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MOSARQP2.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME MOSARQP2
* Problem :
* *********
* A convex quadratic problem, with variable dimensions.
* In this problem, a third of the linear constraints are active at the
* solution.
* Source:
* J.L. Morales-Perez and R.W.H. Sargent,
* "On the implementation and performance of an interior point method for
* large sparse convex quadratic programming",
* Centre for Process Systems Engineering, Imperial College, London,
* November 1991.
* SIF input: Ph. Toint, August 1993.
* minor correction by Ph. Shott, Jan 1995.
* classification QLR2-AN-V-V
* Problem variants: these are distinguished by the triplet ( N, M, COND ),
* where: - N (nb of variables) must be a multiple of 3
* and have an integer square root
* - M (nb of constraints) must be at least sqrt(N)
* and at most N - sqrt(N)
* - COND (problem conditioning) is a positive integer
* Except for the first, the instances suggested are those used by Morales
* and Sargent.
*IE N 36 $-PARAMETER
*IE M 10 $-PARAMETER
*RE COND 2.0 $-PARAMETER
*IE N 100 $-PARAMETER original value
*IE M 10 $-PARAMETER original value
*RE COND 3.0 $-PARAMETER original value
*IE N 900 $-PARAMETER
*IE M 30 $-PARAMETER
*RE COND 1.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 30 $-PARAMETER
*RE COND 2.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 30 $-PARAMETER
*RE COND 3.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 60 $-PARAMETER
*RE COND 1.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 60 $-PARAMETER
*RE COND 2.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 60 $-PARAMETER
*RE COND 3.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 90 $-PARAMETER
*RE COND 1.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 90 $-PARAMETER
*RE COND 2.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 90 $-PARAMETER
*RE COND 3.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 120 $-PARAMETER
*RE COND 1.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 120 $-PARAMETER
*RE COND 2.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 120 $-PARAMETER
*RE COND 3.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 300 $-PARAMETER
*RE COND 1.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 300 $-PARAMETER
*RE COND 2.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 300 $-PARAMETER
*RE COND 3.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 600 $-PARAMETER
*RE COND 1.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 600 $-PARAMETER
*RE COND 2.0 $-PARAMETER
*IE N 900 $-PARAMETER
*IE M 600 $-PARAMETER
*RE COND 3.0 $-PARAMETER
*IE N 2500 $-PARAMETER
*IE M 700 $-PARAMETER
*RE COND 1.0 $-PARAMETER
*IE N 2500 $-PARAMETER
*IE M 700 $-PARAMETER
*RE COND 2.0 $-PARAMETER
IE N 2500 $-PARAMETER
IE M 700 $-PARAMETER
RE COND 3.0 $-PARAMETER
* Constants
IE 1 1
IE 2 2
IE 3 3
IA N-1 N -1
IA N-2 N -2
RI RN-1 N-1
RI RN N
IA M-1 M -1
RA RNP RN 0.1
R( RRTN SQRT RNP
IR RTN RRTN
IA RTN-1 RTN -1
IA RTN-2 RTN -2
IA RTN+1 RTN 1
I+ 2RTN-1 RTN RTN-1
I+ 2RTN RTN RTN
I- M-RTN+1 M RTN-1
* Determination of the quadratic center
* according to the second proposal of Morales and Sargent.
* The proportion of 1.0 in this vector is the proportion of linear
* constraints active at the solution.
