dual of rank constrained SDPs
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
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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
New contributor
Maharshi Ray is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
Jan 4 at 14:53
add a comment |
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
New contributor
Maharshi Ray is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
New contributor
Maharshi Ray is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
New contributor
Maharshi Ray is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
New contributor
Maharshi Ray is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
asked Jan 4 at 12:15
Maharshi RayMaharshi Ray
1
1
New contributor
Maharshi Ray is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Maharshi Ray is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Maharshi Ray is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
Jan 4 at 14:53
add a comment |
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
Jan 4 at 14:53
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
Jan 4 at 14:53
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
Jan 4 at 14:53
add a comment |
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The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
Jan 4 at 14:53