Efficient algorithms for the smallest enclosing ball problem

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1 Efficient algorithms for the smallest enclosing ball roblem Guanglu Zhou, Kim-Chuan Toh, Jie Sun November 27, 2002; Revised August 4, 2003 Abstract. Consider the roblem of comuting the smallest enclosing ball of a set of m balls in R n. Existing algorithms are known to be inefficient when n>30. In this aer we develo two algorithms that are articularly suitable for roblems where n is large. The first algorithm is based on log-exonential aggregation of the maximum function and reduces the roblem into an unconstrained convex rogram. The second algorithm is based on a second-order cone rogramming formulation, with secial structures taken into consideration. Our comutational exeriments show that both methods are efficient for large roblems, with the roduct mn in the order of Using the first algorithm, we are able to solve roblems with n = 100 and m = in about 1 hour. Keywords: Comutational geometry, smoothing aroximation, second order cone rogramming. AMS subject classification: 90C30. 1 Introduction A ball B i in R n with center c i and radius r i 0 is the closed set B i = {x R n : x c i r i }. Given a set of balls B = {B 1,B 2,,B m } in R n, the smallest enclosing ball mb(b) of the set B is the ball with the smallest radius that encloses all the balls in B. Given a set of oints P = { 1, 2,, m } in R n, the smallest enclosing ball m(p) of the oints is the ball with the smallest radius that encloses all the oints in P. Clearly, the roblem of smallest enclosing ball of oints is a secial case of the roblem Singaore-MIT Alliance, National University of Singaore, 4 Engineering Drive 3, Singaore smazgl@nus.edu.sg. Deartment of Mathematics, National University of Singaore, 2 Science Drive 2, Singaore mattohkc@nus.edu.sg. Research suorted in art by the Singaore-MIT Alliance. Deartment of Decision Sciences, National University of Singaore, Singaore. jsun@nus.edu.sg. 1

2 of smallest enclosing ball of balls. In what follows, unless otherwise stated, we will term the roblem of smallest enclosing ball of balls to be the smallest enclosing ball roblem. This roblem can be easily formulated as the following convex otimization roblem: min x R n max { x c i + r i }. (1) 1 i m Let f i (x) = x c i + r i, i =1, 2,,m, (2) and f(x) = max {f i(x)}. (3) 1 i m Then roblem (1) can be rewritten as min f(x). (4) x Rn It is readily seen that the solution to (1) exists since f(x) is coercive. Moreover, the solution is unique, for otherwise there would exist two different balls, C 1 and C 2,of the same radius, with m j=1 B j C i,i =1, 2. Then one can construct a smaller ball containing C 1 C 2 and thus B as well. Problem (1) arises in alications such as location analysis and military oerations and it is itself of interest as a roblem in comutational geometry; see [2, 3, 6, 9, 12, 13, 18, 20] for details. Many algorithms have been develoed for (1). In articular, for the roblem of the smallest enclosing ball of oints, Megiddo [9] resented a deterministic O(m) algorithm for the case where n 3. Welzl [18] develoed a simle randomized algorithm with exected linear time in m for the case where n is small, and Gärtner described a C++ imlementation thereof in [4]. For the roblem of the smallest enclosing ball of balls, we are aware of only a C++ rogram develoed by David White in [19], which is based on Welzl s algorithm [18] and Gärtner s imlementation [4]. The existing software ackages [4, 19], however, are not efficient for solving roblems with n>30. The goal of this aer is to resent efficient algorithms that can be used to solve roblems where n and m are large (say, in the order of thousands). Problem (1) is a non-differentiable (nonsmooth) convex otimization roblem. Due to the non-differentiability of the objective function, gradient-based algorithms cannot be used to solve this roblem. To overcome this difficulty, in Section 2 we give a smooth aroximation for the maximum function f. This aroximation is based on the socalled log-exonential aggregation function studied, for instance, in [8]. Based on this smooth aroximate function, in Section 3 we roose a limited-memory BFGS algorithm for solving the roblem. In Section 4, we reformulate (1) as a second order cone rogramming (SOCP) roblem. Thus, we can also aly interior-oint methods to solve (1). In Section 5, we reort some numerical results, which show the algorithms roosed here are able to solve large roblems. In this aer, unless otherwise stated, all vectors are column vectors. We let I d denote the d d identity matrix. To reresent a large matrix with several small matrices, we use semicolons ; for column concatenation and commas, for row concatenation. These notations also aly to vectors. We denote the convex hull of a set S R n by co(s). For a convex function f : R n R,welet f(x) denote the subdifferential of f at x. Note that f is minimized at x over R n if and only if 0 f(x). For calculus rules on subdifferentials of convex functions, lease refer to Chater VI in [7]. 2

