Large-scale packing of ellipsoids

Transcription

1 Large-scale packng of ellpsods E. G. Brgn R. D. Lobato September 7, 017 Abstract The problem of packng ellpsods n the n-dmensonal space s consdered n the present work. The proposed approach combnes heurstc technques wth the resoluton of recently ntroduced nonlnear programmng models n order to construct solutons wth a large number of ellpsods. Numercal experments llustrate that the ntroduced approach delvers good qualty solutons wth a computatonal cost that scales lnearly wth the number of ellpsods; and a soluton wth more than a mllon ellpsods n the three-dmensonal space s exhbted. Keywords: Packng, ellpsods, nonlnear programmng, models, algorthms. 1 Introducton The problem of packng ellpsods has a number of mportant applcatons, whch nclude the desgn of hgh-densty ceramc materals, the formaton and growth of crystals [8, 5], the structure of lquds, crystals and glasses [3], the flow and compresson of granular materals [14, 15, 16], the thermodynamcs of lqud to crystal transton [1, 7, 4], the chromosome organzaton n human cell nucle [7], and the modelng of vascular network formaton []. See also [9, 10, 11, 1, 13, 1] and the references theren for more applcatons. In the last years, several works [4, 5, 17, 18, 19, 0, 3, 6] addressed the problem of packng ellpsods usng nonlnear programmng models and technques. On the one hand, n [17, 18], global solutons to small-szed nstances (up to three ellpses or ellpsods) were sought. On the other hand, good qualty solutons to medum- and large-szed nstances were obtaned n [4, 5] by seekng local mnmzers of nonlnear programmng models. The models proposed n [4] have a number of varables and constrants that s quadratc n the number of ellpsods beng packed; ths beng the man lmtaton for obtanng good qualty solutons for nstances wth more than a hundred ellpsods. Models wth a number of varables and constrants that s expected to be lnear wth respect to the number of ellpsods beng packed were ntroduced n [5]. Usng those Ths work was supported by FAPESP (grants 01/3916-8, 013/ , 013/ , 013/ , 015/ , 016/ , and 017/ ) and CNPq (grant /014-1). Department of Computer Scence, Insttute of Mathematcs and Statstcs, Unversty of São Paulo, Rua do Matão 1010, Cdade Unverstára, , São Paulo, SP, Brazl. E-mal: egbrgn@me.usp.br Department of Appled Mathematcs, Insttute of Mathematcs, Statstcs, and Scentfc Computng, State Unversty of Campnas, Rua Sérgo Buarque de Holanda 651, , Campnas, SP, Brazl. E-mal: lobato@me.usp.br 1

2 models, solutons wth up to fve hundred ellpsods can be obtaned. Anyway, the nonlnear programmng models that need to be solved are hghly nonconvex and, n general, exstent state-of-the-art methods are not capable of fndng good qualty local mnmzers of nstances wth, say, a thousand ellpsods. In the present work, we consder the problem of packng the maxmum number of ellpsods wthn a gven contaner. We present an heurstc approach based on the nonlnear programmng models ntroduced n [4, 5]. The computatonal cost of the proposed method scales lnearly wth the number of ellpsods and, therefore, huge nstances can be consdered. The aforementoned works consdered the problem of packng a gven collecton of ellpsods wthn a volume-mnmzng contaner. If the number of ellpsods m s very small, global mnmzers (wth a certfcate of optmalty) of the contnuous and dfferentable nonlnear programmng models proposed n [4, 5], as well as the models consdered n [17, 18, 19, 0, 3, 6], can be obtaned consderng state-of-the-art global optmzaton software. For medum- and large-szed nstances (m up to, say, a thousand), state-of-the-art solvers for nonlnear programmng may be able to fnd statonary ponts assocated wth good qualty solutons to the packng problem. For nstances wth larger values of m, the nonconvexty of the models make almost mpractcable to fnd statonary ponts assocated wth reasonable solutons to the packng problem. The rest of ths paper s organzed as follows. In Secton, we state the problem consdered n ths work and ntroduce some notaton. In Secton 3, we present the model ntroduced n [5] to avod the overlappng between ellpsods. In Secton 4.1, we present a smple and general algorthm to solve the problem of packng the largest possble number of ellpsods nsde a gven contaner. In Secton 4., we propose some strateges that can be used to compose the general algorthm. To deal wth the case where the number of ellpsods to be packed s large, we present what we call the solaton constrants n Secton 4.3. These are addtonal constrants to the model to prevent large groups of ellpsods from overlappng and thus reducng the total number of varables and constrants of the model. The complete nonlnear programmng model and algorthm are presented n Secton 5. Some mplementaton detals are dscussed n Secton 6. Fnally, we present numercal experments n Secton 7 and draw some conclusons n Secton 8. The computer mplementaton of the method ntroduced n the present work and the solutons reported n Secton 7 are freely avalable at Problem defnton and notaton We represent an ellpsod n R n by the set E = {x R n (x c) QP 1 Q (x c) 1}, where c R n s the center of the ellpsod, Q R n n s an orthogonal matrx that determne the prncpal axes of the ellpsod, and P R n n s a postve defnte dagonal matrx so that the egenvalues of P 1 are the lengths of the sem-axes of the ellpsod. We denote by nt(e) the nteror of E,.e., nt(e) = {x R n (x c) QP 1 Q (x c) < 1}. Also, we denote by E the fronter of E,.e., E = {x R n (x c) QP 1 Q (x c) = 1}. The k-th standard bass vector (.e., the vector whose k-th components s equal to one and have all the other components equal to zero) s denoted by e k. The largest egenvalue of a matrx M s denoted by λ max (M). In ths paper, we consder the problem of packng the maxmum number of ellpsods wthn a gven contaner. The ellpsods must not overlap each other and they must be entrely nsde the

