Split cuts are arguably the most effective class of cutting planes within a branch-and-cut framework for solving general Mixed-Integer Programs (MIP). Sparsity, on the other hand, is a common characteristic of MIP problems, and it is an important part of why the simplex method works so well inside branch-and-cut. In this work, we evaluate the … Read more
Given a point x inside a polytope P and a direction d, the projection of x along d asks to find the maximum step length t such that x+td is feasible; we say x+td is a pierce point because it belongs to the boundary of P. We address this projection sub-problem with arbitrary interior points … Read more
We give a variant of Basu–Hildebrand–Molinaro’s approximation theorem for continuous minimal valid functions for Gomory–Johnson’s infinite group problem by piecewise linear two-slope extreme functions [Minimal cut-generating functions are nearly extreme, IPCO 2016]. Our theorem is for piecewise linear minimal valid functions that have only rational breakpoints (in 1/q Z for some q ∈ N) and … Read more
We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that defines the convex hull of the integer points in K that are not lexicographically smaller than x. The family of … Read more
We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of a set of binary points satisfying a number of multilinear equations. Running intersection inequalities are a family of facet-defining inequalities for the … Read more
A 0/1 matrix has the Consecutive Ones Property if a permutation of its columns exists such that the ones appear consecutively in each row. In many applications, one has to find a matrix having the consecutive ones property and optimizing a linear objective function. For this problem, literature proposes, among other approaches, an Integer Linear … Read more
In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1–20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. … Read more
The application of decision diagrams in combinatorial optimization has proliferated in the last decade. In recent years, authors have begun to investigate how to utilize not one, but a set of diagrams, to model constraints and objective function terms. Optimizing over a collection of decision diagrams, the problem we refer to as the consistent path … Read more
As an alternative to traditional integer programming (IP), decision diagrams (DDs) provide a new solution technology for discrete problems based on their combinatorial structure and dynamic programming representation. While the literature mainly focuses on the competitive aspects of DDs as a stand-alone solver, we investigate their complementary role by studying IP techniques that can be … Read more
We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which input materials are mixed in intermediate pools, with the outputs of these pools further mixed to make output products meeting given attribute percentage requirements. Our relaxations are derived by considering a set which arises from the formulation by … Read more