Improved Balas and Mazzola Linearization for Quadratic 0-1 Programs with Application in a New Cutting Plane Algorithm

Balas and Mazzola linearization (BML) is widely used in devising cutting plane algorithms for quadratic 0-1 programs. In this article, we improve BML by first strengthening the primal formulation of BML and then considering the dual formulation. Additionally, a new cutting plane algorithm is proposed

A polyhedral approach to quadratic assignment problem

Ankara : Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent University, 1994.Thesis (Master's) -- Bilkent University, 1994.Includes bibliographical references.In this thesis, Quadratic Assignment Problem is considered. Since Quadratic Assignment Problem is JVP-bard, no polynomial time exact solution method exists. Proving optimality of solutions to Quadratic Assignment Problems has been limited to instances of small dimension. In this study, Quadratic Assignment Problem is handled from a polyhedral point of view. A graph theoretic formulation of the problem is presented. Later, Quadratic Assignment Polytope is defined and subsets of valid equalities and inequalities for Quadratic Assignment Polytope is given. Finally, results of the experiments with a polyhedral cutting plane algorithm using the new formulation is also presented.Köksaldı, Ahmet Sertaç MuratM.S

Continuous Optimization Methods for the Quadratic Assignment Problem

In this dissertation we have studied continuous optimization techniques as they are applied in nonlinear 0-1 programming. Specifically, the methods of relaxation with a penalty function have been carefully investigated. When the strong equivalence properties hold, we are guaranteed an integer solution to the original 0-1 problem. The quadratic assignment problem (QAP) possesses such properties and consequently we have developed an algorithm for the QAP based on the method of relaxation using the quadratic penalty function. In our algorithm we have applied two pre-conditioning techniques that enables us to devise a scheme to find a good initial point and hence obtain good solutions to the QAP. Furthermore, we have shown how quadratic cuts can be used to improve on the current solutions. Extensive numerical results on several sets of QAP test problems (including the QAPLIB) have been reported and these results show our algorithm produces good solutions for certain classes of problems in a small amount of time

Quadratic assignment problem : linearizations and polynomial time solvable cases

Cataloged from PDF version of article.The Quadratic Assignment Problem (QAP) is one of the hardest combinatorial optimization problems known. Exact solution attempts proposed for instances of size larger than 15 have been generally unsuccessful even though successful implementations have been reported on some test problems from the QAPLIB up to size 36. In this dissertation, we analyze the binary structure of the QAP and present new IP formulations. We focus on “flow-based” formulations, strengthen the formulations with valid inequalities, and report computational experience with a branch-and-cut algorithm. Next, we present new classes of instances of the QAP that can be completely or partially reduced to the Linear Assignment Problem and give procedures to check whether or not an instance is an element of one of these classes. We also identify classes of instances of the Koopmans-Beckmann form of the QAP that are solvable in polynomial time. Lastly, we present a strong lower bound based on Bender’s decomposition.Erdoğan, GüneşPh.D

PGAGrid: A Parallel Genetic Algorithm of Fine-Grained implemented on GPU to find solutions near the optimum to the Quadratic Assignment Problem (QAP)

This work consists in implementing a fine-grained parallel genetic algorithm improved with a greedy 2-opt heuristic to find near-optimal solutions to the Quadratic Assignment Problem (QAP). The proposed algorithm was fully implemented on Graphics Processing Units (GPUs). A two-dimensional GPU grid of size 8x8 defines the population of the genetic algorithm (set of permutations of the QAP), and each GPU block consists of n GPU threads, where n is the size of the QAP. Each GPU block was used to represent the chromosome of a single individual, and each GPU thread represents a gene of such chromosome. The proposed algorithm was tested on a subset of the standard QAPLIB data set. Results show that this implementation is able to find good solutions for large QAP instances in few parallel iterations of the evolutionary process.Resumen: Este trabajo consiste en implementar un algoritmo genético paralelo de grano fino mejorado con una heurística 2-opt voraz para encontrar soluciones cercanas al óptimo al problema de Asignación Cuadrática (QAP). El algoritmo propuesto fue completamente implementado sobre Unidades de Procesamiento Gráfico (GPUs). Una retícula GPU bidimensional de tamaño 8×8 define la población del algoritmo genético (conjunto de permutaciones del QAP) y cada bloque GPU consiste de n hilos GPU donde n es el tamaño del QAP. Cada bloque GPU fue utilizado para representar el cromosoma de un solo individuo y cada hilo GPU representa un gen de tal cromosoma. El algoritmo propuesto fue comprobado sobre un subconjunto de problemas de la librería estándar QAPLIB. Los resultados muestran que esta implementación es capaz de encontrar buenas soluciones para grandes instancias del QAP en pocas iteraciones del proceso evolutivo.Doctorad