3 contaner. Formally, gven a set C R n, whch we call the contaner, and a sequence of (n n)- dmensonal postve defnte dagonal matrces {P } =1, the objectve s to fnd the maxmum nonnegatve number m and the ellpsods E = {x R n (x c ) Q P 1 Q (x c ) 1}, for I = {1,..., m }, n such a way that 1. nt(e ) nt(e j ) = for all, j I wth j;. E C for all I. By fndng an ellpsod E we mean determnng a vector c R n and an orthogonal matrx Q R n n. If P = P for all then the problem reduces to the problem of packng as many dentcal ellpsods (wth sem-axs lengths gven by the square roots of the dagonal entres of P ) as possble wthn the contaner C. 3 Non-overlappng and contanment models In [4] and [5], nonlnear programmng models for the non-overlappng of ellpsods were ntroduced. Snce they are the foundaton of the methodology proposed n ths paper, we brefly summarze these models n Secton 3.1. Besdes avodng the overlap between the ellpsods, t s requred the ellpsods to be nsde a gven contaner. In Secton 3., t s presented a model to nclude an ellpsod wthn a half-space, whch was ntroduced n [4]. Ths model wll be used to buld a cubodal contaner and also to construct the so called solaton constrants that wll be presented n Secton Non-overlappng model and Consder the ellpsods E and E j n R n defned as E = {x R n (x c ) Q P 1 Q (x c ) 1} E j = {x R n (x c j ) Q j P 1 j Q j (x c j ) 1}, where c and c j n R n are ther centers and Q and Q j are orthogonal matrces n R n n that determne ther orentaton. For example, for n =, we can represent Q as ( ) cos θ Q = sn θ, sn θ cos θ whereas, for n = 3, we can represent Q as Q = cos θ cos ψ sn φ sn θ cos ψ cos φ sn ψ sn φ sn ψ + cos φ sn θ cos ψ cos θ sn ψ cos φ cos ψ + sn φ sn θ sn ψ cos φ sn θ sn ψ sn φ cos ψ sn θ sn φ cos θ cos φ cos θ The parameters θ (when n = ) and θ, φ, and ψ (when n = 3) are called rotaton angles of the ellpsod. We denote by Ω R q the vector of rotaton angles of the -th ellpsod (q = 1 when. 3

4 n = and q = 3 when n = 3). The dea of the model presented n [4] s to transform one of the ellpsods nto a ball so that the problem of avodng the overlap between two ellpsods becomes the problem of preventng the overlap between a ball and an ellpsod. By lettng T j : R n R n be the lnear transformaton defned by T j (x) = P 1 Q (x c j ) (1) and applyng ths transformaton to the ellpsods E and E j, we obtan { and E j = respectvely, where The result s that E j x R n [ ] [ ] } x P 1 Q (c c j ) x P 1 Q (c c j ) 1 E j j = {x R n x S j x 1}, S j = P 1 Q Q j P 1 j Q j Q P 1. s a unt-radus ball and E j j between E and E j s equvalent to requrng that the ball E j (see Lemma 3.1 n [5]). Snce E j s an ellpsod, so that avodng the overlap and the ellpsod E j j do not overlap s a unt-radus ball, t does not overlap the ellpsod E j f and only f ts center s outsde of the ellpsod and ts dstance to the fronter of the ellpsod s at least one. Let c j denote the center of the ball E j. By Proposton 4.1 n [4], f cj / nt(e j j ), then we can wrte for a unque x j R n n the fronter of E j j the projecton of c j x j E j j onto E j j and some µ j 0, then c j the dstance between c j for E j of ellpsods: and E j j not to overlap wth E j µ j c j = x j + µ j S j x j () and a unque scalar µ j 0. Moreover, x j R n s. By Proposton 4. n [4], f cj j / nt(e j j ) and x j s the projecton of c j can be wrtten as n () for some onto E j j. Hence, s gven by c j x j = µ j S j x j, whch must be at least one. Hence, we obtan the followng model for the non-overlappng x js j x j = 1,, j I such that < j (3) S j x j 1,, j I such that < j (4) P 1 Q (c c j ) = x j + µ j S j x j,, j I such that < j (5) µ j 0,, j I such that < j (6) where I = {1,..., m} s the set of ndces of the ellpsods beng packed. Snce ths model has a quadratc number of varables and constrants on the number of ellpsods to be packed, t becomes rapdly hard to be solved as the number of ellpsods grows. In order to allevate ths complexty, a model wth a lnear number of varables and constrants was ntroduced n [5]. j 4

5 To reduce the number of constrants, the constrants of model (3) (6) are frst replaced by ther respectve squared nfeasblty measures where o(c, c j, Ω, Ω j, x j, µ j ; P, P j ) = 0,, j I such that < j, ( ) o(c, c j, Ω, Ω j, x j, µ j ; P, P j ) = x j( P 1 Q ) (c c j ) x j µj + max{0, ɛ j µ j } + x j + µ j S j x j P 1 Q (c c j ) + { } max 0, 1 P 1 Q (c c j ) x j, and then combned nto m 1 constrants as follows: m j=+1 o(c, c j, Ω, Ω j, x j, µ j ; P, P j ) = 0, I \ {m}. To reduce the number of varables, for each, j I such that < j, the varables x j and µ j are replaced wth X j X (c, c j, Ω, Ω j ; P, P j ) and U j U(c, c j, Ω, Ω j ; P, P j ), respectvely, where X j s a soluton to the problem mnmze x x c j subject to x S j x = 1, and U j s the correspondng Lagrange multpler. Therefore, the non-overlappng constrants can be fnally wrtten as m j=+1 f(c, c j, Ω, Ω j ; P, P j ) = 0, I \ {m}, (7) where { } f(c, c j, Ω, Ω j ; P, P j ) = max 0, 1 P 1 Q (c c j ) X (c, c j, Ω, Ω j ; P, P j ) + (8) max{0, ɛ j U(c, c j, Ω, Ω j ; P, P j )} and ɛ j = ɛ(p, P j ) = λ mn (P 1 )λ mn (P 1 )λ mn (P j )λ mn (P 1 j ) > 0; see [4, Prop. 4.3] for detals. It s worth notcng that, f the ellpsods E and E j are far from each other then f(c, c j, Ω, Ω j ; P, P j ) s null and, thus, the quanttes X (c, c j, Ω, Ω j ; P, P j ) and U(c, c j, Ω, Ω j ; P, P j ) do not need to be computed. 5

