p}}_j, \theta _j, \gamma _j)\\ =&\sum _{i=1}^k \beta _i \left( \sum _{j=1}^m\alpha _{ij}(1,\textit{\textbf{p}}_j, \theta _j, \gamma _j) \right) = \sum _{i=1}^k \beta _i(1,\textit{\textbf{q}}_i, H(\textit{\textbf{q}}_i), J^*(\textit{\textbf{q}}_i)). \end{aligned} \end{aligned}$$

We continue the computation using Eq. (55) and get

$$\begin{aligned} \begin{aligned}&\sum _{j=1}^m c_j(1,\textit{\textbf{p}}_j) = \sum _{i=1}^k \beta _i(1,\textit{\textbf{q}}_i) = (1,\textit{\textbf{p}});\\&\sum _{j=1}^m c_j\theta _j=\sum _{i=1}^k \beta _iH(\textit{\textbf{q}}_i)\leqslant -\sum _{i=1}^k\beta _iE_i^- = -E^-; \\&\sum _{j=1}^m c_j\gamma _j=\sum _{i=1}^k \beta _iJ^*(\textit{\textbf{q}}_i)\leqslant \sum _{i=1}^k\beta _ir_i=r.\\ \end{aligned} \end{aligned}$$

Therefore, we conclude that \((c_1,\dots , c_m)\in \varLambda _m\) and

$$\begin{aligned} {\left\{ \begin{array}{ll} \textit{\textbf{p}}= \sum _{j=1}^m c_j\textit{\textbf{p}}_j,\\ E^- \leqslant -\sum _{j=1}^m c_j\theta _j,\\ r \geqslant \sum _{j=1}^m c_j\gamma _j. \end{array}\right. } \end{aligned}$$

As a consequence, \({{\mathrm {co}}~}({\mathrm {epi}~}F)\subseteq {{\mathrm {co}}~}\left( \cup _{j=1}^m \left( \{\textit{\textbf{p}}_j\}\times (-\infty , -\theta _j]\times [\gamma _j, +\infty )\right) \right) \). Now, Lemmas 3.1(iii) and 3.2(iii) imply \(\{\textit{\textbf{p}}_j\}\times (-\infty , -\theta _j]\times [\gamma _j, +\infty ) \subseteq {\mathrm {epi}~}F\) for each \(j\in \{1,\dots ,m\}\). Therefore, we have

$$\begin{aligned} \begin{aligned} {{\mathrm {co}}~}({\mathrm {epi}~}F) = \Bigg \{(\textit{\textbf{p}}, E^-,r)\in \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}: \text {there exists }(c_1,\dots , c_m)\in \varLambda _m\text { s.t. }\quad \quad \\ \textit{\textbf{p}}= \sum _{j=1}^m c_j\textit{\textbf{p}}_j,\ E^- \leqslant -\sum _{j=1}^m c_j\theta _j, \ r \geqslant \sum _{j=1}^m c_j\gamma _j.\Bigg \}. \end{aligned} \end{aligned}$$
(57)

By [68, Def. IV.2.5.3 and Prop. IV.2.5.1], we have

$$\begin{aligned} \begin{aligned} {{\mathrm {co}}~}F(\textit{\textbf{p}},E^-) = \inf \{r\in \mathbb {R}: (\textit{\textbf{p}}, E^-,r)\in {{\mathrm {co}}~}({\mathrm {epi}~}F)\}. \end{aligned} \end{aligned}$$
(58)

The conclusion then follows from Eqs. (57) and (58). \(\square \)

B.4 Proof of Theorem 3.1

Proof of (i): First, the neural network f is the pointwise maximum of m affine functions in \((\textit{\textbf{x}},t)\) and therefore is jointly convex in these variables. Second, as the function H is continuous and bounded in \({\mathrm {dom}~}J^*\) by Lemma 3.2(ii), there exists a continuous and bounded function defined in \(\mathbb {R}^n\) whose restriction to \({\mathrm {dom}~}J^*\) coincides with H [57, Thm. 4.16]. Then, statement (i) follows by substituting this function for \(\tilde{H}\) in statement (ii), and so it suffices to prove the latter.

Proof of (ii) (sufficiency): Suppose \(\tilde{H}(\textit{\textbf{p}}_i) = H(\textit{\textbf{p}}_i)\) for every \(i\in \{1,\dots ,m\}\) and \(\tilde{H}(\textit{\textbf{p}})\geqslant H(\textit{\textbf{p}})\) for every \(\textit{\textbf{p}}\in {\mathrm {dom}~}J^*\). Since \(\tilde{H}\) is continuous on \(\mathbb {R}^n\) and J is convex and Lipschitz continuous with Lipschitz constant \(L= \max _{i\in \{1,\dots ,m\}} \Vert \textit{\textbf{p}}_i\Vert \), [10, Thm. 3.1] implies that \((\textit{\textbf{x}},t) \mapsto \sup _{\textit{\textbf{p}}\in {\mathrm {dom}~}J^*}\left\{ \langle \textit{\textbf{p}},\textit{\textbf{x}}\rangle - t\tilde{H}(\textit{\textbf{p}}) - J^*(\textit{\textbf{p}})\right\} \) is the unique uniformly continuous viscosity solution to the HJ equation (16). But this function is equivalent to the neural network f by Lemma B.1, and therefore, both sufficiency and statement (i) follow.

Proof of (ii) (necessity): Suppose the neural network f is the unique uniformly continuous viscosity solution to (16). First, we prove that \(\tilde{H}(\textit{\textbf{p}}_k) = H(\textit{\textbf{p}}_k)\) for every \(k\in \{1,\dots , m\}\). Fix \(k\in \{1,\dots , m\}\). By Lemma B.2, there exist \(\textit{\textbf{x}}\in \mathbb {R}^n\) and \(t>0\) satisfying \(\partial _{\textit{\textbf{x}}} f(\textit{\textbf{x}},t) = \{\textit{\textbf{p}}_k\}\). Use Lems. 3.1(iii) and 3.2(iii) to write the maximization problem in Eq. (8) as

$$\begin{aligned} f(\textit{\textbf{x}},t) = \max _{\textit{\textbf{p}}\in \{\textit{\textbf{p}}_1,\dots ,\textit{\textbf{p}}_m\}} \{\langle \textit{\textbf{p}}, \textit{\textbf{x}}\rangle - tH(\textit{\textbf{p}})-J^*(\textit{\textbf{p}})\}, \end{aligned}$$
(59)

where \((\textit{\textbf{p}},t) \mapsto \langle \textit{\textbf{p}}, \textit{\textbf{x}}\rangle - tH(\textit{\textbf{p}})-J^*(\textit{\textbf{p}})\) is continuous in \((\textit{\textbf{p}},t)\) and differentiable in t. As the feasible set \(\{\textit{\textbf{p}}_1,\dots ,\textit{\textbf{p}}_m\}\) is compact, f is also differentiable with respect to t [21, Prop. 4.12], and its derivative equals

$$\begin{aligned} \frac{\partial f}{\partial t}(\textit{\textbf{x}},t) = \min \left\{ -H(\textit{\textbf{p}}):\ \textit{\textbf{p}}\text { is a maximizer in Eq. (59)}\right\} . \end{aligned}$$

Since \(\textit{\textbf{x}}\) and t satisfy \(\partial _{\textit{\textbf{x}}} f(\textit{\textbf{x}},t)=\{\textit{\textbf{p}}_k\}\), [67, Thm. VI.4.4.2] implies that the only maximizer in Eq. (59) is \(\textit{\textbf{p}}_k\). As a result, there holds

$$\begin{aligned} \frac{\partial f}{\partial t}(\textit{\textbf{x}},t) = -H(\textit{\textbf{p}}_k). \end{aligned}$$
(60)

Since f is convex on \(\mathbb {R}^n\), its subdifferential \(\partial f(\textit{\textbf{x}},t)\) is non-empty and satisfies

$$\begin{aligned} \partial f(\textit{\textbf{x}},t)\subseteq \partial _{\textit{\textbf{x}}} f(\textit{\textbf{x}},t)\times \partial _t f(\textit{\textbf{x}},t) = \{(\textit{\textbf{p}}_k, -H(\textit{\textbf{p}}_k))\}. \end{aligned}$$

In other words, the subdifferential \(\partial f(\textit{\textbf{x}},t)\) contains only one element, and therefore, f is differentiable at \((\textit{\textbf{x}},t)\) and its gradient equals \((\textit{\textbf{p}}_k, -H(\textit{\textbf{p}}_k))\) [133, Thm. 21.5]. Using (16) and (60), we obtain

$$\begin{aligned} 0 = \frac{\partial f}{\partial t}(\textit{\textbf{x}},t) + \tilde{H}(\nabla _{\textit{\textbf{x}}} f(\textit{\textbf{x}},t)) = -H(\textit{\textbf{p}}_k) + \tilde{H}(\textit{\textbf{p}}_k). \end{aligned}$$

As \(k\in \{1,\dots , m\}\) is arbitrary, we find that \(H(\textit{\textbf{p}}_k)=\tilde{H}(\textit{\textbf{p}}_k)\) for every \(k\in \{1,\dots , m\}\).

