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IE417: Nonlinear Programming: Lecture 15
Jeﬀ Linderoth
Department of Industrial and Systems Engineering
Lehigh University
28th March 2006
Jeﬀ Linderoth
IE417:Lecture 15
Variational Inequality?
Do you remember Nicholas Steir’s Talk?
He used
First Order Conditions
!
He also mentioned
variational inequalities
.
Let
Ω
⊂
R
n
be a closed convex set, and consider the
function/mapping
f
: Ω
→
R
n
. The
variational inequality
problem
involves ﬁnding a vector
x
*
∈
R
n
such that
F
(
x
*
)
T
(
x

x
*
)
≥
0
∀
x
∈
Ω
What does this have to do with optimization? If
x
*
is a local
minimizer of
f
in
Ω
, then
x
*
solves the following VIP:
∇
f
(
x
*
)
T
(
x

x
*
)
≥
0
∀
x
∈
Ω
Recall normal cone, for
x
∈
Ω
N
Ω
(
x
) =
{
v

v
T
(
w

x
)
≤
0
∀
w
∈
Ω
}
Geometrically, you want that
∇
f
(
x
)
∈
N
(
x
)
Jeﬀ Linderoth
IE417:Lecture 15
Loose Categories
min
x
∈
R
n
f
(
x
)
subject to
c
i
(
x
) = 0
∀
i
∈ E
,
c
i
(
x
)
≥
0
∀
i
∈ I
1
Quadratic Programming:
f
(
x
)
def
=
x
T
Qx
+
d
T
i
x
+
b
c
i
(
x
)
def
=
a
T
i
x
+
b
i
∀
i
∈ E ∪ I
2
Bound Constraint Optimization:
c
i
(
x
)
def
=
x
i
+
b
i
∀
i
∈ I
,
(
E
=
∅
)
3
Convex Programming:
f
(
x
)
is convex,
c
i
(
x
)
is linear for all
i
∈ E
, and
c
i
(
x
)
is concave for all
i
∈ I
Jeﬀ Linderoth
IE417:Lecture 15
Loose Categories: Solution Methods
1
Penalty Function: (Say for
I
=
∅
)
f
(
x
) +
1
2
μ
±
i
∈E
c
2
i
(
x
)
2
Barrier Method: (Say for
E
=
∅
)
f
(
x
) +
μ
±
i
∈I
log
c
i
(
x
)
3
Augmented Lagrangian: (Say for
I
=
∅
)
L
A
(
x, λ, μ
) =
f
(
x
)

±
i
∈E
λ
i
c
i
(
x
) +
1
2
μ
±
i
∈E
c
2
i
(
x
)
4
Sequential Quadratic Programming. Iterate via solving a
sequence of
quadratic
approximations to the Lagrangian that
satisfy linear approximation to constraints
Jeﬀ Linderoth
IE417:Lecture 15
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 Linderoth

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