An Adaptive Max-Plus Eigenvector Method for Continuous Time Optimal Control Problems

Abstract

An adaptive max-plus eigenvector method is proposed for approximating the solution of continuous time nonlinear optimal control problems. At each step of the method, given a set of quadratic basis functions, a standard max-plus eigenvector method is applied to yield an approximation of the value function of interest. Using this approximation, an approximate level set of the back substitution error defined by the Hamiltonian is tessellated according to where each basis function is active in approximating the value function. The polytopes obtained, and their vertices, are sorted according to this back substitution error, allowing “worst-case” basis functions to be identified. The locations of these basis functions are subsequently evolved to yield new basis functions that reduce this worst-case. Basis functions that are inactive in the value function approximation are pruned, and the aforementioned steps repeated. Underlying algebraic properties associated with max-plus linearity, dynamic programming, and semiconvex duality are provided as a foundation for the development, and the utility of the proposed method is illustrated by example.

Appendix

In the statement of Theorem 3, the semiconvex transform D _φ is as specified by (12), and its candidate inverse is as per (13). For convenience, formally define

(54)

in which ${{\mathcal {R}}_-^{{\mathsf {{K}}}}}$ is as per (14).

Lemma 16

With −K ∈ Σ _M ∪{M} fixed, D _φ and ${\mathsf {{D}}}_\varphi ^{\sharp }$ of (12) and (54) satisfy the following properties:

• 1) ${\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^\sharp ) {{ \ = {\mathrm{ran}\,}({\mathsf {{D}}}_\varphi ) = {{{{\mathcal {R}}_-^{{\mathsf {{K}}}}}}} \subset {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}}}}$;

• 2) ${\mathrm{ran}\,}({\mathsf {{D}}}_\varphi ^{\sharp }) {{ \ = {\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ) = {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}} \subset {{\mathcal {S}}_+^{{\,\mathsf {{-M}}}}}}}$;

• 3) ${\mathsf {{D}}}_\varphi \, {\mathsf {{D}}}_\varphi ^{\sharp } = {\mathsf {{I}}}$ on ${\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^{\sharp }) = {{\mathcal {R}}_-^{{\mathsf {{K}}}}}$;

• 4) ${\mathsf {{D}}}_\varphi ^{\sharp }\, {\mathsf {{D}}}_\varphi = {\mathsf {{I}}}$ on ${\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ) = {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$;

• 5) ${{{\mathcal {R}}}_-^{{-{\mathsf {{M}}}}}} = {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}}$;

• 6) If −K > M then ${{\mathcal {S}}_+^{{\,\mathsf {{-M}}}}}\setminus {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}} \ne \emptyset$ and ${{{\mathcal {R}}}_-^{{-{\mathsf {{M}}}}}}\setminus {{\mathcal {R}}_-^{{\mathsf {{K}}}}} \ne \emptyset$.

The following observations are useful in establishing Lemma 16.

• 1. Given invertible M ∈ Σ,

  $$\displaystyle \begin{aligned} & \langle x,\, z \rangle + {\,{\frac{1}{2}}}\, \langle x,\, {\mathsf{{M}}}\, x\rangle = {\,{\frac{1}{2}}}\, \langle x - \zeta(z),\, {\mathsf{{M}}} \, (x - \zeta(z)) \rangle - {\,{\frac{1}{2}}}\, \langle \zeta(z),\, {\mathsf{{M}}}\, \zeta(z) \rangle {} \end{aligned} $$

  (55)

  for all $x,z\in {\mathbb {R}}^n$, where .

