Euler Polytopes and Convex Matroid Optimization

The simplex and primal-dual interior point methods are the most computationally successful algorithms for linear optimization. The behaviour of the largest possible diameter of a polytope is closely connected with the theory of the simplex method. We introduce a novel family of polytopes strengthening bounds relevant to combinatorial optimization and convex matroid optimization. Del Pia and Michini recently improved the upper bound of kd due to Kleinschmidt and Onn for the largest possible diameter of the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. We introduce Euler polytopes which include a family of lattice polytopes with diameter (k+1)d/2, and thus reduce the gap between the lower and upper bounds. In addition, we highlight connections between Euler polytopes and a parameter studied in convex matroid optimization, and strengthen the lower and upper bounds for this parameter. 

Based on joint work with George Manoussakis, Orsay, and Shmuel Onn, Technion.