Is first term of my cost function convex?

Question

I have an optimization problem in the form of

[\begin{array}{l} \mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - }}Tr\left( {{\bf{AX}}} \right)\\ {\rm{Subject to Tr}}\left( {\bf{X}} \right){\rm{ - }}{P_{th}} \le 0\\ & \,\,\,\,\,\,\,\,\,\,{\bf{X}} \succ {\bf{0}} \end{array}]

Where ${\bf{X}} = {\bf{W}}{{\bf{W}}^H}$ and ${\bf{A}} = {\bf{F}}{{\bf{F}}^H}$ which ${\bf{W}} \in {C^{N \times N}}$ and ${\bf{F}} \in {C^{M \times N}}$ , $\delta \in C$ and M, N are scalars. First of all I want to know whether my cost function is convex or concave, specifically the first term. Secondly, how can I convert my optimization problem in a convex form that I could use CVX package to solve it?

Answer 1

Your objective is convex, and can be written in conic form (to be precise, a mixed SDP and SOCP). By cyclicity of the trace, and since $X$ is Hermitian $$ Tr(AX^2)=Tr(XAX^H)=Tr(XF(XF)^H) $$ Now, the square root of this term is nothing but the Euclidean norm of $vec(XF)$, which can be written in second-order conic form $$ (z,t)\in L^{N^2},\quad z=vec(XF). $$ Replace the first term in the objective by $2|\delta|t$ and that is a valid input for CVX!