NOVEL GRAVITATIONAL SEARCH ALGORITHMS AND THEIR APPLICATIONS

Abstract

In many real world optimization problems it is often desired to determine a global optimal solution rather than a local optimal solution. Determining the global optimal solution of a nonlinear optimization problem is generally more difficult as compared to determining a local optimal solution. In fact whereas it is easy to check mathematically whether a local optimal solution has been achieved or not, it is not so in the case of a global optimal solution. However because of the practical necessity the search for point of global optimum often becomes necessary. Conventional computing paradigms usually face difficulty in dealing with such real world problems. Natural systems have emerged over millennia to solve such problems. They have inspired several natural computing paradigms that can be used where conventional computing techniques perform unsatisfactorily. Several optimization methods are available in literature to solve above problems. These methods can be categorized into deterministic methods and probabilistic methods. Deterministic methods are applicable only to a restricted class of problems but probabilistic methods are more general. Nature is one of the source of inspiration of these methods. Therefore, they are also known as Nature Inspired Algorithm (NIA). In literature many nature inspired optimization algorithms are available to solve unconstrained optimization problem but there is no single method which can solve all the problems. Therefore new algorithms and existing algorithms are being developed with the hope that they have some advantage over existing ones. Hence searching a new heuristic algorithm is an open problem. Particle Swarm Optimization (PSO), Genetic Algorithm, Artificial Bee Colony, Differential Evolution, etc. are some example of Nature Inspired Algorithms. Gravitational Search Algorithm (GSA) is one of the newly developed Nature Inspired Algorithm. This algorithm is based on the law of gravity and law of motion. The law of gravity state that “each particle attracts every other particle and the gravitational force between two particles is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.” Hence this attraction force is the cause of movement in the particles. The law of motion state that “the velocity of a particle is equal to the sum of the fraction of its previous velocity and the variation in the velocity. Variation in the velocity or acceleration of any particle is equal to the force acted on the system divided by mass of inertia.” ii In GSA, agents are considered as particles or objects and their performance is measured by their masses. The solution of the problem is represented by the position of the particle at specified dimension and the quality of the solution is represented by mass of the particle, higher the mass better the solution. The aim of this PhD Thesis is to improve the efficiency and reliability of GSA, with a view to solve real life optimization problems. To achieve this objective, GSA is hybridized with two well-known real coded genetic algorithm operators namely Laplace Crossover and Power Mutation. As a result, three hybridized variants of GSA are proposed. The first variant is obtained by hybridizing GSA with Laplace Crossover. It is called LXGSA. The second hybridized variant is obtained by hybridizing GSA with Power Mutation. It is called PMGSA. The third version is obtained by hybridizing GSA with Laplace Crossover as well as Power Mutation. It is called LXPMGSA. The performance of GSA and the three proposed variants are evaluated on benchmark problems available in literature not only for unconstrained optimization problems but also for constrained optimization as well as integer and mixed integer optimization problems. Based on the extensive numerical and graphical analysis, including statistical tests it is shown that one of the proposed variant outperforms the other variants as well as the original GSA. Next, the algorithms developed in this Thesis are implemented for the solution of a number of real life optimization problems. The first application problem is that of reconstruction of 3D curves and surfaces which is modelled as a nonlinear optimization problem in which the objective function to be minimized is the error function between given data points and data points on the generated curve/surface. The problem is solved using the original GSA and the three proposed variants. The approach is illustrated with the help of two examples- a helix and a dumbbell. The results are demonstrated and it is concluded that the approach is very promising in the area. Also, one of the variants outperforms all other variants considered in this study. The second application is an unconstrained application problem of curve fitting. All the algorithms proposed in the Thesis are used to solve it. In many experimental studies in scientific applications a set of given data is to be approximated. This can be performed either by minimizing the Least Absolute Deviation or by minimizing the Least Square Error. The objective of this problem is to demonstrate the use of Gravitational Search Algorithm and its proposed hybridized variants to fit polynomials of degree 1, 2, 3 or 4 to a set of N iii points. It is concluded that one of the hybridized version outperforms all other variants for this problem. Further, three well known constrained engineering design optimization problems are considered. The solution methodologies proposed in the earlier chapters of the Thesis are used to compare the existing results available in literature. It is concluded that the performance of one of the variants is a definite improvement over the other variants considered. The Thesis concludes with the overall conclusions and outlines the limitations and scope of the proposed algorithms. Later it suggests future scope and new directions of research in this area.