<s>
In	O
mathematical	O
optimization	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
any	O
of	O
a	O
family	O
of	O
algorithms	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Variants	O
of	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
also	O
solve	O
more	O
general	O
problems	O
with	O
linear	B-Algorithm
inequality	I-Algorithm
constraints	I-Algorithm
and	O
nonlinear	B-Algorithm
objective	O
functions	O
;	O
there	O
are	O
criss-cross	B-Algorithm
algorithms	I-Algorithm
for	O
linear-fractional	B-Algorithm
programming	I-Algorithm
problems	O
,	O
quadratic-programming	B-Algorithm
problems	I-Algorithm
,	O
and	O
linear	O
complementarity	O
problems	O
.	O
</s>
<s>
Like	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
of	O
George	O
B	O
.	O
Dantzig	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
not	O
a	O
polynomial-time	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Both	O
algorithms	O
visit	O
all2Dcorners	O
of	O
a	O
(	O
perturbed	O
)	O
cube	O
in	O
dimensionD	O
,	O
the	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
(	O
after	O
Victor	O
Klee	O
and	O
George	O
J	O
.	O
Minty	O
)	O
,	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
.	O
</s>
<s>
However	O
,	O
when	O
it	O
is	O
started	O
at	O
a	O
random	O
corner	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
on	O
average	O
visits	O
onlyD	O
additional	O
corners	O
.	O
</s>
<s>
Thus	O
,	O
for	O
the	O
three-dimensional	O
cube	O
,	O
the	O
algorithm	O
visits	O
all8	O
corners	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
and	O
exactly3	O
additional	O
corners	O
onaverage	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
was	O
published	O
independently	O
by	O
Tamas	O
Terlaky	O
and	O
by	O
Zhe-Min	O
Wang	O
;	O
related	O
algorithms	O
appeared	O
in	O
unpublished	O
reports	O
by	O
other	O
authors	O
.	O
</s>
<s>
In	O
linear	B-Algorithm
programming	I-Algorithm
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
pivots	O
between	O
a	O
sequence	O
of	O
bases	O
but	O
differs	O
from	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
first	O
finds	O
a	O
(	O
primal	O
-	O
)	O
feasible	O
basis	O
by	O
solving	O
a	O
"	O
phase-one	O
problem	O
"	O
;	O
in	O
"	O
phase	O
two	O
"	O
,	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
pivots	O
between	O
a	O
sequence	O
of	O
basic	O
feasible	O
solutions	O
so	O
that	O
the	O
objective	O
function	O
is	O
non-decreasing	O
with	O
each	O
pivot	O
,	O
terminating	O
with	O
an	O
optimal	O
solution	O
(	O
also	O
finally	O
finding	O
a	O
"	O
dual	O
feasible	O
"	O
solution	O
)	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
simpler	O
than	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
because	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
only	O
has	O
one	O
phase	O
.	O
</s>
<s>
Its	O
pivoting	O
rules	O
are	O
similar	O
to	O
the	O
least-index	B-Algorithm
pivoting	I-Algorithm
rule	I-Algorithm
of	I-Algorithm
Bland	I-Algorithm
.	O
</s>
<s>
Bland	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
uses	O
only	O
signs	O
of	O
coefficients	O
rather	O
than	O
their	O
(	O
real-number	O
)	O
order	O
when	O
deciding	O
eligible	O
pivots	O
.	O
</s>
<s>
Bland	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
selects	O
an	O
entering	O
variables	O
by	O
comparing	O
values	O
of	O
reduced	O
costs	O
,	O
using	O
the	O
real-number	O
ordering	O
of	O
the	O
eligible	O
pivots	O
.	O
</s>
<s>
Unlike	O
Bland	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
"	O
purely	O
combinatorial	O
"	O
,	O
selecting	O
an	O
entering	O
variable	O
and	O
a	O
leaving	O
variable	O
by	O
considering	O
only	O
the	O
signs	O
of	O
coefficients	O
rather	O
than	O
their	O
real-number	O
ordering	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
