TY - JOUR

T1 - Constructing lattice-free gradient polyhedra in dimension two

AU - Paat, Joseph

AU - Schlöter, Miriam

AU - Speakman, Emily

N1 - Publisher Copyright:
© 2021, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2022/3

Y1 - 2022/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85106271723&partnerID=8YFLogxK

U2 - 10.1007/s10107-021-01658-7

DO - 10.1007/s10107-021-01658-7

M3 - Article

AN - SCOPUS:85106271723

SN - 0025-5610

VL - 192

SP - 293

EP - 317

JO - Mathematical Programming

JF - Mathematical Programming

IS - 1-2

ER -