Semidefinite Programming and Graph Equipartition

Semidefinite relaxations are used to approximate the problem of partitioning a graph into equally sized components. The relaxations extend previous eigenvalue based models, and combine semidefinite and polyhedral approaches. Computational results on graphs with several hundred vertices are given, and indicate that semidefinite relaxations approximate the equipartition problem quite well

Lower Bounds for the Quadratic Assignment Problem via Triangle Decompositions

We consider transformations of the (metric) Quadratic Assignment Problem (QAP), that exploit the metric structure of a given instance. We show in particular, how the structural properties of rectangular grids can be used to improve a given lower bound. Our work is motivated by previous research of G.S. Palubetskes, and it extends a bounding approach proposed by J. Chakrapani and J. Skorin-Kapov. Our computational results indicate that the present approach is practical and it has been applied to problems of dimension up to n = 150 . Moreover, the new approach yields by far the best lower bounds on most of the instances of metric QAPs that we considered

A Simulation Study of Fishway Design: An Example of Simulation in Environmental Problem Solving

The paper argues that modelling provides a convenient and cost effective means of characterizing and predicting the behaviour of environmental systems disturbed by man. A series of modelling criteria are suggested that aim at compelling modelling to work with current data, explicitly represent the current understanding of a system, include considerations of variability and address predictive uncertainty. Models which do so provide a rigorous framework for the comparison of alternative courses of action. As an example of the methodology, the problems associated with fishway design, highlighting the required statistical analysis and insights, are discussed. Comparison of the modelling results to those gained from operational experience with real fishways were used to validate model results and ensure model credibility. The models were then used to examine the effect of design on interactions between pools in fishways as a means of selecting the designs which minimized migration stress. ..

Trust Regions and Relaxations for the Quadratic Assignment Problem

General quadratic matrix minimization problems, with orthogonal constraints, arise in continuous relaxations for the (discrete) quadratic assignment problem (QAP). Currently, bounds for QAP are obtained by treating the quadratic and linear parts of the objective function, of the relaxations, separately. This paper handles general objectives as one function. The objectives can be both nonhomogeneous and nonconvex. The constraints are orthogonal or Loewner partial order (positive semidefinite) constraints. Comparisons are made to standard trust region subproblems. Numerical results are obtained using a parametric eigenvalue technique

Semidefinite Programming Relaxations For The Quadratic Assignment Problem

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..