The proposed research is to establish a class of polynomially bounded algorithms for solving convex programming problems efficiently. The work is to focus on classical barrier methods, such as the Sequential Unconstrained Minimization Technique (SUMT) and it's variants. Specifically, " Projective SUMT " is one such variant that is derived from the barrier function method by utilizing the differential equation characterizing the trajectory of unconstrained minimizers. The assumptions for convergence of these barrier-type methods are considerably weaker than those proposed by Karmarkar. The concept of factorable functions wil be used in the analysis of running time. Representation of a function as factorable enables the automatic calculation of higher order derivatives, and can be useful for encoding the data and function calculations. The running time also depends upon the time required to solve a system of equations with respect to the Hessian of the suitable factorable barrier function, the solution of an approximate step-size problem, and the number of iterations required to find the solution to prescribed accuracy. These issues will be explored in the research.