applied	O
to	O
furnish	O
constructive	O
proofs	O
of	O
basic	O
results	O
in	O
linear	B-Language
algebra	I-Language
,	O
such	O
as	O
the	O
lemma	O
of	O
Farkas	O
.	O
</s>
<s>
While	O
most	O
simplex	O
variants	O
are	O
monotonic	O
in	O
the	O
objective	O
(	O
strictly	O
in	O
the	O
non-degenerate	O
case	O
)	O
,	O
most	O
variants	O
of	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
lack	O
a	O
monotone	O
merit	O
function	O
which	O
can	O
be	O
a	O
disadvantage	O
in	O
practice	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
works	O
on	O
a	O
standard	O
pivot	O
tableau	O
(	O
or	O
on-the-fly	O
calculated	O
parts	O
of	O
a	O
tableau	O
,	O
if	O
implemented	O
like	O
the	O
revised	O
simplex	B-Algorithm
method	I-Algorithm
)	O
.	O
</s>
<s>
For	O
example	O
,	O
Gaussian	B-Algorithm
elimination	I-Algorithm
requires	O
on	O
the	O
order	O
of	O
D3	O
operations	O
,	O
and	O
so	O
it	O
is	O
said	O
to	O
have	O
polynomial	O
time-complexity	O
,	O
because	O
its	O
complexity	O
is	O
bounded	O
by	O
a	O
cubic	O
polynomial	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
generalization	O
of	O
Gaussian	B-Algorithm
elimination	I-Algorithm
called	O
Buchberger	O
's	O
algorithm	O
has	O
for	O
its	O
complexity	O
an	O
exponential	O
function	O
of	O
the	O
problem	O
data	O
(	O
the	O
degree	O
of	O
the	O
polynomials	O
and	O
the	O
number	O
of	O
variables	O
of	O
the	O
multivariate	O
polynomials	O
)	O
.	O
</s>
<s>
Several	O
algorithms	O
for	O
linear	O
Khachiyan	O
's	O
ellipsoidal	B-Algorithm
algorithm	I-Algorithm
,	O
Karmarkar	O
's	O
projective	B-Algorithm
algorithm	I-Algorithm
,	O
and	O
central-path	O
algorithms	O
—	O
have	O
polynomial	O
time-complexity	O
(	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
and	O
thus	O
on	O
average	O
)	O
.	O
</s>
<s>
The	O
ellipsoidal	O
and	O
projective	O
algorithms	O
were	O
published	O
before	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
However	O
,	O
like	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
of	O
Dantzig	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
not	O
a	O
polynomial-time	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Terlaky	O
's	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
visits	O
all	O
the2Dcorners	O
of	O
a	O
(	O
perturbed	O
)	O
cube	O
in	O
dimensionD	O
,	O
according	O
to	O
a	O
paper	O
of	O
Roos	O
;	O
Roos	O
's	O
paper	O
modifies	O
the	O
Klee	O
–	O
Minty	O
construction	O
of	O
a	O
cube	O
on	O
which	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
takes2Dsteps	O
.	O
</s>
<s>
Like	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
visits	O
all8	O
corners	O
of	O
the	O
three-dimensional	O
cube	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
.	O
</s>
<s>
When	O
it	O
is	O
initialized	O
at	O
a	O
random	O
corner	O
of	O
the	O
cube	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
visits	O
onlyD	O
additional	O
corners	O
,	O
however	O
,	O
according	O
to	O
a1994	O
paper	O
by	O
Fukuda	O
and	O
Namiki	O
.	O
</s>
<s>
Trivially	O
,	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
takes	O
on	O
averageD	O
steps	O
for	O
a	O
cube	O
.	O
</s>
<s>
Like	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
visits	O
exactly3	O
additional	O
corners	O
of	O
the	O
three-dimensional	O
cube	O
onaverage	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
extended	O
to	O
solve	O
more	O
general	O
problems	O
than	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
There	O
are	O
variants	O
of	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
for	O
linear	B-Algorithm
programming	I-Algorithm
,	O
for	O
quadratic	B-Algorithm
programming	I-Algorithm
,	O
and	O
for	O
the	O
linear-complementarity	O
problem	O
with	O
"	O
sufficient	O
