39,339 research outputs found
Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks
Bayesian Networks (BNs) represent conditional probability relations among a
set of random variables (nodes) in the form of a directed acyclic graph (DAG),
and have found diverse applications in knowledge discovery. We study the
problem of learning the sparse DAG structure of a BN from continuous
observational data. The central problem can be modeled as a mixed-integer
program with an objective function composed of a convex quadratic loss function
and a regularization penalty subject to linear constraints. The optimal
solution to this mathematical program is known to have desirable statistical
properties under certain conditions. However, the state-of-the-art optimization
solvers are not able to obtain provably optimal solutions to the existing
mathematical formulations for medium-size problems within reasonable
computational times. To address this difficulty, we tackle the problem from
both computational and statistical perspectives. On the one hand, we propose a
concrete early stopping criterion to terminate the branch-and-bound process in
order to obtain a near-optimal solution to the mixed-integer program, and
establish the consistency of this approximate solution. On the other hand, we
improve the existing formulations by replacing the linear "big-" constraints
that represent the relationship between the continuous and binary indicator
variables with second-order conic constraints. Our numerical results
demonstrate the effectiveness of the proposed approaches
Decomposition, Reformulation, and Diving in University Course Timetabling
In many real-life optimisation problems, there are multiple interacting
components in a solution. For example, different components might specify
assignments to different kinds of resource. Often, each component is associated
with different sets of soft constraints, and so with different measures of soft
constraint violation. The goal is then to minimise a linear combination of such
measures. This paper studies an approach to such problems, which can be thought
of as multiphase exploitation of multiple objective-/value-restricted
submodels. In this approach, only one computationally difficult component of a
problem and the associated subset of objectives is considered at first. This
produces partial solutions, which define interesting neighbourhoods in the
search space of the complete problem. Often, it is possible to pick the initial
component so that variable aggregation can be performed at the first stage, and
the neighbourhoods to be explored next are guaranteed to contain feasible
solutions. Using integer programming, it is then easy to implement heuristics
producing solutions with bounds on their quality.
Our study is performed on a university course timetabling problem used in the
2007 International Timetabling Competition, also known as the Udine Course
Timetabling Problem. In the proposed heuristic, an objective-restricted
neighbourhood generator produces assignments of periods to events, with
decreasing numbers of violations of two period-related soft constraints. Those
are relaxed into assignments of events to days, which define neighbourhoods
that are easier to search with respect to all four soft constraints. Integer
programming formulations for all subproblems are given and evaluated using ILOG
CPLEX 11. The wider applicability of this approach is analysed and discussed.Comment: 45 pages, 7 figures. Improved typesetting of figures and table
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