<s>
The	O
Frank	B-Algorithm
–	I-Algorithm
Wolfe	I-Algorithm
algorithm	I-Algorithm
is	O
an	O
iterative	B-Algorithm
first-order	B-Algorithm
optimization	O
algorithm	O
for	O
constrained	B-Application
convex	O
optimization	O
.	O
</s>
<s>
Also	O
known	O
as	O
the	O
conditional	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
,	O
reduced	O
gradient	O
algorithm	O
and	O
the	O
convex	O
combination	O
algorithm	O
,	O
the	O
method	O
was	O
originally	O
proposed	O
by	O
Marguerite	O
Frank	O
and	O
Philip	O
Wolfe	O
in1956	O
.	O
</s>
<s>
In	O
each	O
iteration	O
,	O
the	O
Frank	B-Algorithm
–	I-Algorithm
Wolfe	I-Algorithm
algorithm	I-Algorithm
considers	O
a	O
linear	B-Algorithm
approximation	I-Algorithm
of	O
the	O
objective	O
function	O
,	O
and	O
moves	O
towards	O
a	O
minimizer	O
of	O
this	O
linear	O
function	O
(	O
taken	O
over	O
the	O
same	O
domain	O
)	O
.	O
</s>
<s>
(	O
Interpretation	O
:	O
Minimize	O
the	O
linear	B-Algorithm
approximation	I-Algorithm
of	O
the	O
problem	O
given	O
by	O
the	O
first-order	B-Algorithm
Taylor	O
approximation	O
of	O
around	O
constrained	B-Application
to	O
stay	O
within	O
.	O
)	O
</s>
<s>
While	O
competing	O
methods	O
such	O
as	O
gradient	B-Algorithm
descent	I-Algorithm
for	O
constrained	B-Application
optimization	I-Application
require	O
a	O
projection	O
step	O
back	O
to	O
the	O
feasible	O
set	O
in	O
each	O
iteration	O
,	O
the	O
Frank	B-Algorithm
–	I-Algorithm
Wolfe	I-Algorithm
algorithm	I-Algorithm
only	O
needs	O
the	O
solution	O
of	O
a	O
linear	B-Algorithm
problem	I-Algorithm
over	O
the	O
same	O
set	O
in	O
each	O
iteration	O
,	O
and	O
automatically	O
stays	O
in	O
the	O
feasible	O
set	O
.	O
</s>
<s>
The	O
convergence	O
of	O
the	O
Frank	B-Algorithm
–	I-Algorithm
Wolfe	I-Algorithm
algorithm	I-Algorithm
is	O
sublinear	O
in	O
general	O
:	O
the	O
error	O
in	O
the	O
objective	O
function	O
to	O
the	O
optimum	O
is	O
after	O
k	O
iterations	O
,	O
so	O
long	O
as	O
the	O
gradient	O
is	O
Lipschitz	O
continuous	O
with	O
respect	O
to	O
some	O
norm	O
.	O
</s>
<s>
The	O
iterates	O
of	O
the	O
algorithm	O
can	O
always	O
be	O
represented	O
as	O
a	O
sparse	O
convex	O
combination	O
of	O
the	O
extreme	O
points	O
of	O
the	O
feasible	O
set	O
,	O
which	O
has	O
helped	O
to	O
the	O
popularity	O
of	O
the	O
algorithm	O
for	O
sparse	O
greedy	O
optimization	O
in	O
machine	O
learning	O
and	O
signal	O
processing	O
problems	O
,	O
as	O
well	O
as	O
for	O
example	O
the	O
optimization	O
of	O
minimum	O
–	O
cost	O
flows	O
in	O
transportation	B-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
If	O
the	O
feasible	O
set	O
is	O
given	O
by	O
a	O
set	O
of	O
linear	O
constraints	O
,	O
then	O
the	O
subproblem	O
to	O
be	O
solved	O
in	O
each	O
iteration	O
becomes	O
a	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
this	O
corresponding	O
duality	B-Algorithm
gap	I-Algorithm
,	O
that	O
is	O
the	O
difference	O
between	O
and	O
the	O
lower	O
bound	O
,	O
decreases	O
with	O
the	O
same	O
convergence	O
rate	O
,	O
i.e.	O
</s>
