The SDP problem is $$ \min_{Z \in S^{n},Y \in S^{m}} {\rm trace}(Z) +{\rm trace}(Y)\\ {\rm s.t.} \begin{bmatrix} Y & X\\ X^T & Z \end{bmatrix} \succeq 0\\ X \in C $$ Where $C$ is a convex set.But I have some difficulty in transforming it to the standard form: $$ \min_{X}\operatorname{trace}(CX) \\ \text{s.t.} \ \operatorname{trace}(A_{i}X)=b_i,i=1,...,p \\ X\succeq 0 $$ If let $\overline X= \begin{bmatrix} Y & X\\ X^T & Z \end{bmatrix}$ ,then the problem transform to $$ \min_{\overline X} {\rm trace}(\overline X)\\ {\rm s.t.} \overline X \succeq 0\\ \overline{X}_{m+i,j} =\overline{X}_{j,m+i},i=1,...,n,j=1,...,m $$ But $\overline{X}_{m+i,j} =\overline{X}_{j,m+i}$ seems can not writen as ${\rm trace}(A_i\overline{X})=b$,where $A_i$ symmetric matrice This question comes form pape: A Rank Minimization Heuristic with Application to Minimum Order System Approximation
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$\begingroup$ Perhaps you need a numerical method to create an X-bar that is diagonalizeable ? Meaning you need to create A with a linear technique within a tolerance you choose, like SVD. I'm out of my depth here but I like to try and see the bigger picture. $\endgroup$– wbgJun 11, 2022 at 16:14
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$\begingroup$ SDP Solvers have an implicit constraint that the SDP matrix is symmetric, so no additional constraints are needed. Your problem seems to be incomplete though, since the solutions is obviously $\bar{X}=0$. $\endgroup$– Brian BorchersJun 11, 2022 at 17:33
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$\begingroup$ @BrianBorchers If I add additional constraint $X \in C$,the solution seems is $\overline X =0$ too $\endgroup$– KimJun 12, 2022 at 1:37
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$\begingroup$ In the complex case, the matrix is constrained to be Hermitian positive semidefinite. $\endgroup$– Brian BorchersJun 12, 2022 at 3:52
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