How to calculate the conditional min-entropy via a semidefinite program?

I am trying to formulate the calculation of conditional min-entropy as a semidefinite program. However, so far I have not been able to do so. Different sources formulate it differently. For example, in this highly influential paper, it has been formulated as:

$$H_{text{min}}(A|B)_rho = – underset{sigma_B} {text{inf}} D_{infty}(rho_{AB} | id_A otimes sigma_B)
$$
Where $$rho_{AB} in mathcal{H_A otimes H_B}, sigma in mathcal{H_B}$$ and
$$D_{infty}(tau | tau’) = text{inf} {lambda in
mathbb{R}: tau leq 2^{lambda} tau’ }$$

How do I formulate it into a semidefinite program? It is possible as is mentioned in this lecture.

A possible SDP program is given in Watrous’s lecture:

$$text{maximize}: <rho, X>$$
$$text{subject to}$$
$$Tr_X{X} == mathcal{1}_Y$$
$$X in text{Pos}(X otimes Y)$$

How do I write it in CVX or any other optimization system?

cvx_begin sdp
variable X(2, 2) hermitian

minimize(trace(id' * X)) % id is eye(2)
subject to

kron(id, X) >= rho_ab % the tensor product of two density matrices a, b
X >= 0

cvx_end