26 research outputs found
Note on the Complexity of the Mixed-Integer Hull of a Polyhedron
We study the complexity of computing the mixed-integer hull
of a polyhedron .
Given an inequality description, with one integer variable, the mixed-integer
hull can have exponentially many vertices and facets in . For fixed,
we give an algorithm to find the mixed integer hull in polynomial time. Given
and fixed, we compute a vertex description of
the mixed-integer hull in polynomial time and give bounds on the number of
vertices of the mixed integer hull
The distributions of functions related to parametric integer optimization
We consider the asymptotic distribution of the IP sparsity function, which
measures the minimal support of optimal IP solutions, and the IP to LP distance
function, which measures the distance between optimal IP and LP solutions. We
create a framework for studying the asymptotic distribution of general
functions related to integer optimization. There has been a significant amount
of research focused around the extreme values that these functions can attain,
however less is known about their typical values. Each of these functions is
defined for a fixed constraint matrix and objective vector while the right hand
sides are treated as input. We show that the typical values of these functions
are smaller than the known worst case bounds by providing a spectrum of
probability-like results that govern their overall asymptotic distributions.Comment: Accepted for journal publicatio
Sparse solutions of linear Diophantine equations
We present structural results on solutions to the Diophantine system
,
with the smallest number of non-zero entries. Our tools are algebraic and
number theoretic in nature and include Siegel's Lemma, generating functions,
and commutative algebra. These results have some interesting consequences in
discrete optimization
Duality for mixed-integer convex minimization
We extend in two ways the standard Karush–Kuhn–Tucker optimality conditions to problems with a convex objective, convex functional constraints, and the extra requirement that some of the variables must be integral. While the standard Karush–Kuhn–Tucker conditions involve separating hyperplanes, our extension is based on mixed-integer-free polyhedra. Our optimality conditions allow us to define an exact dual of our original mixed-integer convex problem
The Support of Integer Optimal Solutions
The support of a vector is the number of nonzero-components. We show that
given an integral matrix , the integer linear optimization
problem has an optimal solution whose support
is bounded by , where
is the largest absolute value of an entry of . Compared to previous bounds,
the one presented here is independent on the objective function. We furthermore
provide a nearly matching asymptotic lower bound on the support of optimal
solutions
Integer convex minimization by mixed integer linear optimization
Minimizing a convex function over the integral points of a bounded convex set is polynomial in fixed dimension (Grötschel et al., 1988). We provide an alternative, short, and geometrically motivated proof of this result. In particular, we present an oracle-polynomial algorithm based on a mixed integer linear optimization oracle
The distributions of functions related to parametric integer optimization
We consider the asymptotic distribution of the integer program (IP) sparsity function, which measures the minimal support of optimal IP solutions, and the IP to linear program (LP) distance function, which measures the distance between optimal IP and LP solutions. We create a framework for studying the asymptotic distributions of general functions related to integer optimization. There has been a significant amount of research focused on the extreme values that these functions can attain. However, less is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right-hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions.Read More: https://epubs.siam.org/doi/10.1137/19M127595