263. Analysis of the primal-dual central path for nonlinear semidefinite optimization without the nondegeneracy condition
Invited abstract in session FB-6: Higher-order Methods in Mathematical Programming II, stream Challenges in nonlinear programming.
Friday, 10:05 - 11:20Room: M:H
Authors (first author is the speaker)
| 1. | Takayuki Okuno
|
| Seikei University |
Abstract
We study properties of the central path underlying a nonlinear semidefinite optimization problem, called NSDP for short. The latest radical work on this topic was contributed by Yamashita and Yabe (2012): they proved that the Jacobian of a certain equation-system derived from the Karush-Kuhn-Tucker (KKT) conditions of the NSDP is nonsingular at a KKT point under the second-order sufficient condition (SOSC), the strict complementarity condition (SC), and the nondegeneracy condition (NC). This yields uniqueness and existence of the central path through the implicit function theorem. In this paper, we consider the following three assumptions on a KKT point: the enhanced SOSC, the SC, and the Mangasarian-Fromovitz constraint qualification. Under the absence of the NC, the Lagrange multiplier set is not necessarily a singleton and the nonsingularity of the above-mentioned Jacobian is no longer valid. Nonetheless, we establish that the central path exists uniquely, and moreover prove that the dual component of the path converges to the so-called analytic center of the
Lagrange multiplier set. As another notable result, we clarify a region around the central path where Newton’s equations relevant to primal-dual interior point methods are uniquely solvable.
Keywords
- Conic and semidefinite optimization
- Linear and nonlinear optimization
Status: accepted
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