Continuous knapsack sets with divisible capacities

We study two continuous knapsack sets (Formula presented.) and (Formula presented.) with (Formula presented.) integer, one unbounded continuous and (Formula presented.) bounded continuous variables in either (Formula presented.) or (Formula presented.) 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 (Formula presented.) polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable. The latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the (Formula presented.) case. We also provide an extended formulation for the (Formula presented.) case. It follows that, given a specific objective function, optimization over both (Formula presented.) and (Formula presented.) can be carried out by solving (Formula presented.) polynomial size linear programs. A further 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 of the convex hull of such sets.