<s>
or	O
,	O
equivalently	O
,	O
if	O
its	O
restriction	O
to	O
any	O
arbitrary	O
line	O
is	O
self-concordant	B-Algorithm
.	O
</s>
<s>
As	O
mentioned	O
in	O
the	O
"	O
Bibliography	O
Comments	O
"	O
of	O
their	O
1994	O
book	O
,	O
self-concordant	B-Algorithm
functions	I-Algorithm
were	O
introduced	O
in	O
1988	O
by	O
Yurii	O
Nesterov	O
and	O
further	O
developed	O
with	O
Arkadi	O
Nemirovski	O
.	O
</s>
<s>
If	O
and	O
are	O
self-concordant	B-Algorithm
with	O
constants	O
and	O
and	O
,	O
then	O
is	O
self-concordant	B-Algorithm
with	O
constant	O
.	O
</s>
<s>
If	O
is	O
self-concordant	B-Algorithm
with	O
constant	O
and	O
is	O
an	O
affine	O
transformation	O
of	O
,	O
then	O
is	O
also	O
self-concordant	B-Algorithm
with	O
parameter	O
.	O
</s>
<s>
If	O
is	O
self-concordant	B-Algorithm
,	O
then	O
its	O
convex	O
conjugate	O
is	O
also	O
self-concordant	B-Algorithm
.	O
</s>
<s>
If	O
is	O
self-concordant	B-Algorithm
and	O
the	O
domain	O
of	O
contains	O
no	O
straight	O
line	O
(	O
infinite	O
in	O
both	O
directions	O
)	O
,	O
then	O
is	O
non-singular	O
.	O
</s>
<s>
Among	O
other	O
things	O
,	O
self-concordant	B-Algorithm
functions	I-Algorithm
are	O
useful	O
in	O
the	O
analysis	O
of	O
Newton	O
's	O
method	O
.	O
</s>
<s>
Self-concordant	B-Algorithm
barrier	O
functions	O
are	O
used	O
to	O
develop	O
the	O
barrier	O
functions	O
used	O
in	O
interior	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
for	O
convex	O
and	O
nonlinear	O
optimization	O
.	O
</s>
<s>
A	O
self-concordant	B-Algorithm
function	I-Algorithm
may	O
be	O
minimized	O
with	O
a	O
modified	O
Newton	O
method	O
where	O
we	O
have	O
a	O
bound	O
on	O
the	O
number	O
of	O
steps	O
required	O
for	O
convergence	O
.	O
</s>
<s>
We	O
suppose	O
here	O
that	O
is	O
a	O
standard	O
self-concordant	B-Algorithm
function	I-Algorithm
,	O
that	O
is	O
it	O
is	O
self-concordant	B-Algorithm
with	O
parameter	O
.	O
</s>
<s>
Linear	O
and	O
convex	O
quadratic	O
functions	O
are	O
self-concordant	B-Algorithm
with	O
since	O
their	O
third	O
derivative	O
is	O
zero	O
.	O
</s>
<s>
positive	O
half-line	O
:	O
with	O
domain	O
is	O
a	O
self-concordant	B-Algorithm
barrier	O
with	O
.	O
</s>