DO I 1 N-2
DI I 3
IA I+1 I 1
IA I+2 I 2
AE XC(I) -1.0
AE XC(I+1) -1.0
AE XC(I+2) 1.0
OD I
AE XC(N-1) -1.0
AE XC(N) 1.0
* Determination of the vector Y
IE NNZ 10 $ number of nonzeros in Y (as in paper)
RE Y1 -0.3569732 $ values of the NNZ nonzero components
RE Y2 0.9871576 $ of Y
RE Y3 0.5619363
RE Y4 -0.1984624
RE Y5 0.4653328
RE Y6 0.7364367
RE Y7 -0.4560378
RE Y8 -0.6457813
RE Y9 -0.0601357
RE Y10 0.1035624
RE NZ1 0.68971452 $ positions of the NNZ nonzero components
RE NZ2 0.13452678 $ (as a fraction of the dimension)
RE NZ3 0.51234678
RE NZ4 0.76591423
RE NZ5 0.20857854
RE NZ6 0.85672348
RE NZ7 0.04356789
RE NZ8 0.44692743
RE NZ9 0.30136413
RE NZ10 0.91367489
* Compute the integer nonzero positions in Y
* and YN2, the square of norm(Y)
RE YN2 0.0
DO I 1 NNZ
A* RKI NZ(I) RN
AA K(I) RKI 1.1
A* TMP Y(I) Y(I)
R+ YN2 YN2 TMP
OD I
* Set some useful coefficients
RD -2/YN2 YN2 -2.0
R* 4/YN4 -2/YN2 -2/YN2
* Determination of the diagonal on which the Hessian of the objective
* is constructed.
DO I 1 N
IA I-1 I -1
RI RI-1 I-1
R/ TMP RI-1 RN-1
R* TMP TMP COND
A( D(I) EXP TMP
OD I
* Compute D * y, y^T * xc, y^T * D * xc and y^T * D * y
RE YDY 0.0
RE YXC 0.0
RE YDXC 0.0
DO I 1 NNZ
A= RKI K(I)
IR KI RKI
A* DY(I) Y(I) D(KI)
A* TMP DY(I) Y(I)
R+ YDY YDY TMP
A* TMP Y(I) XC(KI)
R+ YXC YXC TMP
A* TMP DY(I) XC(KI)
R+ YDXC YDXC TMP
OD I
R* AA -2/YN2 YXC
R* DD 4/YN4 YDY
R* BB DD YXC
R* CC -2/YN2 YDXC
R+ BB+CC BB CC
RM DD/2 DD 0.5
* Compute C, the quadratic's gradient at the origin
DO I 1 N
A* C(I) D(I) XC(I)
OD I
DO I 1 NNZ
A= RKI K(I)
IR KI RKI
A* TMP DY(I) AA
A+ C(KI) C(KI) TMP
A* TMP Y(I) BB+CC
A+ C(KI) C(KI) TMP
OD I
VARIABLES
DO I 1 N
X X(I)
OD I
GROUPS
* Objective linear coefficients
DO I 1 N
ZN OBJ X(I) C(I)
OD I
* The matrix A of the linear constraints consists of the M first lines
* of the matrix corresponding to the discretized 5-points Laplacian
* operator in the unit square.
XG CS(1) X(1) 4.0
XG CS(1) X(RTN+1) -1.0 X(2) -1.0
DO I 2 RTN-1
IA I+1 I 1
IA I-1 I -1
I+ I+RTN I RTN
XG CS(I) X(I) 4.0 X(I+RTN) -1.0
XG CS(I) X(I-1) -1.0 X(I+1) -1.0
OD I
XG CS(RTN) X(RTN) 4.0
XG CS(RTN) X(RTN-1) -1.0 X(2RTN) -1.0
I= JS RTN
DO J RTN+1 M-RTN+1
DI J RTN
IA J+1 J 1
I+ JS J RTN-1
IA JS-1 JS -1
I- J-RTN J RTN
I+ J+RTN J RTN
XG CS(J) X(J) 4.0 X(J+1) -1.0
XG CS(J) X(J-RTN) -1.0 X(J+RTN) -1.0
DO I J+1 JS-1
IA I+1 I 1
IA I-1 I -1
I+ I+RTN I RTN
I- I-RTN I RTN
XG CS(I) X(I) 4.0
XG CS(I) X(I-1) -1.0 X(I+1) -1.0