6 3. Contanment model The dea to nclude an ellpsod wthn a half-space s smlar to that of avodng the overlap between ellpsods. A transformaton s appled to the ellpsod so that t becomes a ball. The same transformaton s then appled to the half-space, whch transforms t nto another halfspace. The problem of ncludng an ellpsod wthn a half-space then becomes the equvalent problem of ncludng a ball wthn a half-space. Consder the ellpsod E = {x R n (x c ) Q P 1 Q (x c ) 1}, where c R n, Q R n n s orthogonal, and P R n n s postve defnte and dagonal. Let T : R n R n be the lnear transformaton defned by T (x) = P 1 Q x. (9) By applyng transformaton T to E, we obtan the unt-radus ball E = {x R n (x P 1 Q c ) (x P 1 Q c ) 1}. Now, consder the half-space H = {x R n w x s}, where w R n, w 0, and s R, and let H be the set obtaned when transformaton T s appled to the half-space H,.e., H = {x R n w Q P 1 x s}. Requrng E H s equvalent to requrng E H. For E to be contaned n H, the center c of E must belong to H, and the dstance between c and the fronter H = {x R n w Q P 1 x = s} of H must be at least one (the radus of the ball E ). Snce the dstance d(c, H ) from c to the fronter of H s gven by these condtons are therefore d(c, H ) = w Q P 1 c s P 1, Q w (w Q P 1 c s) P 1 Q w Snce c = P 1 Q c, condtons (10) can be equvalently wrtten as ( w c s ) P 1 1 and w c s. Q w 1 and w Q P 1 c s. (10) 6

7 4 Incremental packng of ellpsods 4.1 Model algorthm Brefly, the algorthm to pack ellpsods nsde a gven contaner s as follows. At each teraton, a certan number of ellpsods (that were packed n prevous teratons) are already arranged wthn the contaner. Once these ellpsods are packed, they are fxed n ther postons (ther centers and rotatons are fxed). Then, a new group of ellpsods s packed, so that they do not overlap each other and do not overlap wth the ellpsods already fxed. At the k-th teraton of the algorthm, let F k = {1,..., m k 1 } be the set formed by the ndces of the ellpsods already packed and fxed n ther postons and let N k = {m k 1 + 1,..., m k } be the set of ndces of the new ellpsods. In order to pack the new ellpsods, we must ensure that () they are arranged nsde the contaner, () do not overlap each other, and () do not overlap wth the ellpsods already fxed. So, consderng a contaner C R n and the models presented n Sectons 3.1 and 3., at the k-th teraton of the algorthm, we must fnd a soluton to the feasblty problem gven by E C, N k, (11) f(c, c j, Ω, Ω j ; P, P j ) = 0, N k F k, (1) j N k j> where f s as defned n (8). The varables of ths model are c R n and Q R n n for each N k. Notce that c and Q for each F k are constants, snce the ellpsods n F k have already been fxed. 4. Packng strategy The algorthm descrbed n the last secton requres the new ellpsods to be nsde the contaner, not to overlap each other, and not to overlap wth the ellpsods already packed. However, those constrants descrbe a feasblty problem and they do not specfy how the new ellpsods should be packed. Snce the goal s to pack as many ellpsods as possble, the ellpsods should stay tghtly grouped wthn the contaner. An attempt to acheve ths result s to mnmze, n some sense, the heghts of the ellpsods to be packed. The dea s that the new ellpsods become n contact wth other ellpsods already packed, so that the ellpsods are well packed nsde the contaner. Gven an ellpsod E, we defne two heghts assocated wth t: the lower and the upper heght. The lower heght s defned as mn{x n x E} and the upper heght s defned as max{x n x E}, where x n s the n-th component of x. Snce the goal s to mnmze these heghts, we need a smple way to model them. One way of dong ths s to model the upper and lower heghts of an ellpsod by supportng hyperplanes. The dea s to consder hyperplanes that support the ellpsod precsely at the ponts that realze the lower and upper heghts. Consder the half-space S = {x R n w x s}, where w R n and s R, and the ellpsod E = {x R n (x c ) Q P 1 Q (x c ) 1}, where c R n, Q R n n s orthogonal, and P R n n s dagonal and postve defnte. We saw n Secton 3. that, n order to ensure that the ellpsod be contaned n the half-space S, we can smply requre the center of the ellpsod to 7

