global optimization approach to unequal sphere packing problems in 3d

8/11/2019 Global Optimization Approach to Unequal Sphere Packing Problems in 3D

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Global Optimization Approach to Unequal Sphere

Packing Problems in 3D

Atsusi Sutou Yang Dai

Abstract: The problem of unequal sphere packing in a 3-dimensional polytope is ana-

lyzed. Given a set of unequal spheres and a polytope, the double goal is the assembly of

the spheres in such a way that (1) they do not overlap with each other and (2) the sum

of the volumes of the spheres packed in the polytope is maximized. This optimization

has an application in automated radiosurgical treatment planning and can be formulated

as a nonconvex optimization problem with quadratic constraints and a linear objectivefunction. On the basis of the special structures associated with this problem, we propose

a variety of algorithms which markedly improve the existing simplicial branch-and-bound

algorithm for the general nonconvex quadratic program. Further, heuristic algorithms

are incorporated to strengthen the efficiency of the algorithm. The computational study

demonstrated that the proposed algorithm can successfully obtain the optimization up to

a limiting size.

keywords: Nonconvex Quadratical Programming, Unequal Sphere Packing Problem,

Simplicial Branch-and-Bound Algorithm, LP-Relaxation, Heuristic

The authors would like to thank two anonymous referees for their valuable comments. The work ofthe second author is partially supported by a Grant-in-Aid (C-13650444) from the Ministry of Education,Science, Sports and Culture of Japan.

Research Staff, Enterprise Server Systems Research Department, Central Research Laboratory, Hi-tachi, Ltd, JAPAN E-mail : [email protected]

Assistant Professor, Department of Mathematical and Computing Sciences, Tokyo Institute of Tech-nology, 2-12-1 O-Okayama, Meguro-ku, Tokyo 152-8552, JAPAN, E-mail :[email protected]

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1 Introduction

The optimization of the packing of unequal spheres in a 3-dimensional polytope is ana-

lyzed. Given a set of unequal spheres and a polytope, the objective is to assemble them

in such a way that (1) the spheres do not overlap with each other and (2) the sum of the

volumes of the packed spheres is maximized. We note that the conventional 2-dimensional

and 3-dimensional packing problems (also called the bin-packing problem), which have

been extensively studied [2, 7, 11], are fundamentally different from the problem consid-

ered in the present work.

The unequal sphere packing problem has important applications in automated radio-

surgical treatment planning [12, 14]. Stereotactic radiosurgery is an advanced medical

technology for treating brain and sinus tumors. It uses the Gamma knife to deliver a set

of extremely high dose ionizing radiations, called shots, to the target tumor area [13]. In

good approximation, these shots can be considered as solid spheres. For large or irregular

target regions, multiple shots are used to cover different parts of the tumor. However,this procedure usually results in (1) large dose inhomogeneities, due to the overlap of the

different shots, and (2) the delivery of large amount of dose to normal tissue arising from

enlargement of the treated region when two or more shots overlap.

Optimizing the number, the position, and the individual sizes of the shots can signifi-

cantly reduce both the inhomogeneities and the dose to normal tissue while simultaneously

achieving the required coverage. Unfortunately, since the treatment planning process is

tedious, the quality of the protocol depends heavily on the experience of the users. There-

fore, an automated planning process is desired. To achieve this goal, Wang and Wu et

al. [12, 14] mathematically formulated this planning problem as the packing of spheres

into a 3D region with a packing density greater than a certain given level. This packing

problem was proved to be NP-complete and an approximate algorithm was proposed [12].

In this work we formulate the question as a nonconvex quadratic optimization and present

solution methods based on the branch-and-bound technique.

Let Kbe the number of different radii of the spheres in the given set and rk (k =

1,...,K) the corresponding radii. There are Lavailable spheres for each radius. Therefore,

the total number of the spheres in the set is KL. Here, we use a single value ofL for

simplicity of the presentation. However, this model can be easily modified for the casewhere different numbers Lk of spheres are available for different radii rk and the total

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number of spheres in the given set isKi=1

Lk.

Let the polytope be given by P = { (x,y,z) R3 : amx+bmy+ cmz dm, m =

1,...,M } for some M >0. Let L be the maximum number of the spheres to be packed.

We designate variables (xi, yi, zi) (i= 1,...,L) as the location of sphereiin a packing. For

each sphere in the packing, a radius has to be assigned. The variables tik(i= 1, ..., L, k=

1,...,K) are used to handle this task:

tik=

1 if sphere i has radiusrk,0 otherwise.

With these preliminaries, the optimization can be formulated as follows.

(P1) max . 4

3

Li=1

Kk=1

r3ktik

s.t. (xixl)2 + (yiyl)

2 + (zizl)2 (

K

k=1 rktik+K

k=1 rktlk)2,i= 1,...,L 1, l= i+ 1,...,L, (1)

|amxi+bmyi+cmzidm|a2m+b

2m+c

2m

Kk=1

rktik,

i= 1,...,L, m= 1,...,M, (2)

amxi+bmyi+cmzidm 0,

i= 1, ...,L, m= 1,...,M, (3)K

k=1 tik1, i= 1,...,L, (4)tik {0, 1}, i= 1,...,L, k= 1,...,K. (5)

Constraints (1) and (3) respectively ensure that no two spheres overlap with each other

and that each sphere is centered within the polytope. Constraints (4) and (5) guarantee

that at most one radius is chosen for each sphere, i.e., if tik = 1, then the sphere i is

packed with radius rk, and if tik = 0, then the sphere i with radius rk is not packed.

