What algorithm does CVX use to find the optimal solution? Is it the interior point method?
Does it support quasiconvex functions?
CVX invokes a solver to solve problems. Interior point is the most common, but not only solution method which they use, depending on the problem.
CVX does not directly support quasiconvex functions, but you can try to implement bisection per section 4.2.5 of https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf and use CVX to solve the feasibility problems.