# Is this parametrized semidefinite program convex?

I am considering an optimization problem of the form: $$\begin{split} f(s) &= \min_{X} \mathrm{tr}(C(s)X) \\ &\;\;\;\;\;\;\;\;\;\;\; X \ge 0, \\ &\;\;\;\;\;\;\;\;\;\;\; \mathrm{tr}(A_iX) = a_i, \;\; 1 \le i \le M, \end{split}$$ where the minimization is over $n\times n$ Hermitian matrices $X$. Further, $A_i$ for $1 \le i \le M$ denote some $n\times n$ Hermitian matrices which together with $a_i \in \mathbb{R}$ determine linear constraints on $X$. Finally, the matrix-valued function $C(s)$ is of the block form: $$C(s) = \left( \begin{array}{cc} C_{1}(s) & 0 \\ 0 & 0\end{array} \right),$$ where the upper left block $C_1(s)$ is of size $(n_1 + 1) \times (n_1 + 1)$ for some $n_1 < n$, and is given by: $$C_1(s) = \left( \begin{array}{ccccc} I_{n_1\times n_1} & -ic \mathbb{I}_{n_1\times n_1} & \cdot & \cdot \\ i c \mathbb{I}_{n_1\times n_1} & \cdot & -i \frac{s}{2} \mathbb{I}_{n_1\times n_1} & \cdot \\ \cdot & i \frac{s}{2} \mathbb{I}_{n_1\times n_1} & \cdot & \cdot \\ \cdot & \cdot & \cdot & s^2\end{array} \right).$$ Here, $c \in \mathbb{R}$, $I_{n_1\times n_1}$ is the $n_1 \times n_1$ matrix of ones and $\mathbb{I}_{n_1\times n_1}$ denotes the $n_1 \times n_1$ identity matrix (whereas all entries indicated by $\cdot$ vanish).

Can it be shown that $f(s)$ is convex?

If not, which further requirements has the optimization to fulfill in order to guarantee convexity of $f(s)$?

• Thanks a lot for your answer! Is it difficult to see that this proofs convexity in $s$? If yes, can you give me a reference for this sort of statements? Does convexity also hold for the dual optimum as a function of $s$ (where the $s$-dependence appears in the constraints)? – Marc May 14 '18 at 12:49