The minimum covering Euclidean ball of a set of Euclidean balls in $I\!\!R^n$

Abstract

Primal and dual algorithms are developed for solving the n-dimensional convex optimization problem of finding the Euclidean ball of minimum radius that covers m given Euclidean balls, each with given center and radius. Each algorithm is based on a directional search method in which a search path may be a ray or a two-dimensional conic section in $I\!\!R^n$. At each iteration, a search path is constructed by the intersection of bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional hyperboloids. The optimal stopping point along each search path is determined explicitly.

Appendix: Results on conic sections and applications to the primal and dual algorithms

Appendix A.1–A.4 present properties on the intersection of hyperplanes with conic sections in $I\!\!R^n$, and on the intersection of a sequence of hyperboloids with a common focal point. The proofs are in Dearing (2017). These properties provide the basis for expressions in algorithm Intersections used to compute the vectors and parameters of two-dimensional conic sections that define the search paths of the primal and dual algorithms. Appendix A.5 presents properties used to prove that the primal and dual algorithms compute the optimal solution in a finite number for iterations.

1.1 A.1 Hyperboloids and ellipsoids

An n-dimensional hyperboloid of two sheets, symmetric about its major axis, is the set

$$\begin{aligned} H = \{ \textbf{x} \in I\!\!R^n: | \Vert \textbf{x} - \textbf{p}_1 \Vert - \Vert \textbf{x} - \textbf{p}_2 \Vert | = 2a \}, \end{aligned}$$

(32)

where $\textbf{p}_1, \textbf{p}_2 \in I\!\!R^n$ are the focal points, and a is a positive constant such that $2a < \Vert \textbf{p}_1 - \textbf{p}_1 \Vert$. H is specified by the following vectors and parameters, all of which are determined by the focal points and the constant a. The center $\textbf{c} = (\textbf{p}_1 + \textbf{p}_2)/2$ is the midpoint of the focal points. The axis vector $\textbf{v} = (\textbf{p}_1 - \textbf{p}_2)/ \Vert \textbf{p}_1 - \textbf{p}_2 \Vert$ orients the hyperboloid and is parallel to the major axis, which is the line through $\textbf{p}_1$ and $\textbf{p}_2$. The focal distance $c = \Vert \textbf{p}_1 - \textbf{p}_2 \Vert / 2$ is the distance from the center to each to the focal points. The eccentricity $\epsilon = c/a$ specifies the shape of the hyperboloid.

The sheet of H closest to $\textbf{p}_1$ is the set $H_1$, and the sheet closest to $\textbf{p}_2$ is the set $H_2$, where

$$\begin{aligned} H_1 =&\{ \textbf{x}: \Vert \textbf{p}_2 - \textbf{x} \Vert - \Vert \textbf{p}_1 - \textbf{x} \Vert = 2a \}&= \{ \textbf{x}: \Vert \textbf{p}_1 - \textbf{x} \Vert = \epsilon \textbf{v}(\textbf{x} - \textbf{d}_1) \} \end{aligned}$$

(33)

$$\begin{aligned} H_2 =&\{ \textbf{x}: \Vert \textbf{p}_2 - \textbf{x} \Vert - \Vert \textbf{p}_1 - \textbf{x} \Vert = -2a \}&= \{ \textbf{x}: \Vert \textbf{p}_2 - \textbf{x} \Vert = \epsilon \textbf{v}(\textbf{d}_2 - \textbf{x}) \}. \end{aligned}$$

(34)

The directrix of $H_1$ is $\{ \textbf{x}: \textbf{v}\textbf{x} = \textbf{v}\textbf{d}_1 \}$, with directrix vector $\textbf{d}_1 = \textbf{c} + d\textbf{v}$ and $d = a^2/c$. The directrix of $H_2$ is $\{ \textbf{x}: \textbf{v}\textbf{x} = \textbf{v}\textbf{d}_2 \}$, with directrix vector $\textbf{d}_2 = \textbf{c} - d\textbf{v}$. The vertex $\textbf{a}_1 = \textbf{c} + a \textbf{v}$ of $H_1$ is the point of intersection between the major axis and $H_1$. The vertex of $H_2$ is $\textbf{a}_2 = \textbf{c} - a \textbf{v}$. Observe that $H = H_1 \cup H_2$.

