A NOVEL CONTRIBUTION TO THE TAMMES PROBLEM BY THE HYPERBOLIC SMOOTHING METHOD Adilson Elias Xavier, Joao Lauro Dorneles Faco´, Jurair Rosa de Paula Junior Federal University of Rio de Janeiro Av.Horacio Macedo, 2030, Centro de Tecnologia, Cidade Universitaria 21941-914, Rio de Janeiro, RJ - Brazil
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[email protected] Abstract. In this paper the problem of finding the maximum diameter of q equal non-overlapping circles on the surface of a sphere is considered. The packing problem resolution method proposed adopts a smoothing strategy using a special completely differentiable function. The final solution is obtained by solving a sequence of differentiable constrained optimization subproblems, which gradually approach the original problem. A simplified algorithm containing only the essentials of the method is presented. A set of computational experiments is shown. Keywords: Spherical Packing, Tammes Problem, Optimization, Nonlinear Programming, Hyperbolic Smoothing, and Hyperbolic Penalty.
1. Introduction This paper addresses the equal circles packing on the surface of the unitary sphere problem. It corresponds to find the center positions of the q non-overlapping equal circles with the maximum possible diameter. Equivalently this problem corresponds to maximize the minimum distance between each pair of points on the surface of the unitary sphere. In general, the sphere belongs to the ℜ d ( d ≥ 3 ) space. In this paper, the particular case d = 3 is considered. This problem is known as the Tammes problem, due to his investigations about orifices on spherical pollen grains (Tammes, 1930). It is an apparently simple problem, however it is associated to a mathematical model whose resolution presents many vicissitudes. First, it is nondifferentiable, besides that, the objective-function is nonlinear and non-convex. Therefore the problem presents multiple local minima, and computing its global minimum is a very hard task, see (Croft et al., 1991), (Conway et al., 1996) or (Pinter, 2001). To solve the packing of equal circles on the surface of the unitary sphere problem, the approach known as Hyperbolic Smoothing (HS) is employed. This technique was developed through the adaptation of the Hyperbolic Penalty Method (HP) originally introduced by Xavier (1982) to solve the general constrained nonlinear programming problem, see (Xavier, 2001). In the application of the Hyperbolic Smoothing technique, the solution is achieved through the resolution of a sequence of C∞ differentiable problems which gradually approaches the original problem, see for example, (Xavier and Oliveira, 2005) or (Xavier, 2005). This paper is organized as follows. The definition of the circles packing problem is presented in Section 2. The methodology is described in Section 3. The illustrative computational results are presented in Section 4. Brief conclusions are drawn in Section 5. 2. Definition of the Tammes Problem The circles packing on a unitary sphere problem can be formulated as: maximize
r
such that
xi − x j
2
≥ 2 r , i = 1,..., q − 1;
xi ∈ S ; i = 1, ..., q;
1
j = i + 1,..., q;
(P1)
where r is the circles ray, the vectors xi∈ℜ 3 , i = 1, ..., q, correspond to the centers of the circles on the surface S of the unitary sphere B in ℜ 3 .
3. Problem approximation by Hyperbolic Smoothing By the Hyperbolic Smoothing approach, the non-differentiable Euclidean distance between two points in ℜn ,
z ij ( x ) = xi − x j
n
2
∑ (x
k i
=
− x kj
)
2
k =1
is approximated by the hyperbolic function: n
∑ (x
φij ( x,τ ) =
k i
− x kj
)
2
+τ 2 .
k =1
The function φ has the following properties: a) φij (x, τ) > zij(x), ∀ τ > 0; b) c) d)
φij (x, τ) is a strictly crescent function in τ , for all τ > 0; lim φ ij ( x,τ ) = z ij ( x ) ; τ →0
∞ φij is a class C function.
Using the smooth function φ , instead of the original one, we have the following problem: maximize d such as
φij ( x,τ ) ≥ d ;
i = 1, ..., q − 1;
j = i + 1, ..., q;
(P2)
3
∑ (xik )2 = 1;
i = 1, ..., q.
k =1
Certainly this problem is a differentiable version of the original P1 problem, but not exactly equivalent. Solving it does not mean solving the original problem. To overcome this difficulty, the Hyperbolic Smoothing technique solves an infinite sequence of smooth problems, l = 1, 2, ..., +∞, with a strictly decreasing sequence of parameters τl going to zero:
τ l +1 < τ l lim τ l = 0
l → +∞
This way the reduction of the parameter τ forces the sequence of smooth problems to converge asymptotically to the original problem.
