Constructing lattice-free gradient polyhedra in dimension two

• 成果类型: Article
• 署名作者: Paat, Joseph; Schloter, Miriam; Speakman, Emily
• 署名单位: University of British Columbia; Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Colorado System; University of Colorado Denver
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01658-7
• 发表日期: 2022
• 页码: 293-317
• 关键词: integer; sets; branch

摘要

Lattice-free gradient polyhedra can be used to certify optimality for mixed integer convex minimization models. We consider how to construct these polyhedra for unconstrained models with two integer variables under the assumption that all level sets are bounded. In this setting, a classic result of Bell, Doignon, and Scarf states the existence of a lattice-free gradient polyhedron with at most four facets. We present an algorithm for creating a sequence of gradient polyhedra, each of which has at most four facets, that finitely converges to a lattice-free gradient polyhedron. Each update requires constantly many gradient evaluations. Our updates imitate the gradient descent algorithm, and consequently, it yields a gradient descent type of algorithm for problems with two integer variables.