Penerapan Flower Pollination Algorithm dengan Teknik Clustering dalam Penyelesaian Masalah Diophantine

Permasalahan Diophantine adalah suatu permasalahan yang diwakili persamaan atau sistem persamaan yang memerlukan bilangan bulat non-negatif sebagai solusi. Permasalahan ini banyak dijumpai di berbagai bidang termasuk Computer engineering seperti pengelolaan jaringan dan sinyal. Akan tetapi belum ada metode umum yang secara efektif dapat menyelesaikan permasalahan Diophantine.  Tujuan utama dalam penelitian ini adalah untuk melakukan penyesuaian metode FPAC agar FPAC tidak hanya dapat digunakan pada permasalahan Multimodal tetapi juga dapat dijadikan sebagai alternatif pada permasalahaan Diophantine. Transformasi persamaan ataupun sistem persamaan ke dalam bentuk fungsi optimasi dan transformasi output bilangan real ke bilangan bulat pada setiap tahapan algoritma merupakan kunci utama FPAC dalam menyelesaikan permasalahan Diophantine. Hasil penelitian ini menunjukan bahwa FPAC dapat menemukan seluruh solusi dari persamaan Diophantine baik persamaan yang memiliki jumlah variabel dan pangkat yang berbeda maupun persamaan dalam bentuk eksponensial.  FPAC juga dapat menemukan seluruh solusi yang tersedia pada sistem persamaan Diophantine baik yang berdimensi rendah (kasus 1) maupun dimensi tinggi (kasus 2 dan 3). Secara umum, FPAC terbukti efektif dalam menyelesaikan permasalahan Diophantine baik dalam bentuk persamaan maupun sistem persamaan yang memiliki solusi tunggal maupun jamak dalam sekali running

Nature-inspired algorithms for solving some hard numerical problems

Optimisation is a branch of mathematics that was developed to find the optimal solutions, among all the possible ones, for a given problem. Applications of optimisation techniques are currently employed in engineering, computing, and industrial problems. Therefore, optimisation is a very active research area, leading to the publication of a large number of methods to solve specific problems to its optimality. This dissertation focuses on the adaptation of two nature inspired algorithms that, based on optimisation techniques, are able to compute approximations for zeros of polynomials and roots of non-linear equations and systems of non-linear equations. Although many iterative methods for finding all the roots of a given function already exist, they usually require: (a) repeated deflations, that can lead to very inaccurate results due to the problem of accumulating rounding errors, (b) good initial approximations to the roots for the algorithm converge, or (c) the computation of first or second order derivatives, which besides being computationally intensive, it is not always possible. The drawbacks previously mentioned served as motivation for the use of Particle Swarm Optimisation (PSO) and Artificial Neural Networks (ANNs) for root-finding, since they are known, respectively, for their ability to explore high-dimensional spaces (not requiring good initial approximations) and for their capability to model complex problems. Besides that, both methods do not need repeated deflations, nor derivative information. The algorithms were described throughout this document and tested using a test suite of hard numerical problems in science and engineering. Results, in turn, were compared with several results available on the literature and with the well-known Durand–Kerner method, depicting that both algorithms are effective to solve the numerical problems considered.A Optimização é um ramo da matemática desenvolvido para encontrar as soluções óptimas, de entre todas as possíveis, para um determinado problema. Actualmente, são várias as técnicas de optimização aplicadas a problemas de engenharia, de informática e da indústria. Dada a grande panóplia de aplicações, existem inúmeros trabalhos publicados que propõem métodos para resolver, de forma óptima, problemas específicos. Esta dissertação foca-se na adaptação de dois algoritmos inspirados na natureza que, tendo como base técnicas de optimização, são capazes de calcular aproximações para zeros de polinómios e raízes de equações não lineares e sistemas de equações não lineares. Embora já existam muitos métodos iterativos para encontrar todas as raízes ou zeros de uma função, eles usualmente exigem: (a) deflações repetidas, que podem levar a resultados muito inexactos, devido ao problema da acumulação de erros de arredondamento a cada iteração; (b) boas aproximações iniciais para as raízes para o algoritmo convergir, ou (c) o cálculo de derivadas de primeira ou de segunda ordem que, além de ser computacionalmente intensivo, para muitas funções é impossível de se calcular. Estas desvantagens motivaram o uso da Optimização por Enxame de Partículas (PSO) e de Redes Neurais Artificiais (RNAs) para o cálculo de raízes. Estas técnicas são conhecidas, respectivamente, pela sua capacidade de explorar espaços de dimensão superior (não exigindo boas aproximações iniciais) e pela sua capacidade de modelar problemas complexos. Além disto, tais técnicas não necessitam de deflações repetidas, nem do cálculo de derivadas. Ao longo deste documento, os algoritmos são descritos e testados, usando um conjunto de problemas numéricos com aplicações nas ciências e na engenharia. Os resultados foram comparados com outros disponíveis na literatura e com o método de Durand–Kerner, e sugerem que ambos os algoritmos são capazes de resolver os problemas numéricos considerados

