  We explore the dual problem of the convex roof construction by identifying it
as a linear semi-infinite programming (LSIP) problem. Using the LSIP theory, we
show the absence of a duality gap between primal and dual problems, even if the
entanglement quantifier is not continuous, and prove that the set of optimal
solutions is non-empty and bounded. In addition, we implement a central
cutting-plane algorithm for LSIP to quantify entanglement between three qubits.
The algorithm has global convergence property and gives lower bounds on the
entanglement measure for non-optimal feasible points. As an application, we use
the algorithm for calculating the convex roof of the three-tangle and
$\pi$-tangle measures for families of states with low and high ranks. As the
$\pi$-tangle measure quantifies the entanglement of W states, we apply the
values of the two quantifiers to distinguish between the two different types of
genuine three-qubit entanglement.