Next, we prove by contradiction that \(\tilde{H}(\textit{\textbf{p}})\geqslant H(\textit{\textbf{p}})\) for every \(\textit{\textbf{p}}\in {\mathrm {dom}~}J^*\). It is enough to prove the property only for every \(\textit{\textbf{p}}\in {\mathrm {ri}}~{\mathrm {dom}~}J^*\) by continuity of both \(\tilde{H}\) and H (where continuity of H is proved in Lemma 3.2(ii)). Assume \(\tilde{H}(\textit{\textbf{p}}) < H(\textit{\textbf{p}})\) for some \(\textit{\textbf{p}}\in {\mathrm {ri}}~{\mathrm {dom}~}J^*\). Define two functions F and \(\tilde{F}\) from \(\mathbb {R}^n\times \mathbb {R}\) to \(\mathbb {R}\cup \{+\infty \}\) by

$$\begin{aligned} F(\textit{\textbf{q}},E^{-}){:}{=}{\left\{ \begin{array}{ll} J^{*}(\textit{\textbf{q}}) &{}\quad \text {if }\;\; {E^{-}+H(\textit{\textbf{q}})\leqslant 0},\\ +\infty &{}\quad \mathrm{otherwise.} \end{array}\right. } \quad \text { and }\quad \tilde{F}(\textit{\textbf{q}},E^{-}){:}{=}{\left\{ \begin{array}{ll} J^{*}(\textit{\textbf{q}}) &{}\quad \text {if }\;\; {E^{-}+\tilde{H}(\textit{\textbf{q}})\leqslant 0},\\ +\infty &{}\quad \mathrm{otherwise.} \end{array}\right. } \end{aligned}$$
(61)

for any \(\textit{\textbf{q}}\in \mathbb {R}^n\) and \(E^-\in \mathbb {R}\). Denoting the convex envelope of F by \({{\mathrm {co}}~}F\), Lemma B.3 implies

$$\begin{aligned} \begin{aligned}&{{\mathrm {co}}~}F(\textit{\textbf{q}},E^-) = \inf _{(c_1,\dots , c_m)\in C(\textit{\textbf{q}},E^-) }\sum _{i=1}^{m}c_{i}\gamma _{i},\text { where }C \text { is defined by}\\&C(\textit{\textbf{q}},E^-){:}{=}\left\{ (c_1, \dots , c_m)\in \varLambda _m:\ \sum _{i=1}^{m}c_{i}\textit{\textbf{p}}_{i}=\textit{\textbf{q}}, \ \sum _{i=1}^{m}c_{i}\theta _{i}\leqslant -E^- \right\} . \end{aligned} \end{aligned}$$
(62)

Let \(E_1^- \in \left( -H(\textit{\textbf{p}}), -\tilde{H}(\textit{\textbf{p}})\right) \). Now, we want to prove that \({{\mathrm {co}}~}F(\textit{\textbf{p}}, E_1^-)\leqslant J^*(\textit{\textbf{p}})\); this inequality will lead to a contradiction with the definition of H.

Using statement (i) of this theorem and the supposition that f is the unique viscosity solution to the HJ equation (16), we have that

$$\begin{aligned} f(\textit{\textbf{x}}, t) = \sup _{\textit{\textbf{q}}\in \mathbb {R}^n} \{\langle \textit{\textbf{q}},\textit{\textbf{x}}\rangle - tH(\textit{\textbf{q}}) - J^*(\textit{\textbf{q}})\} = \sup _{\textit{\textbf{q}}\in \mathbb {R}^n} \{\langle \textit{\textbf{q}},\textit{\textbf{x}}\rangle - t\tilde{H}(\textit{\textbf{q}}) - J^*(\textit{\textbf{q}})\}. \end{aligned}$$

Furthermore, a similar calculation as in the proof of [39, Prop. 3.1] yields

$$\begin{aligned} f = F^* = \tilde{F}^*, \text { which implies } f^* = \overline{{\mathrm {co}}}~F = \overline{{\mathrm {co}}}~\tilde{F}. \end{aligned}$$

where \(\overline{{\mathrm {co}}}~F\) and \(\overline{{\mathrm {co}}}~\tilde{F}\) denotes the convex lower semicontinuous envelopes of F and \(\tilde{F}\), respectively. On the one hand, since \(f^* = \overline{{\mathrm {co}}}~\tilde{F}\), the definition of \(\tilde{F}\) in Eq. (61) implies

$$\begin{aligned} f^*\left( \textit{\textbf{p}}, -\tilde{H}(\textit{\textbf{p}})\right) \leqslant \tilde{F}\left( \textit{\textbf{p}}, -\tilde{H}(\textit{\textbf{p}})\right) = J^*(\textit{\textbf{p}}) \quad \text { and } \quad \{\textit{\textbf{p}}\}\times \left( -\infty , -\tilde{H}(\textit{\textbf{p}})\right] \subseteq {\mathrm {dom}~}\tilde{F} \subseteq {\mathrm {dom}~}f^*. \end{aligned}$$
(63)

Recall that \(\textit{\textbf{p}}\in {\mathrm {ri}}~{\mathrm {dom}~}J^*\) and \(E_1^-<-\tilde{H}(\textit{\textbf{p}})\), so that \((\textit{\textbf{p}}, E_1^-) \in {\mathrm {ri}}~{\mathrm {dom}~}f^*\). As a result, we get

$$\begin{aligned} \left( \textit{\textbf{p}}, \alpha E_1^- +(1-\alpha )(-\tilde{H}(\textit{\textbf{p}}))\right) \in {\mathrm {ri}}~{\mathrm {dom}~}f^* \text { for all }\alpha \in (0,1). \end{aligned}$$
(64)

On the other hand, since \(f^* = {{\mathrm {co}}~}F\), we have \({\mathrm {ri}}~{\mathrm {dom}~}f^* = {\mathrm {ri}}~{\mathrm {dom}~}({{\mathrm {co}}~}F)\) and \(f^* = {{\mathrm {co}}~}F\) in \({\mathrm {ri}}~{\mathrm {dom}~}f^*\). Taken together with Eq. (64) and the continuity of \(f^*\), there holds

$$\begin{aligned} \begin{aligned} f^*\left( \textit{\textbf{p}}, -\tilde{H}(\textit{\textbf{p}})\right)&= \lim _{\begin{array}{c} \alpha \rightarrow 0\\ 0<\alpha<1 \end{array}} f^*\left( \textit{\textbf{p}}, \alpha E_1^- +(1-\alpha )(-\tilde{H}(\textit{\textbf{p}}))\right) \\&= \lim _{\begin{array}{c} \alpha \rightarrow 0\\ 0<\alpha <1 \end{array}} {{\mathrm {co}}~}F\left( \textit{\textbf{p}}, \alpha E_1^- +(1-\alpha )(-\tilde{H}(\textit{\textbf{p}}))\right) . \end{aligned} \end{aligned}$$
(65)

Note that \({{\mathrm {co}}~}F(\textit{\textbf{p}},\cdot )\) is monotone non-decreasing. Indeed, if \(E_2^-\) is a real number such that \(E_2^- > E_1^-\), by the definition of the set C in Eq. (62) there holds \(C(\textit{\textbf{p}}, E_2^-)\subseteq C(\textit{\textbf{p}}, E_1^-)\), which implies \({{\mathrm {co}}~}F(\textit{\textbf{p}},E_2^-)\geqslant {{\mathrm {co}}~}F(\textit{\textbf{p}},E_1^-)\). Recalling that \(E_1^- < -\tilde{H}(\textit{\textbf{p}})\), monotonicity of \({{\mathrm {co}}~}F(\textit{\textbf{p}},\cdot )\) and Eq. (65) imply

$$\begin{aligned} \begin{aligned} f^*\left( \textit{\textbf{p}}, -\tilde{H}(\textit{\textbf{p}})\right)&\geqslant \lim _{\begin{array}{c} \alpha \rightarrow 0\\ 0<\alpha <1 \end{array}} {{\mathrm {co}}~}F\left( \textit{\textbf{p}}, \alpha E_1^- +(1-\alpha )E_1^-\right) = {{\mathrm {co}}~}F(\textit{\textbf{p}}, E_1^-). \end{aligned} \end{aligned}$$
(66)

Combining Eqs. (63) and (66), we get

$$\begin{aligned} {{\mathrm {co}}~}F(\textit{\textbf{p}}, E_1^-)\leqslant J^*(\textit{\textbf{p}}) < +\infty . \end{aligned}$$
(67)