3 2 Smooth aroximation For any > 0, define the smoothing log-exonential aggregation function of f in (3) as ( m ) f(x; ) = ln ex (g i (x; )/) where g i (x; ) =r i + x c i (5) Lemma 1 The function f(x; ) has the following roerties: (i) For any x R n, and 1, 2 satisfying 0 < 1 < 2, we have f(x; 1 ) <f(x; 2 ); (ii) For any x R n and >0, f(x) f(x; ) f(x)+(1 + ln m); (iii) For any > 0, f(x; ) is continuously differentiable and strictly convex. Proof. (i) For any x R n and 1, 2 satisfying 0 < 1 < 2, by Jensen s inequality [5], [ m 2 [ m ] (ex(g i (x; 2 ))) 2] 1/ 1 > (ex(g i (x; 2 ))) 1/ 1 [ m 1 > (ex(g i (x; 1 ))) 1] 1/. Hence, f(x; 1 ) <f(x; 2 ). (ii) Fix = 2 and let 1 0 in (i), we have f(x) <f(x; ). Let g (x; ) = max 1 i m {g i(x; )}. (6) It is readily roven that f(x) g (x; ) f(x)+. Thus, from (5), we have f(x; ) =g (x; )+ln ex [(g i (x; ) g (x; )) /] g (x; )+ln m. Hence f(x) f(x; ) f(x)+(1 + ln m). (iii) For any >0, clearly, f(x; ) is continuously differentiable. Now we rove that f(x; ) is strictly convex. From (5), where h i (x; ) = f(x; ) = x c i 2 + 2, τ(x; ) = λ i (x; ) h i (x; ) (x c i), (7) ex(g i (x; )/), (8) λ i (x; ) = ex(g i(x; )/). (9) τ(x; ) 3

4 Let Q ij = (x c i)(x c j ) T h i (x; )h j (x; ), i,j =1, 2,,m. From (7), we get 2 f(x; ) = λ i(x; ) h i (x; ) (I n Q ii )+ λ i(x; ) Q ii j=1 λ i (x; )λ j (x; ) Q ij. For any z R n with z 0, by the Cauchy-Schwartz inequality, Thus, z 2 z T Q ii z z 2 z 2 (x c i )/h i (x; ) 2 > 0, i =1,...,m. z T 2 f(x; )z = > λ i (x; ) ( z 2 z T Q ii z ) h i (x; ) + λ i(x; ) z T Q ii z λ i(x; ) z T Q ii z j=1 j=1 λ i (x; )λ j (x; ) λ i (x; )λ j (x; ) z T Q ij z z T Q ij z = 1 λ i (x; )a 2 i 1 ( m ) 2 λ i (x; )a i 0, (10) where a i = z T (x c i )/h i (x; ). This shows that 2 f(x; ) is ositive definite. Therefore, f(x; ) is strictly convex. Note that the inequality (10) follows from the fact that λ i (x; )a i m λ i (x; ) m λ i (x; )a 2 i, and m λ i (x; ) =1,λ i (x; ) 0 for i =1,...,m. 3 Log-exonential aggregation algorithm We now give an algorithm for roblem (1) based on Lemma 1, followed by a global convergence result. Algorithm 1 Let σ (0, 1), x 0 R n and 0 > 0 be given, and set k := 0. 4

