Research

This thesis focuses on applying nature inspired algorithms for finding roots of systems of nonlinear equations. The developments have been done to solve single variable nonlinear equations, systems of nonlinear equations and in applying a self-tuning framework on tuning the parameters of the algorithms that are used for problem solving.

Fields including Engineering, Mathematics, Chemistry, Computer Science and Economics often encounter applications of univariate as well as systems of nonlinear equations. Providing solutions for such is challenging and the common method of solving them is the use of numerical methods. Numerical methods often have requirements to be fulfilled to begin with the process of finding approximations. The use of different optimization techniques in such situations have been widely applied in all fields of Engineering as the capabilities of computers continue to increase. The remarkable performance of nature inspired algorithms over other optimization techniques encourages researchers to apply them to various optimization problems. Recently developed algorithms like firefly algorithm, bat algorithm and artificial bee colony have shown their success over many difficult optimization tasks where other optimization techniques fail.

From the initial study, strengths and weaknesses of such algorithms were identified. Particularly, the firefly algorithm was identified as a suitable algorithm for the problem. This was later improved to solve univariate nonlinear equations having complex roots. The main consideration was paid on two important tasks; finding almost all real and complex roots within a reasonable range and omit the necessity of the continuity and differentiability of the functions which is essential for many numerical methods. While applying the firefly algorithm as the suitable metaheuristic algorithm, modifications to the original algorithm have been proposed also to identify solutions simultaneously (through archiving) and to identify the poorly performed populations (through a counter variable). Once completing a moving round by all fireflies, better ones (whose fitness is easured against a predefined threshold) are noted and are put into an archive. Poor populations (which have not contributed to the archive) are identified at a predefined point and new fireflies are introduced to the population in a random manner. This random replacements enhance the exploration property. The proposed new firefly algorithm is named as Modified Firefly Algorithm (MODFA) to solve nonlinear equations.

Finding roots of a univariate nonlinear equation can be considered as a single objective problem, while finding roots of a nonlinear system can handle in either; as single objective or multi- objective. The current approach for handling multi-objective optimization problems is to employ the concept of Pareto optimality. But with the MODFA, finding roots of a system of nonlinear equations is handled as a single objective optimization problem. The new concept of archiving is introduced and with that, within a single run of the algorithm, many solutions can be obtained simultaneously. Another concept used here is a self-tuning framework which is used to tune the parameters of the used nature inspired algorithms. The purpose of using it for the study is to let the users use the algorithm without having knowledge about algorithm specific parameters. This is named as Self Tuning Modified Firefly Algorithm (STMODFA).

The performance of the newly proposed algorithm, is evaluated by comparing it with other algorithms such as genetic algorithms, particle swarm optimization Algorithm, differential evolution , harmony search, cuckoo search algorithm. According to the results obtained with these, it has been shown that the MODFA demonstrates better performance than the other nature inspired algorithms. It is hardly seen the ability of finding roots simultaneously by other algorithms.

Since representations can be easily handled, all implementations were implemented using MATLAB. For almost all univariate and systems of nonlinear equations, the accuracy of an approximation is set as 10⁻². Because the study focuses more on finding as many roots as possible within a single run. To increase the accuracy, later in this research, the concept of hybridization has been introduced. Hybrids of MODFA have been built with numerical as well as natural optimization techniques. This concept gave successful results enabling MODFA to find roots simultaneously with a high accuracy around 10⁻¹².

Clustering is a fundamental problem in data analysis, often used to partition datasets into meaningful groups. However, the effectiveness of clustering methods largely depends on the nature of the dataset and the specific objectives of the task. Traditional clustering methods, such as K-Means, have limitations in handling datasets with varying distributions, cluster shapes, and densities. Moreover, these methods often require pre-specifying the number of clusters (K), which may not align with real-world data complexities. This thesis addresses these limitations by proposing a modified Firefly Algorithm (FA) for automatic clustering with centroid movements.

The proposed algorithm integrates meta-heuristic principles to overcome the challenges associated with traditional clustering methods. Unlike canonical approaches that rely solely on proximity, our method introduces a multi-objective fitness function that considers compactness, separation, and a novel total Traveling Salesman Problem (TSP) penalty to ensure both efficient clustering and optimal navigation within clusters. The algorithm employs a self-tuning mechanism to adaptively determine the optimal number of clusters (K) and uses a unique centroid movement strategy to refine cluster boundaries dynamically. These innovations allow the algorithm to achieve robust clustering outcomes without reliance on initial cluster center selection or manual specification of K.

The practical application of the proposed algorithm is also explored in the thesis in the context of a robotic sensor network for the persistent monitoring of large areas. The dataset used for experiments consists of two-dimensional spatial data, reflecting the challenges of clustering in real-world scenarios. Computational experiments compare the performance of the modified Firefly Algorithm with the K-Means algorithm, revealing that the proposed approach not only mitigates initialization dependency but also produces clusters that facilitate shorter navigation paths within clusters. For instance, the total path distance for clusters formed using the proposed method was reduced significantly compared to K-Means, emphasizing its suitability for navigation-centric applications.

The findings demonstrate the effectiveness of the Firefly Algorithm in addressing clustering problems characterized by complex data distributions and multi-objective goals. Future work will explore extending the algorithm's capabilities to higher-dimensional data and developing self-tuning mechanisms for algorithmic parameters, further enhancing its adaptability and applicability in diverse clustering scenarios.

The processes of optimization can be defined simply as an attempt of making something better or finding the best solution for a maximization or minimization problem.

The basic two approaches of optimization are classical and natural where in some problems classical approach works better and for some other problems natural methods are good.

Natural optimizing techniques, which are extracted from the behavior of natural world, are known as Nature Inspired optimization techniques. Ant colonies which mimic the natural food finding behavior of ants, particle swarm optimizations algorithms which takes the advantage of schooling behavior of fish or flocking behavior of birds are some examples for them.

In this research, the main purpose is to measure the optimizing performance of such nature inspired algorithms with one new nature inspired algorithm known as firefly inspired algorithm, which came to the stage, extracting the flashing behavior of fireflies.

Two major areas of nature inspired algorithms are evolutionary strategies and swarm intelligence. An evolutionary algorithm (EA) is a generic population-based metaheuristic optimization algorithm. An EA uses tools motivated by biological evolution: reproduction, mutation, recombination, and selection.

Swarm intelligence is another problem solving behavior, inspired by nature that emerges from the interaction of individual agents (e.g., bacteria, ants, termites, bees, spiders, fish, and birds) which communicate with other agents by acting on their local environments.

For this research, genetic algorithms is taken as an algorithm from evolutionary strategies and Ant colonies, particle swarm optimization from swarm intelligence to make the comparison with the firefly inspired algorithm, which is also an algorithm that belongs to swarm intelligence.

Travelling salesman problem, which is a representative of NP hard problems, was taken as the bench mark problem to employ all these algorithms.

Each algorithm was used to solve four TSP instances with 16, 29, 51 and 100 cities taken from the TSPLIB and statistics were taken appropriately. Another 5 instances of 29 cities were generated randomly and results were calculated for all four algorithms. The results of the study were manipulated using Matlab 2008.

For all 9 TSP instances, firefly algorithm gave the best results and sometimes ant colony systems too. Particle swarm optimization algorithm always scores the third place and Genetic algorithm performs last.

With the results obtained, it can be clearly said that the firefly algorithm is remarkably successful and better than other three algorithms in its discrete version.