The triangle inequality implies $2a = | \Vert \textbf{p}_2 - \textbf{x} \Vert - \Vert \textbf{p}_1- \textbf{x} \Vert | \le \Vert \textbf{p}_2 - \textbf{p}_1 \Vert = 2c$, so that $a \le c$ and $\epsilon \ge 1$. By definition, H is a hyperboloid if $a < c$, or $\epsilon > 1$. If $a = c$, so that $\epsilon = 1$, H is a degenerate hyperboloid where $H_1 = \{ \textbf{x} = \textbf{p}_1 + \alpha \textbf{v}, \alpha \ge 0 \}$, the ray along the major axis from $\textbf{p}_1$ in the direction $\textbf{v}$, and $H_2 = \{ \textbf{x} = \textbf{p}_2 - \alpha \textbf{v}, \alpha \ge 0 \}$, the ray from $\textbf{p}_2$ in the direction $-\textbf{v}$. If $a > c$, $H = \emptyset$.

An n-dimensional ellipsoid, symmetric about its major axis, is the set of points $\textbf{x} \in I\!\!R^n$ such that the sum of the distances from $\textbf{x}$ to two given points $\textbf{p}_1$ and $\textbf{p}_2$ equals a positive constant 2a. That is,

$$\begin{aligned} E = \{ \textbf{x}: \parallel \textbf{p}_{2} - \textbf{x} \parallel + \parallel \textbf{p}_{1} - \textbf{x} \parallel = 2a \}. \end{aligned}$$

(35)

An ellipsoid E is specified by the same vectors and parameters that specify a hyperboloid, all of which are determined by the focal points $\textbf{p}_1$ and $\textbf{p}_2$ and the positive constant a.

The triangle inequality implies $2c = \parallel \textbf{p}_{1} - \textbf{p}_{2}\parallel \le \parallel \textbf{p}_{1} - \textbf{x}\parallel + \parallel \textbf{p}_{2} - \textbf{x}\parallel = 2a$, so that $c \le a$. If $a = c$, E is the line segment between $\textbf{p}_1$ and $\textbf{p}_2$, and is a degenerate ellipsoid. The application to finding a minimum covering ball is concerned only with the case $c < a$, or $\epsilon < 1$.

The next property gives an equivalent representation of a hyperboloid or an ellipsoid as a quadratic form. A version of this representation for hyperboloids in $I\!\!R^3$ is reported in Leva (1996). The proof for hyperboloids (or ellipsoids) expands expressions (32), (33), and (34) (or (35) for ellipsoids) and applies the definition of related vectors and parameters to obtain (36).

Property 13

Given the focal points $\textbf{p}_1$ and $\textbf{p}_2$ and a positive constant a, with corresponding axis vector $\textbf{v}$, center point $\textbf{c}$, and eccentricity $\epsilon = c/a$, let H be the hyperboloid determined by these vectors and parameters if $c > a$, and let E be the ellipsoid determined by these vectors and parameters if $c< a$. Define the set Q by

$$\begin{aligned} Q = \{ \textbf{x}: ( \textbf{x} - \textbf{c})^T [ I - \epsilon ^2 \textbf{v} \textbf{v}^T] ( \textbf{x} - \textbf{c}) = a^2 - c^2 \} = \{ \textbf{x}: ( \textbf{x} - \textbf{c})^2 - \epsilon ^2 (\textbf{v} ( \textbf{x} - \textbf{c}))^2 = a^2 - c^2 \}. \end{aligned}$$

(36)

Then $Q = H$ if $c > a$, and $Q = E$ if $c < a$.

Property 14

Suppose H is a hyperboloid in $I\!\!R^n$ with focal points $\textbf{p}_1$ and $\textbf{p}_2$, center $\textbf{c}$, axis vector $\textbf{v}$, eccentricity $\epsilon$, and sheets $H_1$ and $H_2$. If $\textbf{u}$ is a unit vector orthogonal to the axis vector $\textbf{v}$, then $H_1 \cap \textrm{aff}(\textbf{v},\textbf{u},\textbf{c}) = \{ \textbf{x}_1(\alpha ) = \textbf{c} + a \sec (\alpha ) \textbf{v} + b \tan (\alpha ) \textbf{u}, \frac{-\pi }{2}< \alpha < \frac{\pi }{2} \}$, where $b = \sqrt{c^2 - a^2}$, is one branch of a two-dimensional hyperbola with the same vectors and parameters as $H_1$. Furthermore, $H_2 \cap \textrm{aff}(\textbf{v},\textbf{u},\textbf{c}) = \{ \textbf{x}_2(\alpha ) = \textbf{c} - a \sec (\alpha ) \textbf{v} + b \tan (\alpha ) \textbf{u}, \frac{-\pi }{2}< \alpha < \frac{\pi }{2} \}$ is one branch of a two-dimensional hyperbola with the same vectors and parameters as $H_2$.