Together with (3), (4) and (5), constraints (2) state that the distance between the center

of a sphere and the boundary of the polytope is at least as large as the radius of that

sphere. Hence, (2) and (3) force all the spheres to be packed inside the polytope.

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Letem=

a2m+b2m+c

2m. By (3), constraint (2) can be rewritten as

amxi+bmyi+cmzidm emKk=1

rktik for eachi, m. (6)

Since the right-hand side of (6) is nonnegative, (6) implies (3). Moreover, the binary 0 1

variables tik in (5) can be replaced by the inequalities tik(tik 1) 0 and 0 tik 1.

Note that tik 1 is implied by the constraint (4). These steps allow restatement of

Problem (P1) as follows.

(P2) max .Li=1

Kk=1

r3ktik

s.t. (xixl)2 (yiyl)

2 (zizl)2 + (

Kk=1

rktik+Kk=1

rktlk)2 0,

i= 1,...,L1, l= i+ 1,...,L, (7)

t2ik+tik0, i= 1,...,L, k= 1,...,K, (8)

amxi+bmyi+cmzidm emKk=1

rktik,

i= 1,...,L, m= 1,...,M, (9)Kk=1

tik1, i= 1,...,L, (10)

tik 0, i= 1,...,L, k= 1,...,K. (11)

Note that the constant 43in the original objective function is omitted here. The numbers

of the variables and the quadratic constraints are (3+ K)Land L(L1)2

+ LK, respectively.

The quadratic function in each constraint (7) is neither convex nor concave.There exist several algorithms [1, 9] that have been developed for solving the gen-

eral nonconvex quadratic program (NQP in short). A NQP can be transformed into a

semidefinite programming problem (SDP) with an additional rank-one constraint. Drop-

ping the rank-one constraint, one obtains a SDP relaxation problem, which is the tightest

relaxation among all others [3]. Hence, (P2) can be solved by using a branch-and-bound

type method based on the SDP relaxation. However, due to the relative slowness and the

instability of the SDP software, this conceptual algorithm is impractical even for problem

having a small size, since it requires many iterations of the SDP relaxation.

Commonly, methods of solution for the NQP class are designed through linear pro-

gramming (LP) relaxation, an approach known as the reformulation-linearization tech-

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nique [10]. Under the assumption that additional box constraints for the variables are

given, new nonlinear constraints are generated from each pair of the box constraints in

order to construct a relaxed convex area of the original nonconvex feasible region. The

nonlinear terms in each nonlinear constraint are accordingly replaced by new variables.

The result of this recasting is a linear program. Based on this technique of linearization,

Al-khayyal et al. [1] proposed a rectangular branch-and-bound algorithm to solve a class

of quadratically constrained programs. Raber [9] proposed another branch-and-bound

algorithm for the same problem based on the use of simplices as the partition elements

and the use of an underestimate affine function, whose value at each vertex of the simplex

agrees with that of the corresponding nonconvex quadratic function. This work [9] demon-

strated that the simplicial algorithm often has a better performance over the rectangular

algorithm with respect to the computational time. Interestingly, Raber [9] mentioned

that both the simplicial and the rectangular algorithms exhibit poor performance for a

packing problem : max {t : | t xi xj220, 1 i < jn, xi, xj [0, 1]

2 }without

giving the experimental details. It is obvious that their packing problem is similar toours, but has much simpler structures for the constraints.

In this paper, we examine a customization of Rabers simplicial branch-and-bound

algorithm [9] tailored to our problem. It is well known that the underlying simplicial sub-

division is a key factor influencing the quality of the relaxation, therefore, the efficiency of

the branch-and-bound algorithm. The investigation of the structure of the optimization

suggests (i) an efficient simplicial subdivision and (ii) different underestimations of the

nonconvex functions. Based on these observations, three variants of the algorithm have

been constructed. The discrete nature of the packing enables a heuristic design for ob-

taining good feasible solutions, an outcome which leads to savings on both computational

time and memory size.

The remainder of this paper is organized as follows. In Section 2 we derive the LP

relaxation of the problem with respect to the simplicial subdivision. The simplicial branch-

and-bound algorithm is presented in Section 3. Section 4 gives the heuristic algorithms

and Section 5 presents three variations of the previous algorithm based on the use of

special structures. Section 6 reports the computational results of the proposed branch-

and-bound algorithm. The conclusions of the work are presented in Section 7.

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2 Linear Programming Relaxation

The construction of the LP relaxation of Problem (P2) is the same as the one developed

in [9]. For the sake of a complete description, we outline this procedure below.

Let n = 3L+KL, vT = (x1, y1, z1,...,xL, yL, zL, t11,...,tLK) Rn. Here aT denotes

the transpose of a vector a. First, we write Problem (P2) in the following form.

(P2) max . cTv

s.t. vTQilv 0, i= 1,...,L1, l= i+ 1,...,L,

vTQikv +dTikv 0, i= 1,...,L, k= 1,...,K,Av b,

where Qil,Qik Rnn,dik Rn, and c Rn is the coefficient vector of the objective

function. vTQilv0 and vT

Qikv +

dT

ikv0 correspond to the constraints (7) and (8),

respectively. Av b represents all linear constraints of (9), (10) and (11). Furthermore,

the matrix Qil can be specified as follows.

