Analytical Approach for Linear Programming Using Barrier and Penalty Function Methods

In order to solve the primal linear programming problems (and its dual) some methods have been used such as simplex method, geometric approach and interior points methods. None of these methods used Lagrangian function as a tool to solve the problem. This raises a question why are we not using this to solve the linear programming problems. Thus, in this research we study and analyze how the behavior and performance of barrier functions and penalty functions methods for solving the linear programming problems. All of these functions are in Lagrangian form. With logarithmic barrier function methods we introduce three types of function; that is, primal logarithmic, dual logarithmic and primal-dual logarithmic functions. There are two mam results obtained from the logarithmic function method. First, we prove that for every value of the barrier parameter, the logarithmic barrier function for the problem has a unique minimizer; and then if the sequence of the values of barrier parameters tends to zero, then the sequence of the minimizers converges to a minimizer of the problem. From these properties, we construct an algorithm for solving the problem using the logarithmic barrier function methods. Second, we give the equivalences between the interior points set, the primal logarithmic barrier function, the dual logarithmic barrier function, the primal-dual logarithmic barrier function and the system of linear equations associated with these functions.

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