5 For k =0, 1, 2,, do End 1. Use an unconstrained minimization method to solve min x R n f(x; k ). (11) Let the minimizer be x k. 2. Set k+1 = σ k, increment k by 1, and return to ste 1. Lemma 2 Let {x k } k 1 be the sequence of oints roduced by Algorithm 1. Then any limit oint of {x k } k 1 is an otimal solution of (1). Proof. Let x be a limit oint of {x k } k 1. Without loss of generality, we suose that x k x as k tends to +. From the fact that x k is a solution of (11), we have f(x k ; k ) = λ k i h i (x k ; k ) (xk c i ) = 0, where λ k i = λ i(x k ; k ), and h i (x; k ) is defined as in (8). Let By noting that I(x ) = {i : f i (x )=f(x ),, 2,,m}. λ k i = ex[(g i (x k ; k ) g (x k ; k ))/ k ] m ex[(g i (x k ; k ) g (x k ; k ))/ k ], we have lim k + λk i =0, i I(x ). (12) By (12) and the fact that m λ k i = 1 and λ k i > 0, i = 1, 2,,m, the sequence {λ k i,i =1, 2,,m} k 1 has a convergent subsequence. Without loss of generality, we suose that lim k + λk i = λ i, i I(x ). Then i I(x ) λ i = 1 and λ i 0, i I(x ). For i I(x ), t i = lim k + x k c i h i (x k ; k ) f i(x ). Thus lim k + f(xk ; k )= i I(x ) λ i t i =0. This shows that 0 co( f i (x ),i I(x )) = f(x ). Therefore, x is an otimal solution of (1). 5

6 Theorem 3 Let {x k } k 1 be the sequence of oints roduced by Algorithm 1 and x be the unique otimal solution of (1). Then lim k + xk = x. Proof. For any k 1, by Lemma 1, Since f(x) is coercive, the level set f(x 1 ; 1 ) >f(x 1 ; k ) f(x k ; k ) f(x k ). (13) L = {x R n : f(x) f(x 1 ; 1 )} is bounded. From (13), we have {x k } k 1 L. Hence {x k } k 1 is bounded. As x is the unique solution of (1), it follows from Lemma 2 that lim k + xk = x. In ractice, we would sto Algorithm 1 once some stoing criteria are met. Moreover, we will use a first-order, or gradient based, unconstrained minimization algorithm for solving (11) in ste 1. In the following algorithm, we will choose a limited-memory BFGS algorithm [1] to solve (11) as this algorithm can solve very large unconstrained otimization roblems efficiently. The following is a more ractical version of Algorithm 1. Algorithm 2 Let σ (0, 1), ɛ 1,ɛ 2 > 0, x 0 R n and 0 > 0 be given, and set k := 0. For k =0, 1, 2,, until k ɛ 1, do End 1. Use a version of the limited-memory BFGS algorithm to solve (11) aroximately, and obtain an x k such that f(x k ; k ) ɛ Set k+1 = σ k, increment k by 1, and return to ste 1. The actual arameter values used for 0,σ,ɛ 1,ɛ 2 are given in Section 5. 4 SOCP reformulation Problem (1) is equivalent to the following roblem. min r s.t. y c i + r i r, i =1,,m, y R n. (14) 6

