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 $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\mathbb{R}^2$ 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 $\mathbb{R}^2$ 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 the convergence of the affine hull of the Chv\'atal-Gomory closures

Given an integral polyhedron P and a rational polyhedron Q living in the same n-dimensional space and containing the same integer points as P, we investigate how many iterations of the Chv\'atal-Gomory closure operator have to be performed on Q to obtain a polyhedron contained in the affine hull of P. We show that if P contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on n. On the other hand, we prove that if P is not full-dimensional and does not contain any integer point in its relative interior, then no finite bound on the number of iterations exists.Comment: 13 pages, 2 figures - the introduction has been extended and an extra chapter has been adde

On convergence in mixed integer programming

Let $${P \subseteq {\mathbb R}^{m+n}}$$ be a rational polyhedron, and let P I be the convex hull of $${P \cap ({\mathbb Z}^m \times {\mathbb R}^n)}$$ . 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 $${{\mathbb R}^m}$$ . 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