5,682 research outputs found
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
Lagrangian Relaxation for MAP Estimation in Graphical Models
We develop a general framework for MAP estimation in discrete and Gaussian
graphical models using Lagrangian relaxation techniques. The key idea is to
reformulate an intractable estimation problem as one defined on a more
tractable graph, but subject to additional constraints. Relaxing these
constraints gives a tractable dual problem, one defined by a thin graph, which
is then optimized by an iterative procedure. When this iterative optimization
leads to a consistent estimate, one which also satisfies the constraints, then
it corresponds to an optimal MAP estimate of the original model. Otherwise
there is a ``duality gap'', and we obtain a bound on the optimal solution.
Thus, our approach combines convex optimization with dynamic programming
techniques applicable for thin graphs. The popular tree-reweighted max-product
(TRMP) method may be seen as solving a particular class of such relaxations,
where the intractable graph is relaxed to a set of spanning trees. We also
consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g.
loops), and a connected tree obtained by ``unwinding'' cycles. In addition, we
propose a new class of multiscale relaxations that introduce ``summary''
variables. The potential benefits of such generalizations include: reducing or
eliminating the ``duality gap'' in hard problems, reducing the number or
Lagrange multipliers in the dual problem, and accelerating convergence of the
iterative optimization procedure.Comment: 10 pages, presented at 45th Allerton conference on communication,
control and computing, to appear in proceeding
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