• 2. Given O ₁, O ₂ ∈ Σ satisfying O ₁ < O ₂,

  $$\displaystyle \begin{aligned} {{{\mathcal{S}}_+^{{{{{\,\mathsf{{O}}}_1}}}}}} \subset {{{\mathcal{S}}_+^{{{{{\,\mathsf{{O}}}_2}}}}}}\,, \qquad {{{\mathcal{S}}_-^{{{{{\,\mathsf{{O}}}_1}}}}}} \subset {{{\mathcal{S}}_-^{{{{{\,\mathsf{{O}}}_2}}}}}}\,. {} \end{aligned} $$

  (56)

Proof of Lemma 16

Assertion 1) By definitions (14) and (54),

$$\displaystyle \begin{aligned} {\mathrm{ran}\,}({\mathsf{{D}}}_\varphi) = {{\mathcal{R}}_-^{{\mathsf{{K}}}}} = {\mathrm{dom}\,}({\mathsf{{D}}}_\varphi^\sharp). {} \end{aligned} $$

Fix an arbitrary $a\in {{\mathcal {R}}_-^{{\mathsf {{K}}}}}$. Applying definition (14), there exists a $\psi \in {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$ such that D _φ ψ = a. Define $\psi _+,\psi _\pm :{\mathbb {R}}^n{\rightarrow }{{{{\overline {{\mathbb {R}}}}}}}$ by

(57)

for all $x\in {\mathbb {R}}^n$, and note that

$$\displaystyle \begin{aligned} \psi_\pm(x) & = \psi(x) - {\,{\frac{1}{2}}}\, \langle x,\, {\mathsf{{M}}}\, x\rangle {} \end{aligned} $$

(58)

for all $x\in {\mathbb {R}}^n$. By inspection of (57), ψ ₊ is convex and lower closed on ${\mathbb {R}}^n$, as $\psi \in {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$. Hence, ψ _± is also convex and lower closed on ${\mathbb {R}}^n$, as − (K + M) ≥ 0 by definition of −K ∈ Σ _M ∪{M}. Hence, the convex conjugate $\psi _\pm ^*:{\mathbb {R}}^n{\rightarrow }{{{{\overline {{\mathbb {R}}}}}}}$ of ψ _± is also convex and lower closed [25, Theorem 5, p. 16], with

(59)

for all $z\in {\mathbb {R}}^n$. Define $\widetilde a:{\mathbb {R}}^n{\rightarrow }{{{{\overline {{\mathbb {R}}}}}}}$ by

(60)

for all $\zeta \in {\mathbb {R}}^n$. Note that $\widetilde a$ is upper closed, as $\psi _\pm ^*$ and ${\, {\frac {1}{2}}}\, \langle \cdot ,\, {\mathsf {{M}}}\, \cdot \rangle$ are lower closed. Furthermore, $\zeta \mapsto \tilde a(\zeta ) - {\, {\frac {1}{2}}}\, \langle \zeta , \, -{\mathsf {{M}}}\, \zeta \rangle$ is concave, as $\psi _\pm ^*$ is convex. Hence, $\widetilde a\in {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}}$. Recalling that $\psi _\pm ^*$ is the convex conjugate of ψ _±, (58), (60) imply that

$$\displaystyle \begin{aligned} \widetilde a(\zeta) & {{\ = - \psi_\pm^*(-{\mathsf{{M}}}\, \zeta) - {\,{\frac{1}{2}}}\, \langle \zeta,\, {\mathsf{{M}}}\, \zeta \rangle}} = - \int_{{\mathbb{R}}^n}^\oplus \langle -{\mathsf{{M}}}\, \zeta,\, x \rangle \otimes (-\psi_\pm(x)) \, dx - {\,{\frac{1}{2}}} \, \langle \zeta,\, {\mathsf{{M}}}\, \zeta \rangle {}\\ & = -\int_{{\mathbb{R}}^n}^\oplus {\,{\frac{1}{2}}} \, \langle x,\, {\mathsf{{M}}}\, x \rangle - \langle {\mathsf{{M}}}\, \zeta,\, x \rangle + {\,{\frac{1}{2}}} \, \langle \zeta,\, {\mathsf{{M}}}\, \zeta \rangle -\psi(x) \, dx {}\\ & = ({\mathsf{{D}}}_\varphi\, \psi)(\zeta) = a(\zeta) {} \end{aligned} $$

(61)

for all $\zeta \in {\mathbb {R}}^n$. That is, $a = \widetilde a\in {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}}$, and as $a\in {{\mathcal {R}}_-^{{\mathsf {{K}}}}}$ is arbitrary, ${{\mathcal {R}}_-^{{\mathsf {{K}}}}} \subset {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}}$.

