Computational Management Science

, Volume 14, Issue 2, pp 293–296

# Erratum to: Polyhedral approximation of ellipsoidal uncertainty sets via extended formulations: a computational case study

• Andreas Bärmann
• Andreas Heidt
• Alexander Martin
• Sebastian Pokutta
• Christoph Thurner
Erratum

## 1 Erratum to: Comput Manag Sci (2016) 13(2):151–193 DOI 10.1007/s10287-015-0243-0

The purpose of this erratum is to correct a signing error in the statement of the inner approximation of the second-order cone $$\mathbb {L}^n$$ presented in Bärmann et al. (2016).

In Bärmann et al. (2016), we developed a construction for the inner approximation of $$\mathbb {L}^n$$ based on the ideas of Ben-Tal and Nemirovski (2001) and Glineur (2000). We showed—using the same decomposition as in the aforementioned papers—that it suffices to find an inner approximation of $$\mathbb {L}^2$$, which in turn can be obtained from an inner approximation of the unit ball $$\mathbb {B}^2 \subset \mathbb {R}^2$$. However, in the statement of the latter two approximations, there was a signing error which we would like to correct here.

Our inner approximation of $$\mathbb {B}^2$$ is a regular m-gon $$\bar{P}_m$$ inscribed into it. Via an extended formulation, we can state this m-gon using a number of variables and constraints logarithmic in m:

### Theorem 2.9

The polyhedron
\begin{aligned} \begin{array}{rcc} \bar{D}_k &{}=&{} \left\{ (p_0, \ldots , p_{k - 1}, d_0, \ldots , d_{k - 1}) \in \mathbb {R}^{2k} \left| \begin{array}{rcll} p_{i - 1} &{}=&{} \gamma _i p_i + \sigma _i d_i, &{}\quad (\forall i = 1, \ldots , k - 1)\\ -d_{i - 1}&{}\le &{}\sigma _i p_i - \gamma _i d_i,&{}\quad (\forall i = 1, \ldots , k - 1)\\ d_{i - 1} &{} \le &{}\sigma _i p_i - \gamma _i d_i,&{}\quad (\forall i = 1, \ldots , k - 1)\\ p_{k - 1} &{} = &{} \gamma _k,\\ -d_{k - 1} &{} \le &{} {\sigma _k},\\ d_{k - 1} &{} \le &{} {\sigma _k} \end{array} \right. \right\} \end{array} \end{aligned}
for $$k \ge 2$$ is an extended formulation for $$\bar{P}_{2^k}$$ with $$\mathop {proj}_{p_0, d_0}(\bar{D}_k) = \bar{P}_{2^k}$$.

### Proof

In the following, we describe the construction of the inner approximation as an iterative procedure. We start by defining the polytope
\begin{aligned} P_{k-1} := \{(p_{k-1},d_{k-1}) \mid p_{k-1} = \gamma _k, -\sigma _k \le d_{k-1} \le \sigma _k\}. \end{aligned}
Now, we construct a sequence of polytopes $$P_{k-1}, P_{k-2}, \ldots , P_0$$. Assume that polytope $$P_i$$ has already been constructed. In order to obtain polytope $$P_{i-1}$$ from polytope $$P_i$$, we perform the following actions which we will translate into mathematical operations below:
1. 1.

Rotate $$P_i$$ counterclockwise by an angle of $$\theta _i = \frac{\pi }{2^i}$$ around the origin to obtain a polytope $$P_i^1$$,

2. 2.

Reflect $$P_i^1$$ at the x-axis to obtain a polytope $$P_i^2$$,

3. 3.

Form the convex hull of $$P_i^1$$ and $$P_i^2$$ to obtain polytope $$P_{i-1}$$.

The first step is a simple rotation and can be represented by the linear map
\begin{aligned} \mathcal {R}_{\theta }: {\mathbb {R}}^2 \mapsto {\mathbb {R}}^2, \left( \begin{array}{c} x \\ y \end{array}\right) \mapsto \left( \begin{array}{ccc} \cos (\theta ) &{}\quad -\sin (\theta )\\ \sin (\theta ) &{}\quad \cos (\theta ) \end{array}\right) \left( \begin{array}{c} x \\ y \end{array}\right) . \end{aligned}
The reflection at the x-axis corresponds to the linear map
\begin{aligned} \mathcal {M}: \mathbb {R}^2 \mapsto \mathbb {R}^2, \left( \begin{array}{c} x \\ y \end{array}\right) \mapsto \left( \begin{array}{ccc} 1 &{}\quad 0\\ 0 &{}\quad -1 \end{array}\right) \left( \begin{array}{c} x \\ y \end{array}\right) . \end{aligned}
Thus, the composition $$\mathcal {M}\mathcal {R}_{\theta _i}$$ which first applies $$\mathcal {R}_{\theta _i}$$ and then $$\mathcal {M}$$, is given by
\begin{aligned} \mathcal {M}\mathcal {R}_{\theta _i}: \mathbb {R}^2 \mapsto \mathbb {R}^2, \left( \begin{array}{c} x \\ y \end{array}\right) \mapsto \left( \begin{array}{ccc} \cos (\theta ) &{}\quad \sin (\theta )\\ \sin (\theta ) &{}\quad -\cos (\theta ) \end{array}\right) \left( \begin{array}{c} x \\ y \end{array}\right) . \end{aligned}
With this, we obtain $$P_i^1 = \mathcal {R}_{\theta _i}(P_i)$$ and $$P_i^2 = (\mathcal {M}\mathcal {R}_{\theta _i})(P_i)$$. Finally, adding the two constraints
\begin{aligned} -d_{i - 1} \,{\le }\, \sigma _i p_i - \gamma _i d_i \end{aligned}
and
\begin{aligned} d_{i - 1} \,{\le }\, \sigma _i p_i - \gamma _i d_i \end{aligned}
yields a polyhedron whose projection onto the variables $$(d_{i - 1}, p_{i - 1})$$ is $$P_{i - 1} = \mathop {conv}(P_i^1, P_i^2)$$. Keeping this correspondence in mind, we show that $$P_0 = \bar{P}_{2^k}$$.