As a result, the set \(C(\textit{\textbf{p}}, E_1^-)\) is non-empty. Since it is also compact, there exists a minimizer in Eq. (62) evaluated at the point \((\textit{\textbf{p}}, E_1^-)\). Let \((c_1,\dots , c_m)\) be such a minimizer. By Eqs. (62) and (67) and the assumption that \(E_1^- \in \left( -H(\textit{\textbf{p}}),-\tilde{H}(\textit{\textbf{p}})\right) \), there holds

$$\begin{aligned} {\left\{ \begin{array}{ll} (c_1, \dots , c_m)\in \varLambda _m,\\ \sum _{i=1}^{m}c_{i}\textit{\textbf{p}}_{i}=\textit{\textbf{p}},\\ \sum _{i=1}^m c_i\gamma _i = {{\mathrm {co}}~}F(\textit{\textbf{p}},E_1^-) \leqslant J^*(\textit{\textbf{p}}),\\ \sum _{i=1}^{m}c_{i}\theta _{i}\leqslant -E_1^- < H(\textit{\textbf{p}}). \end{array}\right. } \end{aligned}$$
(68)

Comparing the first three statements in Eq. (68) and the formula of \(J^*\) in Eq. (12), we deduce that \((c_1,\dots , c_m)\) is a minimizer in Eq. (12), i.e., \((c_1,\dots , c_m)\in \mathcal {A}(\textit{\textbf{p}})\). By definition of H in Eq. (14), we have

$$\begin{aligned} H(\textit{\textbf{p}}) =\inf _{\varvec{\alpha }\in \mathcal {A}(\textit{\textbf{p}})} \sum _{i=1}^m \alpha _i \theta _i \leqslant \sum _{i=1}^m c_i\theta _i, \end{aligned}$$

which contradicts the last inequality in Eq. (68). Therefore, we conclude that \(\tilde{H}(\textit{\textbf{p}})\geqslant H(\textit{\textbf{p}})\) for any \(\textit{\textbf{p}}\in {\mathrm {ri}}~{\mathrm {dom}~}J^*\) and the proof is finished.

C Connections between the neural network (17) and the viscous HJ PDE (18)

Let \(f_\epsilon \) be the neural network defined by Eq. (17) with parameters \(\{(\textit{\textbf{p}}_{i}, \theta _i, \gamma _{i})\}_{i=1}^m\) and \(\epsilon > 0\), which is illustrated in Fig. 3. We will show in this appendix that when the parameter \(\theta _{i} = -\frac{1}{2}\left\| \textit{\textbf{p}}_{i}\right\| _{2}^{2}\) for \(i\in \{1,\dots ,m\}\), then the neural network \(f_\epsilon \) corresponds to the unique, jointly convex smooth solution to the viscous HJ PDE (18). This result will follow immediately from the following lemma.

Lemma C.1

Let \(\{(\textit{\textbf{p}}_i,\gamma _i)\}_{i=1}^{m}\subset \mathbb {R}^n \times \mathbb {R}\) and \(\epsilon > 0\). Then, the function \(w_{\epsilon } : \mathbb {R}^n \mapsto \mathbb {R}\) defined by

$$\begin{aligned} w_{\epsilon }(\textit{\textbf{x}},t){:}{=}\sum _{i=1}^{m}e^{\left( \left\langle \textit{\textbf{p}}_{i},\textit{\textbf{x}}\right\rangle +\frac{t}{2}\left\| \textit{\textbf{p}}_{i}\right\| _{2}^{2}-\gamma _{i}\right) /\epsilon } \end{aligned}$$
(69)

is the unique, jointly log-convex and smooth solution to the Cauchy problem

(70)

Proof

A short calculation shows that the function \(w_\epsilon \) defined in Eq. (69) solves the Cauchy problem (70), and uniqueness holds by strict positiveness of the initial data (see [147, Chap. VIII, Thm. 2.2] and note that the uniqueness result can easily be generalized to \(n > 1\)).

Now, let \(\lambda \in [0,1]\) and \((\textit{\textbf{x}}_{1},t_{1})\) and \((\textit{\textbf{x}}_{2},t_{2})\) be such that \(\textit{\textbf{x}}=\lambda \textit{\textbf{x}}_{1}+(1-\lambda )\textit{\textbf{x}}_{2}\) and \(t=\lambda t_{1}+(1-\lambda )t_{2}\). Then, the Hölder’s inequality (see, e.g., [57, Thm. 6.2]) implies

$$\begin{aligned}&\sum _{i=1}^{m}e^{\left( \left\langle \textit{\textbf{p}}_{i},\textit{\textbf{x}}\right\rangle +\frac{t}{2}\left\| \textit{\textbf{p}}_{i}\right\| _{2}^{2}-\gamma _{i}\right) /\epsilon } = \sum _{i=1}^{m}\left( e^{\lambda \left( \left\langle \textit{\textbf{p}}_{i},\textit{\textbf{x}}_{1}\right\rangle +\frac{t_{1}}{2}\left\| \textit{\textbf{p}}_{i}\right\| _{2}^{2}-\gamma _{i}\right) /\epsilon }e^{(1-\lambda )\left( \left\langle \textit{\textbf{p}}_{i},\textit{\textbf{x}}_{2}\right\rangle +\frac{t_{2}}{2}\left\| \textit{\textbf{p}}_{i}\right\| _{2}^{2}-\gamma _{i}\right) /\epsilon }\right) \\&\quad \leqslant \left( \sum _{i=1}^{m}e^{\left( \left\langle \textit{\textbf{p}}_{i},\textit{\textbf{x}}_{1}\right\rangle +\frac{t_{1}}{2}\left\| \textit{\textbf{p}}_{i}\right\| _{2}^{2}-\gamma _{i}\right) /\epsilon }\right) ^{\lambda }\left( \sum _{i=1}^{m}e^{\left( \left\langle \textit{\textbf{p}}_{i},\textit{\textbf{x}}_{2}\right\rangle +\frac{t_{2}}{2}\left\| \textit{\textbf{p}}_{i}\right\| _{2}^{2}-\gamma _{i}\right) /\epsilon }\right) ^{1-\lambda }, \end{aligned}$$

and we find \(w_{\epsilon }(\textit{\textbf{x}},t)\leqslant \left( w_{\epsilon }(\textit{\textbf{x}}_{1},t_{1})\right) ^{\lambda }\left( w_{\epsilon }(\textit{\textbf{x}}_{2},t_{2})\right) ^{1-\lambda }\), which implies that \(w_\epsilon \) is jointly log-convex in \((\textit{\textbf{x}},t)\). \(\square \)

Thanks to Lemma C.1 and the Cole–Hopf transformation \(f_\epsilon (\textit{\textbf{x}},t) = \epsilon \log \left( w_\epsilon (\textit{\textbf{x}},t)\right) \) (see, e.g., [47], Sect. 4.4.1), a short calculation immediately implies that the neural network \(f_\epsilon \) solves the viscous HJ PDE (18), and it is also its unique solution because \(w_{\epsilon }\) is the unique solution to the Cauchy problem (70). Joint convexity in \((\textit{\textbf{x}},t)\) follows from log-convexity of \((\textit{\textbf{x}},t)\mapsto w_{\epsilon }(\textit{\textbf{x}},t)\) for every \(\epsilon >0\).

D Proof of Proposition 3.1

To prove this proposition, we will use three lemmas whose statements and proofs are given in Sect. D.1, D.2, and D.3, respectively. The proof of Prop. 3.1 is given in Sect. D.4.

D.1 Statement and proof of Lemma D.1

Lemma D.1

Consider the one-dimensional case, i.e., \(n=1\). Let \(p_1,\dots ,p_m\in \mathbb {R}\) satisfy \(p_1<\dots <p_m\) and define the function J using Eq. (10). Suppose assumptions (A1)-(A2) hold. Let \(x\in \mathbb {R}\), \(p\in \partial J(x)\), and suppose \(p\ne p_i\) for any \(i\in \{1,\dots ,m\}\). Then, there exists \(k\in \{1,\dots ,m\}\) such that \(p_k< p < p_{k+1}\) and

$$\begin{aligned} k,k+1 \in {\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{i\in \{1,\dots ,m\}}}\{xp_{i} -\gamma _{i}\}. \end{aligned}$$
(71)

Proof

Let \(I_x\) denotes the set of maximizers in Eq. (11) at x. Since \(p\in \partial J(x)\), \(p \ne p_i\) for \(i \in \{1,\dots ,m\}\), and \(\partial J(x) = {{\mathrm {co}}~}\{p_i: i\in I_i\}\) by [67, Thm. VI.4.4.2], there exist \(j,l\in I_x\) such that \(p_j< p < p_l\). Moreover, there exists k with \(j\leqslant k<k+1 \leqslant l\) such that \(p_j \leqslant p_k< p <p_{k+1}\leqslant p_l\). We will show that \(k,k+1 \in I_x\). We only prove \(k\in I_x\); the case for \(k+1\) is similar.