7 It in turns can be reformulated as follows: max r s.t. t i + r = r i, s i + y = c i, t i s i, i =1,...,m. (15) Let N = m(n +1), K := {(t; s) R n+1 : t s }, y =(r; y) R n+1, c = (r 1 ; c 1 ; r 2 ; c 2 ; ; r m ; c m ) R N, A = (I n+1,,i n+1 ) R (n+1) N, b = (1; 0; ;0) R n+1, s =(t 1 ; s 1 ; t 2 ; s 2 ; ; t m ; s m ) R N, where I n+1 is an identity matrix. For later urose, we use the notation γ(α; β) = α 2 β 2 for (α; β) K. The roblem (15) can be written as a standard dual second order cone rogram as follows: (D) max b T y, s.t. s = c A T y, s K, where K = K K K = K m. Let The corresonding rimal roblem is x =(τ 1 ; x 1 ; τ 2 ; x 2 ; ; τ m ; x m ) R N. (P ) min c T x, s.t. Ax = b, x K. This air of roblems (P ) and (D) are studied in Nesterov and Nemirovskii [10], Nesterov and Todd [11], and Xue and Ye [21]. We can solve these roblems by using a rimal-dual interior-oint method (IPM). There are IPM based general urose softwares for solving SOCPs, examles are SeDuMi [14], SDPT3[16], and LOQO [17]. But the roblems (P ) and (D) have a very secial constraint matrix A, and it is beneficial to analyze how such a structure can be exloited when comuting the search direction in each IPM iteration. Suose that at a articular IPM iteration, the current iterate is (x, y, s) and the target barrier arameter is µ. To obtain the next iterate, the most exensive comutation lies in assembling and solving a linear system of n + 1 equations for the search direction y. By adating the results from Section 2.3 of [16] for the so-called Nesterov-Todd search direction, such a linear system takes the following form for our resent SOCPs: ( 1 m ) σ J + 1 g 2 wi 2 i g T i y = h, (16) }{{} M 7

8 where [ ] 1 J =, wi 2 I = γ(t i; s i ) n γ(τ i ; x i ), 1 σ = m 1, 2 wi 2 2 g i = γ(ρ i ; ξ i ) ( ρ i; ξ i ), (ρ i ; ξ i ) = 1 (t i ; s i ) w i ( τ i ; x i ), w i and h = b + 1 wi 2 ( J + gi g T i ) ((ri ; c i )+(t i ; s i ) y) µ γ(t i ; s i ) 2 ( t i; s i ). Note that the matrix M is symmetric ositive definite. In the above, we give a full descrition of the linear system (16) for the sake of comleteness. But our main aim is to study the matrix M, and exlore the ossibility of solving (16) via a Krylov subsace method such as the reconditioned conjugate gradient (PCG) method when n is large. Let I = {i {1,...,m} : σ g i w i > 1}, J = {1,...,m}\I. We can write the matrix M in (16) as follows: M = 1 ( I + σ 2 ) g σ 2 i I wi 2 i g T i + 1 σ 2 g σ 2 j J w 2 j g T j j }{{}}{{} M 0 M 1 2 σ 2 e 1e T 1, where M 0 is ositive definite, M 1 is ositive semidefinite, and e 1 = (1; 0; ;0) R n+1. Numerical exeriments have shown that the cardinality of I is usually much smaller than n when the barrier arameter µ is small. Thus M 0 is a diagonal matrix lus a lowrank matrix, and its inverse can be found readily via the Sherman-Morrison-Woodbury formula: 0 = σ [ 2 I G(I + G T G) 1 G ] T, G = }{{} B M 1 ( ) σ g i i I. wi Suose M 0 is used as a reconditioner in (16), then the reconditioned matrix 0 M has the form: M 1 M 1 0 M = I + 1 σ 2 M 1 0 M 1 2 σ 2 M 1 0 e 1e T 1 = I + BM 1 2Be 1 e T 1. (17) To analyze the sectrum of M 1 0 M 1, we make use of the following lemma. Lemma 4 Let = I. If<n+1, then the matrix B is ositive semidefinite and B 1 8

