Integral Farkas type lemmas for systems with equalities and inequalities

A central result in the theory of integer optimization states that a system of linear diophantine equations Ax = b has no integral solution if and only if there exists a vector in the dual lattice, y T A integral such that y T b is fractional. We extend this result to systems that both have equations and inequalities {Ax = b, Cx d}. We show that a certificate of integral infeasibility is a linear system with rank(C) variables containing no integral point.

Genetic Predictors of Reproduction

Genetic predictions in the form of expected progeny differences (EPDs) represent the beef industry\u27s most powerful source of information for selection and genetic improvement. While EPDs are widely available for traits associated with calving ease, growth, milk and carcass traits, EPDs for reproductive traits are limited. Given the relative economic importance, development of EPDs for reproductive traits should be a priority for the beef industry. Fortunately, recent breakthroughs in analytical procedures have opened the door for potential development of genetic predictions for reproductive traits

An analysis of mixed integer linear sets based on lattice point free convex sets

Split cuts are cutting planes for mixed integer programs whose validity is derived from maximal lattice point free polyhedra of the form $S:=\{x : \pi_0 \leq \pi^T x \leq \pi_0+1 \}$ called split sets. The set obtained by adding all split cuts is called the split closure, and the split closure is known to be a polyhedron. A split set $S$ has max-facet-width equal to one in the sense that $\max\{\pi^T x : x \in S \}-\min\{\pi^T x : x \in S \} \leq 1$. In this paper we consider using general lattice point free rational polyhedra to derive valid cuts for mixed integer linear sets. We say that lattice point free polyhedra with max-facet-width equal to $w$ have width size $w$. A split cut of width size $w$ is then a valid inequality whose validity follows from a lattice point free rational polyhedron of width size $w$. The $w$-th split closure is the set obtained by adding all valid inequalities of width size at most $w$. Our main result is a sufficient condition for the addition of a family of rational inequalities to result in a polyhedral relaxation. We then show that a corollary is that the $w$-th split closure is a polyhedron. Given this result, a natural question is which width size $w^*$ is required to design a finite cutting plane proof for the validity of an inequality. Specifically, for this value $w^*$, a finite cutting plane proof exists that uses lattice point free rational polyhedra of width size at most $w^*$, but no finite cutting plane proof that only uses lattice point free rational polyhedra of width size smaller than $w^*$. We characterize $w^*$ based on the faces of the linear relaxation

Wildlife friendly agriculture: which factors do really matter? A genetic study on field vole

The distribution of genetic differentiation and the directions of gene flow were determined mainly by landscape factors: thus the expectation that organic fields act as genetic reservoir was not met. The fact that agricultural area presented more sub-populations than the undisturbed one, together with the importance of connectivity and habitat size in shaping gene flow and genetic differentiation, shows that switching to organic farming might not be enough to ensure the conservation of species in the agricultural environment. These results emphasise the need to include landscape structure in management policies