Long-step strategies in interior-point primal-dual methods
(1997) Faculty Research Seminar on Optimization in Theory and Practice — Location: UNIV IOWA, OBERMANN CTR ADV STUDIES, IOWA CITY (Ia) (1.August.1994)
Files
document.pdf
Restricted Access - Adobe PDF
- 1.71 MB
Details
- Authors
- Abstract
- 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(root n ln(1/epsilon)) 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.
- Affiliations
Citations
Nesterov, Y. (1997). Long-step strategies in interior-point primal-dual methods. Mathematical Programming, 76(1), 47-94. https://doi.org/10.1007/BF02614378 (Original work published 1997)