New Interior Point Algorithms in Linear Programming †

In this paper the abstract of the thesis ”New Interior Point Algorithms in Linear Programming” is presented. The purpose of the thesis is to elaborate new interior point algorithms for solving linear optimization problems. The theoretical complexity of the new algorithms are calculated. We also prove that these algorithms are polynomial. The thesis is composed of seven chapters. In the first chapter a short history of interior point methods is discussed. In the following three chapters some variants of the affine scaling, the projective and the path-following algorithms are presented. In the last three chapters new path-following interior point algorithms are defined. In the fifth chapter a new method for constructing search directions for interior point algorithms is introduced, and a new primal-dual pathfollowing algorithm is defined. Polynomial complexity of this algorithm is proved. We mention that this complexity is identical with the best known complexity in the present. In the sixth chapter, using a similar approach with the one defined in the previous chapter, a new class of search directions for the self-dual problem is introduced. A new primal-dual algorithm is defined for solving the self-dual linear optimization problem, and polynomial complexity is proved. In the last chapter the method proposed in the fifth chapter is generalized for target-following methods. A conceptual target-following algorithm is defined, and this algorithm is particularized in order to obtain a new primal-dual weighted-path-following method. The complexity of this algorithm is computed.