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POLYGON.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME POLYGON
* Problem :
* *********
* Find the polygon of maximal area, among polygons with nv sides and
* diameter d <= 1
* This is problem 1 in the COPS (Version 2) collection of
* E. Dolan and J. More'
* see "Benchmarking Optimization Software with COPS"
* Argonne National Labs Technical Report ANL/MCS-246 (2000)
* SIF input: Nick Gould, December 2000
* classification OOR2-AN-V-V
* The number of vertices
*IE NV 25 $-PARAMETER
*IE NV 50 $-PARAMETER
*IE NV 75 $-PARAMETER
IE NV 100 $-PARAMETER
* approximation of pi
RF PI/4 ARCTAN 1.0
RM PI PI/4 4.0
* Other useful values
IE 1 1
IA NV-1 NV -1
RI NV NV
RA NV+1 NV 1.0
R* (NV+1)^2 NV+1 NV+1
RD RATR (NV+1)^2 4.0
R/ RATT PI NV
VARIABLES
DO I 1 NV
X R(I) $ polar radius (distance to fixed vertex)
X THETA(I) $ polar angle (measured from fixed direction)
ND
GROUPS
* Objective function: polygon area
N AREA
* Constraints: the thetas are ordered
DO I 1 NV-1
IA I1 I 1
XG O(I) THETA(I1) 1.0 THETA(I) -1.0
ND
* The distances are bounded
DO I 1 NV-1
IA I+1 I 1
DO J I+1 NV
XL D(I,J)
ND
CONSTANTS
DO I 1 NV-1
IA I+1 I 1
DO J I+1 NV
X POLYGON D(I,J) 1.0
ND
BOUNDS
DO I 1 NV-1
XU POLYGON R(I) 1.0
ZU POLYGON THETA(I) PI
ND
XX POLYGON R(NV) 0.0
ZX POLYGON THETA(NV) PI
START POINT
DO I 1 NV-1
RI I I
R- RATRI NV+1 I
R* RATRI RATRI I
R* RATRI RATRI RATR
R* RATTI RATT I
Z POLYGON R(I) RATRI
Z POLYGON THETA(I) RATTI
ND
ELEMENT TYPE
EV SQR X
EV SI R1 R2
EV SI T1 T2
IV SI R1 R2
IV SI T
EV CO R1 R2
EV CO T1 T2
IV CO R1 R2
IV CO T
ELEMENT USES
DO I 1 NV
XT R(I) SQR
ZV R(I) X R(I)
ND
DO I 1 NV-1
IA I+1 I 1
XT S(I) SI
ZV S(I) R1 R(I)
ZV S(I) R2 R(I+1)
ZV S(I) T1 THETA(I+1)
ZV S(I) T2 THETA(I)
ND
DO I 1 NV-1
IA I+1 I 1
DO J I+1 NV
XT C(I,J) CO
ZV C(I,J) R1 R(I)
ZV C(I,J) R2 R(J)
ZV C(I,J) T1 THETA(J)
ZV C(I,J) T2 THETA(I)
ND
GROUP USES
* Objective: minimize - polygon_area:
* - 0.5*sum{i in 1..nv-1} r[i+1]*r[i]*sin(theta[i+1] - theta[i])
DO I 1 NV-1
XE AREA S(I) -0.5
ND
* Distance constraints {i in 1..nv-1,j in i+1..nv}:
* r[i]^2 + r[j]^2 - 2*r[i]*r[j]*cos(theta[j] - theta[i]) <= 1
DO I 1 NV-1
IA I+1 I 1
DO J I+1 NV
XE D(I,J) R(I) 1.0 R(J) 1.0
XE D(I,J) C(I,J) -2.0
ND
OBJECT BOUND
* Solution
*LO SOLUTION -7.7974D-01 $ (NH=25)
*LO SOLUTION -7.7839D-01 $ (NH=50)
*LO SOLUTION -7.7847D-01 $ (NH=100)
*LO SOLUTION -7.7850D-01 $ (NH=200)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS POLYGON
TEMPORARIES
R SINT
R COST
M SIN
M COS
INDIVIDUALS
T SQR
F X * X
G X X + X
H X X 2.0
T SI
R R1 R1 1.0
R R2 R2 1.0
R T T1 1.0 T2 -1.0
A SINT SIN( T )
A COST COS( T )
F R1 * R2 * SINT
G R1 R2 * SINT
G R2 R1 * SINT
G T R1 * R2 * COST
H R1 R2 SINT
H R1 T R2 * COST
H R2 T R1 * COST
H T T - R1 * R2 * SINT
T CO
R R1 R1 1.0
R R2 R2 1.0
R T T1 1.0 T2 -1.0
A SINT SIN( T )
A COST COS( T )
F R1 * R2 * COST
G R1 R2 * COST
G R2 R1 * COST
G T - R1 * R2 * SINT
H R1 R2 COST
H R1 T - R2 * SINT
H R2 T - R1 * SINT
H T T - R1 * R2 * COST
ENDATA