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BLOWEYB.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME BLOWEYB
* Problem :
* *********
* A nonconvex quadratic program proposed by
* James Blowey (University of Durham)
* Given function v(s) and f(s) = v(s) + A(inv) v(s), s in [0,1],
* minimize
* (u(s) - v(s))(trans) ( A + A(inv) ) u(s) - (u(s) - v(s))(trans)f(s)
* where
* u(s) in [-1,1] and int[0,1] u(s) ds = int[0,1] v(s) ds
* and A is the
* "- Laplacian with Neumann boundary conditions on a uniform mesh"
* The troublesome term A(inv) u(s) is replaced by the additional
* variable w(s) and the constraint A w(s) = u(s)
* The function v(s) is chosen to be
0 a b c d 1
* 1 ---- ----
* \ /
* \ /
* \ /
* -1 ---
* Thus the problem is formulated as the nonconvex QP
* minimize
* u(s) (trans) A u(s) + u(s) (trans) w(s) -
* v(s)(trans) A u(s) - 2.0 v(s)(trans) w(s) -
* u(s)(trans) v(s) + constant (ignored)
* subject to A w(s) = u(s),
* u(s) in [-1,1],
* and int[0,1] u(s) ds = 1 + a + b - c - d
* Case B: a = 0.1, b = 0.4, c = 0.6 and d = 0.9.
* Source: a simplification of
* J.F. Blowey and C.M. Elliott,
* "The Cahn-Hilliard gradient theory for phase separation with
* non-smooth free energy Part II: Numerical analysis",
* European Journal of Applied Mathematics (3) pp 147-179, 1992.
* SIF input: Nick Gould, August 1996
* classification QLR2-MN-V-V
* The number of discretization intervals
*IE N 10 $-PARAMETER n = 22, m = 12
*IE N 100 $-PARAMETER n = 202, m = 102
*IE N 1000 $-PARAMETER n = 2002, m = 1002
IE N 2000 $-PARAMETER n = 4002, m = 2002
* Other useful values
IE 0 0
IE 1 1
IE 2 2
IE 3 3
IE 5 5
IE 9 9
IE 10 10
RE ONE 1.0
RE -ONE -1.0
RE TWO 2.0
RE -TWO -2.0
RI RN N
R* N**2 RN RN
IA N-1 N -1
R/ 1/N**2 ONE N**2
R/ -1/N**2 -ONE N**2
RM -2/N**2 -1/N**2 2.0
RM 2N**2 N**2 2.0
RM -2N**2 N**2 -2.0
I/ N/10 N 10
I/ N/5 N 5
I= NA N/10
RI A NA
R/ A A RN
IA NA+1 NA 1
I* NB N/5 2
RI B NB
R/ B B RN
IA NB+1 NB 1
I* NC N/5 3
RI C NC
R/ C C RN
IA NC+1 NC 1
I* ND N/10 9
RI D ND
R/ D D RN
IA ND+1 ND 1
R= INT ONE
R+ INT INT A
R+ INT INT B
R- INT INT C
R- INT INT D
R* INT INT RN
* The values of v
DO I 0 NA
AE V(I) 1.0
ND
R- STEP B A
R* STEP STEP RN
R/ STEP TWO STEP
DO I NA+1 NB
I- J I NA
RI RJ J
R* VAL RJ STEP
R- VAL ONE VAL
A= V(I) VAL
ND
DO I NB+1 NC
AE V(I) -1.0
ND
R- STEP D C
R* STEP STEP RN
R/ STEP TWO STEP
DO I NC+1 ND
I- J I NC
RI RJ J
R* VAL RJ STEP
R+ VAL -ONE VAL
A= V(I) VAL
ND
DO I ND N
AE V(I) 1.0
ND
VARIABLES
DO I 0 N
X U(I)
X W(I)
ND
GROUPS
* Objective function terms
DO I 0 N
A* VAL V(I) -1/N**2
ZN OBJ U(I) VAL
A* VAL V(I) -2/N**2
ZN OBJ W(I) VAL
ND
A- VAL V(1) V(0)
ZN OBJ U(0) VAL
DO I 1 N-1
IA I+1 I 1
IA I-1 I -1
AM VAL V(I) -2.0
A+ VAL VAL V(I-1)
A+ VAL VAL V(I+1)
ZN OBJ U(I) VAL
ND
A- VAL V(N-1) V(N)
ZN OBJ U(N) VAL
* Integral constraint using the trapezoidal rule
XE INT U(0) 0.5
* A w(s) = u(s) constraints
XE CON(0) U(0) 1.0 U(1) -1.0
ZE CON(0) W(0) -1/N**2
DO I 1 N-1
IA I+1 I 1
IA I-1 I -1
XE CON(I) U(I) 2.0
XE CON(I) U(I+1) -1.0 U(I-1) -1.0
ZE CON(I) W(I) -1/N**2
XE INT U(I) 1.0
ND
XE CON(N) U(N) 1.0 U(N-1) -1.0
ZE CON(N) W(N) -1/N**2
XE INT U(N) 0.5
CONSTANTS
ZE BLOWEYB INT 0.2 INT
BOUNDS
XR BLOWEYB 'DEFAULT'
DO I 0 N
XL BLOWEYB U(I) -1.0
XU BLOWEYB U(I) 1.0
ND
START POINT
XV BLOWEYB 'DEFAULT' 0.0
DO I 0 N
ZV BLOWEYB U(I) V(I)
ND
ELEMENT TYPE
EV SQ Z
EV PROD X Y
ELEMENT USES
DO I 0 N
XT C(I) PROD
ZV C(I) X U(I)
ZV C(I) Y W(I)
ND
DO I 0 N-1
IA I+1 I 1
XT D(I) SQ
ZV D(I) Z U(I)
XT O(I) PROD
ZV O(I) X U(I)
ZV O(I) Y U(I+1)
ND
XT D(N) SQ
ZV D(N) Z U(N)
GROUP USES
ZE OBJ O(0) -TWO
ZE OBJ D(0) ONE
DO I 1 N-1
ZE OBJ O(I) -TWO
ZE OBJ D(I) TWO
ND
ZE OBJ D(N) ONE
DO I 0 N
ZE OBJ C(I) 1/N**2
ND
OBJECT BOUND
* Solution
*XL SOLUTION -4.93517D+02 $ N = 10
*XL SOLUTION -5.30009D+03 $ N = 100
*XL SOLUTION -5.33674D+04 $ N = 1000
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS BLOWEYB
INDIVIDUALS
T SQ
F Z * Z
G Z Z + Z
H Z Z 2.0
T PROD
F X * Y
G X Y
G Y X
H X Y 1.0
ENDATA