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

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. 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 $P_{i-1}$.

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 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.