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

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## 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

### Proof

- 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.
Reflect \( P_i^1 \) at the

*x*-axis to obtain a polytope \( P_i^2 \), - 3.
Form the convex hull of \( P_i^1 \) and \( P_i^2 \) to obtain polytope \( P_{i-1} \).

*x*-axis corresponds to the linear map

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.\(\square \)

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

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

## References

- Ben-Tal A, Nemirovski A (2001) On polyhedral approximations of the second-order cone. Math Oper Res 26:193–205CrossRefGoogle Scholar
- 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
- Glineur F (2000) Computational experiments with a linear approximation of second-order cone optimization. Image Technical Report 001, Faculté Polytechnique de MonsGoogle Scholar