If \(p_j=p_k\), then \(k=j\in I_x\) and the conclusion follows directly. Hence suppose \(p_j< p_k < p_l\). Then, there exists \(\alpha \in (0,1)\) such that \(p_k = \alpha p_j + (1-\alpha ) p_l\). Using that \(j,l\in I_x\), assumption (A2), and Jensen inequality, we get

$$\begin{aligned} \begin{aligned} xp_k - \gamma _k&= xp_k - g(p_k) = (\alpha p_j + (1-\alpha )p_l)x - g(\alpha p_j + (1-\alpha )p_l)\\&\geqslant \alpha xp_j + (1-\alpha )xp_l - \alpha g(p_j) - (1-\alpha )g(p_l)\\&= \alpha (xp_j - \gamma _j) + (1-\alpha )(xp_l - \gamma _l)\\&= \max _{i\in \{1,\dots ,m\}}\{xp_{i} -\gamma _{i}\}, \end{aligned} \end{aligned}$$

which implies that \(k \in I_x\). A similar argument shows that \(k+1\in I_x\), which completes the proof. \(\square \)

D.2 Statement and proof of Lemma D.2

Lemma D.2

Consider the one-dimensional case, i.e., \(n=1\). Let \(p_1,\dots ,p_m\in \mathbb {R}\) satisfy \(p_1<\cdots <p_m\) and define the function H using Eq. (14). Suppose assumptions (A1)–(A3) hold. Let \(u_0\in \mathbb {R}\) and \(p_k< u_0< p_{k+1}\) for some index k. Then, there holds

$$\begin{aligned} H(u_0) = \beta _k\theta _k + \beta _{k+1} \theta _{k+1}, \end{aligned}$$
(72)

where

$$\begin{aligned} \beta _k{:}{=}\frac{p_{k+1} - u_0}{p_{k+1}-p_k} \quad \text { and } \quad \beta _{k+1} {:}{=}\frac{u_0- p_k}{p_{k+1}-p_k}. \end{aligned}$$
(73)

Proof

Let \(\varvec{\beta }{:}{=}(\beta _1,\dots ,\beta _m) \in \varLambda _{m}\) satisfy

$$\begin{aligned} \beta _k{:}{=}\frac{p_{k+1} - u_0}{p_{k+1}-p_k} \quad \text { and } \quad \beta _{k+1} {:}{=}\frac{u_0- p_k}{p_{k+1}-p_k}, \end{aligned}$$

and \(\beta _i = 0\) for every \(i\in \{1,\dots ,m\}\setminus \{k,k+1\}\). We will prove that \(\varvec{\beta }\) is a minimizer in Eq. (14) evaluated at \(u_0\), that is,

$$\begin{aligned} \varvec{\beta } \in {\mathop {{{\,\mathrm{arg\,min}\,}}}\limits _{\varvec{\alpha }\in \mathcal {A}(u_0)}} \left\{ \sum _{i=1}^m \alpha _i \theta _i\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathcal {A}(u_0) {:}{=}{\mathop {{{\,\mathrm{arg\,min}\,}}}\limits _{\begin{array}{c} (\alpha _{1},\dots \alpha _{m})\in \varLambda _m\\ \sum _{i=1}^{m}\alpha _{i}p_{i}=u_0 \end{array}} }\left\{ \sum _{i=1}^{m}\alpha _{i}\gamma _{i}\right\} . \end{aligned}$$

First, we show that \(\varvec{\beta } \in \mathcal {A}(u_0)\). By definition of \(\varvec{\beta }\) and Lemma 3.1(ii) with \(p = u_0\), the statement holds provided \(k,k+1\in I_x\), where the set \(I_x\) contains the maximizers in Eq. (10) evaluated at \(x \in \partial J^*(u_0)\). But if \(x\in \partial J^*(u_0)\), we have \(u_0\in \partial J(x)\), and Lemma D.1 implies \(k,k+1\in I_x\). Hence, \(\varvec{\beta } \in \mathcal {A}(u_0)\).

Now, suppose that \(\varvec{\beta }\) is not a minimizer in Eq. (14) evaluated at \(u_0\). By Lemma 3.2(i), there exists a minimizer in Eq. (14) evaluated at the point \(u_0\), which we denote by \((\alpha _1,\dots , \alpha _m)\). Then there holds

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum _{i=1}^m \alpha _i = \sum _{i=1}^m \beta _i = 1,\\ \sum _{i=1}^m \alpha _i p_i = \sum _{i=1}^m \beta _i p_i = u_0,\\ \sum _{i=1}^m \alpha _i \gamma _i = \sum _{i=1}^m \beta _i \gamma _i = J^*(u_0),\\ \sum _{i=1}^m \alpha _i \theta _i < \sum _{i=1}^m \beta _i \theta _i. \end{array}\right. } \end{aligned}$$
(74)

Since \(\alpha _i \geqslant 0\) for every i and \(\beta _i = 0\) for every \(i\in \{1,\dots ,m\}\setminus \{k,k+1\}\), we have \(\alpha _k + \alpha _{k+1} \leqslant 1 = \beta _k + \beta _{k+1}\). As \(\varvec{\alpha }\ne \varvec{\beta }\), then one or both of the inequalities \(\alpha _k < \beta _k\) and \(\alpha _{k+1} < \beta _{k+1}\) hold. This leaves three possible cases, and we now show that each case leads to a contradiction.

Case 1: Let \(\alpha _k < \beta _k\) and \(\alpha _{k+1} \geqslant \beta _{k+1}\). Define the coefficient \(c_i\) by

The following equations then hold

$$\begin{aligned} {\left\{ \begin{array}{ll} (c_1,\dots , c_m)\in \varDelta _m \text { with } c_k = 0,\\ \sum _{i\ne k}c_i p_i = p_k,\\ \sum _{i\ne k}c_i \gamma _i = \gamma _k,\\ \sum _{i\ne k}c_i \theta _i < \theta _k. \end{array}\right. } \end{aligned}$$

These equations, however, violate assumption (A3), and so we get a contradiction.

Case 2: Let \(\alpha _k \geqslant \beta _k\) and \(\alpha _{k+1} < \beta _{k+1}\). A similar argument as in case 1 can be applied here by exchanging the indices k and \(k+1\) to derive a contradiction.

Case 3: Let \(\alpha _k < \beta _k\) and \(\alpha _{k+1} < \beta _{k+1}\). From Eq. (74), we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} \beta _k - \alpha _k + \beta _{k+1} - \alpha _{k+1} = \sum _{i\ne k,k+1} \alpha _i,\\ (\beta _k - \alpha _k) p_k + (\beta _{k+1} - \alpha _{k+1}) p_{k+1} = \sum _{i\ne k,k+1} \alpha _i p_i,\\ (\beta _k - \alpha _k) \gamma _k + (\beta _{k+1} - \alpha _{k+1}) \gamma _{k+1} = \sum _{i\ne k,k+1} \alpha _i \gamma _i,\\ (\beta _k - \alpha _k)\theta _k + (\beta _{k+1} - \alpha _{k+1})\theta _{k+1} > \sum _{i\ne k,k+1} \alpha _i \theta _i. \end{array}\right. } \end{aligned}$$
(75)

Define two numbers \(q_k\) and \(q_{k+1}\) by

$$\begin{aligned} \begin{aligned} q_k{:}{=}\frac{\sum _{i<k} \alpha _ip_i}{\sum _{i<k}\alpha _i} \quad \text { and } \quad q_{k+1}{:}{=}\frac{\sum _{i>k+1} \alpha _ip_i}{\sum _{i>k+1}\alpha _i}. \end{aligned} \end{aligned}$$
(76)

Note that from the first two equations in (74) and the assumption that \(\alpha _k < \beta _k\) and \(\alpha _{k+1} < \beta _{k+1}\), there exist \(i_1<k\) and \(i_2>k+1\) such that \(\alpha _{i_1}\ne 0\) and \(\alpha _{i_2}\ne 0\), and hence, the numbers \(q_k\) and \(q_{k+1}\) are well-defined. By definition, we have \(q_k<p_k<p_{k+1}<q_{k+1}\). Therefore, there exist \(b_k,b_{k+1}\in (0,1)\) such that

$$\begin{aligned} p_k = b_kq_k+ (1-b_k)q_{k+1}\quad \text { and } \quad p_{k+1} = b_{k+1}q_k+ (1-b_{k+1})q_{k+1}. \end{aligned}$$
(77)

A straightforward computation yields

$$\begin{aligned} b_k= \frac{q_{k+1}- p_k}{q_{k+1}- q_k} \quad \text { and }\quad b_{k+1}= \frac{q_{k+1}- p_{k+1}}{q_{k+1}- q_k}. \end{aligned}$$
(78)

Define the coefficients \(c_i^k\) and \(c_i^{k+1}\) as follows

(79)

These coefficients satisfy \(c_i^k, c_i^{k+1}\in [0,1]\) for any i and \(\sum _{i=1}^m c_i^k= \sum _{i=1}^m c_i^{k+1}=1\). In other words, we have

$$\begin{aligned} (c_1^k,\dots , c_m^k)\in \varDelta _m \text { with }c_k^k= 0\quad \text { and }\quad (c_1^{k+1},\dots , c_m^{k+1})\in \varDelta _m \text { with }c_{k+1}^{k+1}= 0. \end{aligned}$$
(80)

Hence, the first equality in Eq. (9) holds for the coefficients \((c_1^k,\dots , c_m^k)\) with the index k and also for the coefficients \((c_1^{k+1}, \dots , c_m^{k+1})\) with the index \(k+1\). We show next that these coefficients satisfy the second and third equalities in (9) and draw a contradiction with assumption (A3).

