6.079/6.975, Fall 200910
S. Boyd & P. Parrilo
(Part
of)
Homework
10:
Standard
form
LP
barrier
method
In the following three exercises, you will implement a barrier method for solving the standard
form LP
minimize
c
T
x
subject to
Ax
=
b,
x
0
,
with variable
x
∈
R
n
, where
A
∈
R
m
×
n
, with
m < n
.
Throughout this exercise we will
assume that
A
is full rank, and the sublevel sets
{
x

Ax
=
b, x
0
, c
T
x
≤
γ
}
are all
bounded. (If this is not the case, the centering problem is unbounded below.)
1.
Centering
step.
Implement Newton’s method for solving the centering problem
minimize
c
T
x
−
i
n
=1
log
x
i
subject to
Ax
=
b,
with variable
x
, given a strictly feasible starting point
x
0
.
Your code should accept
A
,
b
,
c
, and
x
0
, and return
x
⋆
, the primal optimal point,
ν
⋆
,
a dual optimal point, and the number of Newton steps executed.
Use the block elimination method to compute the Newton step. (You can also compute
the Newton step via the KKT system, and compare the result to the Newton step
computed via block elimination. The two steps should be close, but if any
x
i
is very
small, you might get a warning about the condition number of the KKT matrix.)
Plot
λ
2
/
2 versus iteration
k
, for various problem data and initial points, to verify that
your implementation gives asymptotic quadratic convergence. As stopping criterion,
you can use
λ
2
/
2
≤
10
−
6
. Experiment with varying the algorithm parameters
α
and
β
,
observing the eﬀect on the total number of Newton steps required, for a ±xed problem
instance. Check that your computed