3.1 The Hyperbolic Penalty Technique The Hyperbolic Penalty method to solve problem (P2) is a very attractive alternative, for it enables a natural coupling with Hyperbolic Smoothing. A brief presentation of the Hyperbolic
2
Penalty method follows , and may be seen in (Xavier, 1982) and (Xavier, 2001). The method solves the general NLP with inequality constraints:
minimize f ( x ) such that where f : ℜn → ℜ
(P3)
gi ( x ) ≥ 0, i = 1,..., m,
and gi : ℜn → ℜ, i = 1, ..., m.
The name Hyperbolic Penalty comes from the use of the function: 2
1 1 P( y, α , ε ) = − tan α y + tan α y 2 + ε 2 , 2 2 where α ∈ [0, π/2) and ε ≥ 0 ; it is a hyperbole with a horizontal asymptote, and another one with an angle α , having an intercept ε. In a more compact form, the hyperbolic penalty may be put in a different way: P ( y, λ , ε ) = −λy + λ 2 y 2 + ε 2 . To solve problem P3 by the HP technique, a sequence of intermediate subproblems is generated, k = 1, 2, ..., defined as: m
(
)
minimize F ( x, λ , ε ) = f ( x ) + ∑ P gi ( x ), λ k , ε k . k
k
i =1
By this approach, as long as the value of a chosen constraint increases, the value of the penalty decreases asymptotically to zero. As long as this value becomes more negative, i.e. increasing the value of the infeasibility, the penalty value increases asymptotically to the line − y ⋅ tan α . So, it is a penalty that acts coherently in the feasible region, as well in the infeasible one. Differently from the other penalty methods that have only one parameter, the Hyperbolic Penalty has two parameters. The algorithm manages these parameters as follows:
Simplified Hyperbolic Penalty Algorithm Step 1: k=0; introduce initial values for:
x 0 , λ1 and ε 1 ;
Step 2: k := k + 1; solve the unconstrained minimization problem: minimize x F ( x, λ k , ε k ) , starting at xk-1, getting an intermediate point xk; Step 3: Feasibility test: if xk is an infeasible point: go to Step 4; Otherwise: go to Step 5; Step 4: Parameter λ increment: Go to Step 2; Step 5: Parameter ε decrement: Go to Step 2.
λik +1 = r λik , ε ik +1 = b ε ik ,
3
r > 1,
0 < b < 1,
The rationality of the HP algorithm may be described briefly in (Xavier, 1982): “The sequence of subproblems is obtained by the controlled variation of two parameters , λ and ε , in two different phases of the algorithm. Firstly, the parameter λ is increased, implying a significant penalty increment outside the feasible region, and, at the same time, a significant penalty reduction for the feasible region interior points. This process goes on until a feasible point is reached. From there on, λ is fixed, and ε will be decreased sequentially. Thus the interior penalty becomes more irrelevant, keeping the same forbidden level in the exterior region.”
3.2 Connecting Hyperbolic Smoothing with Hyperbolic Penalty Combining HS and HP techniques is a very natural and attractive strategy, for both consider the resolution of an infinite sequence of subproblems. In the smoothing procedure, this sequence is generated by the parameter τ continuous decreasing to zero. In the HP method, at the algorithm second phase, by the parameter ε decreasing to zero. The connection of HS and HP consists in the generation of a unique infinite sequence of smooth problems by the simultaneous decreasing of both parameters. This may be achieved by a simple linear coupling of the smoothing parameter τ with the penalty parameter ε, as performed by the HSCPS Algorithm: . 3.3 Hyperbolic Smoothing Circles Packing on Sphere Algorithm Initialization: Choice of the initial point x0, and the smoothing and penalty parameters: ε1 , τ1. Choice of the reduction rate ρ : 0 < ρ < 1 and the stop rule tolerance: δ. Fix the HP parameter λ: λ = 1; Do k = 1; Main step: Repeat until the stop rule is attained: | f(xk) – f(xk-1) | < δ 10. Solve the differentiable problem P2 with smoothing parameter τ = τk, using the HP method with penalty parameters λ=1 and ε = εk; starting at initial point xk-1, computing the solution xk. 20. Do: τk+1 = ρ⋅τk,
εk+1 = ρ⋅εk,
k := k + 1.