Corollary 1

Suppose E is an ellipsoid in $I\!\!R^n$ with focal points $\textbf{p}_1$ and $\textbf{p}_2$, center $\textbf{c}$, axis vector $\textbf{v}$, eccentricity $\epsilon$. If $\textbf{u}$ is a unit vector orthogonal to the axis vector $\textbf{v}$, then $E \cap \textrm{aff}(\textbf{v},\textbf{u},\textbf{c}) = \{ \textbf{x}(\alpha ) = \textbf{c} + a \cos (\alpha ) \textbf{v} + b \sin (\alpha ) \textbf{u}, - \pi< \alpha < \pi \}$, where $b = \sqrt{a^2 - c^2}$, is a two-dimensional ellipse with the same vectors and parameters as E.

An n-dimensional right circular cone C, with center $\textbf{c}$, axis vector $\textbf{v}$, and eccentricity $\epsilon$, is also expressed in terms of the quadratic form Q with $\epsilon > 1$, but with a right hand side value of zero. That is,

$$\begin{aligned} C = \{ \textbf{x} : (\textbf{x} - \textbf{c})^T[I - \epsilon ^2 \textbf{v} \textbf{v}^T ] (\textbf{x} - \textbf{c}) = 0 \}. \end{aligned}$$

(37)

The cone C has two “sheets”: $C_1 = \{ \textbf{x} : \parallel \textbf{x} - \textbf{c} \parallel = \epsilon \textbf{v}(\textbf{x} - \textbf{c}) \}$ is the subset of C that is closest to the focal point $\textbf{p}_1$, and $C_2 = \{ \textbf{x} : \parallel \textbf{x} - \textbf{c} \parallel = \epsilon \textbf{v}(\textbf{c} - \textbf{x}) \}$ is the subset of C closest to the focal point $\textbf{p}_2$. Observe that $C = C_1 \cup C_2$.

For any point $\textbf{x} \in C$, let $\gamma$ be the angle between the vector $\textbf{x} - \textbf{c}$ and the axis vector $\textbf{v}$. Then the expression for $C_1$ shows that $\textbf{v}(\textbf{x}-\textbf{c})/\Vert \textbf{x}-\textbf{c} \Vert = \cos (\gamma ) = 1/\epsilon$, or $\sec (\gamma ) = \epsilon = c/a$. The next Property and its proof are analogous to Property 14.

Property 15

Given a cone C in $I\!\!R^n$ with center $\textbf{c}$, axis vector $\textbf{v}$, and sheets $C_1$ and $C_2$, if $\textbf{u}$ is a unit vector orthogonal to $\textbf{v}$, then $C_1 \cap \textrm{aff}(\textbf{u}, \textbf{v}, \textbf{c}) = \{ \bar{\textbf{x}}_1(\beta ) = \textbf{c} + a|\beta | \textbf{v} + b \beta \textbf{u}, - \infty< \beta < \infty \}$, where $b = \sqrt{c^2 - a^2}$, is one branch of a two-dimensional cone with the same center and axis vector as C. Furthermore, $C_2 \cap \textrm{aff}(\textbf{u}, \textbf{v}, \textbf{c}) = \{ \bar{\textbf{x}}_2(\beta ) = \textbf{c} - a|\beta | \textbf{v} + b \beta \textbf{u}, - \infty< \beta < \infty \}$ is one branch of a two-dimensional cone with the same vectors and parameters as C.

Property 16

If a cone C and a hyperboloid H have the same axis vector $\textbf{v}$, center $\textbf{c}$, and eccentricity $\epsilon$, then C is the asymptotic approximation of H.

1.2 A.2 Paraboloids

An n-dimensional paraboloid, symmetric about its major axis, is the set P of all points $\textbf{x} \in I\!\!R^n$ such that the distance from $\textbf{x}$ to a given point $\textbf{p}_1$ on the major axis, equals the distance from $\textbf{x}$ to the hyperplane that is orthogonal to the major axis and contains a point $\textbf{p}_2$ on the major axis. A paraboloid is specified by the two points $\textbf{p}_1$ and $\textbf{p}_2$ only. The point $\textbf{p}_1$ is the focal point of the paraboloid. The major axis is the line through the points $\textbf{p}_1$ and $\textbf{p}_2$. The axis vector $\textbf{v} = (\textbf{p}_1 - \textbf{p}_2)/ \Vert \textbf{p}_1 - \textbf{p}_2 \Vert$ is the unit vector parallel to the major axis. A paraboloid P, defined by $\textbf{p}_1, \textbf{p}_2 \in I\!\!R^n$, is the set

$$\begin{aligned} P = \{ \textbf{x}: \Vert \textbf{p}_{1} - \textbf{x} \Vert = \textbf{v}(\textbf{x} - \mathbf {p_2}) \}. \end{aligned}$$

