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I am considering an optimization problem of the form: \begin{equation} \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} \end{equation} 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: \begin{equation} C(s) = \left( \begin{array}{cc} C_{1}(s) & 0 \\ 0 & 0\end{array} \right), \end{equation} 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: \begin{equation} 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). \end{equation} 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 $f(s)$?

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Yes, this is convex because the objective function and all constraints are convex.

The objective function is affine (linear), which is convex. The semidefinjite constraint on X is convex. The trace equality constraint on X is affine (linear), and therefore is convex.

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  • $\begingroup$ 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)? $\endgroup$ – Marc May 14 '18 at 12:49
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    $\begingroup$ I answered a different question than you intended. I addressed the question of whether the optimization problem, for a fixed value of s, is convex. You apparently are viewing f(s) as the optimal objective value of the optimization problem for that value of s; and then inquiring as to the convexity of f(s) as a function of s. $\endgroup$ – Mark L. Stone May 14 '18 at 13:09

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