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MINPERM.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME MINPERM
* Problem :
* *********
* Minimize the permanent of a doubly stochastic matrix.
* Source: ??
* SIF input: N. Gould and Ph. Toint, December 1990.
* minor correction by Ph. Shott, Jan 1995.
* classification LQR2-AN-V-V
* Size of matrix
*IE N 2 $-PARAMETER
*IE N 3 $-PARAMETER
*IE N 4 $-PARAMETER
*IE N 5 $-PARAMETER original value
*IE N 6 $-PARAMETER
*IE N 7 $-PARAMETER
*IE N 8 $-PARAMETER
*IE N 9 $-PARAMETER
IE N 10 $-PARAMETER
* Define constants
IE 0 0
IE 1 1
IE 2 2
IA N+1 N 1
* Compute the number of sub-permanents
IE 2**N 1
DO I 1 N
I- N-I+1 N+1 I
IA I-1 I - 1
RI R2**N 2**N
AA S(N-I+1) R2**N 0.1
AA T(I-1) R2**N 0.1
I* 2**N 2**N 2
ND
IA N-1 N - 1
IA 2**N-1 2**N - 1
VARIABLES
* Sub permanents
DO M 1 N-1
A= RK1 T(M)
IR K1 RK1
IM K2 K1 2
IA K1 K1 1
IA K2 K2 - 1
DO K K1 K2
X P(K)
ND
* Entries in the doubly stochastic matrix
DO I 1 N
DO J 1 N
X A(I,J)
ND
GROUPS
* Define objective function group
XN OBJ P(2**N-1) 1.0
* Linear terms in the sub-permanent constraints
DO M 1 N-1
A= RK1 T(M)
IR K1 RK1
IM K2 K1 2
IA K1 K1 1
IA K2 K2 - 1
DO K K1 K2
XE PE(K) P(K) - 1.0
ND
* Doubly stochastic matrix constraints
DO I 1 N
DO J 1 N
XE R(I) A(I,J) 1.0
XE C(J) A(I,J) 1.0
ND
CONSTANTS
* Doubly stochastic matrix constraints
DO I 1 N
X MINPERM R(I) 1.0 C(I) 1.0
ND
BOUNDS
* Entries in the doubly stochastic matrix
DO I 1 N
DO J 1 N
XU MINPERM A(I,J) 1.0
ND
START POINT
ELEMENT TYPE
EV 2PR A P
ELEMENT USES
* Set up the elements associated with sub-permanent constraint K
DO M 1 N-1
A= RK1 T(M)
IR K1 RK1
IM K2 K1 2
IA K1 K1 1
IA K2 K2 - 1
DO K K1 K2
IE ID 0
IE PT 1
I= KK K
* Construct the I-th component of the binary representation of K.
DO I 1 N
A= SI S(I)
IR ISI SI
I/ BI KK ISI
I+ ID ID BI
I* BISI BI ISI
I- KK KK BISI
RI RI I
AA RNZ(PT) RI 0.1
I+ PT PT BI
OD I
* Associate elements with nonzero entries in the binary string
* This corresponds to finding the sub-permanents which occur
* in the usual expansion of the sub-permanent in terms of its
* sub-sub-permanents.
I= I1 0
I= I2 1
IA ID-2 ID - 2
DO I 1 ID-2
I= I1 ID
I= I2 0
OD I
DO I 1 I1
A= RJ RNZ(I)
IR J RJ
A= SI S(J)
IR ISI SI
I- IPP K ISI
XT E(K,I) 2PR
ZV E(K,I) A A(ID,J)
ZV E(K,I) P P(IPP)
OD I
DO I 1 I2
A= RJ RNZ(1)
IR J RJ
A= RJJ RNZ(2)
IR JJ RJJ
XT E(K,1) 2PR
ZV E(K,1) A A(2,J)
ZV E(K,1) P A(1,JJ)
XT E(K,2) 2PR
ZV E(K,2) A A(2,JJ)
ZV E(K,2) P A(1,J)
OD I
RI RD ID
AA D(K) RD 0.1
ND
GROUP USES
DO M 1 N-1
A= RK1 T(M)
IR K1 RK1
IM K2 K1 2
IA K1 K1 1
IA K2 K2 - 1
DO K K1 K2
A= RD D(K)
IR ID RD
DO I 1 ID
XE PE(K) E(K,I)
ND
OBJECT BOUND
LO MINPERM 0.0
* Solution
*LO SOLTN(2) 5.0D-1
*LO SOLTN(3) 2.22222222D-1
*LO SOLTN(4) 9.375-2
*LO SOLTN(5) 3.84D-2
*LO SOLTN(6) 1.54321098D-2
*LO SOLTN(7) 6.11989902D-3
*LO SOLTN(8) 2.40325927D-3
*LO SOLTN(9) 9.36656708D-4
*LO SOLTN(10) 3.6288D-4
*LO SOLTN(11) 1.39905948D-4
*LO SOLTN(12) 5.37232170D-5
*LO SOLTN(13) 2.05596982D-5
*LO SOLTN(14) 7.84541375D-6
*LO SOLTN(15) 2.98628137D-6
*LO SOLTN(16) 1.13422671D-6
*LO SOLTN(17) 4.29968709D-7
*LO SOLTN(18) 1.62718123D-7
*LO SOLTN(19) 6.14859946D-8
*LO SOLTN(20) 2.32019615D-8
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS MINPERM
INDIVIDUALS
* product of A and P
T 2PR
F A * P
G A P
G P A
H A P 1.0
ENDATA