Lecture 9
Feasible Directions for Inequality
Constraints
UCSD Math 171A: Numerical Optimization
Philip E. Gill
(
pgill@ucsd.edu
)
Monday, January 26, 2009
Recap: Linear programming with equality constraints
LP’s with equality constraints
ELP
minimize
x
∈
R
n
c
T
x
subject to
Ax
=
b
,
UCSD Center for Computational Mathematics
Slide 2/34, Monday, January 26, 2009
Optimality conditions for ELP
Result
Consider minimizing
‘
(
x
) =
c
T
x
subject to
Ax
=
b
.
(a)
If
Ax
=
b
is incompatible, no solution exists;
(b)
If
Ax
=
b
is compatible and
c
does not lie in the range of
A
T
,
the objective function is unbounded below at feasible points;
(c)
If
Ax
=
b
is compatible and
c
lies in the range of
A
T
(so that
c
=
A
T
λ
*
for some vector
λ
*
), then:
(i)
‘
*
, the optimal value of
‘
, is ﬁnite and unique;
(ii)
Every feasible point is a minimizer
x
*
;
(iii)
x
*
is unique
if and only if
the columns of
A
are linearly
independent;
(iv)
λ
*
is unique
if and only if
the rows of
A
are linearly
independent.
UCSD Center for Computational Mathematics
Slide 3/34, Monday, January 26, 2009
x = solve(A,b)
If incompatible, no feasible
point exists
If compatible,
x
is feasible
lambda = solve(A’,c)
If incompatible, no bounded
solution
If compatible,
x
is a minimizer
UCSD Center for Computational Mathematics
Slide 4/34, Monday, January 26, 2009