2) By definitions (12) and (54), ${\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ) = {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$ and ${\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^\sharp ) = {{\mathcal {R}}_-^{{\mathsf {{K}}}}}$. Fix an arbitrary $a\in {\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^\sharp ) = {{\mathcal {R}}_-^{{\mathsf {{K}}}}}$. Following the proof of Assertion 1), there exists a $\psi \in {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$ such that D _φ ψ = a, defining a convex and lower closed $\psi _\pm :{\mathbb {R}}^n\rightarrow {{{{\overline {{\mathbb {R}}}}}}}$ as per (57), (58). Note further that $a = \widetilde a\in {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}}$, where $\widetilde a$ is defined as per (60). As $\psi _\pm = \psi _\pm ^{**}$, see [25, Theorem 5, p. 16], (54), (58) and (60) imply that

$$\displaystyle \begin{aligned} \psi(x) & = (\psi_\pm^*)^*(x) + {\,{\frac{1}{2}}}\, \langle x,\, {\mathsf{{M}}}\, x\rangle = \int_{{\mathbb{R}}^n}^\oplus \langle x,\, z\rangle + {\,{\frac{1}{2}}}\, \langle x,\, {\mathsf{{M}}}\, x\rangle - \psi_\pm^*(z)\, dz {}\\ & = \int_{{\mathbb{R}}^n}^\oplus {\,{\frac{1}{2}}}\, \langle x - \zeta,\, {\mathsf{{M}}} \, (x - \zeta) \rangle + \left[ - \psi_\pm^*(-{\mathsf{{M}}}\, \zeta) - {\,{\frac{1}{2}}}\, \langle \zeta,\, {\mathsf{{M}}}\, \zeta \rangle \right] \, d\zeta {}\\ & \,{{ = \int_{{\mathbb{R}}^n}^\oplus {\,{\frac{1}{2}}}\, \langle x - \zeta,\, {\mathsf{{M}}} \, (x - \zeta) \rangle \otimes \widetilde a(\zeta)\, d\zeta = ({\mathsf{{D}}}_\varphi^{\sharp}\, \widetilde a)(x) = ({\mathsf{{D}}}_\varphi^{\sharp}\, a)(x)}} {} \end{aligned} $$

(62)

where the third equality uses (55) and the change of variable . Hence, ${\mathrm{ran}\,}({\mathsf {{D}}}_\varphi ^\sharp ) \subset {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$, as $\psi = {\mathsf {{D}}}_\varphi ^\sharp \, a\in {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$, and $a\in {\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^\sharp )$ is arbitrary.

Alternatively, fix an arbitrary $\psi \in {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$, and construct ψ _± directly using (57), (58). Subsequently define $a = \widetilde a\in {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}}$ via (60), and note that $\psi = {\mathsf {{D}}}_\varphi ^\sharp \, a$ as per (62). That is, ${{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}\subset {\mathrm{ran}\,}({\mathsf {{D}}}_\varphi ^\sharp )$, as $\psi \in {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$ is arbitrary. Recalling the earlier conclusion ${\mathrm{ran}\,}({\mathsf {{D}}}_\varphi ^\sharp ) \subset {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$ yields ${{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}={\mathrm{ran}\,}({\mathsf {{D}}}_\varphi ^\sharp )$. Finally, as −K ∈ Σ _M ∪{M}, i.e. K ≤−M, (56) implies that ${{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}\subset {{{\mathcal {S}}_+^{{\,\mathsf {{-{\mathsf {{M}}}}}}}}}$ as required.

