Linear Algebra Formula Sheet Page 2
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1
2Linear least-squares problem:
1
2
(
)
min
Ax
b
A Aˆ x = A b
2
Minimum norm solution
1
2
min
x
2
(
)
A A¯ x = A b
AA ˆ u = A¯ x
ˆ x = A ˆ u
s.t. Ax = A¯ x
Newton’s method for
2
( )
( +1)
( )
( )
(
)
min (x)
(x
) x
x
=
(x
)
Gauss-Newton’s method for nonlinear least-squares problems
1
( )
( )
( +1)
( )
( )
( )
2
(
)
min
h(x)
h(x
)
h(x
) x
x
=
h(x
) h(x
)
2
Nonlinear programming with equality constraints:
min
(x)
(
)
=
s.t. h(x) = 0
Lagrange’s conditions:
(ˆ x ) + u
h(ˆ x ) = 0
h(ˆ x ) = 0
Nonlinear programming with inequality constraints:
min
(x)
(
)
s.t. g(x)
0
1
(ˆ x ) + ˆ y
g(ˆ x ) = 0
2 g(ˆ x )
0
KKT-conditions:
3 ˆ y
0
4 ˆ y g(ˆ x ) = 0
Lagrange relaxation:
min
(x)
( )
s.t. g(x)
0
x
(x y) = (x) + y g(x) where x
and y
.
(y) = min (x y) = (ˆ x (y) y)
max
(y)
( )
s.t. y
0
ˆ x = ˆ x (ˆ y )
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