9 Proof. Let the singular value decomosition of G R (n+1) be G = UΣV T, where U R (n+1) has orthonormal columns, Σ R is diagonal, and V R is orthogonal. It is easily shown that Thus G(I + G T G)G T = UΣ(I +Σ 2 ) 1 ΣU T. B = I UΣ(I +Σ 2 ) 1 ΣU T (18) = (I UU T )+U(I +Σ 2 ) 1 U T, which is the sum of two ositive semidefinite matrices, and hence B is also ositive semidefinite. From (18), it is clear that x T Bx x T x for any x R n+1. Hence, B 1. Notice that BM 1 is similar to a ositive semidefinite matrix and thus all its eigenvalues lie in the interval [0, BM 1 ] [0, M 1 ]. For j J, σ g j /w j 1, thus M 1 J, and we would exect M 1 to be a moderate number. Hence, we can exect the reconditioned matrix M0 1 M to have a favorable sectral distribution (with all its eigenvalues clustered around 1 excet ossibly for an outlier due to the resence of a rank-1 erturbation in (17)) for the PCG to attain fast convergence. So far, we have discussed the reconditioning strategy to solve (16) when the barrier arameter µ is small. When µ is not small, the cardinality of I may be comarable to n, and it would be comutationally exensive to use M 0 as a reconditioner. Fortunately when µ is not small, the matrix M itself is quite well-conditioned and reconditioning by the diagonal of M would be good enough for PCG to obtain fast convergence. In ractice, we switch from the diagonal reconditioner to M 0 when 5 Comutational results max( g i /w i ) min( g i /w i ) (19) We imlemented Algorithm 2 described in Section 3 and the numerical exeriments were done by using a Pentium IV 2.2GHz ersonal comuter with 4GB of memory. To solve (11), we choose a limited-memory BFGS algorithm with strong Wolfe-Powell line-search and 7 limited-memory vector udates [1]. We imlemented our code in Fortran 77 and comared it with SeDuMi (version 1.05) [14] and SDPT3 (version 3.02) [16] (two high quality software ackages with Matlab interface for solving semidefinite-quadraticlinear rograms). We also imlemented an adatation of SDPT3 to exloit the secial structure resent in our SOCPs in (P) and (D). The main change lies in relacing the direct solution method in comuting the direction y in SDPT3 by the PCG iterative solution method discussed in the last section. For convenience, we will call the modified algorithm SDPT3-PCG. 9

10 Throughout the comutational exeriments, we use the following arameters in Algorithm 2: 0 =0.01, σ =0.1, ɛ 1 =10 6, and ɛ 2 =10 5. The starting oint is x 0 = 0. For SDPT3, we sto the interior-oint algorithm when φ is smaller than Here ( b Ax φ = max 1+ b, c s AT y x T ) s, 1+ c 1+0.5( b T. y + c T x ) We use the same stoing criterion for SDPT3-PCG. When solving the system (16), we sto the PCG iteration when the relative residual norm, h M y h, is less than For SeDuMi, we use all the default values excet that the arameter es is set to be The test roblems are generated randomly. We use the following seudo-random sequence: ψ 0 =7, ψ i+1 = (445ψ i + 1) mod 4096, i =1, 2,, ψ i = ψ i 40.96, i =1, 2,. (20) The elements of c i,, 2,,m, are successively set to ψ 1, ψ 2,, in the order: r 1,c 1 (1), c 1 (2),,c 1 (n), r 2, c 2 (1),,c 2 (n),,r m,c m (1),,c m (n). The numerical results we obtained are summarized in Tables 1 4. In these tables, n and m denote the dimension of the Euclidean sace and the number of balls, resectively, Obj Value denotes the value of the objective function at the final iteration, Iter denotes the number of iterations, Time denotes the CPU time in second for solving each roblem, and RAM denotes the random access memory usage in MB. The results reorted in Tables 1 and 2 show that algorithms resented in this aer erform very well. We can see from Table 1 that all algorithms are able to obtain good accuracy for moderately large roblems. Moreover, it aears from Table 2 that Algorithm 2 consistently takes more iterations than the SOCP algorithm and that for all tested roblems, Algorithm 2 and SDPT3-PCG consistently use less CPU time than SeDuMi and SDPT3. It is interesting to note that for the last 2 roblems, SeDuMi, SDPT3, and Alg. 2 are about 30, 8.5, and 2.8 times more exensive (in CPU time) than SDPT3-PCG, resectively. From the erformance data in Table 2, it is clear that for the smallest enclosing ball roblem, Alg.2 and SDPT3-PCG are much more efficient than the general urose algorithms in SeDuMi or SDPT3. Thus we shall concentrate only on Alg. 2 and SDPT3-PCG for the rest of this section. Next, we look at the erformance of Alg. 2 and SDPT3-PCG for roblems with large n but m is moderate. For each air of (n, m), we run the algorithms on 5 random roblems generated from (20) with the integer 445 relaced by 437, 441, 445, 449, and 453. The results (each number is the average over 5 roblems) reorted in Table 3 show that Algorithm 2 can solve roblems with very large n in a reasonable amount of CPU time that grows linearly with n. Its memory demand is just slightly more than the 10

