A dual simplex-type algorithm for the smallest enclosing ball of balls

Abstract

We develop a dual simplex-type algorithm for computing the smallest enclosing ball of a set of balls and other closely related problems. Our algorithm employs a pivoting scheme resembling the simplex method for linear programming, in which a sequence of exact curve searches is performed until a new dual feasible solution with a strictly smaller objective function value is found. We utilize the Cholesky factorization and procedures for updating it, yielding a numerically stable implementation of the algorithm. We show that our algorithm can efficiently solve instances of dimension 5000 with 100000 points, often within minutes.

A proof of theorem 2

The proof of the correctness of Algorithm 3 follows a similar approach to [18], and relies on the following definition and lemmas.

Definition 3

The triple $(x, \mathcal {S}, p^*)$ is said to be a (violated) V-triple if:

• (a) $\overline{\mathcal {S}}\cup \{\overline{p}^*\}$ is affinely independent,

• (b) $\left\| {\overline{p}^*-\overline{x}}\right\| \ge (p^*_0-x_0)$,

• (c) $\left\| {\overline{p}_i-\overline{x}}\right\| =(p_{i0}-x_0)$, for all $p_i\in \mathcal {S}$,

• (d) $\overline{x}\in {\text {conv}}(\overline{\mathcal {S}}\cup \{\overline{p}^*\})$.

Lemma 2

Assume that, at the beginning of a curve search, we have a V-triple $(x^j, \mathcal {S}^j, p^*)$ such that $|\mathcal {S}^j|>1$. Suppose a full step is taken (${x}_0^{ F } > x_0^{ P })$. Then,

• i. $(\mathcal {S}^j\cup \{p^*\},x^{ F })$ is a dual feasible S-pair,

• ii. ${x}_0^{ F }\le x_0^j$.

Proof

It is easy to see that $(\mathcal {S}^j\cup \{p^*\}, x^{ F })$ is a dual feasible S-pair. First, $\overline{\mathcal {S}}^j\cup \{\overline{p}^*\}$ is affinely independent by assumption. Additionally, $\overline{x}^{ F }\in {\text {ri}}{\text {conv}}(\overline{\mathcal {S}}^j)$, because otherwise there would exist an $\alpha _i(x_0^{ F })=0$, meaning a partial step would have been taken instead, contradicting the assumption that ${x}_0^{ F } > x_0^{ P }$. Finally, recall that the full step finds a point $x^{ F }$ such that $\left\| {\overline{p}^*-\overline{x}^{ F }}\right\| = p^*_0-x_0^{ F }$, $\left\| {\overline{p}_{j_i}-\overline{x}^{ F }}\right\| = p_{{j_i}0} -x_0^{ F }$, $\forall \, p_{j_i}\in \mathcal {S}^j$, and $\overline{x}\in {\text {aff}}(\overline{\mathcal {S}}\cup \{\overline{p}^*\})$, whenever these conditions are feasible. Lastly, ${x}_0^{ F }\le x_0^j$ follows from the definition of full step. $\square$

Lemma 3

Assume that, at the beginning of a curve search, we have a V-triple $(x^j, \mathcal {S}^j, p^*)$ such that $|\mathcal {S}^j|>1$. Suppose a partial step was taken (${x}_0^{ F } \le x_0^{ P }$). Let $p_{j_k}\in \mathcal {S}^j$ be the point selected to be dropped from $\mathcal {S}^j$. Then,

• i. $({x}^{ P }, \mathcal {S}^j\setminus \{p_{j_k}\}, p^*)$ is a V-triple,

• ii. ${x}_0^{ P }\le x_0^j$.

Proof

Item ii., ${x}_0^{ P }\le x_0^j$, follows from the definition of partial step. Let us now prove properties (a-d) from Definition 3 for $({x}^{ P }, \mathcal {S}^j\setminus \{p_{j_k}\}, p^*)$. Property (a) follows from the assumption that $(x^j, \mathcal {S}^j, p^*)$ is a V-triple. Property (b) is proved by a continuity argument: since the curve search starts at $x^j$ such that $\Vert \overline{p}^*-\overline{x}^j\Vert \ge p_{0}^* -x_0^j$ and it ends at $x^{ P }$ such that $x_0^{ F }\le x_0^{ P }\le x_0^j$, then $\Vert \overline{p}^*-\overline{x}^{ P }\Vert \ge p_{0}^* -x_0^{ P }$ (recall that $x_0$ is decreasing and $x^{ F }$ corresponds to the point of smallest value $x_0\le x_0^j$ for which $\Vert \overline{p}^*-\overline{\varGamma _j}^+(x_0)\Vert \ge p_{0}^* -x_0$). Property (c) is maintained at any point of the curve so it holds for $x^{ P }$, and property (d) also holds since $\alpha _i(x_0^{ P })\ge 0$ and $\alpha ^*(x_0^{ P })\ge 0$. $\square$

Lemma 4

Suppose $|\mathcal {S}^j|=1$. Let $x^{j+1}$ and $\mathcal {S}^{j+1}$ be the optimal solution and basis, respectively, of ${\rm{Inf}}_{{\rm{Q}}} (S^{j} \cup \{ p^{*} \} )$ obtained from Proposition 4. Then $(\mathcal {S}^{j+1}, x^{j+1})$ is a dual feasible S-pair.

