16 research outputs found
Finding Multiple Solutions in Nonlinear Integer Programming with Algebraic Test-Sets
We explain how to compute all the solutions of a nonlinear
integer problem using the algebraic test-sets associated to a suitable
linear subproblem. These test-sets are obtained using Gröbner bases. The
main advantage of this method, compared to other available alternatives,
is its exactness within a quite good efficiency.Ministerio de Economía y Competitividad MTM2016-75024-PMinisterio de Economía y Competitividad MTM2016-74983-C2- 1-RJunta de Andalucía P12-FQM-269
Algorithmic Invariants for Alexander Modules
Let be a group given by generators and relations. It is
possible to compute a presentation matrix of a module over a ring
through Fox's differential calculus. We show how to use Gröbner
bases as an algorithmic tool to compare the chains of elementary
ideals defined by the matrix. We apply this technique to classical
examples of groups and to compute the elementary ideals of
Alexander matrix of knots up to crossings with the same
Alexander polynomial
An exact algebraic ϵ-constraint method for bi-objective linear integer programming based on test sets
A new exact algorithm for bi-objective linear integer problems is presented, based on the classic - constraint method and algebraic test sets for single-objective linear integer problems. Our method pro- vides the complete Pareto frontier N of non-dominated points and, for this purpose, it considers exactly |N | single-objective problems by using reduction with test sets instead of solving with an optimizer. Al- though we use Gröbner bases for the computation of test sets, which may provoke a bottleneck in princi- ple, the computational results are shown to be promising, especially for unbounded knapsack problems,for which any usual branch-and-cut strategy could be much more expensive. Nevertheless, this algorithmcan be considered as a potentially faster alternative to IP-based methods when test sets are available.Ministerio de Economía y Competitividad MTM2016-74983-C2-1-RMinisterio de Economía y Competitividad MTM2016-75024-PJunta de Andalucía P12-FQM-269
A vanishing theorem for a class of logarithmic D-modules
Let OX (resp. DX) be the sheaf of holomorphic functions (resp. the
sheaf of linear differential operators with holomorphic coefficients) on X =
Cn. Let D X be a locally weakly quasi-homogeneous free divisor defined
by a polynomial f. In this paper we prove that, locally, the annihilating
ideal of 1/fk over DX is generated by linear differential operators of order
1 (for k big enough). For this purpose we prove a vanishing theorem for
the extension groups of a certain logarithmic DX–module with OX. The
logarithmic DX–module is naturally associated with D (see Notation 1.1).
This result is related to the so called Logarithmic Comparison Theorem
Comparison of theoretical complexities of two methods for computing annihilating ideals of polynomials
Let f1, . . . , fp be polynomials in C[x1, . . . , xn] and let D = Dn be the n-th Weyl algebra. We provide upper bounds for the complexity of computing the annihilating ideal of f s = f s1 1 · · · f sp p in D[s] = D[s1, . . . , sp]. These bounds provide an initial explanation on the differences between the running times of the two methods known to obtain the so-called BernsteinSato ideals.Ministerio de Ciencia y Tecnología MTM2004-01165Junta de Andalucía FQM-33
Nouvelle Cuisine for the Computation of the Annihilating Ideal of
Let be polynomials in
and let be the -th Weyl algebra. The annihilating
ideal of in
is a necessary step for the computation
of the Bernstein-Sato ideals of .
We point out experimental differences among the efficiency of the
available methods to obtain this annihilating ideal and provide
some upper bounds for the complexity of its computation
An improved test set approach to nonlinear integer problems with applications to engineering design
Many problems in engineering design involve the use of nonlinearities
and some integer variables. Methods based on test sets have been
proposed to solve some particular problems with integer variables, but they
have not been frequently applied because of computation costs. The walk-back
procedure based on a test set gives an exact method to obtain an optimal point
of an integer programming problem with linear and nonlinear constraints, but
the calculation of this test set and the identification of an optimal solution
using the test set directions are usually computationally intensive.
In problems for which obtaining the test set is reasonably fast, we show
how the effectiveness can still be substantially improved. This methodology
is presented in its full generality and illustrated on two specific problems: (1)
minimizing cost in the problem of scheduling jobs on parallel machines given
restrictions on demands and capacity, and (2) minimizing cost in the series
parallel redundancy allocation problem, given a target reliability. Our computational
results are promising and suggest the applicability of this approach
to deal with other problems with similar characteristics or to combine it with
mainstream solvers to certify optimalityJunta de Andalucía FQM- 5849Ministerio de Ciencia e Innovación MTM2010-19336Ministerio de Ciencia e Innovación MTM2010-19576Ministerio de Ciencia e Innovación MTM2013-46962- C2-1-PFEDE
Exact cost minimization of a series-parallel reliable system with multiple component choices using an algebraic method
The redundancy allocation problem is formulated minimizing the design cost for a series-parallel system with multiple component choices while ensuring a given system reliability level. The obtained model is a nonlinear integer programming problem with a nonlinear, nonseparable constraint. We propose a method based on the construction of a test set of an integer linear problem, which allows us to obtain an exact solution of the problem. It is compared to other approaches in the literature and standard nonlinear solvers.FQM-5849, MTM2010-19336, MTM2010-19576 and FEDE
An algebraic approach to Integer Portfolio problems
Integer variables allow the treatment of some portfolio optimization problems in a more realistic way and introduce the possibility of adding some natural features to the model.
We propose an algebraic approach to maximize the expected return under a given admissible level of risk measured by the covariance matrix. To reach an optimal portfolio it is an essential ingredient the computation of different test sets (via Gr\"obner basis) of linear subproblems that are used in a dual search strategy.Universidad de Sevilla P06-FQM-01366Junta de Andalucía (Plan Andaluz de Investigación) FQM-333Ministerio de Ciencia e Innovación (España) MTM2007-64509Instituto de Matemáticas de la Universidad de Sevilla MTM2007-67433-C02-0
Slopes of hypergeometric systems of codimension one.
We describe the slopes, with respect to the coordinates hyperplanes, of the hypergeometric systems of codimension one, that is when the toric ideal is generated by one element