Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence rate of alternating projection method for the intersection of an affine subspace and the second-order cone

Hiroyuki Ochiai, Yoshiyuki Sekiguchi, Hayato Waki

Abstract

We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is $O(k^{-1/2})$, where $k$ is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example.

Convergence rate of alternating projection method for the intersection of an affine subspace and the second-order cone

Abstract

We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is , where is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example.
Paper Structure (26 sections, 18 theorems, 83 equations, 1 figure)

This paper contains 26 sections, 18 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Preliminary
  3. Notation and symbols
  4. Intersection of an affine subspace and the second-order cone
  5. Facial structure of the second-order cone and the singularity degree
  6. Lemma for exact convergence rate
  7. Alternating projection method and the convergence rate
  8. Case 1: $H\cap\mathop{\mathrm{bd}}\nolimits\mathcal{K}=\{\bm{0}\}$
  9. Case 2: $H\cap\mathop{\mathrm{bd}}\nolimits\mathcal{K}=\{\bm{u}_*\}$ and $\bm{u}_{*}\neq \bm{0}$
  10. Case 3: $H\cap\mathop{\mathrm{bd}}\nolimits\mathcal{K}=C(\bm{d})$
  11. Examples of the intersection of an affine subspace and the product set of two second-order cones
  12. Example 1: the convergence rate $O(k^{-1/6})$ is tight
  13. Example 2: the convergence rate $O(k^{-1/6})$ is NOT tight
  14. Proofs of Lemmas \ref{['lemma:socp0']}, \ref{['lemma:matrix']} and \ref{['lemma:socp']}
  15. Proof of Lemma \ref{['lemma:socp0']}
  16. ...and 11 more sections

Key Result

Lemma 2.1

Assume that $\bm{x}\in\mathop{\mathrm{bd}}\nolimits\mathcal{K}\setminus\{\bm{0}\}$. If $\bm{y}\in\mathcal{K}$ is orthogonal to $\bm{x}$, then $\bm{y}$ is formed as $\bm{y} = \alpha \hat{\bm{x}}$ for some $\alpha \ge 0$.

Figures (1)

  • Figure 1: Transition diagram of the recurrence formulas \ref{['Proj3']}, \ref{['Proj4']} and \ref{['Proj5']}: (15A) means the recurrence formula \ref{['Proj5']} with $Y_0 < Z_0$ and $Z_0>0$, and (15B) means \ref{['Proj5']} with $Y_0 > Z_0$ and $Z_0<0$. The solid line from \ref{['Proj4']} to (15B) means that the recurrence formula changes in one iteration. A dotted line means that the recurrence formula changes in a few iterations.

Theorems & Definitions (28)

  • Lemma 2.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.1
  • Theorem 2.2
  • Proposition 2.1
  • Lemma 2.4
  • Theorem 3.1
  • Proposition 3.1
  • ...and 18 more