Description
This model solves the Cardinality Constraint Quadratic Knapsack Problem (kQKP) using a SDP confexification methods. The convexification method requires the solution of a semidefinite program. The communication with the SDP solver is done through ASCII files.
References
Small Model of Type : RMIQCP
Category : GAMS Model library
Main file : kqkpsdp.gms includes : runcsdp.inc 50_25.inc
$title SDP Convexifications of the Cardinality Constraint Quadratic Knapsack Problem (KQKPSDP,SEQ=355)
$onText
This model solves the Cardinality Constraint Quadratic Knapsack Problem
(kQKP) using a SDP confexification methods.
The convexification method requires the solution of a semidefinite
program. The communication with the SDP solver is done through ASCII
files.
Plateau M.C., Reformulations quadratiques convexes pour la
programmation quadratique en variables 0-1. PhD thesis,
Conservatoire National des Arts et Metiers, CEDRIC, 2006.
Guignard, M., Extension to Plateau Convexification Method for
Nonconvex Quadratic 0-1 Programs. Tech. rep., The Wharton School, 2008.
Keywords: relaxed mixed integer quadratic constraint programming, quadratic
knapsack problem, confexification methods, semidefinite programming
$offText
$onEchoV > kQKP.awk
/^$/ {} # do nothing for empty lines
!/^$/ {
if (1==startweight) {
printf("\nParameter w(i) weigths /\n");
for (i=1; i<=n; i++) printf("n%d %d\n",i,$i);
printf("$offdelim\n/\n");
startweight = 0;
}
if (1==startprofit) {
printf("Table p(i,i) profits \n$ondelim\n");
for (i=0; i<=n; i++) printf("n%d ",i);
printf("\n"); startprofit = 2; i=1;
}
if (2==startprofit) {
printf("n%d %s\n",i,$0);
if (n==i)
startprofit = 0;
else
i++;
}
if ($2 == "#n:") {
n = $1;
printf("$setglobal n %d\nset i /n1*n%d/;\n", n, n);
}
if ($2 == "#capacity")
printf("scalar cap capacity /%d/;\n", $1);
if ($2 == "#k:")
printf("scalar ncard cardinality /%d/;\n", $1);
if ($1 == "#Profits:") startprofit = 1;
if ($1 == "#Weights:") startweight = 1;
}
$offEcho
$if not set instance $set instance 50_25
$call awk -f kQKP.awk %instance%.inc > kQKP%instance%.gms
$ifE errorLevel<>0 $abort problems with awk
Set i 'knapsack items', dummy / z /;
Alias (i,j);
Parameter
p(i,j) 'profits of item pairs'
w(i) 'weigths of items'
cap 'capacity of knapsack'
ncard 'cardinality of knapsack';
$offListing
$include kQKP%instance%.gms
$onListing
$onText
Setup of SDP problem to get u and v
max sum((i,j), p(i,j)*X(i,j)
s.t. -ncard*x(i) + sum(j, X(i,j)) =e= 0 for all i (SDP1)
X(i,i) = x(i) (SDP2)
sum(i, x(i)) =e= ncard; (SDP3)
sum(i, w(i)*x(i)) =l= cap (SDP4)
z =e= 1 (SDP5)
(z x^t)
(x X ) is PSD
CSDP gets the problem in SDPA format:
(D) max tr(F0*Y)
st tr(Fi*Y) = ci i = 1,2,...,m
Y >= 0
$offText
Set
n / z,#i /
m / k1*k%n%, i1*i%n%, ncard, cap, zdef /
mk(m) / k1*k%n% /
mi(m) / i1*i%n% /
mkmap(m,i) / #mk:#i /
mimap(m,i) / #mi:#i /;
Parameter
c(m) 'rhs'
F(m,*,*) 'constraint matrix'
F0(*,*) 'objective term'
mLE(m) 'SDP constraints with =l='
xvec(m) 'dual solution';
F0(i,j)$(ord(i) > ord(j)) = p(i,j);
F0('z',i) = p(i,i)/2;
* SDP1
F(mk,'z',i)$mkmap(mk,i) = -ncard;
F(mk,i,j)$mkmap(mk,i) = 1 + 1$sameas(i,j);
c(mk) = 0;
* SDP2
F(mi,'z',i)$mimap(mi,i) = -1.0;
F(mi,i,i)$mimap(mi,i) = 2;
c(mi) = 0;
* SDP3
F('ncard','z',i) = 1.0;
c('ncard') = 2*ncard;
* SDP4
F('cap','z',i) = w(i);
c('cap') = 2*cap;
mLE('cap') = yes;
* SDP5
F('zdef','z','z') = 1;
c('zdef') = 1;
execute_unload 'csdpin.gdx' n, m, mLE, c, F, F0;
execute 'gams runcsdp.inc lo=%gams.lo% --strict=1';
abort$errorLevel 'Problems running RunCSDP. Check listing file for details.';
execute_load 'csdpout.gdx', xvec;
Parameter a(i), u(i);
a(i) = sum(mkmap(mk,i), xvec(mk));
u(i) = sum(mimap(mi,i), xvec(mi));
display a, u;
* Simple MIQP model
Binary Variable x(i) 'select item in knapsack';
Variable obj 'objective';
Equation
defobj 'profit of knapsack'
defcobj 'profit of knapsack'
defcap 'capacity limitation of knapsack'
defcard 'cardinality requirement of knapsack';
defcobj.. obj =e= sum(i, p(i,i)*x(i)) + sum((i,j)$(ord(i) > ord(j)), 2*x(i)*p(i,j)*x(j))
- sum(i, 2*a(i)*x(i)*(sum(j, x(j)) - ncard))
- sum(i, 2*(u(i) + 0.001)*x(i)*(x(i) - 1));
defobj.. obj =e= sum(i, p(i,i)*x(i)) + sum((i,j)$(ord(i) > ord(j)), 2*x(i)*p(i,j)*x(j));
defcap.. sum(i, w(i)*x(i)) =l= cap;
defcard.. sum(i, x(i)) =e= ncard;
Model
kQKP / defobj, defcap, defcard /
ckQKP / defcobj, defcap, defcard /;
solve ckQKP max obj using rmiqcp;