We study two continuous knapsack sets $$\mathcal {Y}_\ge $$Y? and $$\mathcal {Y}_\le $$Y≤ with $$n$$n integer, one unbounded continuous and $$m$$m bounded continuous variables in either $$\ge $$? or $$\le $$≤ 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 $$2^m$$2m 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 $$\ge $$? case. We also provide an extended formulation for the $$\le $$≤ case. It follows that, given a specific objective function, optimization over both $$\mathcal {Y}_\ge $$Y? and $$\mathcal {Y}_\le $$Y≤ can be carried out by solving $$m$$m 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.
['Laurence A. Wolsey', 'Hande Yaman']
Continuous knapsack sets with divisible capacities