(38)

The vertex of a paraboloid is the center $\textbf{c} = (\textbf{p}_1 + \textbf{p}_2)/2$, and is the intersection of the paraboloid with the major axis. The focal distance $c = \Vert \textbf{p}_1 - \textbf{p}_2 \Vert /2$. The directrix of a paraboloid P is the hyperplane with normal vector $\textbf{v}$ containing the point $\textbf{p}_{2}$. Observe that $\textbf{c} - \textbf{p}_2 = c \textbf{v}$, and $\textbf{c} - \textbf{p}_1 = -c\textbf{v}$. All paraboloids have the same shape, so there is no parameter $\epsilon$ or a. The next property characterizes a two-dimensional parabola that is a subset of a given paraboloid P with the same parameters and vectors.

Property 17

Given a paraboloid P in $I\!\!R^n$, with focal points $\textbf{p}_1$ and $\textbf{p}_2$, if $\textbf{u}$ is a unit vector orthogonal to the axis vector $\textbf{v}$, then $P\; \cap \; \textrm{aff}(\textbf{v}, \textbf{c}, \textbf{u}) = \{ \textbf{x}(\alpha ) = \textbf{c} + c \alpha ^2 \textbf{v} + 2c \alpha \textbf{u}, - \infty< \alpha < \infty \}$ is a two-dimensional parabola that is a subset of P and has the same vectors and parameters as P.

Paraboloids may also be expressed in terms of a quadratic form similar to (36). However, for a paraboloid, there is no eccentricity, and the right hand side is a linear expression of $\textbf{x}$.

Property 18

The paraboloid P has the equivalent expression

$$\begin{aligned} P = \{ \textbf{x}: ( \textbf{x} - \textbf{c})^T [I - \textbf{v} \textbf{v}^T] ( \textbf{x} - \textbf{c}) = 4c\textbf{v}(\textbf{x} - \textbf{c}) \}. \end{aligned}$$

(39)

1.3 A.3 Intersections of hyperplanes with conic sections in $I\!\!R^{n}$

From the classical studies of conic sections in $I\!\!R^3$, it is well known that if a plane and a cone intersect at an appropriate angle, measured between the axis vector of the cone and the normal vector of the plane, the intersection is either a two-dimensional hyperbola, ellipse, or parabola. These results extend to the intersection of a hyperplane with each conic section in $I\!\!R^n$, and are reported below. For the intersection of a hyperplane with a hyperboloid, or a cone, conditions are given for identifying the resulting intersection as a hyperboloid, an ellipsoid or a paraboloid of dimension $n-1$. For the intersection of a hyperplane and an ellipsoid, the resulting intersection is always an ellipsoid of dimension $n-1$. For the intersection of a hyperplane and paraboloid, conditions are given for identifying the resulting intersection as a paraboloid or an ellipsoid of dimension $n-1$. In each case, expressions are given for computing the vectors and parameters of the resulting intersection. These expressions are used in algorithm Intersections to compute the search paths for the primal and dual algorithms.

Property 19

Suppose $Q = \{ \textbf{x}: (\textbf{x} - \textbf{c})^T[ I - \epsilon ^2 \textbf{v} \textbf{v}^T ] (\textbf{x} - \textbf{c}) = a^2 -c^2 \}$ is a hyperboloid in $I\!\!R^n$, centered at $\textbf{c} = (c_1, \dots , c_n)^T$, with axis vector $\textbf{v}$ of unit length, eccentricity $\epsilon > 1$, and parameters a and c, and suppose $H\!P = \{ \textbf{x}: \textbf{h} \textbf{x} = \textbf{h}(\textbf{c} + \hat{h}\textbf{h}) \}$ is a hyperplane with $\Vert \textbf{h} \Vert =1$. Let $\rho = \sqrt{1-(\textbf{h} \textbf{v})^2}$. Then $Q \cap H\!P$ is a hyperboloid of dimension $n-1$ iff $\epsilon \rho > 1$, or an ellipsoid of dimension $n-1$ iff $\epsilon \rho < 1$ and $\hat{h}^2 \ge a^2(1-\rho ^2\epsilon ^2)$, or a paraboloid of dimension $n-1$ iff $\epsilon \rho = 1$.

