# Primal-Dual pair in SDP

Let's we have a primal model like

$\max~~ x + \langle I, Z \rangle$

$s.t. ~~~Ax + y I - Z \preceq B$

$~~~~~~~~~Z \succeq 0, ~X \geq 0, ~~y ~free$

where $A, B \in {\mathbb R^{n \times n}}$. The capital letters means they are in the matrix form and small letters means a vector form. Introducing the (matrix) variable $Y$ for the dual, one can find the dual model as

$\min ~~\langle B, Y \rangle$

$s.t.~~~\langle A, Y \rangle \geq 1$

$~~~~~~~~~\langle I, Y \rangle = 0$

$~~~~~~~~~-Y \geq 1$

$~~~~~~~~~Y \succeq 0$

The last constraints and the positive semidefiniteness of variable $Y$ does not make sense together. Have I written the dual in the right form and if not what would be the correct one?

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Actually, your primal model is ill-posed. Your objective function is not a scalar. –  Michael Grant Apr 23 '13 at 20:19