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Semidefinite Programming Relaxations For The Quadratic Assignment Problem
, 1998
"... Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP re ..."
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Cited by 82 (23 self)
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Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e. the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primal-dual interior-point methods. For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting th...
Branch-and-Cut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem
- Comput. Optim. Appl
, 1999
"... The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of the control and system theory in the last few years. This inequality permits to reduce in a elegant way various problems of robust control into its form. However, on ..."
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Cited by 20 (1 self)
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The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of the control and system theory in the last few years. This inequality permits to reduce in a elegant way various problems of robust control into its form. However, on the contrary of the Linear Matrix Inequality (LMI) which can be solved by interior-point-methods, the BMI is a computationally difficult object in theory and in practice. This article improves the branch-and-bound algorithm of Goh, Safonov and Papavassilopoulos (1995) by applying a better convex relaxation of the BMI Eigenvalue Problem (BMIEP), and proposes new Branch-and-Bound and Branch-and-Cut Algorithms. Numerical experiments were conducted in a systematic way over randomly generated problems, and they show the robustness and the efficiency of the proposed algorithms. Keywords: Bilinear Matrix Inequality, Branch-and-Cut Algorithm, Convex Relaxation, Cut Polytope. y Author supported b...
INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION
, 1995
"... Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivale ..."
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Cited by 16 (9 self)
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Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.
A hybrid genetic algorithm for the quadratic assignment problem
- in GECCO-2000: Proceedings of the Genetic and Evolutionary Computation Conference
, 2000
"... A heuristic technique that combines a genetic algorithm with a Tabu Search algorithm is applied to the Quadratic Assignment Problem (QAP). The hybrid algorithm improves the results obtained through the application of each of these algorithms separately. The QAP is a NP-hard problem and instances of ..."
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Cited by 14 (0 self)
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A heuristic technique that combines a genetic algorithm with a Tabu Search algorithm is applied to the Quadratic Assignment Problem (QAP). The hybrid algorithm improves the results obtained through the application of each of these algorithms separately. The QAP is a NP-hard problem and instances of size n> 15 are still considered intractable. The results of our experiments suggest that CHC combined with TS (CHC+TS), and a TS with elitist backtracking algorithm are able to obtain good near optimal solutions within 0.75 % of the best-known solutions. CHC+TS produces the best-known solution in 12 of the 16 QAPLIB problems tested, where n ranges from 10 to 256. 1
Tight QAP Bounds Via Linear Programming
, 1998
"... . Lower bounds for the quadratic assignment problem (QAP) tend to deteriorate rapidly with the size of the QAP. Recently, Resende, Ramakrishnan, and Drezner (1995) computed a linear programming based lower bound for the QAP using an interior point algorithm for linear programming to solve the linear ..."
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Cited by 4 (0 self)
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. Lower bounds for the quadratic assignment problem (QAP) tend to deteriorate rapidly with the size of the QAP. Recently, Resende, Ramakrishnan, and Drezner (1995) computed a linear programming based lower bound for the QAP using an interior point algorithm for linear programming to solve the linear programming relaxation of a classical integer programming formulation of the QAP. That linear program can be viewed as a two body interaction formulation. Those bounds were found to be the tightest for a large number of instances from QAPLIB, a library of QAP test problems. In this paper, we apply the same interior point approach to compute lower bounds derived from the three body interaction formulation of Ramachandran and Pekny (1996). All instances from QAPLIB, having dimension up to n = 12, were solved. The new approach produces tight lower bounds (lower bounds equal to the optimal solution) for all instances tested. Attempts to solve the linear programming relaxations with CPLEX (prima...
Solution Methods for the Balancing of Jet Turbines
- Computers & Operations Research
, 1998
"... Turbine balancing is an important and regular maintenance operation at airline companies. Because of manufacturing inaccuracies, variations occur in the weights of the blades that can, in turn, lead to significant out-of-balance forces on the engine. The overall time required for balancing can be si ..."
