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Analysis of the primal-dual central path for nonlinear semidefinite optimization without the nondegeneracy condition

Takayuki Okuno

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 strong 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the Lagrange multiplier set is not necessarily a singleton and the nonsingularity of the above-mentioned Jacobian is no longer valid.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.

Analysis of the primal-dual central path for nonlinear semidefinite optimization without the nondegeneracy condition

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 strong 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the Lagrange multiplier set is not necessarily a singleton and the nonsingularity of the above-mentioned Jacobian is no longer valid.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.
Paper Structure (32 sections, 19 theorems, 192 equations)

This paper contains 32 sections, 19 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. KKT conditions for NSDP
  4. Other fundamental properties of the complementarity condition
  5. Second-order optimality conditions and relevant properties
  6. Second-order necessary optimality for the NSDP bonnans2013perturbation
  7. Second-order sufficient optimality for the NSDP bonnans2013perturbation
  8. BKKT conditions and central path
  9. Main analysis
  10. Assumptions and outline of analysis
  11. Goal and outline of the analysis:
  12. Additional notations and symbols used hereafter
  13. Existence of analytic center for NSDP
  14. Proof of Claim (i): convergence of BKKT triplets to KKT triplet with analytic center
  15. Proof of Claim (ii): convergence of BKKT points along a specific direction
  16. ...and 17 more sections

Key Result

Lemma 1

For ${Y}\in \mathbb{S}^m_+$ such that $G_{\ast} {Y}=O$ and a direction $d\in \mathbb{R}^n$, it holds that where $Y^{\rm EE}$ and $G_{\ast}^{\rm FF}$ are defined in eq:anotherform, and moreover $\Delta {G}^{\rm FE}$ and $\Delta {G}^{\rm EF}$ in eqn:1212-2.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Definition 3
  • Remark 3
  • Lemma 1
  • Definition 4: Analytic center for NSDP \ref{['al:nsdp']}
  • Proposition 1
  • Proposition 2
  • ...and 18 more