Projective transformations for interior-point algorithms, and a superlinearly convergent algorithm for the w-center problem

The purpose of this study is to broaden the scope of projective transformation methods in mathematical programming, both in terms of theory and algorithms. We start by generalizing the concept of the analytic center of a polyhedral system of constraints to the w-center of a polyhedral system, which stands for weighted center, where there are positive weights on the logarithmic barrier terms for reach inequality constraint defining the polyhedron X . We prove basic results regarding contained and containing ellipsoids centered at the w-center of the system X . We next shift attention to projective transformations, and we exhibit an elementary projective transformation that transforms the polyhedron X to another polyhedron Z , and that transforms the current interior point to the w-center of the transformed polyhedron Z . We work throughout with a polyhedral system of the most general form, namely both inequality and equality costraints. This theory is then applied to the problem of finding the w-center of a polyhedral system X . We present a projective transformation algorithm, which is an extension of Karmarkar's algorithm, for finding the w-center of the system X . At each iteration, the algorithm exhibits either a fixed constant objective function improvement, or converges superlinearly to the optimal solution. The algorithm produces upper bounds on the optimal value at each iteration. The direction chosen at each iteration is shown to be a positively scaled Newton direction. This broadens a result of Bayer and Lagarias regarding the connection between projective transformation methods and Newton's method. Furthermore, the algorithm specializes to Vaidya's algorithm when used with a line-search, and so shows that Vaidya's algorithm is superlinearly convergent as well. Finally, we show how the algorithm can be used to construct well-scaled containing and contained ellipsoids at near-optimal solutions to the w-center problem.