matlab - Optimization with inverse and pseudoinverse
$begingroup$
Let us assume I have to optimize this system:
$$min_{xin S} left|left|left(Ebegin{bmatrix}
I_n\
A'(x)^{-1}C'(x)\
end{bmatrix}right)^+ a -bright|right|^{2}$$
Where x is the vector of my unknowns, A', B' are matrices depending on x while a and b are constant vectors with correct dimensions and E is a matrix with correct dimensions too.
S are the constaints on my x variables which are simple linear inequalities.
Which is the best approach in order to solve this problem? Suggestions?
How to handle both pseudo inverse and inverse?
optimization inverse matlab pseudoinverse
asked Dec 14 '18 at 10:50
iacopoiacopo
1127
$endgroup$
add a comment |
$begingroup$
Let us assume I have to optimize this system:
$$min_{xin S} left|left|left(Ebegin{bmatrix}
I_n\
A'(x)^{-1}C'(x)\
end{bmatrix}right)^+ a -bright|right|^{2}$$
Where x is the vector of my unknowns, A', B' are matrices depending on x while a and b are constant vectors with correct dimensions and E is a matrix with correct dimensions too.
S are the constaints on my x variables which are simple linear inequalities.
Which is the best approach in order to solve this problem? Suggestions?
How to handle both pseudo inverse and inverse?
optimization inverse matlab pseudoinverse
asked Dec 14 '18 at 10:50
iacopoiacopo
1127
$endgroup$
$begingroup$
@snulty ok, done
$endgroup$
– iacopo
Dec 14 '18 at 10:57
$begingroup$
I was just suggesting it because it might tidy up some algebra, and at the end you can probably sub back in for the A' and C'
$endgroup$
– snulty
Dec 14 '18 at 10:59
$begingroup$
@snulty yes, this is a good idea. Thank you but still i can't figure out how to handle this problem in matlab
$endgroup$
– iacopo
Dec 14 '18 at 11:02
$begingroup$
My only guess would be to try expanding out as something like this $ left(a^dagger Ebegin{bmatrix} I_n \ A'(x)^{-1}C'(x)\ end{bmatrix} -b^daggerright)left(begin{bmatrix} I_n & (A'(x)^{-1}C'(x))^dagger\ end{bmatrix}E^dagger a -bright)$ assuming $a$ and $b$ are column vectors, and then trying to differentiate and set it equal to zero to solve for a critical point/minimum. Maybe you can do a type of root finding on that equation with the parameters in $x$? Or maybe someone else has some better suggestions
$endgroup$
– snulty
Dec 14 '18 at 11:15
$begingroup$
The $(cdot)^+$ function is convex but not linear, and the norm is not nondecreasing, so the objective is not convex. This is a hard problem.
$endgroup$
– LinAlg
Dec 14 '18 at 16:43
add a comment |
$begingroup$
Let us assume I have to optimize this system:
$$min_{xin S} left|left|left(Ebegin{bmatrix}
I_n\
A'(x)^{-1}C'(x)\
end{bmatrix}right)^+ a -bright|right|^{2}$$
Where x is the vector of my unknowns, A', B' are matrices depending on x while a and b are constant vectors with correct dimensions and E is a matrix with correct dimensions too.
S are the constaints on my x variables which are simple linear inequalities.
Which is the best approach in order to solve this problem? Suggestions?
How to handle both pseudo inverse and inverse?
optimization inverse matlab pseudoinverse
asked Dec 14 '18 at 10:50
iacopoiacopo
1127
$endgroup$
Let us assume I have to optimize this system:
$$min_{xin S} left|left|left(Ebegin{bmatrix}
I_n\
A'(x)^{-1}C'(x)\
end{bmatrix}right)^+ a -bright|right|^{2}$$
Where x is the vector of my unknowns, A', B' are matrices depending on x while a and b are constant vectors with correct dimensions and E is a matrix with correct dimensions too.
S are the constaints on my x variables which are simple linear inequalities.
Which is the best approach in order to solve this problem? Suggestions?
How to handle both pseudo inverse and inverse?
optimization inverse matlab pseudoinverse
optimization inverse matlab pseudoinverse
asked Dec 14 '18 at 10:50
iacopoiacopo
1127
asked Dec 14 '18 at 10:50
iacopoiacopo
1127
edited Dec 14 '18 at 10:57
iacopo
asked Dec 14 '18 at 10:50
iacopoiacopo
1127
asked Dec 14 '18 at 10:50
iacopoiacopo
1127
asked Dec 14 '18 at 10:50
iacopoiacopo
1127
1127
$begingroup$
@snulty ok, done
$endgroup$
– iacopo
Dec 14 '18 at 10:57
$begingroup$
I was just suggesting it because it might tidy up some algebra, and at the end you can probably sub back in for the A' and C'
$endgroup$
– snulty
Dec 14 '18 at 10:59
$begingroup$
@snulty yes, this is a good idea. Thank you but still i can't figure out how to handle this problem in matlab
$endgroup$
– iacopo
Dec 14 '18 at 11:02
$begingroup$
My only guess would be to try expanding out as something like this $ left(a^dagger Ebegin{bmatrix} I_n \ A'(x)^{-1}C'(x)\ end{bmatrix} -b^daggerright)left(begin{bmatrix} I_n & (A'(x)^{-1}C'(x))^dagger\ end{bmatrix}E^dagger a -bright)$ assuming $a$ and $b$ are column vectors, and then trying to differentiate and set it equal to zero to solve for a critical point/minimum. Maybe you can do a type of root finding on that equation with the parameters in $x$? Or maybe someone else has some better suggestions
$endgroup$
– snulty
Dec 14 '18 at 11:15
$begingroup$
The $(cdot)^+$ function is convex but not linear, and the norm is not nondecreasing, so the objective is not convex. This is a hard problem.
