13,494 research outputs found
Cutting plane methods for general integer programming
Integer programming (IP) problems are difficult to solve due to the integer restrictions imposed on them. A technique for solving these problems is the cutting plane method. In this method, linear constraints are added to the associated linear programming (LP) problem until an integer optimal solution is found. These constraints cut off part of the LP solution space but do not eliminate any feasible integer solution. In this report algorithms for solving IP due to Gomory and to Dantzig are presented. Two other cutting plane approaches and two extensions to Gomory's algorithm are also discussed. Although these methods are mathematically elegant they are known to have slow convergence and an explosive storage requirement. As a result cutting planes are generally not computationally successful
Mechanism Design via Dantzig-Wolfe Decomposition
In random allocation rules, typically first an optimal fractional point is
calculated via solving a linear program. The calculated point represents a
fractional assignment of objects or more generally packages of objects to
agents. In order to implement an expected assignment, the mechanism designer
must decompose the fractional point into integer solutions, each satisfying
underlying constraints. The resulting convex combination can then be viewed as
a probability distribution over feasible assignments out of which a random
assignment can be sampled. This approach has been successfully employed in
combinatorial optimization as well as mechanism design with or without money.
In this paper, we show that both finding the optimal fractional point as well
as its decomposition into integer solutions can be done at once. We propose an
appropriate linear program which provides the desired solution. We show that
the linear program can be solved via Dantzig-Wolfe decomposition. Dantzig-Wolfe
decomposition is a direct implementation of the revised simplex method which is
well known to be highly efficient in practice. We also show how to use the
Benders decomposition as an alternative method to solve the problem. The
proposed method can also find a decomposition into integer solutions when the
fractional point is readily present perhaps as an outcome of other algorithms
rather than linear programming. The resulting convex decomposition in this case
is tight in terms of the number of integer points according to the
Carath{\'e}odory's theorem
Efficient algorithms for conditional independence inference
The topic of the paper is computer testing of (probabilistic) conditional independence (CI) implications by an algebraic method of structural imsets. The basic idea is to transform (sets of) CI statements into certain integral vectors and to verify by a computer the corresponding algebraic relation between the vectors, called the independence implication. We interpret the previous methods for computer testing of this implication from the point of view of polyhedral geometry. However, the main contribution of the paper is a new method, based on linear programming (LP). The new method overcomes the limitation of former methods to the number of involved variables. We recall/describe the theoretical basis for all four methods involved in our computational experiments, whose aim was to compare the efficiency of the algorithms. The experiments show that the LP method is clearly the fastest one. As an example of possible application of such algorithms we show that testing inclusion of Bayesian network structures or whether a CI statement is encoded in an acyclic directed graph can be done by the algebraic method
Foam: A General-Purpose Cellular Monte Carlo Event Generator
A general purpose, self-adapting, Monte Carlo (MC) event generator
(simulator) is described. The high efficiency of the MC, that is small maximum
weight or variance of the MC weight is achieved by means of dividing the
integration domain into small cells. The cells can be -dimensional
simplices, hyperrectangles or Cartesian product of them. The grid of cells,
called ``foam'', is produced in the process of the binary split of the cells.
The choice of the next cell to be divided and the position/direction of the
division hyper-plane is driven by the algorithm which optimizes the ratio of
the maximum weight to the average weight or (optionally) the total variance.
The algorithm is able to deal, in principle, with an arbitrary pattern of the
singularities in the distribution. As any MC generator, it can also be used for
the MC integration. With the typical personal computer CPU, the program is able
to perform adaptive integration/simulation at relatively small number of
dimensions (). With the continuing progress in the CPU power, this
limit will get inevitably shifted to ever higher dimensions. {\tt Foam} is
aimed (and already tested) as a component in the MC event generators for the
high energy physics experiments. A few simple examples of the related
applications are presented. {\tt Foam} is written in fully object-oriented
style, in the C++ language. Two other versions with a slightly limited
functionality, are available in the Fortran77 language. The source codes are
available from http://jadach.home.cern.ch/jadach
Random evolutionary dynamics driven by fitness and house-of-cards mutations. Sampling formulae
We first revisit the multi-allelic mutation-fitness balance problem,
especially when mutations obey a house of cards condition, where the
discrete-time deterministic evolutionary dynamics of the allelic frequencies
derives from a Shahshahani potential. We then consider multi-allelic
Wright-Fisher stochastic models whose deviation to neutrality is from the
Shahsha-hani mutation/selection potential. We next focus on the weak selection,
weak mutation cases and, making use of a Gamma calculus, we compute the
normalizing partition functions of the invariant probability densities
appearing in their Wright-Fisher diffusive approximations. Using these results,
Generalized Ewens sampling formulae (ESF) from the equilibrium distributions
are derived. We start treating the ESF in the mixed mutation/selection
potential case and then we restrict ourselves to the ESF in the simpler
house-of-cards mutations only situation. We also address some issues concerning
sampling problems from infinitely-many alleles weak limits.Comment: \`a paraitre: Journal of Statistical Physic
Fitting aerodynamic forces in the Laplace domain: An application of a nonlinear nongradient technique to multilevel constrained optimization
A technique which employs both linear and nonlinear methods in a multilevel optimization structure to best approximate generalized unsteady aerodynamic forces for arbitrary motion is described. Optimum selection of free parameters is made in a rational function approximation of the aerodynamic forces in the Laplace domain such that a best fit is obtained, in a least squares sense, to tabular data for purely oscillatory motion. The multilevel structure and the corresponding formulation of the objective models are presented which separate the reduction of the fit error into linear and nonlinear problems, thus enabling the use of linear methods where practical. Certain equality and inequality constraints that may be imposed are identified; a brief description of the nongradient, nonlinear optimizer which is used is given; and results which illustrate application of the method are presented
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