Finite semidefinite primal-dual pair

Fix integers $n\ge 0$ and $m\ge 0$, Hermitian matrices $C,A_1,\dots ,A_m\in {\mathbb{C}}^{n\times n}$, and $b=(b_1,\dots ,b_m)\in {\mathbb{R}}^m$. The primal semidefinite program is $\sup \{\mathrm{tr}(CX):X\in {\mathbb{C}}^{n\times n},X⪰0,\mathrm{tr}(A_{j}X)\le b_j(1\le j\le m)\},$ and its declared dual program is $\inf \{b\cdot y:y\in {\mathbb{R}}^m,y_j\ge 0,{\sum }_{j=1}^m{y}_j{A}_j-C⪰0\}\mathrm{.}$ Here $X⪰0$ and $D⪰0$ mean positive semidefinite in the finite-dimensional quadratic-form sense. By Trace of a product of Hermitian matrices is real, all displayed trace objectives and constraints are real. Primal and dual feasibility, boundedness, attainment, trace bounds, sparsity, and normalizations such as $A_1=I,b_1=R$ are separate promises and do not follow from this definition.

The sign convention is maximize for the primal, minimize for the dual, $\le$ for primal scalar constraints, nonnegative dual multipliers, and dual slack ${\sum }_j{y}_j{A}_j-C⪰0$.