Problems that semidefinit program(SDP) can solve while convex optimization cannot?

Question

In this link I found that

it(SDP) admits a new class of problem previously unsolvable by convex optimization techniques

I wonder if there are some examples that convex optimization cannot solve while SDP can solve?

Answer 1

No, it is a vague statement as (linear) semidefinite programming is a special case of convex optimization.

What you could say is that the emergence of semidefinite programming solvers allows you to solve a broader class of problems in practice, as solvers previously were limited to problem classes such as linear programs, quadratic programs, or general nonlinear convex problems over the standard positive orthant.