Continuous knapsack sets with divisible capacities

We study two continuous knapsack sets Y≥ and Y≤ with n integer, one unbounded continuous and m bounded continuous variables in either ≥ or ≤ form. When the coefficients of the integer variables are integer and divisible, we show in both cases that the convex hull is the intersection of the bound constraints and 2m polyhedra arising from a continuous knapsack set with a single unbounded continuous variable. The latter polyhedra are in turn completely described by an exponential family of partition inequalities. A polynomial size extended formulation is known in the ≥ case. We provide an extended formulation for the ≤ case. It follows that, given a specific objective function, optimization over both Y≥ and Y≤ can be carried out by solving a polynomial size linear program. A consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 (after scaling) in any non-trivial facet-defining inequality.