3), 4) Applying Assertions 1) and 2), the compositions ${\mathsf {{D}}}_\varphi \, {\mathsf {{D}}}_\varphi ^\sharp :{{\mathcal {R}}_-^{{\mathsf {{K}}}}}{\rightarrow }{{\mathcal {R}}_-^{{\mathsf {{K}}}}}$ and ${\mathsf {{D}}}_\varphi ^\sharp \, {\mathsf {{D}}}_\varphi :{{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}{\rightarrow }{{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$ are well-defined. Applying (61) and (62) yields $a = {\mathsf {{D}}}_\varphi \, \mathsf {D}_\varphi ^\sharp \, a$ and $\psi = \mathsf {D}_\varphi ^\sharp \, \mathsf {D}_\varphi \, \psi$ for any $a\in {{\mathcal {R}}_-^{\mathsf {K}}} = {\mathrm{dom}\,}(\mathsf {D}_\varphi ^\sharp )$, $\psi \in {{\mathcal {S}}_+^{\,\mathsf {K}}} = {\mathrm{dom}\,}(\mathsf {D}_\varphi )$, i.e.,

$$\displaystyle \begin{aligned} & {\mathsf{{D}}}_\varphi\, {\mathsf{{D}}}_\varphi^\sharp = {\mathsf{{I}}}\,, \qquad && {\mathrm{dom}\,}({\mathsf{{D}}}_\varphi\, {\mathsf{{D}}}_\varphi^\sharp) = {{\mathcal{R}}_-^{{\mathsf{{K}}}}}\,, {}\\ & {\mathsf{{D}}}_\varphi^\sharp\, {\mathsf{{D}}}_\varphi = {\mathsf{{I}}}\,, \qquad && {\mathrm{dom}\,}({\mathsf{{D}}}_\varphi^\sharp\, {\mathsf{{D}}}_\varphi) = {{\mathcal{S}}_+^{{\,\mathsf{{K}}}}}\,. {} \end{aligned} $$

5) Fix any $a\in {{{\mathcal {S}}_-^{{\,\mathsf {{-{\mathsf {{M}}}}}}}}}$. Define $a_\mp :{\mathbb {R}}^n{\rightarrow }{{{{\overline {{\mathbb {R}}}}}}}$ by

$$\displaystyle \begin{aligned} a_\mp(\zeta) & = - \left[ a(\zeta) - {\,{\frac{1}{2}}}\, \langle \zeta,\, -{\mathsf{{M}}}\, \zeta\rangle \right]\,, {} \end{aligned} $$

for all $\zeta \in {\mathbb {R}}^n$. Observe that a _∓ is convex and lower closed by definition of a, so that $a_\mp = a_\mp ^{**}$, see [10, Theorem 5, p. 16]. Hence,

$$\displaystyle \begin{aligned} a(y) & = -(a_\mp^*)^*(y) - {\,{\frac{1}{2}}}\, \langle y,\, {\mathsf{{M}}}\, y\rangle = - \int_{{\mathbb{R}}^n}^\oplus \langle y,\, z\rangle + {\,{\frac{1}{2}}}\, \langle y,\, {\mathsf{{M}}}\, y\rangle - a_\mp^*(z)\, dz {}\\ & = - \int_{{\mathbb{R}}^n}^\oplus {\,{\frac{1}{2}}}\, \langle y - \zeta, \, {\mathsf{{M}}}\, (y - \zeta) \rangle - \left[ {\,{\frac{1}{2}}}\, \langle \zeta,\, {\mathsf{{M}}}\, \zeta \rangle + a_\mp^*({\mathsf{{M}}}\, \zeta) \right] \, d\zeta {} \end{aligned} $$

(63)

where the final equality follows by application of (55). Define $\psi _\mp :{\mathbb {R}}^n{\rightarrow }{{{{\overline {{\mathbb {R}}}}}}}$ by