8 belong to that half-space and the dstance between the center of the ball E and the fronter of the half-space S, obtaned by transformaton T defned n (9), be at least one. To ensure that S supports the ellpsod E, we can just change the mnmum dstance condton and requre t to be exactly one. Therefore, the condtons ( w c s ) P 1 = 1 and w c s (13) Q w guarantee that the hyperplane S supports the ellpsod E. Moreover, f we take w = e n, the n- th standard bass vector, then S wll support the ellpsod E at the pont arg max{x n x E }, and we wll necessarly have s = max{x n x E }. If we take w = e n, then S wll support the ellpsod E at the pont arg mn{x n x E }, and we wll have s = mn{x n x E }. In order to mnmze the upper heght of the ellpsod, we can then consder the problem of mnmzng s subject to (11,1,13) wth w = e n n (13). In an analogous way, n order to mnmze the lower heght of the ellpsod, t s enough to consder the problem of mnmzng s subject to (11,1,13) wth w = e n n (13). As we wll see n Secton 7, experments n the three-dmensonal space show that the packed ellpsod tends to have ts sem-major axs parallel to the upper plane when ts upper heght s mnmzed (the ellpsod s standng ). On the other hand, when the lower heght s mnmzed, the tendency s that the sem-mnor axs remans parallel to the upper plane (the ellpsod s lyng ). To avod ths knd of behavor, whch can result n poor qualty solutons, we can consder the mnmzaton of a convex combnaton of the lower and upper heghts. Let s nf and s sup denote the lower and upper heghts of ellpsod E, respectvely. For a gven ξ [0, 1], we defne an ntermedate heght as ξs nf + (1 ξ)s sup. Snce [c ] n, the n-th component of the center of the ellpsod, s equal to 1 (s nf + s sup), we can wrte s nf = [c ] n s sup. Then, ξs nf + (1 ξ)s sup = ξ[c ] n + (1 ξ)s sup. Hence, to mnmze the ntermedate heght, we can add the varable s sup and the constrants ( e n c s sup) P 1 Q e n = 1 and e n c s sup (14) to the model. For ξ = 1, we have the mnmzaton of the upper heght of the ellpsod beng packed. For ξ = 0, we have the mnmzaton of the lower heght of the ellpsod. For ξ = 1, we have the mnmzaton of [c ] n, the n-th component of the center of the ellpsod (whch we call the mddle heght). Notce that when ξ = 1, the varable s sup and the constrants (14) are not necessary. When N k > 1,.e., when there are more than one ellpsod beng packed at teraton k, we can mnmze the sum of the heghts of the ellpsods: N k ξ[c ] n + (1 ξ)s sup. 8

9 4.3 The solaton constrants In addton to ensurng that the new ellpsods (to be packed) do not overlap each other, we have to make sure that these ellpsods do not overlap wth the ellpsods prevously packed. Thus, the number of pars of ellpsods whose overlappng should be avoded grows as the number of prevously packed ellpsods ncreases. Ths makes the complexty of the evaluaton of the constrants of each subproblem to ncrease, makng each subproblem more and more dffcult to be solved. On the other hand, assumng that a suffcently large number of ellpsods has been packed, t s expected that there s no possblty for the new ellpsods to be n contact wth most of the fxed ellpsods, snce the latter should be surrounded by several other ellpsods. Let N be the set of the new ellpsods and F be the set formed by the ellpsods already packed and that cannot touch the new ellpsods n a feasble soluton. By addng constrants to ensure that the ellpsods n N are suffcently dstant from the ellpsods n F, we can remove the nonoverlappng constrants between these two groups of ellpsods. For ths change n the model to have the desred effect (makng the subproblems smpler), t s clear that the new constrants should be easer than the orgnal non-overlappng constrants. By easy constrants we mean constrants that are smaller n number, defned by smpler functons, and/or nvolve a small number of varables. We wll call these new constrants the solaton constrants. We say that an ellpsod s solated f t s possble to easly nfer that the solaton constrants ensure that the new ellpsods do not overlap wth the ellpsod n queston. We present Fgure 4.1 to llustrate the solaton of ellpsods. Consder the packng of ellpses nsde a rectangle. In Fgure 4.1(a), t s shown some ellpses already packed nsde the rectangle. Now consder the problem of packng a new ellpse. Due to the non-overlappng constrants, ths new ellpse could touch only the blue ellpses. The set F s formed by the green ellpses n Fgure 4.1(a). Now, consder the solaton constrant that requres the new ellpse to le above the lne llustrated n Fgure 4.1(b). Thus, the green ellpses are solated and the orgnal nonoverlappng constrants assocated wth these ellpses can be removed. Because of the smplcty of the solaton constrants, these constrants may solate ellpsods that could touch the new ellpsods n a feasble soluton (as t s the case for some green ellpses n Fgure 4.1(b)). Anyway, t s mportant to pont out that the solaton constrants ensure that the new ellpsods do not overlap wth the solated ellpsods. Even f the solaton constrants are not able to solate all ellpsods of F, the expectaton s that most of these ellpsods are solated and the subproblems have very low numbers of constrants and varables. 5 Complete model and algorthm Consder the case where the contaner C s the followng hypercube wth sde length l: C = {x R n l x l, {1,..., n}}. Ths hypercube can be modeled by n half-spaces, each one correspondng to a dfferent sde of the hypercube. Each sde of the hypercube can then be modeled accordng to the model presented n Secton 3.. Hence, the ncluson of ellpsod E = {x R n (x c ) Q P 1 Q (x c ) 1} 9

10 (a) (b) Fgure 4.1: Illustraton of the solaton constrants. (a) Ellpses already packed and fxed n ther postons. (b) The solaton constrant requres the new ellpse to be packed to le above the hghlghted lne. Only the red ellpses are consdered n the non-overlappng model. wthn C can be modeled by the followng constrants: ( ξe l c l/ ) P 1 Q e l 1, l {1,..., n}, ξ { 1, 1}, ξe l c l/, l {1,..., n}, ξ { 1, 1}. In our experments, we consdered two types of solaton constrants. The frst one constrans the new ellpsods to reman wthn a certan hyperrectangle R centered at u R n and whose sdes have length s > 0, wth the excepton of the sde along the n-th dmenson, whch has nfnty length: R = {x R n s/ x u s/, {1,..., n 1}}. (15) Smlarly to the hypercube model, the ncluson of ellpsod E wthn R can be modeled as: ( ξe l (c u) s/ ) P 1 Q e l 1, l {1,..., n 1}, ξ { 1, 1}, ξe l (c u) s/, l {1,..., n 1}, ξ { 1, 1}. 10

