Mixing
dual of rank constrained SDPs
$begingroup$
The standard rank constrained SDP is as follows:
$$
min , Tr(CX),\ text{ s.t } Tr(A_iX)= b_i text{ for } i = 1,2,...,m ,,\ X >= 0,\ Rank(X) <= r
$$
I was interested in a lower bounding a particular rank constrained SDP. I was trying out the following : 1) Formulate above using the non-linear formulation of Burer-Monteiro . 2) Find a dual feasible solution and report obj value.
The non-linear version of the above would be
$$
min , Tr(CVV^T),\ text{ s.t } Tr(A_iVV^T)= b_i text{ for } i = 1,2,...,m \V in mathbb{R}^{n times r}.
$$
However, taking the Lagrangian Dual of this doesn't seem to be fruitful, at least for my SDP of interest.
Is there another way out? If not analytically, but say numerically ? When I use a nonlinear solver like 'fmincon' on MATLAB, I end up getting a large duality gap and a bad lower bound.
optimization nonlinear-optimization duality-theorems semidefinite-programming
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
$endgroup$
add a comment |
$begingroup$
The standard rank constrained SDP is as follows:
$$
min , Tr(CX),\ text{ s.t } Tr(A_iX)= b_i text{ for } i = 1,2,...,m ,,\ X >= 0,\ Rank(X) <= r
$$
I was interested in a lower bounding a particular rank constrained SDP. I was trying out the following : 1) Formulate above using the non-linear formulation of Burer-Monteiro . 2) Find a dual feasible solution and report obj value.
The non-linear version of the above would be
$$
min , Tr(CVV^T),\ text{ s.t } Tr(A_iVV^T)= b_i text{ for } i = 1,2,...,m \V in mathbb{R}^{n times r}.
$$
However, taking the Lagrangian Dual of this doesn't seem to be fruitful, at least for my SDP of interest.
Is there another way out? If not analytically, but say numerically ? When I use a nonlinear solver like 'fmincon' on MATLAB, I end up getting a large duality gap and a bad lower bound.
optimization nonlinear-optimization duality-theorems semidefinite-programming
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
$endgroup$
$begingroup$
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
$endgroup$
– LinAlg
Jan 4 at 14:53
add a comment |
$begingroup$
The standard rank constrained SDP is as follows:
$$
min , Tr(CX),\ text{ s.t } Tr(A_iX)= b_i text{ for } i = 1,2,...,m ,,\ X >= 0,\ Rank(X) <= r
$$
I was interested in a lower bounding a particular rank constrained SDP. I was trying out the following : 1) Formulate above using the non-linear formulation of Burer-Monteiro . 2) Find a dual feasible solution and report obj value.
The non-linear version of the above would be
$$
min , Tr(CVV^T),\ text{ s.t } Tr(A_iVV^T)= b_i text{ for } i = 1,2,...,m \V in mathbb{R}^{n times r}.
$$
However, taking the Lagrangian Dual of this doesn't seem to be fruitful, at least for my SDP of interest.
Is there another way out? If not analytically, but say numerically ? When I use a nonlinear solver like 'fmincon' on MATLAB, I end up getting a large duality gap and a bad lower bound.
optimization nonlinear-optimization duality-theorems semidefinite-programming
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
$endgroup$
The standard rank constrained SDP is as follows:
$$
min , Tr(CX),\ text{ s.t } Tr(A_iX)= b_i text{ for } i = 1,2,...,m ,,\ X >= 0,\ Rank(X) <= r
$$
I was interested in a lower bounding a particular rank constrained SDP. I was trying out the following : 1) Formulate above using the non-linear formulation of Burer-Monteiro . 2) Find a dual feasible solution and report obj value.
The non-linear version of the above would be
$$
min , Tr(CVV^T),\ text{ s.t } Tr(A_iVV^T)= b_i text{ for } i = 1,2,...,m \V in mathbb{R}^{n times r}.
$$
However, taking the Lagrangian Dual of this doesn't seem to be fruitful, at least for my SDP of interest.
Is there another way out? If not analytically, but say numerically ? When I use a nonlinear solver like 'fmincon' on MATLAB, I end up getting a large duality gap and a bad lower bound.
optimization nonlinear-optimization duality-theorems semidefinite-programming
optimization nonlinear-optimization duality-theorems semidefinite-programming
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
1
$begingroup$
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
$endgroup$
– LinAlg
Jan 4 at 14:53
add a comment |
$begingroup$
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
$endgroup$
– LinAlg
Jan 4 at 14:53
$begingroup$
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
$endgroup$
– LinAlg
Jan 4 at 14:53
$begingroup$
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
$endgroup$
– LinAlg
Jan 4 at 14:53
add a comment |
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$begingroup$
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
$endgroup$
– LinAlg
Jan 4 at 14:53