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

Un Proyecto docente basado en una auténtica evaluación continua utilizando una herramienta softwar

La reciente creación de la titulación de Grado en Ingeniería Informática propició la adecuación de sus asignaturas al Espacio Europeo de Educación Superior, poniendo especial énfasis en la promoción del aprendizaje continuado a lo largo de la vida. En este trabajo se presenta y analiza un proyecto de innovación y mejora docente aplicado en la asignatura Cálculo Infinitesimal y Numérico, perteneciente al módulo de Formación Básica. Se ha tomado para ello un grupo piloto dentro de la titulación. La principal característica del proyecto reside en el diseño de dinámicas docentes diferenciadas entre clases teóricas, prácticas de aula y prácticas de laboratorio y al mismo tiempo integradoras de un proceso continuado de aprendizaje y de aplicación práctica de lo aprendido, garantizando que sea compatible con una auténtica evaluación continua de todo el proceso. La viabilidad del proyecto dependía en gran medida de la carga de trabajo que ello suponía para el profesorado implicado. Para ello, se diseñó una herramienta software que permite la generación y asignación de actividades personalizadas para el alumnado así como una evaluación semi-automática de las mismas. Un primer análisis de los resultados obtenidos muestra una mejora significativa respecto de otros proyectos basados en planteamientos más clásicos.SUMMARY -- The recent creation of the Degree in Computer Engineering led the adaptation of their subjects to the European Higher Education Area, stressing on lifelong learning throughout life. This paper presents and analyzes a proposed educational plan of innovation. It has been applied to the subject Cálculo Infinitesimal y Numérico (Differential Calculus and Numerical Analysis) as part of basic training modules. It has been tested in a pilot group within the degree. The main feature of the project lies in the design of teaching dynamics. It has been scheduled in theoretical classes, classroom practices and laboratory practices. These modules achieve continuous process of learning and practical application of learning, ensuring an authentic continuous assessment throughout the process. The viability of the project depended heavily on the workload for the involved teachers. For this, we designed a software tool that allows the generation and assignment of custom activities for students and semi-automatic evaluation of them. A first analysis of the results shows a significant improvement over previous results with other projects based on classic approaches.Peer Reviewe

Un proyecto docente basado en una auténtica evaluación continua utilizando una herramienta software

La reciente creación de la titulación de Grado en Ingeniería Informática propició la adecuación de sus asignaturas al Espacio Europeo de Educación Superior, poniendo especial énfasis en la promoción del aprendizaje continuado a lo largo de la vida. En este trabajo se presenta y analiza un proyecto de innovación y mejora docente aplicado en la asignatura Cálculo Infinitesimal y Numérico, perteneciente al módulo de Formación Básica. Se ha tomado para ello un grupo piloto dentro de la titulación. La principal característica del proyecto reside en el diseño de dinámicas docentes diferenciadas entre clases teóricas, prácticas de aula y prácticas de laboratorio y al mismo tiempo integradoras de un proceso continuado de aprendizaje y de aplicación práctica de lo aprendido, garantizando que sea compatible con una auténtica evaluación continua de todo el proceso. La viabilidad del proyecto dependía en gran medida de la carga de trabajo que ello suponía para el profesorado implicado. Para ello, se diseñó una herramienta software que permite la generación y asignación de actividades personalizadas para el alumnado así como una evaluación semi-automática de las mismas. Un primer análisis de los resultados obtenidos muestra una mejora significativa respecto de otros proyectos basados en planteamientos más clásicos.The recent creation of the Degree in Computer Engineering led the adaptation of their subjects to the European Higher Education Area, stressing on lifelong learning throughout life. This paper presents and analyzes a proposed educational plan of innovation. It has been applied to the subject Cálculo Infinitesimal y Numérico (Differential Calculus and Numerical Analysis) as part of basic training modules. It has been tested in a pilot group within the degree. The main feature of the project lies in the design of teaching dynamics. It has been scheduled in theoretical classes, classroom practices and laboratory practices. These modules achieve continuous process of learning and practical application of learning, ensuring an authentic continuous assessment throughout the process. The viability of the project depended heavily on the workload for the involved teachers. For this, we designed a software tool that allows the generation and assignment of custom activities for students and semi-automatic evaluation of them. A first analysis of the results shows a significant improvement over previous results with other projects based on classic approaches.La experiencia ha sido financiada por el I Plan Propio de Docencia de la Universidad de Sevilla, bajo el proyecto titulado “Desarrollo e implementación de un sistema automatizado de evaluación continua para la asignatura de Cálculo Infinitesimal y Numérico”. Los autores son parcialmente financiados por los grupos de investigación “Plan de Apoyo a las enseñanzas de Máster Universitario 2012”, FQM-333 y FQM-164

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

Algorithmic Invariants for Alexander Modules

Let $G$ 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 $11$ 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

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 fsf^s

Let $f_1,\ldots, f_p$ be polynomials in ${\bf C}[x_1,\ldots, x_n]$ and let $D = D_n$ be the $n$-th Weyl algebra. The annihilating ideal of $f^s=f_1^{s_1}\cdots f_p^{s_p}$ in $D[s]=D[s_1,\ldots,s_p]$ is a necessary step for the computation of the Bernstein-Sato ideals of $f_1,\ldots, f_p$. 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