11 The second type of solaton constrant requres the new ellpsods to le wthn the followng half-space H: H = {x R n x n h}, (16) where h R. Therefore, the ncluson of ellpsod E wthn H can be modeled as: ( e n c h ) P 1 Q e n 1 and e n c h. Fnally, the non-overlappng can be modeled as n (7) and the upper heght of ellpsod E as n (14). Now, consder an teraton k of the algorthm. Let F k be the set of ndces of the ellpsods packed n prevous teratons, N k be the set of ndces of the ellpsods that must be packed at ths teraton, and F k F k be the set of ndces of fxed ellpsods that should be consdered n the non-overlappng constrants. After determnng the solaton constrants (parameters s > 0, u R n, and h R) and, consequently, the set F k, the problem that must be solved at ths teraton s the followng: mnmze ξ[c ] n + (1 ξ)s sup (17) N k subject to f(c, c j, Ω, Ω j ; P, P j ) = 0, N k F k, (18) j N k j> ( ξe l c l/ ) P 1 Q e l 1, N k, l {1,..., n}, ξ { 1, 1}, (19) ξe l c l/, N k, l {1,..., n}, ξ { 1, 1}, (0) ( ξe l (c u) s/ ) P 1 Q e 1, N k, l {1,..., n 1}, ξ { 1, 1}, (1) l ξe l (c u) s/, N k, l {1,..., n 1}, ξ { 1, 1}, () ( e n c h ) P 1 Q e 1, N k, (3) n e n c h, N k, (4) ( e n c s sup) P 1 Q e = 1, N k, (5) n e n c s sup, N k. (6) 11

12 Consderng that the problem (17) (6) may be nfeasble or that a local optmzaton solver may fal n fndng a feasble pont dependng on the ntal guess, we apply a mult-start strategy startng up to τ tmes from dfferent ntal guesses. The algorthm stops when, at a gven teraton k, t s not possble to solve the problem (17) (6) wthn τ trals. Therefore, we can summarze the algorthm as follows: Algorthm 1. Input: The contaner C and the lengths of the sem-axes of the ellpsods gven by the matrces {P } =1. Output: m (the number of ellpsods packed) and Q and c for {1,..., m }. Step 1. Let k 0. Step. Let k k + 1 and t 0. Step 3. Let t t + 1. If t > τ, stop. Step 3.1. Determne the set N k. Step 3.. Determne the solaton constrants. Step 3.3. Determne the set F k. Step 3.4. Determne the ntal soluton. Step 3.5. Try to solve the subproblem (17) (6). Step 3.6. Analyze the soluton found. Step 4. If the subproblem was solved, go to Step. Otherwse, go to Step 3. 6 Implementaton detals 6.1 Determnng the solaton constrants and the set F k The hyperrectangle R s defned by u R n and s > 0. The parameter s > 0 can be fxed snce the begnnng of the algorthm, but u must vary at each teraton of Step 3 of Algorthm 1 so that we can fll up the whole contaner wth ellpsods. We decded to choose each coordnate of u unformly random on the nterval [ l/, l/] at Step 3.. Once u s determned, we compute the set F k 0, whch wll be used to determne the second type of solaton constrants (constrants (3) and (3)). Ths s the set of ndces of ellpsods that were packed n prevous teratons of the algorthm and that could perhaps overlap wth an ellpsod that would be contaned n R. Ideally, F 0 k should be the set { F k E nt(r) }. (7) But snce t may be computatonally costly to fnd the set (7), we check for suffcent condtons that guarantee that E nt(r) =. The set F 0 k wll then be formed by ndces F k for whch t was not possble show that E nt(r) =. Hence, F 0 k wll be a (potentally proper) superset of (7). Let a denote the largest sem-axs length of ellpsod E,.e., a = λ max (P 1 ). Let B be the mnmal boundng sphere of E,.e., B = {x R n (x c ) (x c ) a }. 1

13 Fgure 6.1: Projecton of the three-dmensonal set R and the three-dmensonal ellpsod onto the x-y plane. It s easy to verfy whether B nt(r) =. And f B nt(r) =, then E nt(r) =. It may happen that E nt(r) = but B nt(r). In ths case, we verfy whether there exst ξ { 1, 1} and l {1,..., n 1} such that ( ξe l (c u) s/ ) P 1 Q e l 1 and ξe l (c u) s/. (8) If (8) s verfed for some ξ { 1, 1} and l {1,..., n 1}, then one of the sdes of R separates E from R and, therefore, E nt(r) =. Notce that t may be the case that E nt(r) = but none of those condtons could be verfed (and then such an ndex would unnecessarly belong to F k 0 ). Fgure 6.1 shows the projecton onto the x-y plane of an ellpsod n the threedmensonal space and the set R. Although ths ellpsod does not ntersect the nteror of R, none of the condtons above can be verfed. The dashed crcle represent the projecton of the mnmal boundng sphere of the ellpsod. Once the set F k 0 s computed, we are ready to defne the second type of solaton constrants. If F k 0 =, then the second type of solaton constrants s not necessary. Suppose that F k 0. As we see n (3) and (4), these solaton constrants are determned by the parameter h R. Let h 0 be the hghest mddle heght of an ellpsod n F k 0,.e., h 0 = max[c ] n, (9) F k 0 13

