A new family of facet defining inequalities for the maximum edge-weighted clique problem

This paper considers a family of cutting planes, recently developed for mixed 0–1 polynomial programs and shows that they define facets for the maximum edge-weighted clique problem. There exists a polynomial time exact separation algorithm for these inequalities. The result of this paper may contribute to the development of more efficient algorithms for the maximum edge-weighted clique problem that use cutting planes

New solution approaches for the quadratic assignment problem

MSc., Faculty of Science, University of the Witwatersrand, 2011A vast array of important practical problems, in many di erent elds, can be modelled and solved as quadratic assignment problems (QAP). This includes problems such as university campus layout, forest management, assignment of runners in a relay team, parallel and distributed computing, etc. The QAP is a di cult combinatorial optimization problem and solving QAP instances of size greater than 22 within a reasonable amount of time is still challenging. In this dissertation, we propose two new solution approaches to the QAP, namely, a Branch-and-Bound method and a discrete dynamic convexized method. These two methods use the standard quadratic integer programming formulation of the QAP. We also present a lower bounding technique for the QAP based on an equivalent separable convex quadratic formulation of the QAP. We nally develop two di erent new techniques for nding initial strictly feasible points for the interior point method used in the Branch-and-Bound method. Numerical results are presented showing the robustness of both methods

A cut-and-branch algorithm for the quadratic knapsack problem

The Quadratic Knapsack Problem (QKP) is a well-known NP-hard combinatorial optimisation problem, with many practical applications. We present a ‘cut-and-branch’ algorithm for the QKP, in which a cutting-plane phase is followed by a branch-and-bound phase. The cutting-plane phase is more sophisticated than the existing ones in the literature, incorporating several classes of cutting planes, two primal heuristics, and several rules for eliminating variables and constraints. Computational results show that the algorithm is competitive

Incorporating Stakeholders’ priorities and preferences in 4D trajectory optimization

A key feature of trajectory based operations (TBO) -- a new concept developed to modernize the air traffic system -- is the inclusion of preferences and priorities of the air traffic management (ATM) stakeholders. In this paper, we present a new mathematical model to optimize flights' 4D-trajectories. This is a multi-objective binary integer programming (IP) model, which assigns a 4D-trajectory to each flight, while explicitly modeling priorities and highlighting the trade off involved with the Airspace Users (AUs) preferences. The scope of the model (to be used at pre-tactical level) is the computation of optimal 4D pre-departure trajectory for each flight to be shared or negotiated with other stakeholders and subsequently managed throughout the flight. These trajectories are obtained by minimising the deviation (delay and re-routing) from the original preferred 4D-trajectories as well as minimizing the air navigation service (ANS) charges subject to the constraints of the system. Computational results for the model are presented, which show that the proposed model has the ability to identify trade-offs between the objectives of the stakeholders of the ATM system under the TBO concept. This can therefore provide the ATM stakeholders with useful decision tools to choose a trajectory for each flight

A dynamic programming heuristic for the quadratic knapsack problem

It is well known that the standard (linear) knapsack problem can be solved exactly by dynamic programming in O(nc) time, where n is the number of items and c is the capacity of the knapsack. The quadratic knapsack problem, on the other hand, is NP-hard in the strong sense, which makes it unlikely that it can be solved in pseudo-polynomial time. We show however that the dynamic programming approach to the linear knapsack problem can be modified to yield a highly effective constructive heuristic for the quadratic version. In our experiments, the lower bounds obtained by our heuristic were consistently within a fraction of a percent of optimal. Moreover, the addition of a simple local search step enabled us to obtain the optimal solution of all instances considered

Cutting planes for RLT relaxations of mixed 0-1 polynomial programs

The Reformulation-Linearization Technique (RLT), due to Sherali and Adams, can be used to construct hierarchies of linear programming relaxations of mixed 0-1 polynomial programs. As one moves up the hierarchy, the relaxations grow stronger, but the number of variables increases exponentially. We present a procedure that generates cutting planes at any given level of the hierarchy, by optimally weakening linear inequalities that are valid at any given higher level. Computational experiments, conducted on instances of the quadratic knapsack problem, indicate that the cutting planes can close a significant proportion of the integrality gap

An optimization model for assigning 4D-trajectories to flights under the TBO concept

The objective of this paper is to present a mathematical model that will contribute to the optimization and optimum configuration of the TBO concept. We develop a binary integer programming model whose aim is to assign a 4D-trajectory to each flight in order to optimize the efficiency of the ATM system. The model considers the preferred 4D-trajectory of all the flights in the pre-tactical planning phase and outputs an optimal predeparture 4D-trajectory for each flight to be shared or negotiated with other stakeholders and subsequently managed throughout the flight. These output trajectories are obtained by minimising the deviation (in terms of time delay, lateral and vertical deviation) from the original preferred trajectories. The particularities of this model are that it considers the complete 4D-trajectory for each flight as well as it incorporates the preferences and priorities of the ATM stakeholders. Some computational results are presented, which show that our optimization model has the ability to identify some trade-offs between the objectives of the stakeholders of the ATM system under the TBO concept.It can also provide the network manager with useful decision tools to choose a trajectory for each flight