Qil=

Q

xyzil O

O Qtil

,

whereO is a matrix having zero for all entries with appropriate size,

Qxyzil =

. . . ...

... ...

.... . . 1 . . . 1 . . .

1 1

. . . 1 . . . 1 . . .... ...

. . . ...

.... . . 1 . . . 1

1 1. . . 1 . . . 1 . . .

... ...

... ...

. . .

R3L3L

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corresponds to the coefficients of (x1, y1, z1,...,xL, yL, zL)T in (7) for iand l, and

Qtil=

. . . ...

... ...

.... . . r21 r1r2 . . . r1rK . . . r

21 r1r2 . . . r1rK . . .

r1r2 r22 . . . r2rK r1r2 r

22 . . . r2rK

... ...

. . . ...

... ...

. . . ...

. . . r1rK r2rK . . . r2

K . . . r1rK r2rK . . . r2

K... ...

. . . ...

... . . .. . . r21 r1r2 . . . r1rK . . . r

21 r1r2 . . . r1rK . . .

r1r2 r22 . . . r2rK r1r2 r

22 . . . r2rK

... ...

. . . ...

... ...

. . . ...

. . . r1rK r2rK . . . r2K . . . r1rK r2rK . . . r

2K . . .

... ...

... ...

. . .

RKLKL

corresponds to those of (t11,...,t1K,...,tL1,...,tLK)T in (7) for i and k. Similarly,

Qik

R3L3L anddik RKLKL can be written as follows:

Qik=

O ... O

0. . . 0 1 0 . . .

0

O ... O

,

dTik= (0, , 0, 1, 0, , 0).Let m be the number of the linear constraints in (P2). Denote the polytope defined by

these linear constraints as U ={v Rn : Av b}, where A Rmn and b Rm.

To construct the LP relaxation problem, we need to represent the matrix Qil (resp.Qik) by the sum of a positive semidefinite matrix Cil(resp.Cik) and a negative semidef-inite matrix Dil (resp.Dik). Usually, a spectrum decomposition achieves this goal.However, we do not need to perform such a task, since the matrices Qil and

Qik possessspecial structures that give the decomposition immediately. It is readily seen that the

decompositions

Qil = Cil+ Dil

= O O

O Qtil

+ Qxyzil O

O O

, (12)

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and

Qik =Cik+Dik= O +Qik (13)

satisfy the desired property.

Now we consider how to construct our linear programming relaxation problem. Let

S= {v0,..., vn}be ann-simplex (U S=), where vi(i= 0,...,n) are its vertices. Then

S= {v Rn :n

j=0

jvj,n

j=0

j = 1, j 0 (j = 0,...,n)}.

Let WS Rnn be a matrix which consists of columns (vj v0) (j = 1,...,n). Then

each point v in Scan be represented as

v= v0+ WS for some B = { Rn :

nj=1

j 1, j 0 (j = 1,...,n)}. (14)

Through substitution ofv in (P2) by (14), we obtain the following equivalent problem.

(P3) max . cTWS + cTv0

s.t. (WS)TQilWS + 2v

T0 QilWS + v

T0 Qilv0 0,

i= 1,...,L1, l= i+ 1,...,L, (15)

(WS)TQikWS + (2Qikv0+dik)TWS + vT0Qikv0+dTikv0 0,

i= 1,...,L, k= 1,...,K, (16)

AWS b Av0, B. (17)

By replacing Qil andQik with (12) and (13), the quadratic term of the left-hand sideof each constraint of (15) and (16) is divided into a convex and a concave functionsby replacing Qil and

Qik with (12) and (13), respectively. The relaxation of the aboveproblem is constructed by ignoring the convex part and replacing the concave part with a

linear underestimate function. For such an underestimation, we use the convex envelope

of a concave function fwith respect to the simplex S, which is an affine function whose

value at each vertex of S coincides with that of f. More precisely, for the quadratic

constraints (15) we have

(WS)TQilWS + 2v

T0 QilWS + v

T0 Qilv0

= (WS)TCilWS + (WS)

TDilWS + 2vT0 QilWS + v

T0 Qilv0

Sil() + 2vT0 QilWS + v

T0 Qilv0, (18)

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where

Sil() =n

j=1

(vj v0)TDil(vj v0)j (19)

is the convex envelope of (WS)TDilWSwith respect toS. In a similar fashion, for the

quadratic constraints (17), we have

(WS)TQikWS + (2Qikv0+dik)TWS + vT0Qikv0+dTikv0

Sik() + (2Qikv0+dik)TWS + vT0Qikv0+dTikv0, (20)

where

Sik() =n

j=1

(vj v0)TDik(vj v0)j, (21)

is the convex envelope of (WS)TDikWSwith respect to S.

Obviously, an upper bound of the objective function of (P3) can be obtained by solving

the following LP relaxation problem.

(LPR)S max . cTWS + cTv0

s.t. Sil() + 2vT0 QilWS + v

T0 Qilv0 0,

i= 1,...,L1, l= i+ 1,...,L, (22)

Sik() + (2Qikv0+dik)TWS + vT0Qikv0+dTikv0 0,

i= 1,...,L, k= 1,...,K, (23)

AWS b Av0, B. (24)

3 The Simplicial Branch-and-Bound AlgorithmThe simplicial branch-and-bound algorithm is presented in this section. As mentioned

above, we use the algorithm in [9] as our prototype. However, two heuristics designed for

obtaining feasible solutions are embedded.

