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KISSING2.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME KISSING2
* Problem: A second formulation of the KISSING NUMBER PROBLEM
*
* Source: This problem is associated to the family of Hard-Spheres
* problem. It belongs to the family of sphere packing problems, a
* class of challenging problems dating from the beginning of the
* 17th century which is related to practical problems in Chemistry,
* Biology and Physics. Given a fixed unit sphere at the origin in R^n,
* the problem consists of arranging a further m unit spheres so that
* sum of the distances to these spheres is as small as possible.
* This problem may be reduced to a nonconvex nonlinear optimization
* problem with a potentially large number of (nonoptimal) points
* satisfying optimality conditions. We have, thus, a class of problems
* indexed by the parameters m and n, that provides a suitable
* set of test problems for evaluating nonlinear programming codes.
* After some algebric manipulations, we can formulate this problem as
*
* m
* Minimize sum <p_i,p_i> - m n
* i=1
*
* subject to
*
* <p_i - p_j, p_i - p_j> >= 4 for all different pair of indices i, j
*
* and
*
* <p_i, p_i> >= 4 for all indices i
*
* as well as n(n-1)/2 normalisation constraints fixing components.
*
* The goal is to find an objective value equal to 0.
*
* [1] "Sphere Packings, Lattices and Groups", J. H. Conway and
* N. J. C. Sloane, Springer-Verlag, NY, 1988.
*
*
* SIF input: Nick Gould, September 2000
* classification QQR2-RN-V-V
***********************************************************************
* Number of points: m
*IE m 24 $-PARAMETER number of points
IE m 25 $-PARAMETER number of points
* Dimension: n
IE n 4 $-PARAMETER dimension of sphere
* Other useful parameters.
IE 1 1
IE 2 2
I- n-1 n 1
RI rm m
RI rn n
R+ RM+N rm rn
R* mn rm rn
RF PI/4 ARCTAN 1.0
RM PI PI/4 4.0
R/ PI/m PI rm
RM 2PI/m PI/m 2.0
VARIABLES
DO I 1 m
DO J 1 n
X P(I,J)
ND
GROUPS
XN OBJ
DO I 1 m
DO J 1 m
XG C(I,J)
ND
CONSTANTS
Z KISSING2 OBJ mn
DO I 1 m
DO J 1 m
X KISSING2 C(I,J) 4.0
ND
BOUNDS
DO I 1 m
DO J 1 n
XR KISSING2 P(I,J)
ND
DO I 2 n
DO J I n
XX KISSING2 P(I,J) 0.0
ND
START POINT
DO I 1 m
RI RI I
R* 2PIi/m 2PI/m RI
R( cos COS 2PIi/m
R( sin SIN 2PIi/m
R+ cos cos cos
R+ sin sin sin
Z KISSING2 P(I,1) cos
DO J 2 n-1
Z KISSING2 P(I,J) sin
ND
Z KISSING2 P(I,n) cos
ELEMENT TYPE
EV PROD1 P
EV PROD2 Q R
IV PROD2 P
ELEMENT USES
DO I 1 m
IA I- I -1
IA I+ I 1
DO J 1 I-
DO K 1 n
XT E(I,J,K) PROD2
ZV E(I,J,K) Q P(I,K)
ZV E(I,J,K) R P(J,K)
OD K
OD J
DO K 1 n
XT E(I,I,K) PROD1
ZV E(I,I,K) P P(I,K)
OD K
DO J I+ m
DO K 1 n
XT E(I,J,K) PROD2
ZV E(I,J,K) Q P(I,K)
ZV E(I,J,K) R P(J,K)
OD K
OD J
OD I
GROUP USES
DO I 1 m
* Objective
DO K 1 n
XE OBJ E(I,I,K)
OD K
* Inequality constraints.
DO J 1 m
DO K 1 n
XE C(I,J) E(I,J,K)
OD K
OD J
OD I
OBJECT BOUND
* Solution
*XL SOLUTION 0.00000D+00 $ n=4, m = 24
*XL SOLUTION 6.48030D+00 $ n=4, m = 25 one of many local solutions
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS KISSING2
INDIVIDUALS
T PROD1
F P * P
G P P + P
H P P 2.0
T PROD2
R P Q 1.0 R -1.0
F P * P
G P P + P
H P P 2.0
ENDATA