Combinatorics and Optimization

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This is the collection for the University of Waterloo's Department of Combinatorics and Optimization.

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Browsing Combinatorics and Optimization by Author "Cook, William"

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    ADMM for SDP Relaxation of GP
    (University of Waterloo, 2016-08-30) Sun, Hao; Cook, William; Wolkowicz, Henry
    We consider the problem of partitioning the set of nodes of a graph G into k sets of given sizes in order to minimize the cut obtained after removing the k-th set. This is a variant of the well-known vertex separator problem that has applications in e.g., numerical linear algebra. This problem is well studied and there are many lower bounds such as: the standard eigenvalue bound; projected eigenvalue bounds using both the adjacency matrix and the Laplacian; quadratic programming (QP) bounds derived from imitating the (QP) bounds for the quadratic assignment problem; and semidefinite programming (SDP) bounds. For the quadratic assignment problem, a recent paper of [8] had great success from applying the ADMM (altenating direction method of multipliers) to the SDP relaxation. We consider the SDP relaxation of the vertex separator problem and the application of the ADMM method in solving the SDP. The main advantage of the ADMM method is that optimizing over the set of doubly non-negative matrices is about as difficult as optimizing over the set of positive semidefinite matrices. Enforcing the non-negativity constraint gives us a clear improvement in the quality of bounds obtained. We implement both a high rank and a nonconvex low rank ADMM method, where the difference is the choice of rank of the projection onto the semidefinite cone. As for the quadratic assignment problem, though there is no theoretical convergence guarantee, the nonconvex approach always converges to a feasible solution in practice.
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    Primal Cutting Plane Methods for the Traveling Salesman Problem
    (University of Waterloo, 2017-04-26) Stratopoulos, Christos; Cook, William
    Most serious attempts at solving the traveling salesman problem (TSP) are based on the dual fractional cutting plane approach, which moves from one lower bound to the next. This thesis describes methods for implementing a TSP solver based on a primal cutting plane approach, which moves from tour to tour with non-degenerate primal simplex pivots and so-called primal cutting planes. Primal cutting plane solution of the TSP has received scant attention in the literature; this thesis seeks to redress this gap, and its findings are threefold. Firstly, we develop some theory around the computation of non-degenerate primal simplex pivots, relevant to general primal cutting plane computation. This theory guides highly efficient implementation choices, a sticking point in prior studies. Secondly, we engage in a practical study of several existing primal separation algorithms for finding TSP cuts. These algorithms are all conceptually simpler, easier to implement, or asymptotically faster than their standard counterparts. Finally, this thesis may constitute the first computational study of the work of Fleischer, Letchford, and Lodi on polynomial-time separation of simple domino parity inequalities. We discuss exact and heuristic enhancements, including a shrinking-style heuristic which makes the algorithm more suitable for application on large-scale instances. The theoretical developments of this thesis are integrated into a branch-cut-price TSP solver which is used to obtain computational results on a variety of test instances.