Articles of semidefinite programming

If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq \eta \leq e} \min_{w,\xi} &\frac{\beta}{2}\Vert w\Vert^2 + e^T \xi+ e^T (e-\eta)\cr \text{s.t} &\xi \geq 0, \xi \geq diag(\eta) (e-YX^T w)\cr =\min_{0\leq \eta \leq e}\max_{0\leq […]

Adding Elements to Diagonal of Symmetric Matrix to Ensure Positive Definiteness.

I have a symmetric matrix $A$, which has zeroes all along the diagonal i.e. $A_{ii}=0$. I cannot change the off diagonal elements of this matrix, I can only change the diagonal elements. I need this matrix to be positive definite. One way to ensure this is as follows: Let $\lambda’$ by the absolute value of […]

Maximize $\langle \mathrm A , \mathrm X \rangle$ subject to $\| \mathrm X \|_2 \leq 1$

Given $\mathrm A \in \mathbb R^{m \times n}$, $$\begin{array}{ll} \text{maximize} & \langle \mathrm A , \mathrm X \rangle\\ \text{subject to} & \| \mathrm X \|_2 \leq 1\end{array}$$ $\| \mathrm X \|_2 \leq 1$ is equivalent to $\sigma_{\max} (\mathrm X ) \leq 1$, which is equivalent to $\lambda_{\max} (\mathrm X^T \mathrm X) \leq 1$. Hence, $$1 […]

dual problem of a Semidefinite programming in a non-standard forme

I have a problem with calculating the dual problem of : $$ \mbox{Minimize } tr(Y) + \frac{1}{\eta} tr(Z) $$ $$ \begin{pmatrix} Y & X \\ X & Z+\varepsilon I \end{pmatrix} \succeq 0 \mbox{, } % \begin{pmatrix} I & X \\ X & Z \end{pmatrix} \succeq 0$$ $$ \langle A_i,X\rangle = 0\mbox{, } i=1,\cdots,m \mbox{, } […]