Using Eqs. (76), (77), and (79) to write the formulas for \(p_k\) and \(p_{k+1}\) via the coefficients \(c_i^k\) and \(c_i^{k+1}\), we find

$$\begin{aligned} \begin{aligned}&p_k = b_k\frac{\sum _{i<k} \alpha _ip_i}{\sum _{i<k}\alpha _i} + (1-b_k)\frac{\sum _{i>k+1} \alpha _ip_i}{\sum _{i>k+1}\alpha _i} = \sum _{i\ne k,k+1}c_i^kp_i= \sum _{i\ne k}c_i^kp_i,\\&p_{k+1} = b_{k+1}\frac{\sum _{i<k} \alpha _ip_i}{\sum _{i<k}\alpha _i} + (1-b_{k+1})\frac{\sum _{i>k+1} \alpha _ip_i}{\sum _{i>k+1}\alpha _i}=\sum _{i\ne k,k+1} c_i^{k+1}p_i = \sum _{i\ne k+1} c_i^{k+1}p_i, \end{aligned} \end{aligned}$$
(81)

where the last equalities in the two formulas above hold because \(c_{k+1}^k=0\) and \(c_k^{k+1} = 0\) by definition. Hence, the second equality in Eq. (9) also holds for both the index k and \(k+1\).

From the third equality in Eq. (75), assumption (A2), Eq. (81), and Jensen’s inequality, we have

$$\begin{aligned} \begin{aligned}&\sum _{i\ne k,k+1} \alpha _i\gamma _i = (\beta _k - \alpha _k)\gamma _k + (\beta _{k+1} - \alpha _{k+1})\gamma _{k+1}\\&\quad = (\beta _k - \alpha _k)g(p_k) + (\beta _{k+1} - \alpha _{k+1})g(p_{k+1})\\&\quad = (\beta _k - \alpha _k)g\left( \sum _{i\ne k,k+1}c_i^kp_i\right) + (\beta _{k+1} - \alpha _{k+1})g\left( \sum _{i\ne k,k+1}c_i^{k+1}p_i\right) \\&\quad \leqslant (\beta _k - \alpha _k)\left( \sum _{i\ne k,k+1}c_i^kg(p_i)\right) + (\beta _{k+1} - \alpha _{k+1})\left( \sum _{i\ne k,k+1}c_i^{k+1}g(p_i)\right) \\&\quad = \sum _{i\ne k,k+1} ((\beta _k - \alpha _k)c_i^k+ (\beta _{k+1} - \alpha _{k+1})c_i^{k+1})g(p_i)\\&\quad = \sum _{i\ne k,k+1} ((\beta _k - \alpha _k)c_i^k+ (\beta _{k+1} - \alpha _{k+1})c_i^{k+1})\gamma _i. \end{aligned} \end{aligned}$$
(82)

We now compute and simplify the coefficients \((\beta _k - \alpha _k)c_i^k+ (\beta _{k+1} - \alpha _{k+1})c_i^{k+1}\) in the formula above. First, consider the case when \(i<k\). Eqs. (78) and (79) imply

$$\begin{aligned}&(\beta _k - \alpha _k)c_i^k+ (\beta _{k+1} - \alpha _{k+1})c_i^{k+1}\\&\quad = (\beta _k - \alpha _k) \frac{b_k\alpha _i}{\sum _{\omega<k}\alpha _\omega } + (\beta _{k+1} - \alpha _{k+1})\frac{b_{k+1}\alpha _i}{\sum _{\omega<k}\alpha _\omega }\\&\quad =\frac{\alpha _i}{\sum _{\omega<k}\alpha _\omega } ((\beta _k-\alpha _k)b_k+ (\beta _{k+1} - \alpha _{k+1})b_{k+1})\\&\quad = \frac{\alpha _i}{\sum _{\omega<k}\alpha _\omega }\left( (\beta _k-\alpha _k)\frac{q_{k+1}- p_k}{q_{k+1}- q_k} + (\beta _{k+1} - \alpha _{k+1})\frac{q_{k+1}- p_{k+1}}{q_{k+1}- q_k}\right) \\&\quad = \frac{\alpha _i}{\sum _{\omega <k}\alpha _\omega }\cdot \frac{1}{q_{k+1}- q_k} ((\beta _k - \alpha _k + \beta _{k+1} - \alpha _{k+1})q_{k+1}\\&\quad - (\beta _k - \alpha _k) p_k - (\beta _{k+1} - \alpha _{k+1}) p_{k+1}). \end{aligned}$$

Applying the first two equalities in Eq. (75) and Eq. (76) to the last formula above, we obtain

$$\begin{aligned} \begin{aligned}&(\beta _k - \alpha _k)c_i^k+ (\beta _{k+1} - \alpha _{k+1})c_i^{k+1}\\&\quad = \frac{\alpha _i}{\sum _{\omega<k}\alpha _\omega }\cdot \frac{1}{q_{k+1}- q_k} \left( \left( \sum _{i\ne k,k+1}\alpha _i\right) q_{k+1}- \sum _{i\ne k,k+1}\alpha _ip_i\right) \\&\quad =\frac{\alpha _i}{\sum _{\omega<k}\alpha _\omega }\cdot \frac{1}{q_{k+1}- q_k} \left( \sum _{i\ne k,k+1}\alpha _iq_{k+1}- \sum _{i< k}\alpha _ip_i - \sum _{i> k+1}\alpha _ip_i\right) \\&\quad =\frac{\alpha _i}{\sum _{\omega<k}\alpha _\omega }\cdot \frac{1}{q_{k+1}- q_k} \left( \sum _{i\ne k,k+1}\alpha _iq_{k+1}- \left( \sum _{i< k}\alpha _i\right) q_k- \left( \sum _{i> k+1}\alpha _i\right) q_{k+1}\right) \\&\quad = \frac{\alpha _i}{\sum _{\omega<k}\alpha _\omega }\cdot \frac{1}{q_{k+1}- q_k} \left( \sum _{i<k}\alpha _i(q_{k+1}- q_k)\right) \\&\quad = \alpha _i. \end{aligned} \end{aligned}$$

The same result for the case when \(i>k+1\) also holds and the proof is similar. Therefore, we have

$$\begin{aligned} (\beta _k - \alpha _k)c_i^k+ (\beta _{k+1} - \alpha _{k+1})c_i^{k+1}= \alpha _i \quad \text { for each } i\ne k, k+1. \end{aligned}$$
(83)

Combining Eqs. (82) and (83), we have

$$\begin{aligned} \begin{aligned}&\sum _{i\ne k,k+1} \alpha _i\gamma _i \leqslant \sum _{i\ne k,k+1} ((\beta _k - \alpha _k)c_i^k+ (\beta _{k+1} - \alpha _{k+1})c_i^{k+1})\gamma _i = \sum _{i\ne k,k+1} \alpha _i\gamma _i. \end{aligned} \end{aligned}$$

Since the left side and right side are the same, the inequality above becomes equality, which implies that the inequality in Eq. (82) also becomes equality. In other words, we have

$$\begin{aligned} \begin{aligned} \gamma _k&= g\left( p_k\right) =\sum _{i\ne k,k+1}c_i^kg(p_i) = \sum _{i\ne k,k+1} c_i^k\gamma _i = \sum _{i\ne k} c_i^k\gamma _i,\\ \gamma _{k+1}&= g\left( p_{k+1}\right) = \sum _{i\ne k,k+1}c_i^{k+1}g(p_i) = \sum _{i\ne k,k+1} c_i^{k+1}\gamma _i = \sum _{i\ne k+1} c_i^{k+1}\gamma _i, \end{aligned} \end{aligned}$$
(84)

where the last equalities in the two formulas above hold because \(c_{k+1}^k=0\) and \(c_k^{k+1} = 0\) by definition. Hence, the third equality in (9) also holds for both indices k and \(k+1\).