11 amount required to store the data oints {c i R n : i =1,...,m}. However, SDPT3- PCG fails for the last 2 larger roblems in Table 3 due to excessive memory requirement. In fact, SDPT3-PCG requires about 18 times more memory sace than Alg. 2. It is interesting to note however, that when there is enough comuter memory, SDPT3-PCG aears to be more efficient (in terms of CPU time) than Alg. 2. In Table 4, we comare the the erformance of Alg. 2 and SDPT3-PCG for roblems with large m but n is moderate. Again, we test the algorithms on 5 randomly generated roblems for each air of (n, m). Both Alg. 2 and SDPT3-PCG can solve these roblems very efficiently. The required CPU time and memory for Alg. 2 grow almost linearly with m. For SDPT3-PCG, the required CPU time grows like m 1.13 with m. Again, its memory demand is about 18 times more than that required by Alg. 2. From the results in Tables 2 to 4, we conclude that Algorithm 2 is better than SeDuMi, SDPT3 and SDPT3-PCG for very large roblems, since it consumes less memory. For moderately large roblems, SDPT3-PCG is more efficient (in CPU time) than Algorithm 2. Problem Obj Value n m SeDuMi SDPT3 Alg2 SDPT3-PCG e e e e e e e e e e e e e e e e e e e e3 Table 1: Objective function value at the final iteration. Problem SeDuMi SDPT3 Alg2 SDPT3-PCG n m Time Iter Time Iter Time Iter Time Iter Table 2: Performance comarison of SeDuMi, SDPT3, Algorithm 2 and SDPT3-PCG. The numbers in the last two columns corresond to average value of I and the average number of PCG iterations in each interior-oint iteration, resectively. References [1] R.H. Byrd, P. Lu, J. Nocedal and C. Zhu, A limited memory algorithm for bound constrained otimization, SIAM J. Sci. Comut., 16 (1995)

12 Problem Alg2 SDPT3-PCG n m Time Iter RAM Time Iter RAM out of memory out of memory Table 3: Performance of Algorithm 2 and SDPT3-PCG on roblems with large n. Problem Alg2 SDPT3-PCG n m Time Iter RAM Time Iter RAM out of memory out of memory Table 4: Performance of Algorithm 2 and SDPT3-PCG on roblems with large m. [2] J. Elzinga and D. Hearn, The minimum covering shere roblem, Management Sci., 19 (1972) [3] B. Gärter, Fast and robust smallest enclosing balls, J. Nestril, editor, Algorithms- ESA 99: 7th Annual Euroean Symosium Proceedings, Lecture Notes in Comuter Science 1643, , Sringer-Verlag. [4] B. Gärter, Smallest enclosing ball - fast and robust in C++, htt:// [5] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, Cambridge University Press, [6] D.W. Hearn and J. Vijan, Efficient algorithms for the minimum circle roblem, Oer. Res., 30 (1982) [7] J.B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithm, Sringer-Verlag Berlin Heidelberg, [8] X.S. Li, An aggregate function method for nonlinear rogramming, Sci. China Ser. A, 34 (1991) [9] N. Megiddo, Linear-time algorithms for linear rogramming in R 3 and related roblems, SIAM J. Comut., 12 (1983)

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