The branching operation is carried out by dividing the current simplex S into two

simplices. Let vi and vj be two vertices ofSsatisfying

vi vj2 = max{vk vk2 | vk, vk S}, (25)

wherea2 denotes the 2-norm of a vector a. Define

vM=1

2vi+

1

2vj. (26)

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The simplex Sis split into two simplices

S1 = [v0, v1,..., vi1, vM, vi+1,..., vn], (27)

and

S2 = [v0, v1,..., vj1, vM, vj+1,..., vn]. (28)

The splitting of the simplices has the property that for each nested sequence {Sq} of

simplices,

2(Sq) 0 (q ), (29)

where2(Sq) = max{vi vj22 | vi, vj Sq}. For further details, see Horst [4, 5]. The

resulting algorithm is presented below.

Branch and Bound Algorithm :

Step 1. Start Heuristic-1 to calculate a possible feasible solution vf and the objective

function value f(vf). If successful, set LB = f(vf).

Step 1.1. Let S = . Construct a simplex S0 which contains the polytope U. Set

S = {S0}. Solve the problem (LPR)S0. If the problem is infeasible, then the

original problem has no solution, stop. Otherwise, let the optimal solution be v0

and the optimal value be (S0). Set U B = (S0). If v0 is a feasible solution of

(P2), stop. Otherwise, start Heuristic-2 withv0 to calculate a feasible solution v

0

and the value f(v0). IfLB < f(v

0), then set LB = f(v

0), vf= v

0. Setk = 0.

Step 1.2. If (UBLB)/UB , then stop.

Step 2 (Branching). Split Sk into S1k and S

2k according to (25) and (28). Set S =

S \ {Sk} {S1k} {S

2k}. Forj = 1, 2 run Step 2.1 - Step 2.3.

Step 2.1. Solve the problem (LPR)Sjk

. If it is infeasible, setS = S \ {Sjk}. Otherwise,

let the optimal solution be vjk and the optimal value be (Sjk).

Step 2.2. If vjk is a feasible solution of (P2), set S = S \ {Sjk}. Furthermore, ifLB 0, go to Step 5.

Step 3. Ifk = 0 and l = K L, then sets = s+ 1. Ifs = N, stop. Otherwise set l=1 and

go to Step 1.

Step 4. Setl = l+ 1, go to Step 1.

Phase II

Step 5. Ifk = 1, stop. Otherwise, for each pair of the packed spheres i, j, set l = 1,

repeat Steps 6 and 7.

Step 6. Ifl > KLstop. Otherwise set M= (xi+xj

2 ,yi+yj

2 ,zi+zj

2 ).

Step 7. Locate the center of the spherel atM. If it satisfies the constraints Pand does

not overlap with the spheres 1,...,k, then pack the sphere l and set k = k + 1. If

k= L stop. Otherwise setl = l+ 1 and go to Step 6.

By the termination of the algorithm ifk >0 a feasible packing is obtained.

Next, we describe Heuristic-2. Let (x1, y

1, z

1 ,...,x

L, y

L, z

L, t

11,...,t

1K) be the solution

of a relaxation subproblem. Suppose that it is not feasible. Then either tik is not integral

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or some spheres overlap with each other if the radius is decided by tik = 1 for each i,

i.e., use rk as the radius of the sphere i (see Figure 3). In the following, we fix the

centers of these spheres and determine their radii so that they do not mutually overlap

(see Figure 4). Note that for a triplet (xi , y

i , z

i ), ift

ik = 0 for all k = 1,...,K, it means

that no sphere is placed there. Let r1> > rK.

Figure 3: Figure 4:

Heuristic-2

Step 1. Setl = 1, k = 1.

Step 2. If the sphere l centered at (xl , y

l, z

l) with radius rk satisfies the polytope con-

straints and does not overlap with spheres 1,...,l1, then set tlk = 1, t

lk = 0 for

k =k (the sphere lcentered at (xl , y

l, z

l) has radius rk), go to Step 4.

Step 3. Ifk < K, set k = k + 1, go to Step 2. Otherwise, sett

lk = 0 for k

= 1,...,K(no sphere is centered at (xl , y

l, z

l)).

Step 4. Ifl < L, setl = l+ 1, go to Step 2. Otherwise, stop.

5 The Improvement of the Algorithm

In this section we discuss the special structures of the optimization and present the im-

provements on the previous algorithm based on these structures. First, we describe how

to construct the initial simplex.

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5.1 Generating the Initial Simplex

To start the branch-and-bound algorithm, we need an initial simplex S0 which contains

the polytope U. It is generated by the following algorithm.

Generate Initial Simplex:

Step 1. Choose a nondegenerate vertex v0 of the polytope U.

Step 2. Construct a matrix A Rnn from the n binding constraints (Ai)Tv0 = b

i at

v0 (A = [A1, A

2,..., A

n]).

Step 3. Compute = max{(n

i=1 A

i)Tv: v U}.

Step 4. Set S0={v : (A)Tv b, (

ni=1 A

i)Tv }.