14 s γb (a) (b) (c) Fgure 6.: Selecton of the ellpsods to be consdered n the non-overlappng constrants. (a) Fxed ellpsods from the set F k. (b) Frst type of solaton constrants and determnaton of set F k 0 formed by the blue ellpsods. (c) Consderng also the second type of solaton constrants, the set F k s then formed by the red ellpsods. and b denote the largest sem-axs length among the new ellpsods,.e., b = max j N k λ max (P 1 j ). For a gven γ 0, we defne h = h 0 γb. Fnally, we let F k = { F 0 k E nt(h) = }. An llustratve example of the constructon of the set F k s gven n Fgure 6.. The ellpsods n F k, that were packed n prevous teratons, are shown n Fgure 6.(a). The hyperrectangle wth sde length s s hghlghted n Fgure 6.(b). A new ellpsod that s placed nsde ths hyperrectangle can only possbly overlap wth the blue ellpsods, whch therefore form the set F k 0. Once F k 0 s found, the second type of solaton constrants s defned. The hyperplane that determnes the half-space H (see (16)) s placed at a dstance γb from the center of the hghest (n the sense of maxmum mddle heght) ellpsod n F k 0 ; see Fgure 6.(c). Then, a new ellpsod placed nsde the hyperrectangle and above ths hyperplane can only overlap wth the red ellpsods, whch consttute the set F k. 6. Removng unnecessary constrants Let l {1,..., n 1} and ξ { 1, 1}. Consder the pars of constrants (19,0) and (1,) assocated wth l and ξ. Notce that only one par among these two are necessary n the model (17) (6), as one wll necessarly mples the other. Snce the objectve of the model s to mnmze the heght of the ellpsods, they wll be as low as possble from the top ld of the cube. In ths case, the constrants (19,0) assocated wth l = n and ξ = 1 would play no role n the model. We then remove these constrants and check whether they are satsfed when we obtan a soluton to the problem. Some advantages of removng these constrants from the model are that we can easly construct an ntal feasble soluton when the contaner s almost full and the number of constrants are reduced. 14

15 6.3 Defnng the ntal soluton The ntal soluton s defned by the centers and rotaton angles of the ellpsods n N k. Each rotaton angle of the ellpsods s unformly randomly chosen on the nterval [ π, π]. The center of the ellpsods are randomly chosen so that the ellpsod are assuredly nsde the contaner and satsfy the solaton constrants. For each N k, we defne the frst n 1 components of c to be [c ] l = max{ l/ + a, mn{l/ a, u l + β(s/ a )}}, for each l {1,..., n 1}, where β s a random varable that follows a unform dstrbuton on the nterval [ 1, 1]. If F k =, let h = l/. Otherwse, let h be defned as follows: h = max j F k {[c j ] n + a j }. Let r = F k + 1 and suppose that N k = {r, r + 1,..., r + N k 1}. For each N k, we defne the last component of the center of E to be [c ] n = h 1 + a + a j. Ths constructon guarantees that the ntal soluton s feasble: every ellpsod s nsde the contaner, satsfy the solaton constrants, and do not overlap wth any other ellpsod. 6.4 Solvng the subproblems and analyzng the soluton found We solve problem (17) (6) wth the nonlnear programmng solver Algencan [, 6] verson As we saw n Secton 6., after a soluton s returned by the solver, we must check whether t satsfes the constrants (19,0) assocated wth l = n and ξ = 1, snce we removed these constrants from the model. If they are not satsfed, then we declare that the soluton s not feasble. Even f the soluton s feasble, we must check whether ths soluton s reasonable. We say that a soluton s reasonable f t s feasble (t satsfes all constrants of the model (17) (6), ncludng (19,0) assocated wth l = n and ξ = 1), and each of the new packed ellpsods s optmally packed. An ellpsod wth ndex N k s optmally packed f at least one of the followng statements s true: 1. t touches the bottom sde of the contaner (.e., the constrant (19) assocated wth, l = n and ξ = 1 holds wth equalty); j=r. t touches an ellpsod packed n prevous teratons; 3. t touches another optmally packed ellpsod. If the soluton found s reasonable, we declare that the subproblem was solved. Otherwse, we declare that the subproblem was not solved. 15

16 6.5 Reducng the sze of N k Consder the stuaton where, at an teraton k of Algorthm 1, we want to pack N k > 1 ellpsods. Suppose that t s not possble to pack N k ellpsods after the τ trals of Step 3 of Algorthm 1. Ths stuaton naturally occurs when the contaner s almost full of ellpsods. However, t could be the case that t s possble to pack less than N k ellpsods. For example, consderng the contaner s almost full, t may not be possble to pack fve more ellpsods, but two new ellpsods could ft n the contaner. In order to consder ths stuaton and mprove Algorthm 1, we modfy Step 3 n the followng way. When t > τ, we stop Algorthm 1 f and only f N k = 1. If t > τ but N k > 1, we reduce the sze of N k by one unt, let t 0, and contnue agan from Step Objectve Gven ξ [0, 1], the objectve of problem (17) (6) s to mnmze the sum of the heghts of the ellpsods: N k ξ[c ] n + (1 ξ)s sup. When ξ = 1, the above expresson becomes smply [c ] n. N k In ths case, the varables s sup, for N k, and the constrants (5) (6) can be removed from the problem. 7 Numercal experments In our numercal experments, we consdered the problem of packng the maxmum number of three-dmensonal ellpsods wthn a cube. We consdered the non-overlappng model presented n Secton 3.1. We mplemented, n Fortran 90, the model (17) (6) and the optmzaton procedure descrbed n Secton 5. To solve the nonlnear programmng problems, we used Algencan [, 6] verson The models, the optmzaton procedure, and Algencan were compled wth the GNU Fortran compler (GCC) wth the -O3 opton enabled. The tests were run on a machne wth Intel R Xeon R Processor X5650, 8GB of RAM memory, and Ubuntu operatng system. Our computer mplementaton of the method and the solutons reported n ths secton are freely avalable at In our experments, we consdered the two types of solaton constrants descrbed n Secton 5. The frst one constrans the new ellpsods to reman wthn a hyperrectangle wth nfnte heght. The second type of solaton constrant requres the new ellpsods to le above a certan plane parallel to the x-y plane. The solaton constrants depend on some parameters. The frst type of solaton constrant depends on the choce of the lengths of the sdes of the hyperrectangle (parameter s R n (15)). As for the second type, we need to decde at whch pont the plane must pass through (parameter h R n (16)). Ideally, the presence of solaton constrants 16

