In this paper we analyze from a unique point of view the behavior of path-following and primal–dual potential reduction methods on nonlinear conic problems. We demonstrate that most interior-point methods with O(√(n) ln (1/ε)) efficiency estimate can be considered as different strategies of minimizing a convex primal–dual potential function in an extended primal–dual space. Their efficiency estimate is a direct consequence of large local norm of the gradient of the potential function along a central path. It is shown that the neighborhood of this path is a region of the fastest decrease of the potential. Therefore the long-step path-following methods are, in a sense, the best potential-reduction strategies. We present three examples of such long-step strategies. We prove also an efficiency estimate for a pure primal–dual potential reduction method, which can be considered as an implementation of a penalty strategy based on a functional proximity measure. Using the convex primal dual potential, we prove efficiency estimates for Karmarkar-type and Dikin-type methods as applied to a homogeneous reformulation of the initial primal–dual problem.