4. Computational results When developing the present methodological proposal implementation - the combination of HS and HP -, the main effort was directed to the harmonization of the two effects, as well, to the choices of the other algorithm parameters. This was empirically achieved by a long process of computing experiments. It is an empirical research to overcome a theoretical gap. The adjusted parameters are: the HS parameter τ initial value in the interval 0.01 ≤ τ ≤ 0.1, the HP parameter ε initial value, ε = 0.0001, and the reduction rate ρ for both parameters at each iteration, ρ = 1/10. The stop rule tolerance δ was set as the fixed value: δ = 1. E-11. The HP algorithm Step 2 consists in the resolution of an unconstrained nonlinear programming problem. The quasi-Newton algorithm with BFGS Hessian matrix update (VA13C, Harwell Library) was employed in the experiments. The programs are coded in FORTRAN, COMPAQ compiler, Visual Fortran, version 6.1. Considering the multiple local minima nature of the circles packing on the unit sphere problem, it was adopted a multi-start strategy by using several initial points.
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Table 1 presents the computational results for packing instances in the range: 31 ≤ q ≤ 50. The columns show: the number of circles, the optimum putative solution rounded to seven decimal digits (SLOANE, 2007), the best solution obtained by the HSCPS algorithm, the ratio between the putative solution and the best one obtained by the HSCPS algorithm, the number of initial points in the multi-start strategy, and the frequency of the best HSCPS solution.
Table 1 – Computational results for packing instances: 31 ≤ q ≤ 50. q
F*
Fsh
F*/Fsh
NT
N-Fsh
31
0.6463457
0.64634570734263
0.999999989
100
10
32
0.6424693
0.64246927572825
1.000000038
100
68
33
0.6222578
0.62225780244392
0.999999996
100
8
34
0.6148425
0.61484252096880
0.999999966
100
17
35
0.6067327
0.60673260749547
1.000000152
100
7
36
0.6045690
0.60456896167342
1.000000063
100
97
37
0.5917897
0.59178969196630
1.000000014
100
7
38
0.5889257
0.58892570087719
0.999999999
100
64
39
0.5762095
0.57620947161290
1.000000049
100
21
40
0.5706802
0.57068016878996
1.000000055
100
6
41
0.5634956
0.56349562016022
0.999999964
100
24
42
0.5597650
0.55976503706401
0.999999934
100
41
43
0.5527949
0.55279496119603
0.999999889
100
6
44
0.5509965
0.55099659108528
0.999999835
100
23
45
0.5399084
0.53990837355349
1.000000049
100
41
46
0.5337899
0.53378990836079
0.999999984
100
65
47
0.5308062
0.53080625201498
0.999999902
100
33
48
0.5304860
0.53048601286428
0.999999976
100
98
49
0.5163497
0.51634972828458
0.999999945
200
2
50
0.5134721
0.51347208467711
1.00000003
200
3
Table 1 shows that, for all instances, the proposed algorithm can obtain the putative optimum value within the specified seven decimal digits precision. Moreover, the number of initial points is relatively small.
5. Conclusions It is possible to observe that the proposal was successful, since the connection of HS with HP techniques, and the latter empirical harmonization of the parameters set produced good numerical results. The desirable robustness and efficiency properties were empirically observed.
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References CONWAY, J.H., HARDIN, R.H. and SLOANE, N.J.A., 1996 “Packing lines, planes, etc.: Packing in Grassmanian Spaces”, Experimental Math., 5(2): 139-159. CROFT, H.T., FALCONER, K.J., GUY, R.K., 1991, “Unsolved Problems in Geometry”, In: Problem Books in Mathematics, v. II, Berlin New York, Springer-Verlag. PINTER, J.D., 2001, “Globally Optimized Spherical Point Arrangements: Model Variants and Illustrative Results”, Annals of Operations Research, 104, pp 213-230. SLOANE, N. J. A. , 2007 Spherical Codes: Nice arrangements of points on a sphere in various dimensions. : http://www.enginemonitoring.org/sphere/index.htm, TAMMES,P.M.L., 1930 “On the origin of number and arrangement of the places of exit on the surface of pollen grains”, Recueil des Travaux Botaniques Néerlandais, 27, 1-84. XAVIER, A.E., 1982, Penalização Hiperbólica: Um Novo Método para Resolução de Problemas de Otimização, M.Sc.Thesis , COPPE/UFRJ, Rio de Janeiro, RJ XAVIER, A.E., 2001, “Hyperbolic Penalty: a new method for nonlinear programming with inequalities”, Intl. Trans. in Op. Res., n. 8, pp. 659-671. XAVIER, A.E., 2005, The Hyperbolic Smoothing Clustering Method, Relatório Técnico 674/05, PESC/COPPE/UFRJ. XAVIER, A.E., OLIVEIRA, A.A.F., 2005, “Optimum Covering of Plane Domes by Circles via Hyperbolic Smoothing Method”, Journal of Global Optimization, v. 31, n. 3, pp. 493-504
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