17 should not affect the qualty of the soluton. Thus, we need to determne what would be good parameters for those constrants. Let b be the largest length of a sem-axs among the new ellpsods to be packed. We shall let s = ηb and h = h 0 γb (where h 0 s gven by (9)) for the factors η and γ varyng n the set {4, 5,..., 10}. Notce that the values of these parameters wll not change durng the executon of Algorthm 1. Another parameter that must be chosen s the sze of the set N k,.e., the number of ellpsods that must be packed at each teraton. We decded to let the sze of ths set be the same for all teratons (unless ths sze s reduced as explaned n Secton 6.5). We consdered sets of szes from 1 to 5. To assess the nfluence of these parameters on the qualty of the soluton, we consdered the packng of ellpsods wth sem-axs lengths 1, 0.75, and 0.5 wthn a cube wth sde length 30. The objectve s to mnmze the mddle heght of the ellpsods (.e., takng ξ = 1/ n (17) and thus mnmzng the sum of the n-th coordnate of the centers of the ellpsods), accordng to Algorthm 1 presented n Secton 5. At each teraton k of Algorthm 1, we use τ = 100,.e., we try to solve the subproblem at most 100 tmes. As explaned n Secton 6.5, f t was not possble to solve the subproblem after τ attempts, we reduce the sze of N k by one unt, and try to solve the subproblem agan wthn τ new attempts. Once the number of ellpsods to be packed s reduced, t s never ncreased agan. Tables 7.1 and 7. show the results we have obtaned when packng the ellpsods one by one, consderng N k = 1 for each teraton k. Each entry n these tables has two numbers and s assocated wth a partcular choce of η and γ. For Table 7.1, each entry shows the number of ellpsods that were packed (left) and the CPU tme n seconds (rght). As expected, the qualty of the soluton mproves as the length of the sde of the hyperrectangle ncreases. On the other hand, the behavor s not clear wth respect to the γ parameter, whch determne the heght of the hyperplane. Ths suggests that even for γ = 4, the hyperplane s low enough not to affect the qualty of the soluton. We can also gauge the mpact of η and γ by checkng whether the solaton constrants were actve at the soluton found at each teraton. Table 7. shows the number of teratons where the frst solaton constrant was actve (left) and the number of teratons where the second solaton constrant was actve (rght). When η = γ = 4, the second type of solaton constrant s actve only n two teratons out of 107, whch s a neglgble amount. For any other combnaton of values for η and γ, the second type of solaton constrant s never actve. Ths suggests that 4 can be a reasonable choce for the value of γ. Nevertheless, the frst type of solaton constrant s actve n a consderable number of teratons. For η = 4, ths constrant s actve around 48% of the teratons. For η = 10, ths fgure drops to 11%. Tables 7.3 and 7.4 show the results when the ellpsods are packed two by two; Tables 7.5 and 7.6 present the results when the ellpsods are packed three at a tme; Tables 7.7 and 7.8 show the results when the ellpsods are packed four at a tme; Tables 7.9 and 7.10 present the results when the ellpsods are packed fve by fve. Let N be the number of ellpsods that are packed at each teraton of the algorthm. We can observe that the CPU tme ncreases when N ncreases. Ths s because the subproblems become harder to solve when there are more ellpsods to pack at the same tme. However, the qualty of the soluton s not consderably mproved when N ncreases; t s almost the same for all N {1,, 3, 4, 5}. 17

18 Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.1: Number of ellpsods packed (left) and CPU tme n seconds (rght) consderng the strategy of packng one ellpsod at a tme. Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.: Number of subproblems n whch the frst type of solaton constrant was actve (left) and number of subproblems n whch the second type of solaton constrant was actve (rght), consderng the strategy of packng one ellpsod at a tme. Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.3: Number of ellpsods packed (left) and CPU tme n seconds (rght) consderng the strategy of packng two ellpsods at a tme. Table 7.1 shows the results when we pack one ellpsod at a tme and mnmze ts mddle heght (ξ = 1/). Now, we also consder the strategy of packng one ellpsod at a tme but mnmzng a dfferent heght. We consder the mnmzaton of the lower (ξ = 1), upper (ξ = 0), and a random heght of the ellpsod at each teraton. For the mnmzaton of the random heght, the value of ξ s determned rght before Step 3.5 of Algorthm 1 and s chosen unformly randomly on the nterval [0, 1]. Table 7.11 shows the results for the mnmzaton of the lower heght. Table 7.1 shows the results for the mnmzaton of the upper heght. Table 7.13 shows 18