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Cited by 3 (0 self)
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Turbine balancing is an important and regular maintenance operation at airline companies. Because of manufacturing inaccuracies, variations occur in the weights of the blades that can, in turn, lead to significant out-of-balance forces on the engine. The overall time required for balancing can be significantly decreased if the best placement of turbine blades is first determined mathematically. This problem is formulated as a variation of the quadratic assignment problem and a number of solution schemes are investigated. A neighbourhood search algorithm is found to significantly outperform the other solution approaches when applied to data from a major South Pacific airline. The neighbourhood search algorithm can be combined with various strategies to initialize starting points. The use of starting points obtained from a Lagrangean dual scheme is shown to improve results for large problems. Scope and Purpose The balancing of turbines is a regular maintenance operation at airlines. Ty...
Optimization-Based Mapping Framework for Parallel Applications
- University of the Basque Country
, 2010
"... The optimal mapping of tasks of a parallel program onto nodes of a parallel computing system has a remarkable impact on application performance. In this paper we propose an optimization framework to solve the mapping problem, which takes into account the communication matrix of the application and a ..."
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Cited by 1 (0 self)
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The optimal mapping of tasks of a parallel program onto nodes of a parallel computing system has a remarkable impact on application performance. In this paper we propose an optimization framework to solve the mapping problem, which takes into account the communication matrix of the application and a cost matrix that depends on the topology of the parallel system. This cost function can be a distance matrix (the classic approach), or can take into account other considerations. We propose a novel definition of the cost criterion, applicable to torus networks, that tries to distribute traffic evenly over the different axes: the Traffic Distribution criterion. As the mapping problem is a particular instance of the Quadratic Assignment Problem, we can apply any QAP solver to the mapping. In particular, in this work we use GRASP. Using simulation, we test the performance of the optimization-based mappings, and compare it with that of trivial mappings (consecutive, random), in two different environments: static (a single applications uses all system resources all the time) and dynamic (applications are managed by a scheduler, and use and free resources as needed), always using a system with a 2D topology and real application traffic. In both environments optimization-based mappings with the TD criterion provides excellent performance. 1 1
Solving the Multi-objective Quadratic Assignment Problem Using a fast messy Genetic Algorithm
"... Abstract- The multi-objective quadratic assignment problem is an NP-complete problem with a multitude of real-world applications. The specific application addressed in this paper is the minimization of communication flows in a heterogenous mix of unmanned aerial vehicles. Developed is a multi-object ..."
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Abstract- The multi-objective quadratic assignment problem is an NP-complete problem with a multitude of real-world applications. The specific application addressed in this paper is the minimization of communication flows in a heterogenous mix of unmanned aerial vehicles. Developed is a multi-objective approach to solving the general mQAP for this UAV application. The combinatoric nature of this problem calls for a stochastic search algorithm; moreover, the Multi-Objective fast messy Genetic Algorithm (MOMGA-II) [22] is used for experimentation. Results indicate that much of the Pareto optimal points are found. 1
Parallelization of the Quadratic Assignment Problem on the Cell
"... Abstract—The mapping problem involves the assig-nation of a set of tasks of a parallel application onto a set of computational nodes. This problem can be for-mulated as an instance of the Quadratic Assignment Problem (QAP), being this one of the most difficult combinatorial optimization problems. Th ..."
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Abstract—The mapping problem involves the assig-nation of a set of tasks of a parallel application onto a set of computational nodes. This problem can be for-mulated as an instance of the Quadratic Assignment Problem (QAP), being this one of the most difficult combinatorial optimization problems. The solution for the QAP is a permutation that minimizes an ob-jective function. This paper proposes two algorithms to generate sequences of permutations with distance one between them, and to map each permutation of size n with a number in the range {1,n!}. This is very important because, once the objective function for a given permutation has been computed, the computa-tion for the next permutation is very simple. Once we have a fast solution to the QAP, we can use it to solve non-trivial cases of mapping problems, in this case checking the complete permutation space.