XG CS(I) X(I-RTN) -1.0 X(I+RTN) -1.0
OD I
I+ JS+RTN JS RTN
I- JS-RTN JS RTN
XG CS(JS) X(JS) 4.0 X(JS-1) -1.0
XG CS(JS) X(JS-RTN) -1.0 X(JS+RTN) -1.0
OD J
IA K JS 1
DO I K M
DI I M
IA K+1 K 1
I+ K+RTN K RTN
I- K-RTN K RTN
XG CS(K) X(K) 4.0 X(K+1) -1.0
XG CS(K) X(K-RTN) -1.0 X(K+RTN) -1.0
OD I
IA K K 1
DO I K M
IA I+1 I 1
IA I-1 I -1
I+ I+RTN I RTN
I- I-RTN I RTN
XG CS(I) X(I) 4.0
XG CS(I) X(I-1) -1.0 X(I+1) -1.0
XG CS(I) X(I-RTN) -1.0 X(I+RTN) -1.0
OD I
CONSTANTS
* The constraints constants are computed as A*x0 - p, where
* both x0 and p are set to 0.5 * e
X MOSARQP2 CS(1) 0.5
X MOSARQP2 CS(RTN) 0.5
I= K RTN+1
DO J RTN+1 M-RTN+1
DI J RTN
IA K K 1
DO I 1 RTN-2
IA K K 1
X MOSARQP2 CS(K) -0.5
OD I
IA K K 1
OD J
IA K K 1
DO J K M
X MOSARQP2 CS(J) -0.5
OD J
START POINT
XV MOSARQP2 'DEFAULT' 0.5
ELEMENT TYPE
EV SQ X
EV 2PR X Y
ELEMENT USES
* The elements corresponding to the squre variables
DO I 1 N
XT XSQ(I) SQ
ZV XSQ(I) X X(I)
OD I
* The mixed products corresponding to the nonzero entries of Y
DO I 1 NNZ
A= RKI K(I)
IR KI RKI
IA I-1 I -1
DO J 1 I-1
A= RKJ K(J)
IR KJ RKJ
XT P(I,J) 2PR
ZV P(I,J) X X(KI)
ZV P(I,J) Y X(KJ)
OD J
OD I
GROUP USES
* The diagonal elements
DO I 1 N
AM TMP D(I) 0.5
ZE OBJ XSQ(I) TMP
OD I
* The elements corresponding to the nonzero entries of Y
DO I 1 NNZ
A= RKI K(I)
IR KI RKI
IA I-1 I -1
DO J 1 I-1
A* TMP DY(I) Y(J)
R* WIJ TMP -2/YN2
A* TMP DY(J) Y(I)
R* TMP TMP -2/YN2
R+ WIJ WIJ TMP
A* TMP Y(I) Y(J)
R* TMP TMP DD
R+ WIJ WIJ TMP
ZE OBJ P(I,J) WIJ
OD J
A* TMP DY(I) Y(I)
R* WII TMP -2/YN2
A* TMP Y(I) Y(I)
R* TMP TMP DD/2
R+ WII WII TMP
ZE OBJ XSQ(KI) WII
OD I
OBJECT BOUND
* Solution
*LO SOLTN( 36, 10,2) -35.69811798
*LO SOLTN( 900, 30,1) -509.8245900
*LO SOLTN( 900, 30,2) -950.8404853
*LO SOLTN( 900, 30,3) -1896.596722
*LO SOLTN( 900, 60,1) -504.3600140
*LO SOLTN( 900, 60,2) -945.1134463
*LO SOLTN( 900, 60,3) -1890.602184
*LO SOLTN( 900, 90,1) -498.9518964
*LO SOLTN( 900, 90,2) -939.2704526
*LO SOLTN( 900, 90,3) -1884.291256
*LO SOLTN( 900,120,1) -493.5058050
*LO SOLTN( 900,120,2) -933.1963138
*LO SOLTN( 900,120,3) -1877.513644
*LO SOLTN( 900,300,1) -457.1185630
*LO SOLTN( 900,300,2) -887.3869230
*LO SOLTN( 900,300,3) -1819.655008
*LO SOLTN( 900,600,1) -377.5813314
*LO SOLTN( 900,600,2) -755.0919955
*LO SOLTN( 900,600,3) -1597.482277
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS MOSARQP2
INDIVIDUALS
T SQ
F X * X
G X X + X
H X X 2.0
T 2PR
F X * Y
G X Y
G Y X
H X Y 1.0
ENDATA