$endgroup$
– LinAlg
Dec 14 '18 at 16:43
add a comment |
$begingroup$
@snulty ok, done
$endgroup$
– iacopo
Dec 14 '18 at 10:57
$begingroup$
I was just suggesting it because it might tidy up some algebra, and at the end you can probably sub back in for the A' and C'
$endgroup$
– snulty
Dec 14 '18 at 10:59
$begingroup$
@snulty yes, this is a good idea. Thank you but still i can't figure out how to handle this problem in matlab
$endgroup$
– iacopo
Dec 14 '18 at 11:02
$begingroup$
My only guess would be to try expanding out as something like this $ left(a^dagger Ebegin{bmatrix} I_n \ A'(x)^{-1}C'(x)\ end{bmatrix} -b^daggerright)left(begin{bmatrix} I_n & (A'(x)^{-1}C'(x))^dagger\ end{bmatrix}E^dagger a -bright)$ assuming $a$ and $b$ are column vectors, and then trying to differentiate and set it equal to zero to solve for a critical point/minimum. Maybe you can do a type of root finding on that equation with the parameters in $x$? Or maybe someone else has some better suggestions
$endgroup$
– snulty
Dec 14 '18 at 11:15
$begingroup$
The $(cdot)^+$ function is convex but not linear, and the norm is not nondecreasing, so the objective is not convex. This is a hard problem.
$endgroup$
– LinAlg
Dec 14 '18 at 16:43
$begingroup$
@snulty ok, done
$endgroup$
– iacopo
Dec 14 '18 at 10:57
$begingroup$
@snulty ok, done
$endgroup$
– iacopo
Dec 14 '18 at 10:57
$begingroup$
I was just suggesting it because it might tidy up some algebra, and at the end you can probably sub back in for the A' and C'
$endgroup$
– snulty
Dec 14 '18 at 10:59
$begingroup$
I was just suggesting it because it might tidy up some algebra, and at the end you can probably sub back in for the A' and C'
$endgroup$
– snulty
Dec 14 '18 at 10:59
$begingroup$
@snulty yes, this is a good idea. Thank you but still i can't figure out how to handle this problem in matlab
$endgroup$
– iacopo
Dec 14 '18 at 11:02
$begingroup$
@snulty yes, this is a good idea. Thank you but still i can't figure out how to handle this problem in matlab
$endgroup$
– iacopo
Dec 14 '18 at 11:02
$begingroup$
My only guess would be to try expanding out as something like this $ left(a^dagger Ebegin{bmatrix} I_n \ A'(x)^{-1}C'(x)\ end{bmatrix} -b^daggerright)left(begin{bmatrix} I_n & (A'(x)^{-1}C'(x))^dagger\ end{bmatrix}E^dagger a -bright)$ assuming $a$ and $b$ are column vectors, and then trying to differentiate and set it equal to zero to solve for a critical point/minimum. Maybe you can do a type of root finding on that equation with the parameters in $x$? Or maybe someone else has some better suggestions
$endgroup$
– snulty
Dec 14 '18 at 11:15
$begingroup$
My only guess would be to try expanding out as something like this $ left(a^dagger Ebegin{bmatrix} I_n \ A'(x)^{-1}C'(x)\ end{bmatrix} -b^daggerright)left(begin{bmatrix} I_n & (A'(x)^{-1}C'(x))^dagger\ end{bmatrix}E^dagger a -bright)$ assuming $a$ and $b$ are column vectors, and then trying to differentiate and set it equal to zero to solve for a critical point/minimum. Maybe you can do a type of root finding on that equation with the parameters in $x$? Or maybe someone else has some better suggestions
$endgroup$
– snulty
Dec 14 '18 at 11:15
$begingroup$
The $(cdot)^+$ function is convex but not linear, and the norm is not nondecreasing, so the objective is not convex. This is a hard problem.
$endgroup$
– LinAlg
Dec 14 '18 at 16:43
$begingroup$
The $(cdot)^+$ function is convex but not linear, and the norm is not nondecreasing, so the objective is not convex. This is a hard problem.
$endgroup$
– LinAlg
Dec 14 '18 at 16:43
add a comment |
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$begingroup$
@snulty ok, done
$endgroup$
– iacopo
Dec 14 '18 at 10:57
$begingroup$
I was just suggesting it because it might tidy up some algebra, and at the end you can probably sub back in for the A' and C'
$endgroup$
– snulty
Dec 14 '18 at 10:59
$begingroup$
@snulty yes, this is a good idea. Thank you but still i can't figure out how to handle this problem in matlab
$endgroup$
– iacopo
Dec 14 '18 at 11:02
$begingroup$
My only guess would be to try expanding out as something like this $ left(a^dagger Ebegin{bmatrix} I_n \ A'(x)^{-1}C'(x)\ end{bmatrix} -b^daggerright)left(begin{bmatrix} I_n & (A'(x)^{-1}C'(x))^dagger\ end{bmatrix}E^dagger a -bright)$ assuming $a$ and $b$ are column vectors, and then trying to differentiate and set it equal to zero to solve for a critical point/minimum. Maybe you can do a type of root finding on that equation with the parameters in $x$? Or maybe someone else has some better suggestions
$endgroup$
– snulty
Dec 14 '18 at 11:15
$begingroup$
The $(cdot)^+$ function is convex but not linear, and the norm is not nondecreasing, so the objective is not convex. This is a hard problem.
$endgroup$
– LinAlg
Dec 14 '18 at 16:43