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KISSING.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME KISSING
* Problem: 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. It consists on maximizing the minimum pairwise
* distance between NP points on a sphere in \R^{MDIM}.
* 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 MDIM and NP, that provides a suitable
* set of test problems for evaluating nonlinear programming codes.
* After some algebric manipulations, we can formulate this problem as
*
* Minimize z
*
* subject to
*
* z \geq <x_i, x_j> for all different pair of indices i, j
*
* ||x_i||^2 = 1 for all i = 1,...,NP
*
* The goal is to find an objective value less than 0.5 (This means
* that the NP points stored belong to the sphere and every distance
* between two of them is greater than 1.0).
*
* Obs: the starting point is aleatorally chosen although each
* variable belongs to [-1.,1.].
*
* References:
* [1] "Validation of an Augmented Lagrangian algorithm with a
* Gauss-Newton Hessian approximation using a set of
* Hard-Spheres problems", N. Krejic, J. M. Martinez, M. Mello
* and E. A. Pilotta, Tech. Report RP 29/98, IMECC-UNICAMP,
* Campinas, 1998.
* [2] "Inexact-Restoration Algorithm for Constrained Optimization",
* J. M. Martinez and E. A. Pilotta, Tech. Report, IMECC-UNICAMP,
* Campinas, 1998.
* [3] "Sphere Packings, Lattices and Groups", J. H. Conway and
* N. J. C. Sloane, Springer-Verlag, NY, 1988.
*
*
* SIF input: September 29, 1998
* Jose Mario Martinez
* Elvio Angel Pilotta
*
* classification LQR2-RN-V-V
***********************************************************************
* Number of points: NP >= 12
*IE NP 12 $-PARAMETER
*IE NP 13 $-PARAMETER
*IE NP 14 $-PARAMETER
*IE NP 15 $-PARAMETER
*IE NP 22 $-PARAMETER
*IE NP 23 $-PARAMETER
*IE NP 24 $-PARAMETER
*IE NP 25 $-PARAMETER
*IE NP 26 $-PARAMETER
*IE NP 27 $-PARAMETER
*IE NP 37 $-PARAMETER
*IE NP 38 $-PARAMETER
*IE NP 39 $-PARAMETER
*IE NP 40 $-PARAMETER
*IE NP 41 $-PARAMETER
IE NP 42 $-PARAMETER
* Dimension: MDIM >= 3
IE MDIM 3 $-PARAMETER
*IE MDIM 4 $-PARAMETER
*IE MDIM 5 $-PARAMETER
* Other useful parameters.
IA N- NP -1
IE 1 1
VARIABLES
DO I 1 NP
DO J 1 MDIM
X X(I,J)
OD J
OD I
X Z
GROUPS
XN OBJ Z 1.0
* Inequality constraints.
DO I 1 N-
IA I+ I 1
DO J I+ NP
XL IC(I,J) Z -1.0
ND
* Equality constraints.
DO I 1 NP
XE EC(I)
ND
CONSTANTS
DO I 1 NP
X KISSING EC(I) 1.0
ND
BOUNDS
DO I 1 NP
DO J 1 MDIM
XR KISSING X(I,J)
ND
XR KISSING Z
START POINT
XV KISSING X1,1 -0.10890604
XV KISSING X1,2 0.85395078
XV KISSING X1,3 -0.45461680
XV KISSING X2,1 0.49883922
XV KISSING X2,2 -0.18439316
XV KISSING X2,3 -0.04798594
XV KISSING X3,1 0.28262888
XV KISSING X3,2 -0.48054070
XV KISSING X3,3 0.46715332
XV KISSING X4,1 -0.00580106
XV KISSING X4,2 -0.49987584
XV KISSING X4,3 -0.44130302
XV KISSING X5,1 0.81712540
XV KISSING X5,2 -0.36874258
XV KISSING X5,3 -0.68321896
XV KISSING X6,1 0.29642426
XV KISSING X6,2 0.82315508
XV KISSING X6,3 0.35938150
XV KISSING X7,1 0.09215152
XV KISSING X7,2 -0.53564686
XV KISSING X7,3 0.00191436
XV KISSING X8,1 0.11700318
XV KISSING X8,2 0.96722760
XV KISSING X8,3 -0.14916438
XV KISSING X9,1 0.01791524
XV KISSING X9,2 0.17759446
XV KISSING X9,3 -0.61875872
XV KISSING X10,1 -0.63833630
XV KISSING X10,2 0.80830972
XV KISSING X10,3 0.45846734
XV KISSING X11,1 0.28446456
XV KISSING X11,2 0.45686938
XV KISSING X11,3 0.16368980
XV KISSING X12,1 0.76557382
XV KISSING X12,2 0.16700944
XV KISSING X12,3 -0.31647534
ELEMENT TYPE
EV PROD X Y
EV QUA V
ELEMENT USES
* Inequality constraints.
DO I 1 N-
IA I+ I 1
DO J I+ NP
DO K 1 MDIM
XT A(I,J,K) PROD
ZV A(I,J,K) X X(I,K)
ZV A(I,J,K) Y X(J,K)
ND
* Equality constraints.
DO I 1 NP
DO K 1 MDIM
XT B(I,K) QUA
ZV B(I,K) V X(I,K)
ND
GROUP USES
* Inequality constraints.
DO I 1 N-
IA I+ I 1
DO J I+ NP
DO K 1 MDIM
XE IC(I,J) A(I,J,K)
ND
* Equality constraints.
DO I 1 NP
DO K 1 MDIM
XE EC(I) B(I,K)
ND
OBJECT BOUND
* Solution
*XL SOLUTION 4.47214D-01
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS KISSING
INDIVIDUALS
* Product of 2 elemental variables.
T PROD
F X * Y
G X Y
G Y X
H X Y 1.0
* Square of an elemental variables.
T QUA
F V * V
G V V + V
H V V 2.0
ENDATA