matrices	O
"	O
;	O
conversely	O
,	O
for	O
linear	O
complementarity	O
problems	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
terminates	O
finitely	O
only	O
if	O
the	O
matrix	O
is	O
a	O
sufficient	O
matrix	O
.	O
</s>
<s>
A	O
sufficient	O
matrix	O
is	O
a	O
generalization	O
both	O
of	O
a	O
positive-definite	B-Algorithm
matrix	I-Algorithm
and	O
of	O
a	O
P-matrix	B-Algorithm
,	O
whose	O
principal	O
minors	O
are	O
each	O
positive	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
adapted	O
also	O
for	O
linear-fractional	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
was	O
used	O
in	O
an	O
algorithm	O
for	O
enumerating	B-Algorithm
all	I-Algorithm
the	I-Algorithm
vertices	I-Algorithm
of	I-Algorithm
a	I-Algorithm
polytope	I-Algorithm
,	O
which	O
was	O
published	O
by	O
David	O
Avis	O
and	O
Komei	O
Fukuda	O
in1992	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
often	O
studied	O
using	O
the	O
theory	O
of	O
oriented	O
matroids	O
(	O
OMs	O
)	O
,	O
which	O
is	O
a	O
combinatorial	O
abstraction	O
of	O
linear-optimization	O
theory	O
.	O
</s>
<s>
However	O
,	O
Bland	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
exhibits	O
cycling	O
on	O
some	O
oriented-matroid	O
linear-programming	O
problems	O
.	O
</s>
<s>
The	O
first	O
purely	O
combinatorial	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
was	O
devised	O
by	O
Michael	O
J	O
.	O
Todd	O
.	O
</s>
<s>
Todd	O
's	O
algorithm	O
was	O
developed	O
not	O
only	O
for	O
linear-programming	O
in	O
the	O
setting	O
of	O
oriented	O
matroids	O
,	O
but	O
also	O
for	O
quadratic-programming	B-Algorithm
problems	I-Algorithm
and	O
linear-complementarity	O
problems	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
and	O
its	O
proof	O
of	O
finite	O
termination	O
can	O
be	O
simply	O
stated	O
and	O
readily	O
extend	O
the	O
setting	O
of	O
oriented	O
matroids	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
adapted	O
for	O
problems	O
that	O
are	O
more	O
complicated	O
than	O
linear	B-Algorithm
programming	I-Algorithm
:	O
There	O
are	O
oriented-matroid	O
variants	O
also	O
for	O
the	O
quadratic-programming	B-Algorithm
problem	O
and	O
for	O
the	O
linear-complementarity	O
problem	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
a	O
simply	O
stated	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
It	O
was	O
the	O
second	O
fully	O
combinatorial	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
partially	O
combinatorial	O
simplex	B-Algorithm
algorithm	I-Algorithm
of	O
Bland	O
cycles	O
on	O
some	O
(	O
nonrealizable	O
)	O
oriented	O
matroids	O
.	O
</s>
<s>
The	O
first	O
fully	O
combinatorial	O
algorithm	O
was	O
published	O
by	O
Todd	O
,	O
and	O
it	O
is	O
also	O
like	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
in	O
that	O
it	O
preserves	O
feasibility	O
after	O
the	O
first	O
feasible	O
basis	O
is	O
generated	O
;	O
however	O
,	O
Todd	O
's	O
rule	O
is	O
complicated	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
not	O
a	O
simplex-like	O
algorithm	O
,	O
because	O
it	O
need	O
not	O
maintain	O
feasibility	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
does	O
not	O
have	O
polynomial	O
time-complexity	O
,	O
however	O
.	O
</s>
<s>
Researchers	O
have	O
extended	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
for	O
many	O
optimization-problems	O
,	O
including	O
linear-fractional	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
can	O
solve	O
quadratic	B-Algorithm
programming	I-Algorithm
problems	O
and	O
linear	O
complementarity	O
problems	O
,	O
even	O
in	O
the	O
setting	O
of	O
oriented	O
matroids	O
.	O
</s>
<s>
Even	O
when	O
generalized	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
remains	O
simply	O
stated	O
.	O
</s>
