4,225 research outputs found
The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with a Single Defect
In this paper, a two-dimensional cutting problem is considered in which a single plate (large object) has to be cut down into a set of small items of maximal value. As opposed to standard cutting problems, the large object contains a defect, which must not be covered by a small item. The problem is represented by means of an AND/OR-graph, and a Branch & Bound procedure (including heuristic modifications for speeding up the search process) is introduced for its exact solution. The proposed method is evaluated in a series of numerical experiments that are run on problem instances taken from the literature, as well as on randomly generated instances.Two-dimensional cutting, defect, AND/OR-graph, Branch & Bound
An interior point algorithm for minimum sum-of-squares clustering
Copyright @ 2000 SIAM PublicationsAn exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the rst time for several fairly large data sets from the literature, including Fisher's 150 iris.This research was supported by the Fonds
National de la Recherche Scientifique Suisse, NSERC-Canada, and FCAR-Quebec
A note on QUBO instances defined on Chimera graphs
McGeoch and Wang (2013) recently obtained optimal or near-optimal solutions
to some quadratic unconstrained boolean optimization (QUBO) problem instances
using a 439 qubit D-Wave Two quantum computing system in much less time than
with the IBM ILOG CPLEX mixed-integer quadratic programming (MIQP) solver. The
problems studied by McGeoch and Wang are defined on subgraphs -- with up to 439
nodes -- of Chimera graphs. We observe that after a standard reformulation of
the QUBO problem as a mixed-integer linear program (MILP), the specific
instances used by McGeoch and Wang can be solved to optimality with the CPLEX
MILP solver in much less time than the time reported in McGeoch and Wang for
the CPLEX MIQP solver. However, the solution time is still more than the time
taken by the D-Wave computer in the McGeoch-Wang tests.Comment: Version 1 discussed computational results with random QUBO instances.
McGeoch and Wang made an error in describing the instances they used; they
did not use random QUBO instances but rather random Ising Model instances
with fields (mapped to QUBO instances). The current version of the note
reports on tests with the precise instances used by McGeoch and Wan
The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with a Single Defect
In this paper, a two-dimensional cutting problem is considered in which a single plate (large object) has to be cut down into a set of small items of maximal value. As opposed to standard cutting problems, the large object contains a defect, which must not be covered by a small item. The problem is represented by means of an AND/OR-graph, and a Branch & Bound procedure (including heuristic modifications for speeding up the search process) is introduced for its exact solution. The proposed method is evaluated in a series of numerical experiments that are run on problem instances taken from the literature, as well as on randomly generated instances
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