8,305 research outputs found
A linear programming reformulation of the standard quadratic optimization problem
The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note,
we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.Accepted versio
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 Machine-Part Cell Formation Problem with Non-Binary Values: A MILP Model and a Case of Study in the Accounting Profession
The traditional machine-part cell formation problem simultaneously clusters machines
and parts in different production cells from a zero–one incidence matrix that describes the existing
interactions between the elements. This manuscript explores a novel alternative for the well-known
machine-part cell formation problem in which the incidence matrix is composed of non-binary values.
The model is presented as multiple-ratio fractional programming with binary variables in quadratic
terms. A simple reformulation is also implemented in the manuscript to express the model as a
mixed-integer linear programming optimization problem. The performance of the proposed model
is shown through two types of empirical experiments. In the first group of experiments, the model
is tested with a set of randomized matrices, and its performance is compared to the one obtained
with a standard greedy algorithm. These experiments showed that the proposed model achieves
higher fitness values in all matrices considered than the greedy algorithm. In the second type of
experiment, the optimization model is evaluated with a real-world problem belonging to Human
Resource Management. The results obtained were in line with previous findings described in the
literature about the case study
Using a conic bundle method to accelerate both phases of a quadratic convex reformulation
We present algorithm MIQCR-CB that is an advancement of method
MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving
mixed-integer quadratic programs and works in two phases: the first phase
determines an equivalent quadratic formulation with a convex objective function
by solving a semidefinite problem , and, in the second phase, the
equivalent formulation is solved by a standard solver. As the reformulation
relies on the solution of a large-scale semidefinite program, it is not
tractable by existing semidefinite solvers, already for medium sized problems.
To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm
within a Lagrangian duality framework for solving that substantially
speeds up the first phase. Moreover, this algorithm leads to a reformulated
problem of smaller size than the one obtained by the original MIQCR method
which results in a shorter time for solving the second phase.
We present extensive computational results to show the efficiency of our
algorithm
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