  It is important to design separation algorithms of low computational
complexity in mixed integer programming. We study the separation problems of
the two continuous knapsack polyhedra with divisible capacities. The two
polyhedra are the convex hulls of the sets which consist of $ n $ nonnegative
integer variables, one unbounded continuous, $ m $ bounded continuous
variables, and one linear constraint in either $ \geq $ or $ \leq $ form where
the coefficients of integer variables are integer and divisible. Wolsey and
Yaman (Math Program 156: 1--20, 2016) have shown that the polyhedra can be
described by adding the two closely related families of partition inequalities.
However, no combinatorial separation algorithm is known yet for the polyhedra.
In this paper, for each polyhedron, we provide a combinatorial separation
algorithm with the complexity of $ \mathcal{O}(nm+m\log m ) $. In particular,
we solve the separation problem for the continuous $ \geq $-knapsack polyhedron
by presenting a polynomial-time separation algorithm for the family of $ \geq
$-partition inequalities. For the continuous $ \leq $-knapsack polyhedron, we
derive the complemented partition inequalities, which dominate the $ \leq
$-partition inequalities. The continuous $ \leq $-knapsack polyhedron is
completely described with the addition of the proposed inequalities. We again
show that the family of the proposed inequalities can be separated in
polynomial time.