Remark 1: Then constraints which decide the nondegenerate vertex v0 can be selected

as follows. For eachiwe choose three constraints from (9) and all constraints of (11). Itis not difficult to see that n (n= 3L+KL) coefficient vectors from these constraints are

linearly independent. Set these linear inequality constraints as linear equations, i.e.,

amxi+bmyi+cmzi+dm= emKk=1

rktik,

i= 1,...,L, m= 1,...,M, (30)

tik= 0, i= 1,...,L, k= 1,...,K. (31)

The above system of linear equations determines an unique solution which can be consid-

ered as v0.

Remark 2: The vertex v0 has K Lzero entries, which are determined by (31). Further-

more, if an equation in (30) is replaced by (ni=1

Ai)Tv = , then the solution obtained

maintainstik = 0 for all i and k. Repeating this process for all equations in (30) yields

3Lsolutions, which we index as v1,..., v3L. All the other solutions generated by replacing

tik= 0 (i= 1,...,L, k= 1,...,K) are denoted by v3L+1, ..., vn. More precisely, we have

vi= (vi,1, , vi,3L, 0, , 0)T, i= 0,..., 3L, (32)

and

vi = (vi,1, , vi,3L, 0, , 0, vi,i, 0, , 0), i= 3L+ 1,...,n. (33)

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5.2 Splitting a Simplex in a Different Way

It is well known that in the simplicial branch-and-bound paradigm both the computational

time and the usage of the memory grow extremely fast as the dimension of the polytope

U increases.

When a simplex S is split into two simplices according to (25) - (28), the length of

vi vj is the longest among all other pairs of the vertices of the simplex S. If thevertex v0 is not replaced by the vertex vM, then the matrix WS1 is the same as WSexcept

the column vi v0, which is replaced by the column vM v0. Similarly, all columns

in the matrix WS2 are the same as WS except the column vj v0, which is replaced

by vM v0. Recall that the entries (i, j) (i, j = 3L+ 1,...,n) are zeros in the matrices

Dil(i= 1,...,L1, l= i +1,...,L) in (12). Hence, the coefficients ofj in Sil() of (22),

only depends on the first 3L coordinates of v0, v1,..., vn. If vM, vi and vj have same

values for the first 3L coordinates, then Sil() remains unchanged in the constraints

(22) of the subproblem with respect to the simplices Sj (j= 1, 2), and the quality of the

relaxation would not be significantly improved. Such a splitting leads to the computation

of subproblems which provide neither good lower nor useful upper bounds. To avoid this

situation, we choose vi, vj such that

3Lk=1

(vi,kvj,k)2

is the maximum among all otheri j pairs. Consequently, the coordinates corresponding

totik(i= 1, ...,L, k= 1,...,K) will not be considered.

One negative effect of this choice is that the quality of the relaxation of the constraint

tik(tik1) 0 is not necessarily improved. However, since the linear constraint 0 tik1

is always considered, this defect is not expected to be serious. Accordingly, the number

of subproblems can be decreased in the whole branch-and-bound process.

There is another potential difficulty with the convergence of the algorithm which must

be addressed. Let 2(Sq) = max{3Lk=1

(vi,kvj,k)2 | vi, vj are vertices ofSq}. Since we

divide the simplex to minimize the largest value3Lk=1

(vi,kvj,k)2 over all vertices, it is

possible that a nested sequence {Sq} of simplices with2(Sq) 0 (q ), but

2(Sq)

does not satisfy (29). In this case, we are not guaranteed that the accumulation point of

an infinite sequence obtained by the algorithm will be an optimal solution.

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5.3 Another Decomposition of the Matrix Qil

As shown in Section 2, to make the underestimation of the quadratic function of the

left-hand side of the quadratic constraint (7), we use the decomposition of (12). From

tik {0, 1}, we have t2ik=tik. Replacingt

2ikbytik in (7) could result in a different matrix

Qil as follows.

Qil= Qxyzil OO Qtil, ,Q

xyzil = Q

xyzil ,

Qtil=

. . . ...

... ...

.... . . 0 r1r2 . . . r1rk . . . r

21 r1r2 . . . r1rk . . .

r1r2 0 . . . r2rk r1r2 r22 . . . r2rk

... ...

. . . ...

... ...

. . . ...

. . . r1rk r2rk . . . 0 . . . r1rk r2rk . . . r2k

... ...

. . . ...

... . . .

. . . r21 r1r2 . . . r1rk . . . 0 r1r2 . . . r1rk . . .r1r2 r

22 . . . r2rk r1r2 0 . . . r2rk

... ...

. . . ...

... ...

. . . ...

. . . r1rk r2rk . . . r2k . . . r1rk r2rk . . . 0 . . .

... ...

... ...

. . .

,

and

dTil = ( , 0, r21, , r

2k, 0, , 0, r

21, , r

2k, 0, ).

A spectral decomposition of the matrix Qil provides positive semidefinite and negative

semidefinite matrices which are different from those given in (12). Following a proceduresimilar to that in Section 2, we have

Qil = C

il+ D

il,

where Cil and D

il are positive semidefinite and negative semidefinite, respectively. For

example, if only two magnitudes of radii r1 and r2 are considered, then Ctil and D

til

RKLKL can be constructed as follows. Let s =

r41+ 14r21r

22+r

42. Then

Qil = C

il+ D

il

= O O

O Ctil

+ Qxyzil O

O Dtil

,

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where

Ctil=

. . . ...