In each iteration, $$P_i$$ is rotated counterclockwise by an angle of $$\theta _i$$ around the origin, such that the vertex of $$P_i$$ with minimal vertical coordinate is rotated to $$(\gamma _k, \sigma _k)$$, therefore $$P_i^1 = \mathcal {R}(P_i)$$. It is $$|\mathcal {V}(P_i^1)| = |\mathcal {V}(P_i)|$$ and $$P_i^1$$ lies strictly above the horizontal axis. Applying $$\mathcal {M}$$, we obtain $$P_i^2 = \mathcal {M}(P_i^1)$$, which satisfies $$|\mathcal {V}(P_i^2)| = |\mathcal {V}(P_i^1)|$$ and lies strictly below the horizontal axis. Then $$P_{i - 1} = \mathop {conv}(P_i^1, P_i^2)$$ satisfies $$|\mathcal {V}(P_{i - 1})| = 2 |\mathcal {V}(P_i)|$$ because all vertices $$v \in \mathcal {V}(P_i^1) \cup \mathcal {V}(P_i^2)$$ remain extreme points of $$P_i$$. We obtain polytope $$P_0$$ after $$k - 1$$ iterations of the above procedure, which has $$|\mathcal {V}(P_0)|= 2^k$$ vertices. As the interior angles at each vertex of $$P_0$$ are of equal size, it follows $$P_0 = \bar{P}_{2^k}$$. This proves the correctness of our construction.$$\square$$

The intermediate steps of the construction are depicted in Fig. 1 for the case $$k = 3$$, which leads to an octagon-approximation. The upper left picture shows the initial polytope $$P_2$$, which is an interval on the line $$x = \gamma _k$$. The upper middle and upper right picture show its rotation by $$45^{\circ }$$ counterclockwise and subsequent reflection at the x-axis, thus representing $$P_2^1$$ and $$P_2^2$$, respectively. The lower left picture shows $$P_1$$ as the convex hull of $$P_2^1$$ and $$P_2^2$$. The lower middle picture contains both $$P_1^1$$ and $$P_1^2$$ as a rotation of $$P_1$$ by $$90^{\circ }$$ counterclockwise and subsequent reflection at the x-axis, respectively. Finally, the lower right picture shows $$P_0 = \bar{P}_{2^3}$$ as the convex hull of $$P_1^1$$ and $$P_1^2$$.

By homogenization, we can obtain an inner $$\epsilon$$-approximation of $$\mathbb {L}^2$$, i.e., a set $$\bar{\mathcal {L}}_{\epsilon }^2$$ with $$\{(r,x) \in \mathbb {R}\times \mathbb {R}^2 \mid ||x || \le \frac{1}{1~+~\epsilon }r\} \subseteq \bar{\mathcal {L}}^2 \subseteq \mathbb {L}^2$$:

### Corollary 2.10

The projection of the set
with $$\epsilon > 0$$ and $$k = \lceil \log (\pi \arccos (\frac{1}{\epsilon + 1})^{-1}) \rceil$$ onto the variables $$(s, p_0, d_0)$$ is an inner $$\epsilon$$-approximation of $$\mathbb {L}^2$$.

We apologize for the incorrect statements of the two approximations in the initial paper.

## References

1. Ben-Tal A, Nemirovski A (2001) On polyhedral approximations of the second-order cone. Math Oper Res 26:193–205
2. Bärmann A, Heidt A, Martin A, Pokutta S, Thurner C (2016) Polyhedral approximation of ellipsoidal uncertainty sets via extended formulations: a computational case study. Comput Manag Sci 13(2):151–193Google Scholar
3. Glineur F (2000) Computational experiments with a linear approximation of second-order cone optimization. Image Technical Report 001, Faculté Polytechnique de MonsGoogle Scholar

## Authors and Affiliations

• Andreas Bärmann
• 1
Email author
• Andreas Heidt
• 1
• Alexander Martin
• 1
• Sebastian Pokutta
• 2
• Christoph Thurner
• 1
1. 1.Lehrstuhl für Wirtschaftsmathematik, FAU Erlangen-NürnbergErlangenGermany
2. 2.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA