Talk:Semidefinite programming

Hello,

welcome to the article on semidefinite programming.
So far (May 3rd 2006) it is only a stub and definitely needs more input.
I prepared a skeleton and some first exterior links.

There is a lot to do on this article, and for its "environment" (e.g. theorem on primal-dual gap, etc).
In addition to the obvious (definition, duality, solvers etc.) it would be nice to have some formulations for interesting SDPs.
Moreover, it would be good to document the links to polynomial optimization and sums of squares (of polynomials).
But: it seems there are no articles for these as well.

Hopefully we get this article (and its environment) up and running fast.
Best, Hartwig

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(@Hartwig) I found Polynomial SOS has its own wikipedia page now (or rather since 2008), so I will do some edits soon to incorporate it into this SDP page. Probably will start a separate section discussing the algebraic properties of SDP. BR, hattoriace March 2013

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I added a "citation needed" tag to the reference to quantum computing, mostly because I would like to know more about this connection. Can someone give an in-depth reference for that?

Thanks! —Preceding unsigned comment added by 128.135.214.108 (talk) 01:57, 3 February 2008 (UTC)

See Barnum, Saks & Szegedy CCC 2003. This technique has been applied in particular to the ordered search problem. I have not added these references to the article because I feel the application to quantum computing, while significant, is still only a minor application. 99.236.146.26 (talk) 00:08, 20 January 2009 (UTC)

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Hello, The reduction from LP with (x^2/y) form objectives to semi-definite program is simply a relaxation or approximation. It is not exact.

The key step is when you are reducing from a constraint det(D) >= 0 to D \succeq 0 (SDP inequality). The implication is strictly one way. For instance, a 2x2 negative semi-definite matrix can also have a positive determinant. It is rather misleading, please consider revising the note.

SS —Preceding unsigned comment added by 69.253.172.250 (talk) 13:04, 22 December 2008 (UTC)

Can the kernel trick be applied in this context to extend the method?

I recall hearing that in many instances where a dot product is used, one can use replace the dot product with a Mercer kernel (http://en.wikipedia.org/wiki/Mercer's_theorem)

Also see (http://en.wikipedia.org/wiki/Kernel_trick).

Furthermore, if this method can be extended with the kernel trick, can the problem still be solved in polynomial time? — Preceding unsigned comment added by 24.251.45.60 (talk) 01:37, 18 November 2012 (UTC)

The objective function

I thought semidefinite programming optimized a linear function of the variable, as in c'x. Why is there a quadratic function of x in the objective in the article? — Preceding unsigned comment added by 129.74.154.207 (talk) 16:42, 20 August 2013 (UTC)

Semidefinite programming is actually quite general, and includes the quadratic case as described (and is in fact equivalent if you cared to write it out in that way) but if you scroll down to the Equivalent Formulations section you will find the objective you are probably more used to. Zfeinst (talk) 17:18, 20 August 2013 (UTC)

Add Dimensionalities

The readability of the article would definitely benefit from adding dimensionalities to all matrices and tensors (e.g. $c$, $A$, $B$, $m$). 139.19.252.11 (talk) 14:29, 1 May 2015 (UTC)

How to solve when...?

So if I have a problem: finding

min <C,X>
subject to <Ai,X> = bi
X >= 0

How do I try to solve it if C, Ai and bi are all real matrices/values? The article says to use Interior point method, but that uses the Hessian and the gradient of some functions, so I'm lost. I also read the Augmented Lagrangian (and the Penalty method), they don't seem to work with matrices. And if the matrices of the problem are big and sparse, what method would be recommended? Thanks! W3ricardo (talk) 19:38, 22 July 2015 (UTC)

Formula error?

In section titled "motivation and definition" subsection titled "equivalent formulations" at paragraph starting with "note that if we add slack variables ... can be converted to one of the form". Problem is the form is identical to the one above it Davidglickman3141 (talk) 11:32, 22 February 2019 (UTC)

Please ignore - my mistake - didn't read carefully :-( Davidglickman3141 (talk) 11:35, 22 February 2019 (UTC)

Notation Issue in "Equivalent formulations" section?

In the "Equivalent formulations", it introduces a matrix M, but then in the equations lower uses X. I believe they are the same matrix, but it's not clear. Is that right? If so, I think we can update the notation so it replaces references to M with X. — Preceding unsigned comment added by AsdEdit (talk • contribs) 05:29, 23 January 2021 (UTC)