... ...

...

. . . r2

1(r21+3r2

2+s)

3r21+r2

2+s

r1r2(r21+3r2

2+s)

3r21+r2

2+s

. . . r2

1(r21+3r2

2+s)

3r21+r2

2+s

0 . . .

. . . r1r2 r22 . . . r1r2 0 . . .

... ...

. . . ...

...

. . . r2

1(r21+3r2

2+s)

3r21

+r22

+s

r1r2(r21+3r2

2+s)

3r21

+r22

+s . . .

r21(r21+3r2

2+s)

3r21

+r22

+s 0 . . .

. . . r1r2 r22 . . . r1r2 0 . . ....

... ...

... . . .

and

Dt

il= Qt

il Ct

il.

It can be shown that the eigenvalues of matrix Ct

il (resp. Dt

il ) are nonnegative (resp.

nonpositive). Hence, the two matrices have the desired properties. Consequently, another

LP relaxation can be constructed based on the new spectral decomposition of

Qil. Note

thatQik remains the same here, so there is no change in its decomposition.5.4 Another Form of the Relaxation

In this section we focus on a different form of relaxation of problem (P2). Let us omit

the quadratic constraints (8), which are corresponding to the 0 1 condition of tik.

Furthermore, we relax the condition that two spheres can not overlap with each other.

Here the distance between two spheres only needs to be greater than a small positive

value . More precisely, consider the problem

(P4) max .Li=1

Kk=1

r3ktik

s.t. (xixl)2 + (yiyl)

2 + (zizl)2 (2)2,

i= 1,...,L1, l= i+ 1,...,L, (34)

amxi+bmyi+cmzi+dm emKk=1

rktik,

i= 1,...,L, m= 1,...,M, (35)K

k=1tik1, i= 1,...,L, (36)

tik0, i= 1,...,L, k= 1,...,K. (37)

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0 tik1, i= 1,...,L, k= 1,...,K, (44)

AxWS + A

tt b Axv0, B . (45)

Since the constraints (8) are ignored and the constraints (7) are relaxed as (42) in the

above problem, the quality of this relaxation may be inferior to the previous methods.

However, the dimension of the simplices kept in the memory is only 3L. Therefore, the

total memory used may be far smaller than that required in the other methods.

6 The Computational Study

In this section we discuss details of the implementation of the algorithm and report the

experimental results.

Reoptimization : Suppose that S is the simplex at a branching node. The simplex S

is divided into S1 and S2 by (25) to (28). If the vertex v0 is not replaced by vM, then

only columns vi

v0 and vj

v0 of matrices WS1

and WS2

, are replaced by vM v0,respectively. All the other columns remain unchanged. Therefore, in this case, when

two new subproblems are generated by splitting S into S1 and S2, we could use the

information of the optimal solution of the LP relaxation problem associated with S. By

the construction of the LP relaxation problem with respect to S1 and S2, only those

coefficients corresponding toi andj in the constraints (22) and (23) are changed due

to the substitutions of the column vM v0 in WS1 and WS2, respectively. Consequently,

by changing these coefficients, we could use the reoptimization technique to solve the LP

relaxation problem starting from the optimal solution corresponding to the simplex S.

Test Problems : The test problems are generated as follows. First, construct a sim-

plex with vertices (0, 0, 10), (10, 0, 0), (0, 10, 0) and (10, 10, 10). Calculate the maxi-

mum inscribed sphere of the simplex (the radius is 2.88675). Then randomly gener-

ate m points on the surface of the sphere. Construct tangent planes {(x,y,z) | aix+

biy + ciz = di} (i = 1,...,m) which pass each of those m points respectively. Let

Hi = {(x,y,z ) | aix+biy+ ciz di} be the halfspace containing the inscribed sphere.

The intersection ofHiwith the simplex is the 3-dimensional polytope in which the sphere

packing problem is considered. Hence, the total number of the linear constraints of the

polytope is M = m+ 4. In our test,m was set at 4. Different pairs of radii of spheresthat we used are shown in Tables 1-2.

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Table 1: Combinations of radii for K= 2

Cr radius1 1.75 0.752 1.50 1.253 1.25 1.004 1.25 0.505 1.00 0.756 1.00 0.507 1.00 0.25

Table 2: Combinations of radii for K= 3

Cr radius1 2.00 1.50 1.002 1.75 1.00 0.25

3 1.50 1.00 0.504 1.20 0.90 0.605 1.00 0.75 0.25

The computational experiments were conducted on a DEC Alpha 21164 Workstation

(600MHz). We used CPLEX 6.5.1 as an LP solver for the relaxation problems. The

limits of the memory and the computational time were set at 512MB and 3600 seconds,

respectively. Besides the prototype algorithm given in Section 3, three variations were

also implemented. They were (1) Algorithm NSS, which uses the simplex splitting givenin Section 5.2, (2) Algorithm NMD, which uses the matrix decomposition given in Section

5.3, and (3) Algorithm XYZ, which uses the simplex in the XYZ space given in Section

5.4. For each algorithm, we tested 5 instances for each L= 2, 3, 4 and each pairCrof radii

from the Table 1. Since the other three algorithms reached either the limit of memory

or the limiting computational time for most of the instances for L= 5, we only present

the results of Algorithm XYZ. The data shown in Tables 4-7 and Table 10 represent the

average of the results. The legends used in the tables are given in Table 3.

