A Study on Conjugate Gradient Methods and their Modifications

In recent years there has been the great deal of interest in the development of optimization algorithms that deal with the problems of finding a global or local minimum of a given problem. Unconstrained optimization problem arise in virtually in areas in Science and Engineering, and in many areas of the Social Sciences. A significant percentage of real world optimization problems are data fitting problem. The size of real world unconstrained optimization problem is widely distributed, varying from small problems to large problems. In many cases, the objective function is a complete routine that is expensive to evaluate so that even small problems are expensive and difficult to solve. The user of an unconstrained optimization problem is expected to provide the function and a starting guess to the solution. The routine is expected to return and estimate of local minimiser (say) of f(x). But in most cases they are not provided and instead is approximated in various ways by the algorithm. Approximating these derivatives is one of the main challenges of creating unconstrained optimization method.