Complex Least Squares With Magnitude Equality Constraints
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For $mathbfx in mathbbC^N$, I'd like to solve the following problem:
$$
mathbfx^ast = arg min_mathbfx Vert mathbfAx-b Vert_2 ,,,,,, mathrms.t. ,,,,, Vert x_i Vert_2 = a_i, ,,,, i = 0, dots, N-1,
$$
where $a_i in mathbbR$. The above is a least-squares problem where the magnitude of the elements of $mathbfx$ are fixed and only their phase may vary.
Can anyone point me in the direction of how to solve this? I have tried adding the equality constraints as a penalty term to the cost function, but had no success. Though I have not found anything yet, I am hoping that is a well-studied problem with a known solution.
Thanks for any help you can provide.
optimization nonlinear-system least-squares
asked May 12 '16 at 3:21
AnonSubmitter85
2,83221420
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up vote
1
down vote
favorite
For $mathbfx in mathbbC^N$, I'd like to solve the following problem:
$$
mathbfx^ast = arg min_mathbfx Vert mathbfAx-b Vert_2 ,,,,,, mathrms.t. ,,,,, Vert x_i Vert_2 = a_i, ,,,, i = 0, dots, N-1,
$$
where $a_i in mathbbR$. The above is a least-squares problem where the magnitude of the elements of $mathbfx$ are fixed and only their phase may vary.
Can anyone point me in the direction of how to solve this? I have tried adding the equality constraints as a penalty term to the cost function, but had no success. Though I have not found anything yet, I am hoping that is a well-studied problem with a known solution.
Thanks for any help you can provide.
optimization nonlinear-system least-squares
asked May 12 '16 at 3:21
AnonSubmitter85
2,83221420
Interesting problem. No idea how to solve it. :-) It's non-convex and in a particularly difficult manner. (For instance, if $x$ were real, one could search the $2^N$ combinations exhaustively at least.)
– Michael Grant
May 12 '16 at 21:30
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
For $mathbfx in mathbbC^N$, I'd like to solve the following problem:
$$
mathbfx^ast = arg min_mathbfx Vert mathbfAx-b Vert_2 ,,,,,, mathrms.t. ,,,,, Vert x_i Vert_2 = a_i, ,,,, i = 0, dots, N-1,
$$
where $a_i in mathbbR$. The above is a least-squares problem where the magnitude of the elements of $mathbfx$ are fixed and only their phase may vary.
Can anyone point me in the direction of how to solve this? I have tried adding the equality constraints as a penalty term to the cost function, but had no success. Though I have not found anything yet, I am hoping that is a well-studied problem with a known solution.
Thanks for any help you can provide.
optimization nonlinear-system least-squares
asked May 12 '16 at 3:21
AnonSubmitter85
2,83221420
For $mathbfx in mathbbC^N$, I'd like to solve the following problem:
$$
mathbfx^ast = arg min_mathbfx Vert mathbfAx-b Vert_2 ,,,,,, mathrms.t. ,,,,, Vert x_i Vert_2 = a_i, ,,,, i = 0, dots, N-1,
$$
where $a_i in mathbbR$. The above is a least-squares problem where the magnitude of the elements of $mathbfx$ are fixed and only their phase may vary.
Can anyone point me in the direction of how to solve this? I have tried adding the equality constraints as a penalty term to the cost function, but had no success. Though I have not found anything yet, I am hoping that is a well-studied problem with a known solution.
Thanks for any help you can provide.
optimization nonlinear-system least-squares
asked May 12 '16 at 3:21
AnonSubmitter85
2,83221420
edited May 12 '16 at 21:24
asked May 12 '16 at 3:21
AnonSubmitter85
2,83221420
asked May 12 '16 at 3:21
AnonSubmitter85
2,83221420
asked May 12 '16 at 3:21
AnonSubmitter85
2,83221420
2,83221420
Interesting problem. No idea how to solve it. :-) It's non-convex and in a particularly difficult manner. (For instance, if $x$ were real, one could search the $2^N$ combinations exhaustively at least.)
– Michael Grant
May 12 '16 at 21:30
add a comment |Â
Interesting problem. No idea how to solve it. :-) It's non-convex and in a particularly difficult manner. (For instance, if $x$ were real, one could search the $2^N$ combinations exhaustively at least.)
– Michael Grant
May 12 '16 at 21:30
Interesting problem. No idea how to solve it. :-) It's non-convex and in a particularly difficult manner. (For instance, if $x$ were real, one could search the $2^N$ combinations exhaustively at least.)
– Michael Grant
May 12 '16 at 21:30
Interesting problem. No idea how to solve it. :-) It's non-convex and in a particularly difficult manner. (For instance, if $x$ were real, one could search the $2^N$ combinations exhaustively at least.)
– Michael Grant
May 12 '16 at 21:30
add a comment |Â
1 Answer
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Really nice and interesting question.
I tried, at first, solving it on the Real domain.
