Continuous knapsack sets with divisible capacities

• 成果类型: Article
• 署名作者: Wolsey, Laurence A.; Yaman, Hande
• 署名单位: Universite Catholique Louvain; Ihsan Dogramaci Bilkent University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-015-0868-3
• 发表日期: 2016
• 页码: 1-20
• 关键词: integer knapsack; polyhedra; inequalities

摘要

We study two continuous knapsack sets and with integer, one unbounded continuous and 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 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 case. We also provide an extended formulation for the case. It follows that, given a specific objective function, optimization over both and can be carried out by solving 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.