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Computational Management Science

, Volume 14, Issue 2, pp 293–296 | Cite as

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

  • Andreas BärmannEmail author
  • Andreas Heidt
  • Alexander Martin
  • Sebastian Pokutta
  • Christoph Thurner
Erratum
  • 373 Downloads

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 \( P_1^2 \).
Fig. 1

Construction of the inner approximation of the unit disc \( \mathbb {B}^2 \) for \( k = 3 \)

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 setwith \( \epsilon > 0 \) 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 26:193–205CrossRefGoogle Scholar
  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

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

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

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