1.4 A.4 Applications to problem M(P)

Problem M(P) assumes a given a set $P = \{ \textbf{p}_1, \dots , \textbf{p}_m \} \subset I\!\!R^n$, of distinct points and the set of balls $\{ [\textbf{p}_i, r_i], \textbf{p}_i \in P \}$, where $r_i$ is a non-negative radius corresponding to each $\textbf{p}_i \in P$. The bisector $B_{j,k}$ of points $\textbf{p}_j, \textbf{p}_k \in P$ is defined by (1). If $\textbf{p}_j, \textbf{p}_k \in P$ are non-redundant and $r_j > r_k$, then the bisector $B_{j,k}$ is one sheet of a hyperboloid in $I\!\!R^n$.

For problem M(P), the next result characterizes the intersection of three bisectors corresponding to a set of three non-redundant and affinely independent points in P. Using this result, the intersection of any two bisectors may be determined by the intersection of either bisectors in the pair with the hyperplane $H_T$. This result leads to a procedure for finding the intersection of bisectors determined by all pairs of points in a subset of P.

Property 20 extends a result in Leva (1996) that assumes only two hyperboloids with a common focal point.

Property 20

Suppose that $T = \{ \textbf{p}_{j}, \textbf{p}_{k}, \textbf{p}_{l} \}$ is a subset of three affinely independent and non-redundant points from P, ordered so that $r_{j} \ge r_{k} \ge r_{l}$, with $r_{j} > r_{l}$, and suppose $\mathcal {B} = \{ B_{j,k}, B_{j,l} , B_{k,l} \}$ is the set of bisectors defined by each pair of points in T. Let $H_T = \{ \textbf{h}_T \textbf{x} = \textbf{h}_T\textbf{d}_T \}$, where $\textbf{h}_T = \epsilon _{j,k}\textbf{v}_{j,k} - \epsilon _{j,l}\textbf{v}_{j,l}$, $\textbf{d}_T = \textbf{d}_{j,l} + \frac{(\textbf{v}_{j,k}(\textbf{d}_{j,k} - \textbf{d}_{j,k})}{\textbf{v}_{j,k}\textbf{u}_{T}} \textbf{u}_{T}$, and $\textbf{u}_{T} \leftarrow$ Projection $(\textbf{h}_{T}, \{\textbf{v}_{j,l} \})$. Then the intersection of each pair of bisectors in $\mathcal {B}$ equals the intersection of $H_T$ with either bisector of the pair. If any of the bisectors in $\mathcal {B}$ is a hyperplane, it is identical to $H_T$.

Given an active set $S = \{ \textbf{p}_{i_1}, \ldots , \textbf{p}_{i_s} \}$, the following results show how to compute the vectors and parameters of $B_S = \cap _{1 \le j < k \le s} B_{i_j,i_k}$. If all the radii of the points in S are equal, each bisector $B_{i_j,i_k}$ is a hyperplane, so that $B_S$ is a linear manifold of dimension $n-s+1$. If some of the radii are unequal, $B_S$ will be a conic section of dimension $n-s+2$.

The next two properties give alternative representations for $B_S$.

Property 21

$B_S = \cap _{j=2}^{s} B_{i_1,i_j}$.

Property 22

$B_S = B_{i_1,i_s} \cap _{j=2}^{s-1} H_j$, where for $j = 2, \dots , s-1$, $H_j = \{ \textbf{h}_j\textbf{x} = \textbf{h}_j\textbf{d}_j \}$, is determined by the set $T_j = \{ \textbf{p}_{i_1}, \textbf{p}_{i_j}, \textbf{p}_{i_s} \}$, with $r_{i_1} \ge r_{i_j} \ge r_{i_s}$, and $r_{i_1} > r_{i_s}$.

1.5 A.5 Properties of search paths

Property 23

At each iteration of either the primal or dual algorithm with active set S, and for either a linear, hyperbolic, elliptic, or parabolic search path $X_S = \{ \textbf{x}(\alpha ): \alpha \in D \}$, S is active for the ball $[\textbf{x}(\alpha ), z(\alpha )]$, for $\alpha \in D$. Furthermore, the complementary slackness conditions (6) of the KKT conditions are satisfied by S and the ball $[ \textbf{x}(\alpha ), z(\alpha ) ]$, for $\alpha \in D$, where $z(\alpha ) = \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1} \Vert + r_{i_1}$.

