Continuous knapsack sets with divisible capacities

Abstract

We study two continuous knapsack sets $$\mathcal {Y}_\ge $$ and $$\mathcal {Y}_\le $$ with $$n$$ integer, one unbounded continuous and $$m$$ 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 $$2^m$$ 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 $$\mathcal {Y}_\ge $$ and $$\mathcal {Y}_\le $$ can be carried out by solving $$m$$ 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.