160 research outputs found
Convex quadratic sets and the complexity of mixed integer convex quadratic programming
In pure integer linear programming it is often desirable to work with
polyhedra that are full-dimensional, and it is well known that it is possible
to reduce any polyhedron to a full-dimensional one in polynomial time. More
precisely, using the Hermite normal form, it is possible to map a non
full-dimensional polyhedron to a full-dimensional isomorphic one in a
lower-dimensional space, while preserving integer vectors. In this paper, we
extend the above result simultaneously in two directions. First, we consider
mixed integer vectors instead of integer vectors, by leveraging on the concept
of "integer reflexive generalized inverse." Second, we replace polyhedra with
convex quadratic sets, which are sets obtained from polyhedra by enforcing one
additional convex quadratic inequality. We study structural properties of
convex quadratic sets, and utilize them to obtain polynomial time algorithms to
recognize full-dimensional convex quadratic sets, and to find an affine
function that maps a non full-dimensional convex quadratic set to a
full-dimensional isomorphic one in a lower-dimensional space, while preserving
mixed integer vectors. We showcase the applicability and the potential impact
of these results by showing that they can be used to prove that mixed integer
convex quadratic programming is fixed parameter tractable with parameter the
number of integer variables. Our algorithm unifies and extends the known
polynomial time solvability of pure integer convex quadratic programming in
fixed dimension and of convex quadratic programming
Mixed-integer Quadratic Programming is in NP
Mixed-integer quadratic programming is the problem of optimizing a quadratic
function over points in a polyhedral set where some of the components are
restricted to be integral. In this paper, we prove that the decision version of
mixed-integer quadratic programming is in NP, thereby showing that it is
NP-complete. This is established by showing that if the decision version of
mixed-integer quadratic programming is feasible, then there exists a solution
of polynomial size. This result generalizes and unifies classical results that
quadratic programming is in NP and integer linear programming is in NP
Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane
We complete the complexity classification by degree of minimizing a
polynomial over the integer points in a polyhedron in . Previous
work shows that optimizing a quadratic polynomial over the integer points in a
polyhedral region in can be done in polynomial time, while
optimizing a quartic polynomial in the same type of region is NP-hard. We close
the gap by showing that this problem can be solved in polynomial time for cubic
polynomials.
Furthermore, we show that the problem of minimizing a homogeneous polynomial
of any fixed degree over the integer points in a bounded polyhedron in
is solvable in polynomial time. We show that this holds for
polynomials that can be translated into homogeneous polynomials, even when the
translation vector is unknown. We demonstrate that such problems in the
unbounded case can have smallest optimal solutions of exponential size in the
size of the input, thus requiring a compact representation of solutions for a
general polynomial time algorithm for the unbounded case
Relaxations of mixed integer sets from lattice-free polyhedra
This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitel
On convergence in mixed integer programming
Let be a rational polyhedron, and let P I be the convex hull of . We define the integral lattice-free closure of P as the set obtained from P by adding all inequalities obtained from disjunctions associated with integral lattice-free polyhedra in . We show that the integral lattice-free closure of P is again a polyhedron, and that repeatedly taking the integral lattice-free closure of P gives P I after a finite number of iterations. Such results can be seen as a mixed integer analogue of theorems by Chvátal and Schrijver for the pure integer case. One ingredient of our proof is an extension of a result by Owen and Mehrotra. In fact, we prove that for each rational polyhedron P, the split closures of P yield in the limit the set P
The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization
With the goal of obtaining strong relaxations for binary polynomial
optimization problems, we introduce the pseudo-Boolean polytope defined as the
convex hull of the set of binary points satisfying a collection of equations
containing pseudo-Boolean functions. By representing the pseudo-Boolean
polytope via a signed hypergraph, we obtain sufficient conditions under which
this polytope has a polynomial-size extended formulation. Our new framework
unifies and extends all prior results on the existence of polynomial-size
extended formulations for the convex hull of the feasible region of binary
polynomial optimization problems of degree at least three
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