Proof

For either the primal or dual algorithm, consider a search path that is the ray $X_S = \{ \textbf{x}(\alpha ) = \textbf{x}_S + \alpha \textbf{d}_{S}, \alpha \in D \}$. By construction, $\alpha _S \in D$, and $\textbf{x}_S = \textbf{x}(\alpha _S) \in X_S \subset B_S = \cap _{\textbf{p}_j, \textbf{p}_k \in S}B_{j,k}$. Then for $\alpha \in D$, and each pair of points $\{ \textbf{p}_{i_j}, \textbf{p}_{i_k} \} \subset S$, $B_{i_j.i_k} = \{ \textbf{x}: (\textbf{p}_{i_j} - \textbf{p}_{i_k})\textbf{x} = (\textbf{p}_{i_j} - \textbf{p}_{i_k})\textbf{x}_S \}$, and $(\textbf{p}_{i_j} - \textbf{p}_{i_k})\textbf{x}(\alpha ) = (\textbf{p}_{i_j} - \textbf{p}_{i_k})\textbf{x}_S + \alpha (\textbf{p}_{i_j} - \textbf{p}_{i_k})\textbf{d}_S = (\textbf{p}_{i_j} - \textbf{p}_{i_k})\textbf{x}_S$ since by construction $(\textbf{p}_{i_j} - \textbf{p}_{i_k})\textbf{d}_S = 0$. Thus for each $\{ \textbf{p}_{i_j}, \textbf{p}_{i_k} \} \subset S$, $\textbf{x}(\alpha ) \in B_{i_j,i_k}$. That is, $\Vert \textbf{x}(\alpha ) - \textbf{p}_{i_j} \Vert + r_{i_j} = \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_k} \Vert + r_{i_k}$ which implies S is active for $[\textbf{x}(\alpha ), z(\alpha )]$.

For either the primal or dual algorithm consider a search path $X_S$ that is either a hyperbola, specified by (13), an ellipse specified by (15), or a parabola specified by (17). In each case, for $\alpha \in D$, substitution of $\textbf{x}(\alpha )$ for $\textbf{x}_S$ satisfies the expression (36) for $B_S$ which shows that $X_S \subset B_S$. Thus, for each $\{ \textbf{p}_{i_j}, \textbf{p}_{i_k} \} \subset S$, $\textbf{x}(\alpha ) \in B_{i_1,i_j}$ which implies $\Vert \textbf{x}(\alpha ) - \textbf{p}_{i_j} \Vert + r_{i_j} = \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_k} \Vert + r_{i_k}$ so that S is active for $[\textbf{x}(\alpha ), z(\alpha )]$ for $\alpha \in D$.

Since the points in S are affinely independent, there exists a solution $\pi _{i_j}$ for $\textbf{p}_{i_j} \in S$ to the linear system (7) and (8) with $\textbf{x}(\alpha )$ replacing $\textbf{x}$. This result along with the result that S is active for $[\textbf{x}(\alpha ), z(\alpha )]$ for all cases of the search path and for $\alpha \in D$, shows that $(z(\alpha ) -\Vert \textbf{x}(\alpha ) - \textbf{p}_{i_j} \Vert -r_{i_j})\pi _{i_j} = 0$ for $\textbf{p}_{i_j} \in S$. For $\textbf{p}_{i} \in P\setminus S$, choose $\pi _i = 0$ so that $(z(\alpha ) -\Vert \textbf{x}(\alpha ) - \textbf{p}_{i} \Vert -r_{i})\pi _{i} = 0$. Thus, the complementary slackness conditions (6) of the KKT are satisfied at each point on the search path. $\square$

Property 24

At each iteration of either the primal or dual algorithm with active set S, and a search path $X_S = \{ \textbf{x}(\alpha ): \alpha \in D \}$ that is either a ray, hyperbola, ellipse, or parabola, the objective function $z(\alpha )$ is decreasing for the primal algorithm and increasing for the dual algorithm.

Proof

For either the primal or dual algorithm the linear search path is the ray $X_S = \{ \textbf{x}(\alpha ) = \textbf{x}_S + \alpha \textbf{d}_S, \alpha \in D \}$. The objective function is $z(\alpha ) = \Vert \textbf{x}_S + \alpha \textbf{d}_S - \textbf{p}_{i_1} \Vert + r_{i_1}$ and $z'(\alpha ) = [(\textbf{x}_S - \textbf{p}_{i_1})\textbf{d}_S + \alpha \textbf{d}^2_S] / \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1}\Vert$.