In summary, Eqs. (80), (81), and (84) imply that Eq. (9) holds for the index k with coefficients \((c_1^k,\dots , c_m^k)\) and also for the index \(k+1\) with coefficients \((c_1^{k+1}, \dots , c_m^{k+1})\). Hence, by assumption (A3), we find

$$\begin{aligned} \sum _{i\ne k} c_i^k\theta _i> \theta _k \quad \text { and } \quad \sum _{i\ne k+1} c_i^{k+1}\theta _i > \theta _{k+1}. \end{aligned}$$

Using the inequalities above with Eq. (83) and the fact that \(c_{k+1}^k=0\) and \(c_k^{k+1} = 0\), we find

$$\begin{aligned} \begin{aligned}&(\beta _k - \alpha _k) \theta _k + (\beta _{k+1} - \alpha _{k+1})\theta _{k+1} < (\beta _k - \alpha _k) \sum _{i\ne k} c_i^k\theta _i + (\beta _{k+1} - \alpha _{k+1} )\sum _{i\ne k+1} c_i^{k+1}\theta _i\\&\quad = \sum _{i\ne k,k+1} ((\beta _k - \alpha _k) c_i^k+ (\beta _{k+1} - \alpha _{k+1} )c_i^{k+1})\theta _i = \sum _{i\ne k, k+1} \alpha _i\theta _i, \end{aligned} \end{aligned}$$

which contradicts the last inequality in Eq. (75).

In conclusion, we obtain contradictions in all the three cases. As a consequence, we conclude that \(\varvec{\beta }\) is a minimizer in Eq. (14) evaluated at \(u_0\) and Eq. (72) follows from the definition of H in (14). \(\square \)

D.3 Statement and proof of Lemma D.3

Lemma D.3

Consider the one-dimensional case, i.e., \(n=1\). Let \(p_1,\dots ,p_m\in \mathbb {R}\) satisfy \(p_1<\dots <p_m\). Suppose assumptions (A1)-(A2) hold. Let \(x\in \mathbb {R}\) and \(t>0\). Assume jkl are three indices such that \(1\leqslant j\leqslant k<l\leqslant m\) and

$$\begin{aligned} j, l \in {\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{i\in \{1,\dots ,m\}}}\{xp_{i} -t\theta _{i}-\gamma _{i}\}. \end{aligned}$$
(85)

Then, there holds

$$\begin{aligned} \frac{\theta _l - \theta _k}{p_l - p_k} \leqslant \frac{\theta _l - \theta _j}{p_l - p_j}. \end{aligned}$$
(86)

Proof

Note that Eq. (86) holds trivially when \(j=k\), so we only need to consider the case when \(j<k<l\). On the one hand, Eq. (85) implies

$$\begin{aligned} xp_j - t\theta _j - \gamma _j = xp_l - t\theta _l - \gamma _l \geqslant xp_k - t\theta _k - \gamma _k, \end{aligned}$$

which yields

$$\begin{aligned} \begin{aligned}&\gamma _l - \gamma _k \leqslant x(p_l - p_k) - t(\theta _l - \theta _k),\\&\gamma _l - \gamma _j = x(p_l - p_j) - t(\theta _l - \theta _j).\\ \end{aligned} \end{aligned}$$
(87)

On the other hand, for each \(i\in \{j,j+1,\dots , l-1\}\) let \(q_i\in (p_i, p_{i+1})\) and \(x_i \in \partial J^*(q_i)\). Such \(x_i\) exists because \(q_i \in {\mathrm {int}~}{{\mathrm {dom}~}J^*}\), so that the subdifferential \(\partial J^*(q_i)\) is non-empty. Then, \(q_i\in \partial J(x_i)\) and Lemma D.1 imply

$$\begin{aligned} x_ip_i - \gamma _i = x_ip_{i+1} - \gamma _{i+1} = \max _{\omega \in \{1,\dots ,m\}} \{x_i p_\omega - \gamma _\omega \}. \end{aligned}$$

A straightforward computation yields

$$\begin{aligned} \begin{aligned}&\gamma _l - \gamma _k = \sum _{i=k}^{l-1} (\gamma _{i+1} - \gamma _i) = \sum _{i=k}^{l-1} x_i(p_{i+1} - p_i),\\&\gamma _l - \gamma _j = \sum _{i=j}^{l-1} (\gamma _{i+1} - \gamma _i) = \sum _{i=j}^{l-1} x_i(p_{i+1} - p_i). \end{aligned} \end{aligned}$$

Combining the two equalities above with Eq. (87), we conclude that

$$\begin{aligned} \begin{aligned}&x(p_l - p_k) - t(\theta _l - \theta _k)\geqslant \sum _{i=k}^{l-1} x_i(p_{i+1} - p_i),\\&x(p_l - p_j) - t(\theta _l - \theta _j)= \sum _{i=j}^{l-1} x_i(p_{i+1} - p_i). \end{aligned} \end{aligned}$$

Now, divide the inequality above by \(t(p_l - p_k) > 0\) (because by assumption \(t>0\) and \(l>k\), which implies that \(p_l>p_k\)), divide the equality above by \(t(p_l - p_j) > 0\) (because \(l>j\), which implies that \(t(p_l - p_j) \ne 0\)), and rearrange the terms to obtain

$$\begin{aligned} \begin{aligned}&\frac{\theta _l - \theta _k}{p_l - p_k} \leqslant \frac{x}{t} - \frac{1}{t} \frac{\sum _{i=k}^{l-1} x_i(p_{i+1} - p_i)}{p_l - p_k},\\&\frac{\theta _l - \theta _j}{p_l - p_j} = \frac{x}{t} - \frac{1}{t} \frac{\sum _{i=j}^{l-1} x_i(p_{i+1} - p_i)}{p_l - p_j}. \end{aligned} \end{aligned}$$
(88)

Recall that \(q_j< q_{j+1}< \dots < q_{l-1}\) and \(x_i \in \partial J^*(q_i)\) for any \(j\leqslant i<l\). Since the function \(J^*\) is convex, the subdifferential operator \(\partial J^*\) is a monotone non-decreasing operator [67, Def. IV.4.1.3, and Prop. VI.6.1.1], which yields \(x_j\leqslant x_{j+1}\leqslant \dots \leqslant x_{l-1}\). Using that \(p_1< p_2< \dots < p_m\) and \(j<k<l\), we obtain

$$\begin{aligned}&\frac{\sum _{i=k}^{l-1} x_i(p_{i+1} - p_i)}{p_l - p_k} \geqslant \frac{\sum _{i=k}^{l-1} x_k(p_{i+1} - p_i)}{p_l - p_k} = x_k\nonumber \\&= \frac{\sum _{i=j}^{k-1} x_k(p_{i+1} - p_i)}{p_k - p_j} \geqslant \frac{\sum _{i=j}^{k-1} x_i(p_{i+1} - p_i)}{p_k - p_j}. \end{aligned}$$
(89)

To proceed, we now use that fact that if four real numbers \(a,c\in \mathbb {R}\) and \(b,d>0\) satisfy \(\frac{a}{b}\geqslant \frac{c}{d}\), then \(\frac{a}{b}\geqslant \frac{a+c}{b+d}\). Combining this fact with inequality (89), we find

$$\begin{aligned}&\frac{\sum _{i=k}^{l-1} x_i(p_{i+1} - p_i)}{p_l - p_k} \geqslant \frac{\sum _{i=k}^{l-1} x_i(p_{i+1} - p_i) + \sum _{i=j}^{k-1} x_i(p_{i+1} - p_i)}{p_l - p_k + p_k - p_j}\\&= \frac{\sum _{i=j}^{l-1} x_i(p_{i+1} - p_i)}{p_l - p_j}. \end{aligned}$$

We combine the inequality above with (88) to obtain

$$\begin{aligned} \frac{\theta _l - \theta _k}{p_l - p_k} \leqslant \frac{\theta _l - \theta _j}{p_l - p_j}. \end{aligned}$$

which concludes the proof. \(\square \)

D.4 Proof of Proposition 3.1

Proof of (i): First, note that u is piecewise constant. Second, recall that J is defined as the pointwise maximum of a finite number of affine functions. Therefore, the initial data \(u(\cdot ,0) = \nabla J(\cdot )\) (recall that here, the gradient \(\nabla \) is taken in the sense of distribution) are bounded and of locally bounded variation (see [48, Chap. 5, page 167] for the definition of locally bounded variation). Finally, the flux function H, defined in Eq. (14), is Lipschitz continuous in \({\mathrm {dom}~}J^*\) by Lemma D.2. It can therefore be extended to \(\mathbb {R}\) while preserving its Lipschitz property [57, Thm. 4.16]. Therefore, we can invoke [36, Prop. 2.1] to conclude that u is the entropy solution to the conservation law (21) provided it satisfies the two following conditions. Let \(\bar{x}(t)\) be any smooth line of discontinuity of u. Fix \(t>0\) and define \(u^-\) and \(u^+\) as

$$\begin{aligned} u^-{:}{=}\lim _{x\rightarrow \bar{x}(t)^-} u(x,t)\quad \text { and } \quad u^+{:}{=}\lim _{x\rightarrow \bar{x}(t)^+} u(x,t). \end{aligned}$$
(90)

Then, the two conditions are:

  1. 1.

