Advances in interior point methods and column generation

In this thesis we study how to efficiently combine the column generation technique (CG) and interior point methods (IPMs) for solving the relaxation of a selection of integer programming problems. In order to obtain an efficient method a change in the column generation technique and a new reoptimization strategy for a primal-dual interior point method are proposed. It is well-known that the standard column generation technique suffers from unstable behaviour due to the use of optimal dual solutions that are extreme points of the restricted master problem (RMP). This unstable behaviour slows down column generation so variations of the standard technique which rely on interior points of the dual feasible set of the RMP have been proposed in the literature. Among these techniques, there is the primal-dual column generation method (PDCGM) which relies on sub-optimal and well-centred dual solutions. This technique dynamically adjusts the column generation tolerance as the method approaches optimality. Also, it relies on the notion of the symmetric neighbourhood of the central path so sub-optimal and well-centred solutions are obtained. We provide a thorough theoretical analysis that guarantees the convergence of the primal-dual approach even though sub-optimal solutions are used in the course of the algorithm. Additionally, we present a comprehensive computational study of the solution of linear relaxed formulations obtained after applying the Dantzig-Wolfe decomposition principle to the cutting stock problem (CSP), the vehicle routing problem with time windows (VRPTW), and the capacitated lot sizing problem with setup times (CLSPST). We compare the performance of the PDCGM with the standard column generation method (SCGM) and the analytic centre cutting planning method (ACCPM). Overall, the PDCGM achieves the best performance when compared to the SCGM and the ACCPM when solving challenging instances from a column generation perspective. One important characteristic of this column generation strategy is that no speci c tuning is necessary and the algorithm poses the same level of difficulty as standard column generation method. The natural stabilization available in the PDCGM due to the use of sub-optimal well-centred interior point solutions is a very attractive feature of this method. Moreover, the larger the instance, the better is the relative performance of the PDCGM in terms of column generation iterations and CPU time. The second part of this thesis is concerned with the development of a new warmstarting strategy for the PDCGM. It is well known that taking advantage of the previously solved RMP could lead to important savings in solving the modified RMP. However, this is still an open question for applications arising in an integer optimization context and the PDCGM. Despite the current warmstarting strategy in the PDCGM working well in practice, it does not guarantee full feasibility restorations nor considers the quality of the warmstarted iterate after new columns are added. The main motivation of the design of the new warmstarting strategy presented in this thesis is to close this theoretical gap. Under suitable assumptions, the warmstarting procedure proposed in this thesis restores primal and dual feasibilities after the addition of new columns in one step. The direction is determined so that the modi cation of small components at a particular solution is not large. Additionally, the strategy enables control over the new duality gap by considering an expanded symmetric neighbourhood of the central path. As observed from our computational experiments solving CSP and VRPTW, one can conclude that the warmstarting strategies for the PDCGM are useful when dense columns are added to the RMP (CSP), since they consistently reduce the CPU time and also the number of iterations required to solve the RMPs on average. On the other hand, when sparse columns are added (VRPTW), the coldstart used by the interior point solver HOPDM becomes very efficient so warmstarting does not make the task of solving the RMPs any easier

Nivel de satisfacción de las clases online por parte de los estudiantes de Educación Física de Chile en tiempos de pandemia.

Introduction: In Chile the first case of COVID-19 was diagnosed on March 3, 2020 and on the eighteenth of the same month the president decreed state of catastrophe, so that classes at school and university level went from the presence to the virtuality. The present study aims to know the level of satisfaction of the virtual classes of the students of Physical Education. Methodology: Quantitative, non experimental, transversal. The sample consisted of 542 students of Physical Education from different study houses in Chile. The Satisfaction scale online classes was adapted and validated. Results: there are significant differences comparing theoretical and practical subjects, with theoretical chairs being better valued. Differences were also found by sex, where males have a more negative perception about virtual classes and when comparing by course, freshmen have a more positive perception about virtual classes in relation to higher courses. Conclusions: there is a resistance on the part of students to virtual classes in Physical Education, because, although significant differences were found between the theoretical and practical subjects, values were always around 3 on a scale of 1 to 5. Future research with other variables such as physical activity, stress levels and strategies for the teaching of Virtual Physical Education are necessary. (English) [ABSTRACT FROM AUTHOR]Introducción: En Chile el primer caso de COVID-19 fue diagnosticado el tres de marzo del 2020 y el día dieciocho del mismo mes el presidente decreto estado de catástrofe, por lo que las clases a nivel escolar y universitario pasaron de la presencialidad a la virtualidad. El presente estudio tiene como objetivo conocer el nivel de satisfacción de las clases virtuales de los estudiantes de Educación Física. Metodología: Cuantitativa, no experimental, transversal. La muestra estuvo constituida por 542 alumnos de Educación Física de diferentes casas de estudio de Chile. Se adaptó y validó la escala Satisfacción clases online. Resultados: existen diferencias significativas comparando las asignaturas teóricas y prácticas, siendo mejor valoradas las cátedras teóricas. También se encontraron diferencias por sexo, donde los varones poseen una percepción más negativa sobre las clases virtuales y al comparar por curso, los alumnos de primer año poseen una percepción más positiva sobre las clases virtuales en relación con los cursos superiores. Conclusiones: existe una resistencia por parte de los alumnos a las clases virtuales en la Educación Física, pues, si bien se encontraron diferencias significativas entre las asignaturas teóricas y prácticas, los valores siempre estuvieron alrededor de 3 en una escala de 1 a 5. Se hacen necesarias futuras investigaciones con otras variables como actividad física, niveles de estrés y estrategias para la enseñanza de la Educación Física virtual

Large-scale optimization with the primal-dual column generation method

The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.Comment: 28 pages, 1 figure, minor revision, scaled CPU time