For the primal algorithm $D = \{ 0 \le \alpha \le \alpha _m \}$, $\alpha _m = (\textbf{p}_{i_1} - \textbf{x}_S)\textbf{d}_S/\Vert \textbf{d}_S\Vert ^2 > 0$, and $\textbf{d}_S \leftarrow$ Projection$((\textbf{p}_{i_1} - \textbf{x}_S), R)$, with $R = \{ \textbf{p}_{i_2} - \textbf{p}_{i_1}, \ldots , \textbf{p}_{i_s} - \textbf{p}_{i_1} \}$. By construction $( \textbf{x}_S - \textbf{p}_{i_1} )\textbf{d}_S < 0$, so that $z'(0) < 0$, for $0 \le \alpha < \alpha _m$, $z'(\alpha ) = 0$, and $z'(\alpha _m) = 0$,

For the dual algorithm $D = \{ 0 \le \alpha \}$, and $\textbf{d}_S \leftarrow$ Projection$((\textbf{p}_{e} - \textbf{x}_{S}), R)$. Then $z'(\alpha ) = \alpha \textbf{d}^2_S] / \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1}\Vert$, since $( \textbf{x}_S - \textbf{p}_{i_1} )\textbf{d}_S = 0$. Thus for $0 < \alpha$, $z'(\alpha ) > 0$, and $z'(0) = 0$,

For the primal or dual algorithm with a hyperbolic search path, $X_S = \{ \textbf{x}_S(\alpha ) = \textbf{c}_S + a_S\sec (\alpha )\textbf{v}_S + b_S\tan (\alpha )\textbf{u}_S, \alpha \in D \}$, with $\textbf{x}'(\alpha ) = a_S\sec (\alpha ) \tan (\alpha )\textbf{v}_S + b_S\sec ^2(\alpha )\textbf{u}_S$. The objective function is $z(\alpha ) = \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1} \Vert + r_{i_1}$, and $z'(\alpha ) = (\textbf{x}(\alpha ) - \textbf{p}_{i_1})\textbf{x}'(\alpha )/\Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1}\Vert = a_S\sec (\alpha )\tan (\alpha )[(\textbf{c}_S - \textbf{p}_{i_1}) \textbf{v}_S +( c^2_S/a_S) \sec (\alpha )]/ \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1}\Vert .$

Since $\textbf{x}(0) = \textbf{x}_S = \textbf{c}_S + a_S\textbf{v}_S \in \text {conv}(S)$, there exists $\pi _j \ge 0$, for each $\textbf{p}_{i_j} \in S$ such that $\sum _{\textbf{p}_{i_j} \in S}\pi _j = 1$ and $\sum _{\textbf{p}_{i_j} \in S}( \textbf{x}(0) - \textbf{p}_{i_j})\pi _j = \sum _{\textbf{p}_{i_j} \in S}( \textbf{c}_S - \textbf{p}_{i_j} + a_S\textbf{v}_S )\pi _j = \textbf{0}$. Multiply the last equation by $\textbf{v}_S$ to get $\sum _{\textbf{p}_{i_j} \in S}( (\textbf{c}_S - \textbf{p}_{i_j} )\textbf{v}_S + a_S )\pi _j = 0$.

If $(\textbf{c}_S - \textbf{p}_{i_j} )\textbf{v}_S + a_S = 0$ for each $\textbf{p}_{i_j} \in S$, then the points in S are on the hyperplane containing the point $\textbf{c}_S + a_S\textbf{v}_S$ with normal vector $\textbf{v}_S$. Thus, the points in S are affinely dependent, which is a contradiction. If the summands $(\textbf{c}_S - \textbf{p}_{i_j} )\textbf{v}_S + a_S$, for each $\textbf{p}_{i_j} \in S$, are either all positive or all negative, the sum is not zero. Therefore, there exists some $\textbf{p}_{i_k} \in S$, such that $(\textbf{c}_S - \textbf{p}_{i_k} )\textbf{v}_S + a_S > 0$. For the hyperbolic search path, $\epsilon ^2_S > 1$, or $c^2_S/a_S > a_S$. Then $0< (\textbf{c}_S - \textbf{p}_{i_k} )\textbf{v}_S + a_S < (\textbf{c}_S - \textbf{p}_{i_k} )\textbf{v}_S + c^2_S/a_S \le (\textbf{c}_S - \textbf{p}_{i_k} )\textbf{v}_S + (c_S^2/a_S)\sec (\alpha )$.