    The curve \(\bar{x}(t)\) is a straight line with the slope

    $$\begin{aligned} \frac{d\bar{x}}{dt} = \frac{H(u^+)-H(u^-)}{u^+- u^-}. \end{aligned}$$
    (91)
  2. 2.

    For any \(u_0\) between \(u^+\) and \(u^-\), we have

    $$\begin{aligned} \frac{H(u^+) - H(u_0)}{u^+- u_0} \leqslant \frac{H(u^+) - H(u^-)}{u^+- u^-}. \end{aligned}$$
    (92)

First, we prove the first condition and Eq. (91). According to the definition of u in Eq. (20), the range of u is the compact set \(\{p_1,\dots , p_m\}\). As a result, \(u^-\) and \(u^+\) are in the range of u, i.e., there exist indices j and l such that

$$\begin{aligned} u^-= p_j\quad \text { and }\quad u^+= p_l. \end{aligned}$$
(93)

Let \((\bar{x}(s),s)\) be a point on the curve \(\bar{x}\) which is not one of the endpoints. Since u is piecewise constant, there exists a neighborhood \(\mathcal {N}\) of \((\bar{x}(s),s)\) such that for any \((x^-,t), (x^+,t)\in \mathcal {N}\) satisfying \(x^-< \bar{x}(t) < x^+\), we have \(u(x^-,t) = u^-= p_j\) and \(u(x^+,t) = u^+= p_l\). In other words, if \(x^-, x^+, t\) are chosen as above, according to the definition of u in Eq. (20), we have

$$\begin{aligned} j\in {\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{i\in \{1,\dots ,m\}}}\{x^-p_{i} -t\theta _{i}-\gamma _{i}\} \quad \text { and }\quad l\in {\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{i\in \{1,\dots ,m\}}}\{x^+p_{i} -t\theta _{i}-\gamma _{i}\}. \end{aligned}$$
(94)

Define a sequence \(\{x^-_k\}_{k=1}^{+\infty } \subset (-\infty ,\bar{x}(s))\) such that \((x^-_k, s)\in \mathcal {N}\) for any \(k\in \mathbb {N}\) and \(\lim _{k\rightarrow +\infty }x^-_k = \bar{x}(s)\). By Eq. (94), we have

$$\begin{aligned} x^-_k p_{j} -s\theta _{j}-\gamma _{j} \ge x^-_k p_{i} -s\theta _{i}-\gamma _{i} \text { for any }i\in \{1,\dots ,m\}. \end{aligned}$$

When k approaches infinity, the above inequality implies

$$\begin{aligned} \bar{x}(s) p_{j} -s\theta _{j}-\gamma _{j} \ge \bar{x}(s) p_{i} -s\theta _{i}-\gamma _{i}\quad \text { for any }i\in \{1,\dots ,m\}. \end{aligned}$$

In other words, we have

$$\begin{aligned} j\in {\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{i\in \{1,\dots ,m\}}}\{\bar{x}(s)p_{i} -s\theta _{i}-\gamma _{i}\}. \end{aligned}$$
(95)

Similarly, define a sequence \(\{x^+_k\}_{k=1}^{+\infty } \subset (\bar{x}(s), +\infty )\) such that \((x^+_k, s)\in \mathcal {N}\) for any \(k\in \mathbb {N}\) and \(\lim _{k\rightarrow +\infty }x^+_k = \bar{x}(s)\). Using a similar argument as above, we can conclude that

$$\begin{aligned} l\in {\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{i\in \{1,\dots ,m\}}}\{\bar{x}(s)p_{i} -s\theta _{i}-\gamma _{i}\}. \end{aligned}$$
(96)

By a continuity argument, Eqs. (95) and (96) also hold for the end points of \(\bar{x}\). In conclusion, for any \((\bar{x}(t), t)\) on the curve \(\bar{x}\), we have

$$\begin{aligned} j, l\in {\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{i\in \{1,\dots ,m\}}}\{\bar{x}(t)p_{i} -t\theta _{i}-\gamma _{i}\}, \end{aligned}$$
(97)

which implies that

$$\begin{aligned} \bar{x}(t)p_{l} -t\theta _{l}-\gamma _{l} = \bar{x}(t)p_{j} -t\theta _{j}-\gamma _{j}. \end{aligned}$$

Therefore, the curve \(\bar{x}(t)\) lies on the straight line

$$\begin{aligned} x(p_l - p_j) - t(\theta _l - \theta _j) -(\gamma _l - \gamma _j) = 0 \end{aligned}$$

and Eq. (93) and Lemma 3.2(iii) imply that its slope equals

$$\begin{aligned} \frac{d\bar{x}}{dt} = \frac{\theta _l - \theta _j}{p_l - p_j} = \frac{H(u^+) - H(u^-)}{u^+- u^-}. \end{aligned}$$

This proves Eq. (91) and the first condition holds.

It remains to show the second condition. Since u equals \(\nabla _x f\) and f is convex by Theorem 3.1, its corresponding subdifferential operator u is monotone non-decreasing with respect to x [67, Def. IV.4.1.3 and Prop. VI.6.1.1]. As a result, \(u^-< u^+\) and \(u_0\in (u^-, u^+)\), where we still adopt the notation \(u^-= p_j\) and \(u^+= p_l\). Recall that Lemma 3.2(iii) implies \(H(p_i) = \theta _i\) for any i. Then, Eq. (92) in the second condition becomes

$$\begin{aligned} \frac{\theta _l - H(u_0)}{p_l - u_0} \leqslant \frac{\theta _l - \theta _j}{p_l - p_j}. \end{aligned}$$
(98)

Without loss of generality, we may assume that \(p_1< p_2< \dots < p_m\). Then, the fact \(p_j=u^-< u^+= p_l\) implies \(j<l\). We consider the following two cases.

First, if there exists some k such that \(u_0= p_k\), then \(H(u_0) = \theta _k\) by Lemma 3.2(iii). Since \(u^-< u_0< u^+\), we have \(j< k < l\). Recall that Eq. (97) holds. Therefore, the assumptions of Lemma D.3 are satisfied, which implies Eq. (98) holds.

Second, suppose \(u_0\ne p_i\) for every \(i\in \{1,\dots ,m\}\). Then there exists some \(k\in \{j, j+1,\dots , l-1\}\) such that \(p_k< u_0< p_{k+1}\). Lemma D.2 then implies that Eqs. (72) and (73) hold, that is,

$$\begin{aligned} H(u_0) = \beta _k \theta _k + \beta _{k+1}\theta _{k+1}, \quad u_0= \beta _k p_k + \beta _{k+1} p_{k+1}, \quad \text { and }\quad \beta _k + \beta _{k+1} = 1. \end{aligned}$$

Using these three equations, we can write the left-hand side of Eq. (98) as

$$\begin{aligned} \frac{\theta _l - H(u_0)}{p_l - u_0} = \frac{\theta _l - \beta _k \theta _k - \beta _{k+1} \theta _{k+1}}{p_l - \beta _kp_k - \beta _{k+1}p_{k+1}} = \frac{\beta _k (\theta _l - \theta _k) + \beta _{k+1} (\theta _l-\theta _{k+1})}{\beta _k(p_l - p_k) + \beta _{k+1}(p_l - p_{k+1})}. \end{aligned}$$
(99)

If \(k+1 = l\), then this equation become

$$\begin{aligned} \frac{\theta _l - H(u_0)}{p_l - u_0} = \frac{\theta _l - \theta _k}{p_l - p_k}. \end{aligned}$$

Since \(j\leqslant k<l\) and Eq. (97) hold, then the assumptions of Lemma D.3 are satisfied. This allows us to conclude that Eq. (98) holds.

If \(k+1\ne l\), then using Eq. (97), the inequalities \(j\leqslant k<k+1< l\), and Lemma D.3, we obtain

$$\begin{aligned} \frac{\beta _k(\theta _l - \theta _k)}{\beta _k(p_l - p_k)} = \frac{\theta _l - \theta _k}{p_l - p_k} \leqslant \frac{\theta _l - \theta _j}{p_l - p_j} \quad \text { and }\quad \frac{\beta _{k+1}(\theta _l - \theta _{k+1})}{\beta _{k+1}(p_l - p_{k+1})} = \frac{\theta _l - \theta _{k+1}}{p_l - p_{k+1}} \leqslant \frac{\theta _l - \theta _j}{p_l - p_j}. \end{aligned}$$

Note that if \(a_i \in \mathbb {R}\) and \(b_i \in (0,+\infty )\) for \(i\in \{1,2,3\}\) satisfy \(\frac{a_1}{b_1} \leqslant \frac{a_3}{b_3}\) and \(\frac{a_2}{b_2} \leqslant \frac{a_3}{b_3}\), then \(\frac{a_1+a_2}{b_1+b_2}\leqslant \frac{a_3}{b_3}\). Then, since \(\beta _k(p_l - p_k)\), \(\beta _{k+1}(p_l - p_{k+1})\) and \(p_l-p_j\) are positive, we have

$$\begin{aligned} \frac{\beta _k (\theta _l - \theta _k) + \beta _{k+1} (\theta _l-\theta _{k+1})}{\beta _k(p_l - p_k) + \beta _{k+1}(p_l - p_{k+1})}\leqslant \frac{\theta _l - \theta _j}{p_l - p_j}. \end{aligned}$$

Hence, Eq. (98) follows directly from the inequality above and Eq. (99).

