Using semidefinite programming to solve the following problem

Question

I am struggling with the following problem, and wonder is SDP can help: $$\mathrm{maximize\ } \alpha_{10}+\alpha_5+(\alpha_2+\alpha_8)/2 \mathrm{\ subjected\ to\ } \mathrm{T_1}\succeq0, \mathrm{T_2}\succeq0, \mathrm{T_3}\succeq0, \mathrm{D-T}\succeq0,$$ where $\mathrm{T_1},\mathrm{T_2},\mathrm{T_3}$ and $\mathrm{T}$ are 3$\times$3, 4$\times$4, 5$\times$5 and 18$\times$18 symmetric matrices, whose components are linear combonations of $\alpha_i,i=1,..,12$, $\mathrm{D}$ is a constant matrix which depends on 9 coefficients, and $\succeq$ denotes positive-semidefiniteness.