For the primal algorithm $D = \{ \alpha _S \le \alpha < 0 \}$. For $\alpha _S< \alpha < 0$, $\tan (\alpha ) < 0$ and $z'(\alpha ) < 0$. For $\alpha = 0$, $z'(\alpha ) = 0$. For the dual algorithm $D = \{ 0 \le \alpha \}$. For $0 < \alpha$, $\tan (\alpha ) > 0$ and $z'(\alpha ) > 0$. For $\alpha = 0$, $z'(\alpha ) = 0$.

For the primal or dual algorithm with a elliptlic search path, $X_S = \{ \textbf{x}_S(\alpha ) = \textbf{c}_S + a_S\cos (\alpha )\textbf{v}_S + b_S\sin (\alpha )\textbf{u}_S, \alpha \in D \}$ with $\textbf{x}'(\alpha ) = - a_S\sin (\alpha )\textbf{v}_S + b_S\cos (\alpha )\textbf{u}_S$. The objective function is $z(\alpha ) = \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1} \Vert + r_i$, and $z'(\alpha ) = (\textbf{x}(\alpha ) - \textbf{p}_{i_1})\textbf{x}'(\alpha )/\Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1}\Vert = a_S\sin (\alpha )[(\textbf{p}_{i_1} - \textbf{c}_S) \textbf{v}_S - ( c^2_S/a_S) \cos (\alpha )]/ \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1}\Vert .$

By the same argument used for the hyperbolic case, since $\textbf{x}(0) = \textbf{c}_S + a_S\textbf{v}_S \in \text {conv}(S)$, there exists some $\textbf{p}_{i_k} \in S$, such that $( \textbf{p}_{i_k} - \textbf{c}_S )\textbf{v}_S - a_S > 0$. For the elliptic search path, $\epsilon ^2_S < 1$, so that $c^2_S/a_S < a_S$. Then $0< ( \textbf{p}_{i_k} - \textbf{c}_S )\textbf{v}_S - a_S < ( \textbf{p}_{i_k} - \textbf{c}_S )\textbf{v}_S - c^2_S/a_S \le ( \textbf{p}_{i_k} - \textbf{c}_S )\textbf{v}_S - (c^2_S/a_S)\cos (\alpha )$.

For the primal algorithm, $D = \{\alpha _S \le \alpha < 0 \}$. For $\alpha _S< \alpha < 0$, $\sin (\alpha ) < 0$, and $z'(\alpha ) < 0$. For $\alpha = 0$, $z'(\alpha ) = 0$. For the dual algorithm with $D = \{ 0 \le \alpha \}$. For $0 < \alpha$, $\sin (\alpha ) > 0$ and $z'(\alpha ) > 0$. For $\alpha = 0$, $z'(\alpha ) = 0$.

For the primal or dual algorithm with a parabolic search path, $X_S = \{ \textbf{x}(\alpha ) = \textbf{c}_S + \tilde{c}_S\alpha ^2\textbf{v}_S + 2\tilde{c}_S\alpha \textbf{u}_S, \alpha \in D \}$ with $\textbf{x}'(\alpha ) = 2\tilde{c}_S\alpha \textbf{v}_S + 2\tilde{c}_S\textbf{u}_S$. The objective function is $z(\alpha ) = \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1} \Vert + r_{i_1}$ and $z'(\alpha ) = (\textbf{x}(\alpha ) - \textbf{p}_{i_1})\textbf{x}'(\alpha )/\Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1}\Vert = 2\tilde{c}\alpha [( \textbf{c}_S - \textbf{p}_{i_1} ) \textbf{v}_S + \tilde{c}(\alpha ^2 + 2) ] / \Vert \textbf{x}(\alpha ) - \textbf{p}_{i_1}\Vert .$

Using the same argument as used for the hyperboloid and ellipsoid cases, $\textbf{x}(0) = \textbf{c}_S \in \text {conv}(S)$ implies there exists some $\textbf{p}_{i_k} \in S$, such that $( \textbf{c}_S - \textbf{p}_{i_k} )\textbf{v}_S > 0$, which implies $( \textbf{c}_S - \textbf{p}_{i_k} )\textbf{v}_S+ \tilde{c}(\alpha ^2 + 2) > 0$.

For the primal algorithm $D = \{ -\infty < \alpha \le 0 \}$, so that $z'(\alpha ) < 0$, and for $\alpha = 0$, $z'(\alpha ) = 0$. For the dual algorithm $D = \{ 0 \le \alpha < \infty \}$, so that $z'(\alpha ) > 0$, and for $\alpha = 0$, $z'(\alpha ) = 0$. $\square$