Therefore, the two conditions, including Eqs. (91) and (92), are satisfied and we apply [36, Prop 2.1] to conclude that the function u is the entropy solution to the conservation law (21).

Proof of (ii) (sufficiency): Without loss of generality, assume \(p_1<p_2<\dots <p_m\). Let \(C\in \mathbb {R}\). Suppose \(\tilde{H}\) satisfies \(\tilde{H}(p_i)=H(p_i)+C\) for each \(i\in \{1,\dots ,m\}\) and \(\tilde{H}(p)\geqslant H(p)+C\) for any \(p\in [p_1,p_m]\). We want to prove that u is the entropy solution to the conservation law (22).

As in the proof of (i), we apply [36, Prop 2.1] and verify that the two conditions hold through Eqs. (91) and (92). Let \(\bar{x}(t)\) be any smooth line of discontinuity of u, define \(u^-\) and \(u^+\) by Eq. (90) (and recall that \(u^-= p_j\) and \(u^+= p_l\)), and let \(u_0\in (u^-,u^+)\). We proved in the proof of (i) that \(\bar{x}(t)\) is a straight line, and so it suffices to prove that

$$\begin{aligned} \frac{d\bar{x}}{dt} = \frac{\tilde{H}(u^+)-\tilde{H}(u^-)}{u^+- u^-}, \quad \text { and }\quad \frac{\tilde{H}(u^+) - \tilde{H}(u_0)}{u^+- u_0} \leqslant \frac{\tilde{H}(u^+) - \tilde{H}(u^-)}{u^+- u^-}. \end{aligned}$$
(100)

We start with proving the equality in Eq. (100). By assumption, there holds

$$\begin{aligned}&\tilde{H}(u^-) = \tilde{H}(p_j) = H(p_j)+C=H(u^-)+C\quad \text { and }\nonumber \\&\quad \tilde{H}(u^+) = \tilde{H}(p_l) = H(p_l)+C=H(u^+)+C. \end{aligned}$$
(101)

We combine Eq. (101) with Eq. (91), (which we proved in the proof of (i)), we obtain

$$\begin{aligned} \frac{d\bar{x}}{dt} = \frac{H(u^+)-H(u^-)}{u^+- u^-} = \frac{H(u^+)+C-(H(u^-)+C)}{u^+- u^-} = \frac{\tilde{H}(u^+)-\tilde{H}(u^-)}{u^+- u^-}. \end{aligned}$$

Therefore, the equality in (100) holds.

Next, we prove the inequality in Eq. (100). Since \(u_0\in (u^-,u^+)\subseteq [p_1,p_m]\), by assumption there holds \(\tilde{H}(u_0) \geqslant H(u_0) +C\). Taken together with Eqs. (92) and (101), we get

$$\begin{aligned}&\frac{\tilde{H}(u^+) - \tilde{H}(u_0)}{u^+- u_0} \leqslant \frac{H(u^+)+C - (H(u_0)+C)}{u^+- u_0}\\&\leqslant \frac{H(u^+) - H(u^-)}{u^+- u^-} = \frac{\tilde{H}(u^+) - \tilde{H}(u^-)}{u^+- u^-}, \end{aligned}$$

which shows that the inequality in Eq. (100) holds.

Hence, we can invoke [36, Prop 2.1] to conclude that u is the entropy solution to the conservation law (22).

Proof of (ii) (necessity): Suppose that u is the entropy solution to the conservation law (22). We prove that there exists \(C\in \mathbb {R}\) such that \(\tilde{H}(p_i)=H(p_i)+C\) for any i and \(\tilde{H}(p)\geqslant H(p)+C\) for any \(p\in [p_1,p_m]\).

By Lemma B.2, for each \(i\in \{1,\dots ,m\}\) there exist \(x\in \mathbb {R}\) and \(t>0\) such that

$$\begin{aligned} f(\cdot ,t) \text { is differentiable at }x, \text { and }\nabla _x f(x,t)=p_i. \end{aligned}$$
(102)

Moreover, the proof of Lemma B.2 implies there exists \(T>0\) such that for any \(0<t<T\), there exists \(x\in \mathbb {R}\) such that Eq. (102) holds. As a result, there exists \(t>0\) such that for each \(i\in \{1,\dots ,m\}\), there exists \(x_i\in \mathbb {R}\) satisfying Eq. (102) at the point \((x_i,t)\), which implies \(u(x_i,t) = p_i\). Note that \(p_i\ne p_j\) implies that \(x_i \ne x_j\). (Indeed, if \(x_i = x_j\), then \(p_i = \nabla _x f(x_i,t) = \nabla _x f(x_j,t) = p_j\) which gives a contradiction since \(p_i \ne p_j\) by assumption (A1).) As mentioned before, the function \(u(\cdot ,t) \equiv \nabla _{x}f\) is a monotone non-decreasing operator and \(p_i\) is increasing with respect to i, and therefore \(x_1<x_2<\dots <x_m\). Since u is piecewise constant, for each \(k\in \{1,\dots ,m-1\}\) there exists a curve of discontinuity of u with \(u = p_k\) on the left-hand side of the curve and \(u=p_{k+1}\) on the right-hand side of the curve. Let \(\bar{x}(s)\) be such a curve and let \(u^-\) and \(u^+\) be the corresponding numbers defined in Eq. (90). The argument above proves that we have \(u^-= p_k\) and \(u^+= p_{k+1}\).

Since u is the piecewise constant entropy solution, we invoke [36, Prop 2.1] to conclude that the two aforementioned conditions hold for the curve \(\bar{x}(s)\), i.e., (100) holds with \(u^-= p_k\) and \(u^+= p_{k+1}\). From the equality in (100) and Eq. (91) proved in (i), we deduce

$$\begin{aligned} \frac{\tilde{H}(p_{k+1})-\tilde{H}(p_k)}{p_{k+1} - p_k} = \frac{\tilde{H}(u^+)-\tilde{H}(u^-)}{u^+- u^-} = \frac{d\bar{x}}{dt} = \frac{H(u^+)-H(u^-)}{u^+- u^-} = \frac{H(p_{k+1})-H(p_k)}{p_{k+1} - p_k}. \end{aligned}$$

Since k is an arbitrary index, the equality above implies that \(\tilde{H}(p_{k+1})-\tilde{H}(p_k) = H(p_{k+1})-H(p_k)\) holds for any \(k\in \{1,\dots ,m-1\}\). Therefore, there exists \(C\in \mathbb {R}\) such that

$$\begin{aligned} \tilde{H}(p_k) = H(p_k)+C \quad \text { for any }k\in \{1,\dots ,m\}. \end{aligned}$$
(103)

It remains to prove \(\tilde{H}(u_0)\geqslant H(u_0)+C\) for all \(u_0\in [p_k,p_{k+1}]\). If this inequality holds, then the statement follows because k is an arbitrary index. We already proved that \(\tilde{H}(u_0)\geqslant H(u_0)+C\) for \(u_0=p_k\) with \(k\in \{1,\dots ,m\}\). Therefore, we need to prove that \(\tilde{H}(u_0)\geqslant H(u_0)+C\) for all \(u_0\in (p_k,p_{k+1})\). Let \(u_0\in (p_k,p_{k+1})\). By Eq. (103) and the inequality in (100), we have

$$\begin{aligned} \frac{H(p_{k+1})+C - \tilde{H}(u_0)}{p_{k+1} - u_0} = \frac{\tilde{H}(u^+) - \tilde{H}(u_0)}{u^+- u_0} \leqslant \frac{\tilde{H}(u^+) - \tilde{H}(u^-)}{u^+- u^-} = \frac{H(p_{k+1}) - H(p_k)}{p_{k+1} - p_k}. \end{aligned}$$
(104)

By Lemma D.2 and a straightforward computation, we also have

$$\begin{aligned} \frac{H(p_{k+1}) - H(u_0)}{p_{k+1} - u_0} = \frac{H(p_{k+1}) - H(p_k)}{p_{k+1} - p_k}. \end{aligned}$$
(105)

Comparing Eqs. (104) and (105), we obtain \(\tilde{H}(u_0)\geqslant H(u_0)+C\). Since k is arbitrary, we conclude that \(\tilde{H}(u_0)\geqslant H(u_0)+C\) holds for all \(u_0\in [p_1,p_m]\) and the proof is complete.

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Darbon, J., Langlois, G.P. & Meng, T. Overcoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architectures. Res Math Sci 7, 20 (2020). https://doi.org/10.1007/s40687-020-00215-6

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