<s>
The	O
quadratic	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
(	O
QKP	O
)	O
,	O
first	O
introduced	O
in	O
19th	O
century	O
,	O
is	O
an	O
extension	O
of	O
knapsack	B-Algorithm
problem	I-Algorithm
that	O
allows	O
for	O
quadratic	O
terms	O
in	O
the	O
objective	O
function	O
:	O
Given	O
a	O
set	O
of	O
items	O
,	O
each	O
with	O
a	O
weight	O
,	O
a	O
value	O
,	O
and	O
an	O
extra	O
profit	O
that	O
can	O
be	O
earned	O
if	O
two	O
items	O
are	O
selected	O
,	O
determine	O
the	O
number	O
of	O
items	O
to	O
include	O
in	O
a	O
collection	O
without	O
exceeding	O
capacity	O
of	O
the	O
knapsack	O
,	O
so	O
as	O
to	O
maximize	O
the	O
overall	O
profit	O
.	O
</s>
<s>
Usually	O
,	O
quadratic	B-Algorithm
knapsack	I-Algorithm
problems	I-Algorithm
come	O
with	O
a	O
restriction	O
on	O
the	O
number	O
of	O
copies	O
of	O
each	O
kind	O
of	O
item	O
:	O
either	O
0	O
,	O
or	O
1	O
.	O
</s>
<s>
This	O
special	O
type	O
of	O
QKP	O
forms	O
the	O
0-1	B-Algorithm
quadratic	I-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
,	O
which	O
was	O
first	O
discussed	O
by	O
Gallo	O
et	O
al	O
.	O
</s>
<s>
The	O
0-1	B-Algorithm
quadratic	I-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
is	O
a	O
variation	O
of	O
knapsack	B-Algorithm
problems	I-Algorithm
,	O
combining	O
the	O
features	O
of	O
unbounded	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
,	O
0-1	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
and	O
quadratic	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Specifically	O
,	O
the	O
0	O
–	O
1	O
quadratic	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
has	O
the	O
following	O
form	O
:	O
</s>
<s>
As	O
one	O
might	O
expect	O
,	O
QKP	O
has	O
a	O
wide	O
range	O
of	O
applications	O
including	O
telecommunication	O
,	O
transportation	O
network	O
,	O
computer	B-General_Concept
science	I-General_Concept
and	O
economics	O
.	O
</s>
<s>
Applications	O
of	O
QKP	O
in	O
the	O
field	O
of	O
computer	B-General_Concept
science	I-General_Concept
is	O
more	O
common	O
after	O
the	O
early	O
days	O
:	O
compiler	B-Language
design	O
problem	O
,	O
clique	O
problem	O
,	O
very	O
large	O
scale	O
integration	O
(	O
VLSI	O
)	O
design	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
decision	O
version	O
of	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
(	O
Can	O
a	O
value	O
of	O
at	O
least	O
V	O
be	O
achieved	O
under	O
a	O
restriction	O
of	O
a	O
certain	O
capacity	O
W	O
?	O
)	O
</s>
<s>
The	O
optimization	O
knapsack	B-Algorithm
problem	I-Algorithm
is	O
NP-hard	O
and	O
there	O
is	O
no	O
known	O
algorithm	O
that	O
can	O
solve	O
the	O
problem	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
As	O
a	O
particular	O
variation	O
of	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
the	O
0-1	B-Algorithm
quadratic	I-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
is	O
also	O
NP-hard	O
.	O
</s>
<s>
While	O
no	O
available	O
efficient	O
algorithm	O
exists	O
in	O
the	O
literature	O
,	O
there	O
is	O
a	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
based	O
on	O
dynamic	B-Algorithm
programming	I-Algorithm
and	O
other	O
heuristic	B-Algorithm
algorithms	I-Algorithm
that	O
can	O
always	O
generate	O
“	O
good	O
”	O
solutions	O
.	O
</s>
<s>
While	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
is	O
one	O
of	O
the	O
most	O
commonly	O
solved	O
operation	O
