In these notes we describe in more detail the analytic center cuttingplane method (AC
CPM) for nondifferentiable convex optimization, prove its convergence, and give some nu
merical examples.
ACCPM was developed by Goffin and Vial [GV93] and analyzed by
Nesterov [Nes95] and Atkinson and Vaidya [AV95].
These notes assume a basic familiarity with convex optimization (see [BV04]), cutting
plane methods (see the EE364b notes
Localization and CuttingPlane Methods
), and subgra
dients (see the EE364b notes
Subgradients
).
1
Analytic center cuttingplane method
The basic ACCPM algorithm is:
Analytic center cuttingplane method (ACCPM)
given
an initial polyhedron
P
0
known to contain
X
.
k
:= 0.
repeat
Compute
x
(
k
+1)
, the analytic center of
P
k
.
Query the cuttingplane oracle at
x
(
k
+1)
.
If the oracle determines that
x
(
k
+1)
∈
X
, quit.
Else, add the returned cuttingplane inequality to
P
.
P
k
+1
:=
P
k
∩ {
z

a
T
z
≤
b
}
If
P
k
+1
=
∅
, quit.
k
:=
k
+ 1.
There are several variations on this basic algorithm. For example, at each step we can add
multiple cuts, instead of just one. We can also prune or drop constraints, for example, after
computing the analytic center of
P
k
. Later we will see a simple but nonheuristic stopping
criterion.
We can construct a cuttingplane
a
T
z
≤
b
at
x
(
k
)
, for the standard convex problem
minimize
f
0
(
x
)
subject to
f
i
(
x
)
≤
0
,
i
= 1
, . . . , m,
as follows. If
x
(
k
)
violates the
i
th constraint,
i.e.
,
f
i
(
x
(
k
)
)
>
0, we can take
a
=
g
i
,
b
=
g
T
i
x
(
k
)
−
f
i
(
x
(
k
)
)
,
(1)
where
g
i
∈
∂f
i
(
x
(
k
)
). If
x
(
k
)
is feasible, we can take
a
=
g
0
,
b
=
g
T
0
x
(
k
)
−
f
0
(
x
(
k
)
) +
f
(
k
)
best
,
(2)
where
g
0
∈
∂f
0
(
x
(
k
)
), and
f
(
k
)
best
is the best (lowest) objective value encountered for a feasible
iterate.
2