Interior point methods or sometimes called Barrier methods are a class of optimization algorithms that are useful in solving inequality constrained linear or nonlinear programming problems. The optimization problem with inequality constraint is solved by using the Newtons method by converting it into series of equality constraints or KKT conditions. This paper presents the comparative study of the classical logarithmic barrier method and the primal-dual interior-point method along with their convergence analysis. A kernel function-based interior point method for second-order cone problems is also examined. The computational complexity of the algorithms is also discussed along with the implementation issues.