research	O
(	O
OR	O
)	O
problems	O
,	O
there	O
are	O
limited	O
efficient	O
algorithms	O
that	O
can	O
solve	O
0-1	B-Algorithm
quadratic	I-Algorithm
knapsack	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Heuristic	B-Algorithm
algorithms	I-Algorithm
based	O
on	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
dynamic	B-Algorithm
programming	I-Algorithm
can	O
give	O
a	O
relatively	O
“	O
good	O
”	O
solution	O
to	O
the	O
0-1	O
QKP	O
efficiently	O
.	O
</s>
<s>
Problems	O
of	O
such	O
form	O
are	O
difficult	O
to	O
solve	O
directly	O
using	O
standard	O
solvers	O
and	O
thus	O
people	O
try	O
to	O
reformulate	O
it	O
as	O
a	O
linear	B-Algorithm
program	I-Algorithm
using	O
auxiliary	O
variables	O
and	O
constraints	O
so	O
that	O
the	O
problem	O
can	O
be	O
readily	O
solved	O
using	O
commercial	O
packages	O
.	O
</s>
<s>
This	O
reformulates	O
the	O
QKP	O
into	O
a	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
which	O
we	O
can	O
then	O
solve	O
optimally	O
using	O
standard	O
solvers	O
.	O
</s>
<s>
We	O
need	O
to	O
make	O
P	O
a	O
positive	B-Algorithm
semi-definite	I-Algorithm
matrix	I-Algorithm
in	O
order	O
to	O
reformulate	O
a	O
convex	O
function	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
we	O
modify	O
the	O
objective	O
function	O
to	O
be	O
by	O
applying	O
results	O
from	O
linear	O
algebra	O
,	O
where	O
P	O
is	O
a	O
diagonally	B-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
and	O
thus	O
a	O
positive	O
semi-definite	O
.	O
</s>
<s>
George	O
Dantzig	O
proposed	O
a	O
greedy	O
approximation	O
algorithm	O
to	O
unbounded	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
which	O
can	O
also	O
be	O
used	O
to	O
solve	O
the	O
0-1	O
QKP	O
.	O
</s>
<s>
Quadknap	O
is	O
an	O
exact	O
branch-and-bound	B-Algorithm
algorithm	I-Algorithm
proposed	O
by	O
Caprara	O
et	O
al.	O
,	O
where	O
upper	O
bounds	O
are	O
computed	O
by	O
considering	O
a	O
Lagrangian	O
relaxation	O
which	O
approximate	O
a	O
difficult	O
problem	O
by	O
a	O
simpler	O
problem	O
and	O
penalizes	O
violations	O
of	O
constraints	O
using	O
Lagrange	O
multiplier	O
to	O
impost	O
a	O
cost	O
on	O
violations	O
.	O
</s>
<s>
While	O
dynamic	B-Algorithm
programming	I-Algorithm
can	O
generate	O
optimal	O
solutions	O
to	O
knapsack	B-Algorithm
problems	I-Algorithm
,	O
dynamic	B-Algorithm
programming	I-Algorithm
approaches	O
for	O
QKP	O
can	O
only	O
yield	O
a	O
relatively	O
good	O
quality	O
solution	O
,	O
which	O
can	O
serve	O
as	O
a	O
lower	O
bound	O
to	O
the	O
optimal	O
objectives	O
.	O
</s>
<s>
While	O
it	O
runs	O
in	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
,	O
it	O
has	O
a	O
large	O
memory	O
requirement	O
.	O
</s>
<s>
Note	O
that	O
by	O
dynamic	B-Algorithm
programming	I-Algorithm
,	O
the	O
solution	O
to	O
a	O
problem	O
arises	O
from	O
the	O
solution	O
to	O
its	O
smaller	O
sub-problems	O
.	O
</s>
<s>
Redefine	O
to	O
be	O
the	O
current	O
value	O
of	O
the	O
most	O
profitable	O
packing	O
found	O
by	O
the	O
heuristic	B-Algorithm
.	O
</s>
<s>
Researchers	O
have	O
studied	O
0-1	B-Algorithm
quadratic	I-Algorithm
knapsack	I-Algorithm
problems	I-Algorithm
for	O
decades	O
.	O
</s>
<s>
One	O
focus	O
is	O
to	O
find	O
effective	O
algorithms	O
or	O
effective	O
heuristics	B-Algorithm
,	O
especially	O
those	O
with	O
an	O
outstanding	O
performance	O
solving	O
real	O
world	O
problems	O
.	O
</s>
