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 16 hours ago
Maharshi 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.
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 16 hours ago
Maharshi 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
13 hours ago
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 16 hours ago
Maharshi 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 16 hours ago
Maharshi 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 16 hours ago
Maharshi 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 16 hours ago
Maharshi 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 16 hours ago
Maharshi Ray
1
asked 16 hours ago
Maharshi 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
13 hours ago
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
13 hours ago
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
13 hours ago
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
13 hours ago
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Maharshi Ray is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061593%2fdual-of-rank-constrained-sdps%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Maharshi Ray is a new contributor. Be nice, and check out our Code of Conduct.
Maharshi Ray is a new contributor. Be nice, and check out our Code of Conduct.
Maharshi Ray is a new contributor. Be nice, and check out our Code of Conduct.
Maharshi Ray is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061593%2fdual-of-rank-constrained-sdps%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
The paper by Burer and Monteiro has some leads on solving the nonlinear version. Can you show the dual formulation that you derived?
– LinAlg
13 hours ago