Lemma 5

Let $(\mathcal {S}^j, x^j)$ be a dual feasible S-pair and $p^*\in \mathcal {P}$ correspond to an infeasible constraint at $x^j$. Suppose $\overline{\mathcal {S}}^j\cup \{\overline{p}^*\}$ is affinely dependent and let $p_{j_k}\in \mathcal {S}^j$ be the point corresponding to the minimizing index of (13), k. Then, $(x^j, \mathcal {S}^j\setminus \{p_{j_k}\}, p^*)$ is a V-triple.

Proof

First note that the affine dependence assumption implies that $|\mathcal {S}^j|>1$. To prove that $(x^j, \mathcal {S}^j\setminus \{p_{j_k}\}, p^*)$ is a V-triple, we need to prove the four properties from Definition 3. Property (b) is trivial, and property (c) follows directly from $(\mathcal {S}^j, x^j)$ being a dual feasible S-pair. Now, in order to prove (a), suppose, by contradiction, that $\overline{\mathcal {S}}^j\setminus \{\overline{p}_{j_k}\}\cup \{\overline{p}^*\}$ is affinely dependent. Since $\overline{\mathcal {S}}^j\setminus \{\overline{p}_{j_k}\}$ is affinely independent, there exists $\beta _i$, $i\ne k$, such that

$$\begin{aligned} \textstyle \overline{p}^* = \sum_{i=1, i\ne k}^{s}{\beta _i\overline{p}_{j_i}},\quad \text {with} \quad \sum _{i=1, i\ne k }^{s}\beta _i=1. \end{aligned}$$

On the other hand, $z=\overline{p}^*-\overline{p}_1+Mw=0$ since $\overline{\mathcal {S}}^j\cup \{\overline{p}^*\}$ is affinely dependent. Thus $\textstyle \overline{p}^* = - \sum _{i=1}^{s}{\sigma _i\overline{p}_{j_i}}$, for $\sigma _i$, $i=1,...,s$, as in (12). As a consequence, we have

$$\begin{aligned} \textstyle \overline{p}_{j_k} = \sum _{i=1, i\ne k}^{s}{\frac{\beta _i+\sigma _i}{-\sigma _k}\overline{p}_{j_i}},\quad \text {with} \; \sum _{ i=1, i\ne k }^{s} \frac{\beta _i+\sigma _i}{-\sigma _k} = 1, \end{aligned}$$

contradicting the affine independence of $\overline{\mathcal {S}}^j$ (note that $\sigma _k< 0$).

Now consider $\rho$ as in (12). Let $\delta =\rho -\frac{\rho _k}{\sigma _k}\sigma$ and $\theta =-\frac{\rho _k}{\sigma _k}$. It is easy to see that $\delta \ge 0$ and, in particular, $\delta _k=0$. Since $z=0$, property (d) now follows from:

$$\begin{aligned} M\delta _{2:s}+\theta (\overline{p}^*-\overline{p}_{j_1})+\overline{p}_{j_1}=M(u+x_0v) + \overline{p}_{j_1}- \frac{\rho _k}{\sigma _k}z=\overline{x}^j. \end{aligned}$$

$\square$

Recall that the algorithm is initialized with a dual feasible S-pair. The lemmas above show that we always have a V-triple before a curve search; and that we always have a dual feasible S-pair at the beginning of a new loop iteration when an infeasible point $p^*$ is selected. We conclude that, starting from a V-triple $(x^j,\mathcal {S}^j,p^*)$ for which $\left\| {\overline{p}^*-\overline{x}^j}\right\| >(p_0^*-x_0^j)$, one can obtain a new dual feasible S-pair, $(\mathcal {S}^{j+1},{x}^{j+1})$, in $|\mathcal {S}^j|-|\mathcal {S}^{j+1}|\le n$ partial steps and one full step. This new feasible S-pair is such that $\mathcal {S}^{j+1}\subseteq \mathcal {S}^j$, and $x^{j+1}_0<x_0^j$, since, even though the value of $x_0$ may not change in some partial steps or in a full step, the value of $x_0$ must decrease either in at least one of the partial steps or the full step. That is because, otherwise, we would have $x^{j+1}=x^j$, contradicting the fact that $p^*$ is infeasible at $x^j$. Therefore, since the value of $x_0$ strictly decreases at each loop iteration, the same dual feasible S-pair can never reoccur. Since the number of dual feasible S-pairs is finite, we conclude the statement of Theorem 2.