Box-inequalities for quadratic assignment polytopes

@article{Jnger2001BoxinequalitiesFQ,
  title={Box-inequalities for quadratic assignment polytopes},
  author={Michael J{\"u}nger and Volker Kaibel},
  journal={Mathematical Programming},
  year={2001},
  volume={91},
  pages={175-197}
}
Abstract.Linear Programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years. Their quality has turned out to be quite good in practice. Investigations of the polytopes underlying the corresponding integer linear programming formulations (the non-symmetric and the symmetric quadratic assignment polytope) have been started during the last decade [34, 31, 21, 22]. They have lead to basic… 

The Quadratic Semi-Assignment Polytope

We study a polytope which arises from a mixed integer programming formulation of the quadratic semi-assignment problem. We introduce an isomorphic projection in order to transform the polytope to

A study of the quadratic semi-assignment polytope

Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

TLDR
The branch-and-cut framework integrated into GloMIQO 2, which addresses mixed-integer quadratically constrained quadratic programs (MIQCQP) to ε-global optimality, is documents.

A Dual Framework for Lower Bounds of the Quadratic Assignment Problem Based on Linearization

TLDR
A new and more general bounding procedure based on the dual of the linearization of Adams and Johnson is proposed and the computational results indicate that the new bound competes well with existing linearization bounds and yields a good trade off between computation time and bound quality.

The QAP-polytope and the graph isomorphism problem

TLDR
A geometric approach to solve the Graph Isomorphism (GI in short) problem and defines a partial ordering on each exponential sized family of facet defining inequalities and shows that if there exists a common minimal violated inequality for all points in the feasible region outside the QAP-polytope, then the problem can be solved in polynomial time.

On the SQAP-Polytope

TLDR
A technique is derived for investigating the SQAP-polytope that is based on projecting the (low-dimensional) polytope into a lower dimensional vector-space, where the vertices have a "more convenient" coordinate structure.

Non-convex mixed-integer nonlinear programming: A survey

Unbounded convex sets for non-convex mixed-integer quadratic programming

TLDR
It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family of unbounded convex sets.

A comprehensive review of quadratic assignment problem: variants, hybrids and applications

TLDR
QAP is NP-hard problem that is impossible to be solved in polynomial time when the problem size increases, hence heuristic and metaheuristic approaches are utilized for solving the problem instead of exact approaches because these approaches achieve quality in the solution in short computation time.

References

SHOWING 1-10 OF 46 REFERENCES

The QAP-polytope and the star transformation

Polyhedral Methods for the QAP

For many combinatorial optimization problems investigations of associated polyhedra have led to enormous successes with respect to both theoretical insights into the structures of the problems as

Polyhedral Combinatorics of QAPs with Less Objects than Locations

TLDR
Besides answering the basic questions for the aane hulls, the dimensions, and the trivial facets of the m n-polytopes, a large class of facet deening inequalities is presented, which enable us to compute optimal solutions for some hard instances from the QAPLIB for the rst time without using branch-and-bound.

On the Applicability of Lower Bounds for Solving Rectilinear Quadratic Assignment Problems in Parallel

TLDR
This work investigates the strength of lower bounds based on decomposition when applied not only at the root node of a search tree but as the bound function used in a Branch-and-Bound code solving large scale QAPs.

Solving Large Quadratic Assignment Problems in Parallel

TLDR
The combination of one of the best bound functions for a Branch-and-Bound algorithm (the Gilmore-Lawler bound) and various testing, variable binding and recalculation of bounds between branchings when used in a parallel Branch- and- Bound algorithm is investigated.

Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming

TLDR
This work compute lower bounds for a wide range of QAPs using a linear programming-based lower bound studied by Z. Drezner 1995, and produces the best known lower bound on 87% of the instances.

Lower Bounds for the Quadratic Assignment Problem Based upon a Dual Formulation

A new bounding procedure for the Quadratic Assignment Problem (QAP) is described which extends the Hungarian method for the Linear Assignment Problem (LAP) to QAPs, operating on the four dimensional

On the SQAP-Polytope

TLDR
A technique is derived for investigating the SQAP-polytope that is based on projecting the (low-dimensional) polytope into a lower dimensional vector-space, where the vertices have a "more convenient" coordinate structure.

Optimal and Suboptimal Algorithms for the Quadratic Assignment Problem

A recent article by Steinberg [6] describes a suboptimal procedure for solving what he calls the backboard wiring problem. This problem in the more general form in which it was discussed by Koopmans