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- Evaluating student behaviour on the mathE platform - clustering algorithms approachesPublication . Azevedo, Beatriz Flamia; Rocha, Ana Maria A.C.; Fernandes, Florbela P.; Pacheco, Maria F.; Pereira, Ana I.The MathE platform is an online educational platform that aims to help students who struggle to learn college mathematics as well as students who wish to deepen their knowledge on subjects that rely on a strong mathematical background, at their own pace. The MathE platform is currently being used by a significant number of users, from all over the world, as a tool to support and engage students, ensuring new and creative ways to encourage them to improve their mathematical skills. This paper is addressed to evaluate the students’ performance on the Linear Algebra topic, which is a specific topic of the MathE platform. In order to achieve this goal, four clustering algorithms were considered; three of them based on different bio-inspired techniques and the k-means algorithm. The results showed that most students choose to answer only basic level questions, and even within that subset, they make a lot of mistakes. When students take the risk of answering advanced questions, they make even more mistakes, which causes them to return to the basic level questions. Considering these results, it is now necessary to carry out an in-depth study to reorganize the available questions according to other levels of difficulty, and not just between basic and advanced levels as it is.
- Parameter estimation of the kinetic α -pinene isomerization model using the MCSfilter algorithmPublication . Amador, Andreia; Fernandes, Florbela P.; Santos, Lino O.; Romanenko, Andrey; Rocha, Ana Maria A.C.This paper aims to illustrate the application of a derivative- free multistart algorithm with coordinate search filter, designated as the MCSFilter algorithm. The problem used in this study is the parameter estimation problem of the kinetic α-pinene isomerization model. This is a well known nonlinear optimization problem (NLP) that has been investigated as a case study for performance testing of most derivative based methods proposed in the literature. Since the MCSFilter algorithm features a stochastic component, it was run ten times to solve the NLP problem. The optimization problem was successfully solved in all the runs and the optimal solution demonstrates that in all runs the MCSFilter provides a good quality solution.
- Multistart hooke and jeeves filter method for mixed variable optimizationPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Fernandes, Edite M.G.P.; Rocha, Ana Maria A.C.In this study, we propose an extended version of the Hooke and Jeeves algorithm that uses a simple heuristic to handle integer and/or binary variables and a filter set methodology to handle constraints. This proposal is integrated into a multistart method as a local solver and it is repeatedly called in order to compute different optimal solutions. Then, the best of all stored optimal solutions is selected as the global optimum. The performance of the new method is tested on benchmark problems. Its effectiveness is emphasized by a comparison with other well-known stochastic solvers.
- Analyzing the mathE platform through clustering algorithmsPublication . Azevedo, Beatriz Flamia; Amoura, Yahia; Rocha, Ana Maria A.C.; Fernandes, Florbela P.; Pacheco, Maria F.; Pereira, Ana I.University lecturers have been encouraged to adopt innovative methodologies and teaching tools in order to implement an interactive and appealing educational environment. The MathE platform was created with the main goal of providing students and teachers with a new perspective on mathematical teaching and learning in a dynamic and appealing way, relying on digital interactive technologies that enable customized study. The MathE platform has been online since 2019, having since been used by many students and professors around the world. However, the necessity for some improvements on the platform has been identified, in order to make it more interactive and able to meet the needs of students in a customized way. Based on previous studies, it is known that one of the urgent needs is the reorganization of the available resources into more than two levels (basic and advanced), as it currently is. Thus, this paper investigates, through the application of two clustering methodologies, the optimal number of levels of difficulty to reorganize the resources in the MathE platform. Hierarchical Clustering and three Bio-inspired Automatic Clustering Algorithms were applied to the database, which is composed of questions answered by the students on the platform. The results of both methodologies point out six as the optimal number of levels of difficulty to group the resources offered by the platform.
- Improving efficiency of a multistart with interrupted hooke-and-jeeves filter search for solving MINLP problemsPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Rocha, Ana Maria A.C.; Fernandes, Edite M.G.P.This paper addresses the problem of solving mixed-integer nonlinear programming (MINLP) problems by a multistart strategy that invokes a derivative-free local search procedure based on a filter set methodology to handle nonlinear constraints. A new concept of componentwise normalized distance aiming to discard randomly generated points that are sufficiently close to other points already used to invoke the local search is analyzed. A variant of the Hooke-and-Jeeves filter algorithm for MINLP is proposed with the goal of interrupting the iterative process if the accepted iterate falls inside an ϵ-neighborhood of an already computed minimizer. Preliminary numerical results are included.
- Multiple solutions of mixed variable optimization by multistart hooke and jeeves filter methodPublication . Costa, M. Fernanda P.; Fernandes, Florbela P.; Fernandes, Edite M.G.P.; Rocha, Ana Maria A.C.In this study, we propose a multistart method based on an extended version of the Hooke and Jeeves (HJ) algorithm for computing multiple solutions of mixed variable optimization problems. The inequality and equality constraints of the problem are handled by a filter set methodology. The basic ideas present in the HJ algorithm, namely the exploratory and pattern moves, are extended to consider two objective functions and to handle continuous and integer variables simultaneously. This proposal is integrated into a multistart method as a local search procedure that is repeatedly invoked to converge to different global and non-global optimal solutions starting from randomly generated points. To avoid repeated convergence to previously computed solutions, the concept of region of attraction of an optimizer is implemented. The performance of the new method is tested on benchmark problems. Its effectiveness is emphasized by a comparison with a well-known solver.