Teachers' and fourth graders' questions during a problem-solving lesson

Peer reviewe

Analysis of physiological signals using machine learning methods

Technological advances in data collection enable scientists to suggest novel approaches, such as Machine Learning algorithms, to process and make sense of this information. However, during this process of collection, data loss and damage can occur for reasons such as faulty device sensors or miscommunication. In the context of time-series data such as multi-channel bio-signals, there is a possibility of losing a whole channel. In such cases, existing research suggests imputing the missing parts when the majority of data is available. One way of understanding and classifying complex signals is by using deep neural networks. The hyper-parameters of such models have been optimised using the process of back propagation. Over time, improvements have been suggested to enhance this algorithm. However, an essential drawback of the back propagation can be the sensitivity to noisy data. This thesis proposes two novel approaches to address the missing data challenge and back propagation drawbacks: First, suggesting a gradient-free model in order to discover the optimal hyper-parameters of a deep neural network. The complexity of deep networks and high-dimensional optimisation parameters presents challenges to find a suitable network structure and hyper-parameter configuration. This thesis proposes the use of a minimalist swarm optimiser, Dispersive Flies Optimisation(DFO), to enable the selected model to achieve better results in comparison with the traditional back propagation algorithm in certain conditions such as limited number of training samples. The DFO algorithm offers a robust search process for finding and determining the hyper-parameter configurations. Second, imputing whole missing bio-signals within a multi-channel sample. This approach comprises two experiments, namely the two-signal and five-signal imputation models. The first experiment attempts to implement and evaluate the performance of a model mapping bio-signals from A toB and vice versa. Conceptually, this is an extension to transfer learning using CycleGenerative Adversarial Networks (CycleGANs). The second experiment attempts to suggest a mechanism imputing missing signals in instances where multiple data channels are available for each sample. The capability to map to a target signal through multiple source domains achieves a more accurate estimate for the target domain. The results of the experiments performed indicate that in certain circumstances, such as having a limited number of samples, finding the optimal hyper-parameters of a neural network using gradient-free algorithms outperforms traditional gradient-based algorithms, leading to more accurate classification results. In addition, Generative Adversarial Networks could be used to impute the missing data channels in multi-channel bio-signals, and the generated data used for further analysis and classification tasks

Uncertain Multi-Criteria Optimization Problems

Most real-world search and optimization problems naturally involve multiple criteria as objectives. Generally, symmetry, asymmetry, and anti-symmetry are basic characteristics of binary relationships used when modeling optimization problems. Moreover, the notion of symmetry has appeared in many articles about uncertainty theories that are employed in multi-criteria problems. Different solutions may produce trade-offs (conflicting scenarios) among different objectives. A better solution with respect to one objective may compromise other objectives. There are various factors that need to be considered to address the problems in multidisciplinary research, which is critical for the overall sustainability of human development and activity. In this regard, in recent decades, decision-making theory has been the subject of intense research activities due to its wide applications in different areas. The decision-making theory approach has become an important means to provide real-time solutions to uncertainty problems. Theories such as probability theory, fuzzy set theory, type-2 fuzzy set theory, rough set, and uncertainty theory, available in the existing literature, deal with such uncertainties. Nevertheless, the uncertain multi-criteria characteristics in such problems have not yet been explored in depth, and there is much left to be achieved in this direction. Hence, different mathematical models of real-life multi-criteria optimization problems can be developed in various uncertain frameworks with special emphasis on optimization problems

Enhanced Harris's Hawk algorithm for continuous multi-objective optimization problems

Multi-objective swarm intelligence-based (MOSI-based) metaheuristics were proposed to solve multi-objective optimization problems (MOPs) with conflicting objectives. Harris’s hawk multi-objective optimizer (HHMO) algorithm is a MOSIbased algorithm that was developed based on the reference point approach. The reference point is determined by the decision maker to guide the search process to a particular region in the true Pareto front. However, HHMO algorithm produces a poor approximation to the Pareto front because lack of information sharing in its population update strategy, equal division of convergence parameter and randomly generated initial population. A two-step enhanced non-dominated sorting HHMO (2SENDSHHMO) algorithm has been proposed to solve this problem. The algorithm includes (i) a population update strategy which improves the movement of hawks in the search space, (ii) a parameter adjusting strategy to control the transition between exploration and exploitation, and (iii) a population generating method in producing the initial candidate solutions. The population update strategy calculates a new position of hawks based on the flush-and-ambush technique of Harris’s hawks, and selects the best hawks based on the non-dominated sorting approach. The adjustment strategy enables the parameter to adaptively changed based on the state of the search space. The initial population is produced by generating quasi-random numbers using Rsequence followed by adapting the partial opposition-based learning concept to improve the diversity of the worst half in the population of hawks. The performance of the 2S-ENDSHHMO has been evaluated using 12 MOPs and three engineering MOPs. The obtained results were compared with the results of eight state-of-the-art multi-objective optimization algorithms. The 2S-ENDSHHMO algorithm was able to generate non-dominated solutions with greater convergence and diversity in solving most MOPs and showed a great ability in jumping out of local optima. This indicates the capability of the algorithm in exploring the search space. The 2S-ENDSHHMO algorithm can be used to improve the search process of other MOSI-based algorithms and can be applied to solve MOPs in applications such as structural design and signal processing

Exploration and exploitation zones in a minimalist swarm optimiser

The trade off between exploration and exploitation is one of the key challenges in evolutionary and swarm optimisers which are led by guided and stochastic search. This work investigates the exploration and exploitation balance in a minimalist swarm optimiser in order to offer insights into the population’s behaviour. The minimalist and vector-stripped nature of the algorithm—dispersive flies optimisation or DFO—reduces the challenges of understanding particles’ oscillation around constantly changing centres, their influence on one another, and their trajectory. The aim is to examine the population’s dimensional behaviour in each iteration and each defined exploration-exploitation zone, and to subsequently offer improvements to the working of the optimiser. The derived variants, titled unified DFO or uDFO, are successfully applied to an extensive set of test functions, as well as high-dimensional tomographic reconstruction, which is an important inverse problem in medical and industrial imaging