(64)

for all $\zeta \in {\mathbb {R}}^n$. As $a_\mp ^*$ is also convex and lower closed [10], $\psi _\mp \in {{{\mathcal {S}}_+^{{\,\mathsf {{-{\mathsf {{M}}}}}}}}}$. Substituting (64) in (63), and recalling (12) with selected in the definition of dom (D _φ),

$$\displaystyle \begin{aligned} a(y) & = - \int_{{\mathbb{R}}^n}^\oplus {\,{\frac{1}{2}}}\, \langle y - \zeta, \, {\mathsf{{M}}}\, (y - \zeta) \rangle \otimes (-\psi_\mp(\zeta))\, d\zeta = ({\mathsf{{D}}}_\varphi\, \psi_\mp)(y) {} \end{aligned} $$

for all $y\in {\mathbb {R}}^n$. Hence, $a\in {{{\mathcal {R}}}_-^{{-{\mathsf {{M}}}}}}$ by inspection of (14). As $a\in {{{\mathcal {S}}_-^{{\,\mathsf {{-{\mathsf {{M}}}}}}}}}$ is arbitrary, it follows immediately that ${{{\mathcal {S}}_-^{{\,\mathsf {{-{\mathsf {{M}}}}}}}}}\subset {{{\mathcal {R}}}_-^{{-{\mathsf {{M}}}}}}$. However, applying Assertions 1) and 2) for K = −M yields that ${{{\mathcal {R}}}_-^{{-{\mathsf {{M}}}}}}\subset {{{\mathcal {S}}_-^{{\,\mathsf {{-{\mathsf {{M}}}}}}}}}$, so that the claimed property holds.

6) Let ${\mathsf {{D}}}_\varphi ^{\mathsf {{M}}}$ and ${\mathsf {{D}}}_\varphi ^{{\mathsf {{M}}}\sharp }$ denote the operators D _φ and ${\mathsf {{D}}}_\varphi ^\sharp$ of (12) and (54) with respective domains given by ${\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^{\mathsf {{M}}}) = {{\mathcal {S}}_+^{{\,\mathsf {{-M}}}}}$ and ${\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^{{\mathsf {{M}}}\sharp }) = {{\mathcal {R}}_-^{{\mathsf {{-M}}}}}$. Fix any −K ∈ Σ _M, i.e. so that −K > M. Observe that ${{\mathcal {S}}_+^{{\,\mathsf {{-M}}}}}\setminus {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}\ne \emptyset$, as it contains ${\, {\frac {1}{2}}}\, \langle \cdot ,\, {\mathsf {{N}}} \, \cdot \rangle$ for any N ∈ Σ satisfying M < N < −K. Fix any $\psi \in {{\mathcal {S}}_+^{{\,\mathsf {{-M}}}}}\setminus {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$. By Assertions 1) through 5) applied to ${\mathsf {{D}}}_\varphi ^{{\mathsf {{M}}}}$ and ${\mathsf {{D}}}_\varphi ^{{\mathsf {{M}}}\sharp }$, there exists an $a\in {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}} = {{{\mathcal {R}}}_-^{{-{\mathsf {{M}}}}}}\supset {{{\mathcal {R}}}_-^{{{\mathsf {{K}}}}}}$ such that ${\mathsf {{D}}}_\varphi ^{{\mathsf {{M}}}}\, \psi = a$ and ${\mathsf {{D}}}_\varphi ^{{\mathsf {{M}}}\sharp }\, a = \psi$. Suppose that $a\in {{{\mathcal {R}}}_-^{{{\mathsf {{K}}}}}} = {\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^\sharp )\subset {\mathrm{dom}\,}({\mathsf {{D}}}_\varphi ^{{\mathsf {{M}}}\sharp })$. By (12) and Assertion 4), applied to D _φ and ${\mathsf {{D}}}_\varphi ^\sharp$,

$$\displaystyle \begin{aligned} & {\mathsf{{D}}}_\varphi^\sharp \, a = {\mathsf{{D}}}_\varphi^\sharp \, {\mathsf{{D}}}_\varphi^{{\mathsf{{M}}}}\, \psi = {\mathsf{{D}}}_\varphi^\sharp \, {\mathsf{{D}}}_\varphi\, \psi = \psi\,. {} \end{aligned} $$

