Description
This model solves the Cardinality Constraint Quadratic Knapsack Problem (kQKP) using a SDP convexification methods. The convexification method requires the solution of a semidefinite program.
Large Model of Type : RMIQCP
Category : GAMS Model library
Main file : kqkpsdp.gms includes : 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 convexification methods.
The convexification method requires the solution of a semidefinite
program.
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)
(X x)
(x^t z) is PSD
$offText
Set n 'SDP variables' / 1*%n%, 0 /;
Set ni(n) / 1*%n% /;
Alias(ni, nj);
Set map(ni,i);
map(ni,i)$(ord(ni)=ord(i)) = yes;
Variables
Xx(n,n) 'PSDMATRIX (X x; x^t z)'
sdpobjvar
;
Parameter
sdpp(n,n) 'p(i,j) but indexed over ni and symmetric'
sdpw(n) 'w(i) but indexed over ni'
;
sdpp(ni,nj) = sum((i,j)$(map(ni,i) and map(nj,j)), p(i,j));
sdpp(ni,nj)$(ord(ni)<ord(nj)) = sdpp(nj,ni);
sdpw(ni) = sum(map(ni,i), w(i));
Equations
sdpobj 'SDP objective function'
sdp1(n) '(SDP1)'
sdp2(n) '(SDP2)'
sdp3 '(SDP3)'
sdp4 '(SDP4)'
;
sdpobj.. sum((ni,nj), (sdpp(ni,nj)+eps) * Xx(ni,nj)) =E= sdpobjvar + eps*Xx('0','0');
sdp1(ni).. -ncard * (Xx(ni,'0') + Xx('0',ni))/2 + sum(nj, Xx(ni,nj) + Xx(nj,ni))/2 =E= 0;
sdp2(ni).. Xx(ni,ni) =e= (Xx(ni,'0') + Xx('0',ni))/2;
sdp3.. sum(ni, Xx('0',ni) + Xx(ni,'0'))/2 =e= ncard;
sdp4.. sum(ni, sdpw(ni) * (Xx(ni,'0') + Xx('0',ni)))/2 =L= cap;
* (SDP5)
Xx.fx('0','0') = 1;
Model sdp / sdpobj, sdp1, sdp2, sdp3, sdp4 /;
option lp = mosek;
Solve sdp max sdpobjvar using lp;
Parameter a(i), u(i);
a(i) = sum(map(ni,i),sdp1.m(ni));
u(i) = sum(map(ni,i),sdp2.m(ni));
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)$(ord(i) > ord(j)), 2*x(i)*p(i,j)*x(j))
- sum(i, a(i)*x(i)*(sum(j, x(j)) - ncard))
- sum(i, (u(i) + 0.001)*x(i)*(x(i) - 1));
defobj.. obj =e= sum(i, p(i,i)*x(i)) + sum((i,j)$(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;