From Tables 4-7 we observe that the computational time grows drastically as the

number of spheres packed is increased, since the dimension of the problem is (3 + K)L.

This outcome is consistent with the observation in [1, 9] that the computational time

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Table 3: Legends used in the tables

legend meaningL the maximum number of the spheres packedCr the combination of the radii#.LP the number of linear programs solved#.T the number of cases terminated caused by the limit of CPU

time (3600 seconds)#.E the number of cases terminated when = 106

#.M the number of cases terminated caused by the limit of memory (512 MB)Time CPU time (seconds)UBLB

UB the ratio of the values (U BLB) and U B, where U B and LB is the

upper and lower bounds of the objective function value, respectivelyAlgorithm ORG the prototype algorithm given in Section 3Algorithm NSS the algorithm using the simplex splitting given in Section 5.2Algorithm NMD the algorithm using the spectral decomposition given in Section 5.3

Algorithm XYZ the algorithm using the simplex in the XYZ space in Section 5.4

increases exponentially as the dimension increases. There is no big difference between

Algorithm ORG and Algorithm NMD, in the sense of their gross behaviors. Algorithm

NSS solves fewer LPs than the two previously mentioned algorithms do, therefore, it needs

less time. This leads to the solution of a greater number of instances without violating

the limits on either memory or time. Since the simplex splitting is based on the length of

the first 3Lcoordinates, it avoids solving unnecessary subproblems.

Among all the algorithms, Algorithm XYZ shows the best performance. All instancesfor L = 2, 3, 4 are solved except the case Cr = 1, L = 4. It even successfully obtained

the solutions for about half of the instances for L = 5. Two reasons for this behavior

are considered. First, since the dimension of simplices is only 3L, much less memory is

needed to keep the information on the simplices. Second, since the simplices involve only

the coordinates of variables (x, y, z), the simplex splitting has the favorable characteristic

of Algorithm NSS that fewer LPs are needed to be solved.

It should be noted that (i) the dimension of the instances solved in the previous

papers [1, 9] range up to 16 and (ii) the computational time and memory demand of the

simplicial branch-and-bound algorithm usually increases exponentially with the dimension

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of the problem. Therefore, a modest increase in the dimension could put the solution out

of range. In contrast, for the sphere packing problem discussed above, the dimension of

the instances solved by Algorithm XYZ extended up to n = 25 (whenL = 5 and K= 2).

Hence, we conclude that Algorithm XYZ gains efficiency, since it takes advantage of the

intrinsic structure of the problem.

Next, let us focus on Algorithm ORG and Algorithm XYZ and consider the influence

of the polytope P on their performance. Five instances of the polytope are generated

randomly and they have same number of constraints. TakingL= 4,Cr = 2 the results are

shown in Tables 8-9. It is observed that even ifLand Cr are identical, the comparative

behaviors of the algorithms are very sensitive to the shape of the polytope. This behavior

arises since (i) the quality of solutions generated by the heuristics depends on the shape

of the polytope and (ii) the initial simplex S0 depends on the polytope. For a skinny

polytope, the volume ofS0\ P0 would be large, a fact which means that the algorithms

copiously waste effort on solving LPs defined on this zone before reaching the optimal

solution.Table 10 shows the results when three values of radii are considered, i.e., K= 3. Note

that (i) the value of K effects the number of the quadratic constraints in (8), and (ii)

increasing the number of the quadratic constraints enhances the difficulty of the problem.

Comparing this case with the results for K= 2 (Table 7), we observed that there is no

big change in the numbers of LPs solved. The algorithm is less sensitive to the value ofK

than to the size of ofL. This occurs, since the numbers of both the quadratic constraints

(7) and (8) grow with increasing ofL.

Finally, all of the algorithms exhibited reduced performance when Cr = 1, namely,

when the differences of the radii are large.

7 Conclusions

In this paper, we considered the optimization of unequal sphere packing and demonstrated

the improvements over the existing simplicial branch-and-bound algorithm through ad-

vantageous use of the intrinsic structure of the problem. Specially, the computational

study showed that the improved Algorithm XYZ could solve instances with much larger

size. We observed that optimal solutions were found for many instances when the algo-rithm reached the limitations of either computational time or memory. This signals that

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Table 4: Results of Algorithm ORG

L Cr #.LP (UBLB)

UB Time #.E #.T #.M

2 1 4809 0.00 8.32 5 0 02 1 0.00 0.00 5 0 03 1 0.00 0.00 5 0 0

4 1 0.00 0.00 5 0 05 2670 0.00 4.42 5 0 06 3164 0.00 8.56 5 0 07 2542 0.00 4.24 5 0 0

3 1 270542 0.13 2505.31 3 2 02 7617 0.00 28.27 5 0 03 2530 0.00 7.63 5 0 04 8511 0.00 27.51 5 0 05 18585 0.00 61.39 5 0 06 13494 0.00 44.06 5 0 07 15625 0.00 49.27 5 0 0

4 1 221050 0.35 3019.95 1 4 02 139077 0.00 865.56 4 1 03 109891 0.07 1564.78 3 2 04 148904 0.14 1947.72 3 2 05 109476 0.08 1554.34 3 2 06 149203 0.14 2235.52 2 3 07 207166 0.15 2730.21 2 3 0

the algorithm spends a large amount of effort verifying the optimality of the solution.

