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I understand that semidefinite programming (SDP) is a subset of conic programming (CP). According to Boyd's Convex Optimization book section 4.6.1, the conic problem in inequality form is written as

$$ \begin{array}{ll} \text{minimize} & c^Tx \\ \text{subject to} & Fx + g \preceq_K 0 \end{array} $$

and in the next section, 4.6.2, a semidefinite program in inequality form is written as

$$ \begin{array}{ll} \text{minimize} & c^Tx \\ \text{subject to} & x_1 A_1 + \cdots + x_n A_n \preceq B \end{array} $$

So if we have set up a problem as a semidefinite program with known $A_i,B$ how can we rewrite it in the conic form?

This wiki page gives an idea of how the SDP is represented by vectors like in the CP. However, it is not clear to me how you rewrite the SDP in the CP form.

Nick
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  • $\sum_i x_i A_i - B \preceq_K 0$ where $K$ is the semidefinite cone of the right dimension. – Michal Adamaszek Oct 13 '23 at 07:30
  • This is one way to rearrange the SDP, but it does not rewrite the problem in the conic form where vector $g$ is used instead of a matrix variable $B$ – Nick Oct 13 '23 at 08:12
  • Of course you might want to flatten the $n\times n$ matrices down to vectors of length $n(n+1)/2$. That's just a matter of notation and how you represent the cone. It is more of a question you might have to consider when you actually want to input the problem into the API of some solver. On the mathematical level you can think of a matrix cone or the flattened cone just as well. – Michal Adamaszek Oct 13 '23 at 08:19
  • When you say flattened cone, what do you mean? The SDP constraints, from what I know, do form a (convex) cone. However, I am not sure how we represent these constraints (the LMI) in the inequality conic form – Nick Oct 13 '23 at 11:58
  • I wish I could help but I really don't understand the question. I think you are going in circles around some imagined notational issue. The notation $X \preceq Y$ means the same thing for the SDP cone as for all other cones. That is a shorthand for $Y-X\in K$. – Michal Adamaszek Oct 14 '23 at 14:36
  • Maybe all you need is that $n\times n $ matrix can be written as a vector of length $n^2$. – Michal Adamaszek Oct 14 '23 at 14:53
  • @MichalAdamaszek I appreciate your patience with me. You may be right that I am going in circles but hopefully this clarifies my question.

    The cone constraint with $Fx+g$ is a vector $\in R^n$. So the constraint checks the order of a vector.

    The SDP constraint is a matrix $\in R^{n \times n}$. So the constraint checks the ordering of the principal minors???

    How does that SDP constraint, which says all principal minors must be positive, get represented by a vector constraint in the conic program?

     – Nick Oct 15 '23 at 06:51
  • @MichalAdamaszek ok one more reference I think might clarify my question. These lectures from Ben-Tal show something related to my question on pg 162. The notational convention is mentioned. But I still don't clearly see how we can go between the conic program and the SDP – Nick Oct 15 '23 at 07:59
  • After reading more of the lectures I mentioned above, I think equation 3.1.1 partially answers my question. – Nick Oct 16 '23 at 00:05

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