Hence, $\psi \in {\mathrm{ran}\,}({\mathsf {{D}}}_\varphi ^\sharp ) = {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$, by assertion 2), which contradicts $\psi \in {{\mathcal {S}}_+^{{\,\mathsf {{-M}}}}}\setminus {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$. That is, $a\in {{\mathcal {R}}_-^{{\mathsf {{-M}}}}}$ and $a\not \in {{{\mathcal {R}}}_-^{{{\mathsf {{K}}}}}}$, so that $a\in {{{\mathcal {R}}}_-^{{-{\mathsf {{M}}}}}}\setminus {{{\mathcal {R}}}_-^{{{\mathsf {{K}}}}}}\ne \emptyset$. $\square$

Corollary 17

Given −K ∈ Σ _M ∪{M}, and φ as per (10), the semiconvex transform ${\mathsf {{D}}}_\varphi :{{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}{\rightarrow }{{\mathcal {R}}_-^{{\mathsf {{K}}}}}$ of (12) has a well-defined inverse ${\mathsf {{D}}}_\varphi ^{-1}:{{\mathcal {R}}_-^{{\mathsf {{K}}}}} {\rightarrow } {{\mathcal {S}}_+^{{\,\mathsf {{K}}}}}$ given by

where ${\mathsf {{D}}}_\varphi ^\sharp$ is as per (54).

Proof of Corollary 17 and Theorem 3

Immediate by inspection of Lemma 16. $\square$

Lemma 18

The asserted forms (16), (17) of the semiconvex transform and its inverse (12), (13) hold.

Proof

Given any $\psi \in {{\mathcal {S}}_+^{{\,\mathsf {{-M}}}}}$, Theorem 3 implies that ${\mathsf {{D}}}_\varphi \, \psi \in {{\mathcal {S}}_-^{{\,\mathsf {{-M}}}}}$. Define $a_\mp :{\mathbb {R}}^n\rightarrow {{{{\overline {{\mathbb {R}}}}}}}$ by

(65)

for all $z\in {\mathbb {R}}^n$, and observe that a _∓ is convex and lower closed, i.e. a _∓ = cl⁻ co a _∓, see [10]. Motivated by the form of the argument of the convex hull operation in the right-hand side of (16), define $\widetilde a_\mp :{\mathbb {R}}^n\rightarrow {{{{\overline {{\mathbb {R}}}}}}}$ by for all $z\in {\mathbb {R}}^n$. Note by definition (8) of $\delta _{z_i}^-$ that

$$\displaystyle \begin{aligned} \widetilde a_\mp(z) & = \left\{ \begin{array}{cl} -[{\mathsf{{D}}}_\varphi\, \psi](z_i) - \varphi(0,z_i), & z = z_i\,, \ i\in{\mathbb{N}}\,, \\ +\infty\,, & \text{otherwise}, \end{array} \right. {} \end{aligned} $$

(66)

for all $z\in {\mathbb {R}}^n$. As $\{z_i\}_{i\in {\mathbb {N}}}$ is dense in ${\mathbb {R}}^n$, inspection of (65) and (66) yields that $a_\mp = {\text{cl}}^-\, {\text{co}}\ \widetilde a_\mp$. Recalling the definition of $\widetilde a_\mp$ subsequently yields (16).

The remaining assertion (17) is an immediate consequence of (13), as $\{z_i\}_{i\in {\mathbb {N}}}$ is dense in ${\mathbb {R}}^n$. $\square$