In turn, this behavior indicates that (1) developing an improved method of relaxation is

necessary for solving problems with a larger size, (2) when the algorithm is terminated,

the solution obtained then can be of high quality, and (3) a better heuristic method, which

could start with a feasible solution obtained from the branch-and-bound algorithm, would

be very important for obtaining the approximate solution to larger problems.

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Table 5: Results of Algorithm NMD

L Cr #.LP (UBLB)

UB Time #. E #. T #.M

2 1 4805 0.00 8.43 5 0 02 1 0.00 0.00 5 0 03 1 0.00 0.00 5 0 04 1 0.00 0.00 5 0 05 2699 0.00 4.55 5 0 06 3226 0.00 8.68 5 0 0

7 2540 0.00 4.22 5 0 03 1 250702 0.23 1710.93 3 2 0

2 7485 0.00 27.33 5 0 03 2523 0.00 7.56 5 0 04 8209 0.00 26.07 5 0 05 17970 0.00 60.48 5 0 06 12376 0.00 45.53 5 0 07 18158 0.00 59.26 5 0 0

4 1 239966 0.37 3201.76 1 4 02 116394 0.02 837.32 4 1 03 122589 0.01 1681.80 3 2 04 162731 0.14 2226.64 3 2 05 130796 0.10 1719.24 3 2 06 151269 0.10 2246.07 2 3 07 230333 0.20 3577.78 1 4 0

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Table 6: Results of Algorithm NSS

L Cr #.LP (UBLB)

UB Time #.E #.T #.M

2 1 1574 0.00 3.08 5 0 02 1 0.00 0.00 5 0 03 1 0.00 0.00 5 0 04 1 0.00 0.00 5 0 05 215 0.00 0.32 5 0 06 228 0.00 0.48 5 0 0

7 299 0.00 0.45 5 0 03 1 212048 0.19 2424.45 2 3 0

2 35 0.00 0.09 5 0 03 1685 0.00 4.52 5 0 04 1067 0.00 2.85 5 0 05 7243 0.00 25.04 5 0 06 7645 0.00 26.36 5 0 07 1671 0.00 4.34 5 0 0

4 1 222788 0.28 3019.95 1 4 02 112190 0.04 1397.28 3 1 13 56887 0.07 710.24 4 0 14 111017 0.05 1305.93 4 0 15 30829 0.08 209.47 5 0 06 65880 0.04 833.36 4 1 07 63520 0.00 520.50 5 0 0

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Table 7: Results of Algorithm XYZ

L Cr #.LP (UBLB)

UB Time #.E #.T #.M2 1 577 0.00 0.70 5 0 0

2 1 0.00 0.00 5 0 03 1 0.00 0.00 5 0 04 1 0.00 0.00 5 0 05 164 0.00 0.19 5 0 06 485 0.00 0.70 5 0 07 293 0.00 0.33 5 0 0

3 1 147821 0.06 1432.87 4 1 02 11 0.00 0.00 5 0 03 793 0.00 1.38 5 0 04 1603 0.00 2.85 5 0 05 1827 0.00 3.04 5 0 06 1734 0.00 2.86 5 0 07 1617 0.00 2.67 5 0 0

4 1 267940 0.32 3600.00 0 5 02 49365 0.00 448.67 5 0 03 33103 0.00 124.17 5 0 04 68588 0.00 532.67 5 0 05 20487 0.00 94.53 5 0 06 22199 0.00 108.72 5 0 0

7 67332 0.00 630.07 5 0 05 1 266880 0.46 3600.00 0 5 0

2 86677 0.03 927.39 4 1 03 199838 0.06 2660.20 2 3 04 178181 0.11 2367.46 2 3 05 156760 0.05 1996.32 3 2 06 164904 0.06 2089.97 3 2 07 165105 0.11 2296.86 2 3 0

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Table 8: The results of Algorithm ORG (L= 4, Cr = 2)

P #.LP (UBLB)UB

Time1 5981 0.00 25.102 28563 0.00 134.32

3 1 0.00 0.024 590915 0.52 3600.035 69929 0.00 568.33

Table 9: The results of Algorithm XYZ (L= 4, Cr = 2)

P #.LP (UBLB)UB

Time1 27453 0.00 88.472 210891 0.00 2578.083 1511 0.00 3.204 71329 0.00 393.705 25477 0.00 86.90

Table 10: The results of Algorithm XYZ (K= 3)

L Cr #.LP (UBLB)UB Time #.E #.T #.M3 1 308570 0.37 3600.00 0 5 0

2 123418 0.05 1122.95 4 1 03 11 0.00 0.02 5 0 04 502 0.00 0.96 5 0 05 1591 0.00 2.81 5 0 0

4 1 218901 0.50 3600.00 0 5 02 261260 0.30 3600.00 0 5 03 46204 0.00 235.28 5 0 04 79014 0.02 124.17 4 1 0

5 47947 0.00 414.87 5 0 0

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Captions of the figures.

Figure 1. An initial packing found at Phase I

Figure 2. A dotted sphere is added at Phase II

Figure 3. An infeasible packing given by the relaxation problem

Figure 4. A packing of solid spheres are found by Heuristic-2

30

global optimization approach to unequal sphere packing problems in 3d

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