A brute force approach is my reference and I tried working with the following cost function:
$$ arg min_x f left( x right) = arg min_x frac12 left| A x - b right|_2^2 + fraclambda2 left| operatornameabs left( x right) - a right|_2^2 $$
Where $ operatornameabs left( x right) $ is element wise.
The derivative is given by:
$$ fracdd x f left( x right) = A^T left( A x - b right) + lambda operatornamesign left( x right) left( operatornameabs left( x right) - a right) $$
I tried Gradient Descent where I raise the value of $ lambda $ at each iteration.
It worked not so bad, but even the sign of the solution wasn't consistent with the optimal solution.
My intermediate code is given here.
I will try another 2 approaches:
- Using Iterative Least Squares like approach.
- Using Orthogonal Matching Pursuit (OMP) like approach.
answered Sep 9 '17 at 9:37
Royi
3,05012046
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Really nice and interesting question.
I tried, at first, solving it on the Real domain.
A brute force approach is my reference and I tried working with the following cost function:
$$ arg min_x f left( x right) = arg min_x frac12 left| A x - b right|_2^2 + fraclambda2 left| operatornameabs left( x right) - a right|_2^2 $$
Where $ operatornameabs left( x right) $ is element wise.
The derivative is given by:
$$ fracdd x f left( x right) = A^T left( A x - b right) + lambda operatornamesign left( x right) left( operatornameabs left( x right) - a right) $$
I tried Gradient Descent where I raise the value of $ lambda $ at each iteration.
It worked not so bad, but even the sign of the solution wasn't consistent with the optimal solution.
My intermediate code is given here.
I will try another 2 approaches:
- Using Iterative Least Squares like approach.
- Using Orthogonal Matching Pursuit (OMP) like approach.
answered Sep 9 '17 at 9:37
Royi
3,05012046
add a comment |Â
up vote
0
down vote
Really nice and interesting question.
I tried, at first, solving it on the Real domain.
A brute force approach is my reference and I tried working with the following cost function:
$$ arg min_x f left( x right) = arg min_x frac12 left| A x - b right|_2^2 + fraclambda2 left| operatornameabs left( x right) - a right|_2^2 $$
Where $ operatornameabs left( x right) $ is element wise.
The derivative is given by:
$$ fracdd x f left( x right) = A^T left( A x - b right) + lambda operatornamesign left( x right) left( operatornameabs left( x right) - a right) $$
I tried Gradient Descent where I raise the value of $ lambda $ at each iteration.
It worked not so bad, but even the sign of the solution wasn't consistent with the optimal solution.
My intermediate code is given here.
I will try another 2 approaches:
- Using Iterative Least Squares like approach.
- Using Orthogonal Matching Pursuit (OMP) like approach.
answered Sep 9 '17 at 9:37
Royi
3,05012046
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Really nice and interesting question.
I tried, at first, solving it on the Real domain.
A brute force approach is my reference and I tried working with the following cost function:
$$ arg min_x f left( x right) = arg min_x frac12 left| A x - b right|_2^2 + fraclambda2 left| operatornameabs left( x right) - a right|_2^2 $$
Where $ operatornameabs left( x right) $ is element wise.
The derivative is given by:
$$ fracdd x f left( x right) = A^T left( A x - b right) + lambda operatornamesign left( x right) left( operatornameabs left( x right) - a right) $$
I tried Gradient Descent where I raise the value of $ lambda $ at each iteration.
It worked not so bad, but even the sign of the solution wasn't consistent with the optimal solution.
My intermediate code is given here.
I will try another 2 approaches:
- Using Iterative Least Squares like approach.
- Using Orthogonal Matching Pursuit (OMP) like approach.
answered Sep 9 '17 at 9:37
Royi
3,05012046
Really nice and interesting question.
I tried, at first, solving it on the Real domain.
A brute force approach is my reference and I tried working with the following cost function:
$$ arg min_x f left( x right) = arg min_x frac12 left| A x - b right|_2^2 + fraclambda2 left| operatornameabs left( x right) - a right|_2^2 $$
Where $ operatornameabs left( x right) $ is element wise.
The derivative is given by:
$$ fracdd x f left( x right) = A^T left( A x - b right) + lambda operatornamesign left( x right) left( operatornameabs left( x right) - a right) $$
I tried Gradient Descent where I raise the value of $ lambda $ at each iteration.
It worked not so bad, but even the sign of the solution wasn't consistent with the optimal solution.
My intermediate code is given here.
I will try another 2 approaches:
- Using Iterative Least Squares like approach.
- Using Orthogonal Matching Pursuit (OMP) like approach.
answered Sep 9 '17 at 9:37
Royi
3,05012046
edited Aug 16 at 7:35
answered Sep 9 '17 at 9:37
Royi
3,05012046
answered Sep 9 '17 at 9:37
Royi
3,05012046
answered Sep 9 '17 at 9:37
Royi
3,05012046
3,05012046
add a comment |Â
add a comment |Â
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Interesting problem. No idea how to solve it. :-) It's non-convex and in a particularly difficult manner. (For instance, if $x$ were real, one could search the $2^N$ combinations exhaustively at least.)
– Michael Grant
May 12 '16 at 21:30