19 Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.4: Number of subproblems n whch the frst type of solaton constrant was actve (left) and number of subproblems n whch the second type of solaton constrant was actve (rght), consderng the strategy of packng two ellpsods at a tme. Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.5: Number of ellpsods packed (left) and CPU tme n seconds (rght) consderng the strategy of packng three ellpsods at a tme. Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.6: Number of subproblems n whch the frst type of solaton constrant was actve (left) and number of subproblems n whch the second type of solaton constrant was actve (rght), consderng the strategy of packng three ellpsods at a tme. the results for the mnmzaton of a random heght. We can observe that the qualty of the solutons s much lower than those found n prevous experments n whch the mddle heght was mnmzed. In Fgure 7.1, we show the graphcal representaton of the best soluton found for the mn- 19

20 Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.7: Number of ellpsods packed (left) and CPU tme n seconds (rght) consderng the strategy of packng four ellpsods at a tme. Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.8: Number of subproblems n whch the frst type of solaton constrant was actve (left) and number of subproblems n whch the second type of solaton constrant was actve (rght), consderng the strategy of packng four ellpsods at a tme. Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.9: Number of ellpsods packed (left) and CPU tme n seconds (rght) consderng the strategy of packng fve ellpsods at a tme. mzaton of each knd of heght of the ellpsod. Fgure 7.1(a) represents the soluton obtaned by mnmzng the mddle heght of the ellpsod to be packed. In ths case, 1073 ellpsods were packed. Fgure 7.1(b) represents the soluton found by mnmzng the lower heght of the ellpsod to be packed. In ths case, 1073 ellpsods were packed. We can notce that the sem-mnor axs of the ellpsods tends to be almost perpendcular to the base of the cube (the ellpsods are almost lyng ). In Fgure 7.1(c), we have the soluton wth 1081 ellpsods obtaned by mnmzng the upper heght of the ellpsod. We observe n ths case another trend: the ellp- 0

21 Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.10: Number of subproblems n whch the frst type of solaton constrant was actve (left) and number of subproblems n whch the second type of solaton constrant was actve (rght), consderng the strategy of packng fve ellpsods at a tme. Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.11: Number of ellpsods packed (left) and CPU tme n seconds (rght) consderng the strategy of packng one ellpsod at a tme and mnmzng the lower heght of the ellpsod. Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.1: Number of ellpsods packed (left) and CPU tme n seconds (rght) consderng the strategy of packng one ellpsod at a tme and mnmzng the upper heght of the ellpsod. sods have ther sem-major axes nearly perpendcular to base of the cube (the ellpsods are almost standng ). Fgure 7.1(d) shows the soluton wth ellpsods found by mnmzng a random heght of the ellpsod. Contrary to what occurred n the mnmzaton of the lower and upper heghts, we cannot notce any postonng trend of the ellpsods when we mnmze the mddle or a random heght. They are postoned n a more vared way (they are messer ), whch should have contrbuted n gettng a hgher qualty soluton. We also consder the problem of packng non-dentcal ellpsods wthn a cube. In ths 1

22 Hyperrectangle sde length factor η Hyperplane heght factor γ Table 7.13: Number of ellpsods packed (left) and CPU tme n seconds (rght) consderng the strategy of packng one ellpsod at a tme and mnmzng a random heght of the ellpsod. experment, we chose the length of each sem-axs of each ellpsod to be unformly random on the nterval [0.1, 1]. The ellpsods were packed one at a tme wth ther mddle heghts beng mnmzed. Consderng a cube wth sde length 30 and usng the parameters η = 10, γ = 6, and τ = 100, we were able to pack 3860 ellpsods n h45m. Fgure 7. llustrates ths soluton. Fnally, n order to show that the computatonal cost of the ntroduced strategy scales lnearly wth the number of ellpsods beng packed, we consder the packng of ellpsods wth sem-axs lengths (1, 0.75, 0.5) wthn a cube wth sde length 140. We have chosen to pack one ellpsod at a tme and to mnmze the mddle heght of the ellpsod. We have also chosen η = 10, γ = 4, and τ = Fgure 7.3 shows the packng of 1,16,474 ellpsods. Ths soluton was found n 4d14h3m. 8 Concludng remarks The problem of packng ellpsods n the n-dmensonal space has been tackled through the applcaton of global and local nonlnear optmzaton technques n recent years. In all cases, only small- and medum-szed problems could be solved due to the nonconvexty of the hghly complex consdered models. In the present work, we ntroduced a methodology that uses nonlnear programmng models and methods for solvng small subproblems. In a constructve way, we were able to fnd solutons to packng problems wth a huge number of ellpsods. Assessng the qualty of the obtaned solutons, n the sense measurng n some way how far they are from a global soluton s an open queston that may be addressed n future research. On the other hand, the presented strategy s the frst one based on nonlnear programmng able to delver solutons to that knd of huge ellpsods packng problems.

23 (a) (b) (c) (d) Fgure 7.1: Packng of ellpsods wth sem-axs lengths (1, 0.75, 0.5) wthn a cube wth sde length 30. (a) 1073 ellpsods obtaned by mnmzng the mddle heght of the ellpsod. (b) 1073 ellpsods obtaned by mnmzng the lower heght. (c) 1081 ellpsods obtaned by mnmzng the upper heght. (c) ellpsods obtaned by mnmzng a random heght. 3

24 Fgure 7.: Packng of 3860 ellpsods wth unformly random sem-axs lengths n the nterval [0.1, 1] wthn a cube wth sde length 30. Ths soluton was found by packng ellpsods one by one, mnmzng the mddle heght, and usng η = 10, γ = 6, and τ =

25 Fgure 7.3: Packng of 1,16,474 ellpsods wth sem-axs lengths (1, 0.75, 0.5) wthn a cube wth sde length 140. Ths soluton was found by packng ellpsods one by one, mnmzng the mddle heght, and usng η = 10, γ = 4, and τ =