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Retractions on closed sets

Guillaume Olikier

TL;DR

This paper shows that the weaker definition of retraction on a closed subset of a Euclidean vector space should be preferred as it inherits the main property of the other while being less restrictive.

Abstract

On a manifold or a closed subset of a Euclidean vector space, a retraction enables to move in the direction of a tangent vector while staying on the set. Retractions are a versatile tool to perform computational tasks such as optimization, interpolation, and numerical integration. This paper studies two known definitions of retraction on a closed subset of a Euclidean vector space, one being weaker than the other. Specifically, it shows that, in the context of constrained optimization, the weaker definition should be preferred as it inherits the main property of the other while being less restrictive.

Retractions on closed sets

TL;DR

This paper shows that the weaker definition of retraction on a closed subset of a Euclidean vector space should be preferred as it inherits the main property of the other while being less restrictive.

Abstract

On a manifold or a closed subset of a Euclidean vector space, a retraction enables to move in the direction of a tangent vector while staying on the set. Retractions are a versatile tool to perform computational tasks such as optimization, interpolation, and numerical integration. This paper studies two known definitions of retraction on a closed subset of a Euclidean vector space, one being weaker than the other. Specifically, it shows that, in the context of constrained optimization, the weaker definition should be preferred as it inherits the main property of the other while being less restrictive.
Paper Structure (9 sections, 3 theorems, 33 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 3 theorems, 33 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Elements of variational analysis
  3. Outer semicontinuity and local boundedness of a correspondence
  4. Projection map
  5. Tangent cone
  6. Projective retractions on closed sets
  7. Projective retractions on the determinantal variety
  8. On a retracted line-search method for optimization on closed sets
  9. Conclusion

Key Result

Proposition 2

If $C$ is geometrically derivable at each of its points, then every function such that, for all $x \in C$ and $v \in {T}_{C}^{}(x)$, is a retraction on $C$ called a projective retraction on $C$.

Figures (1)

  • Figure 1: Illustration of the proof of Proposition \ref{['prop:ProjectiveRetractionDeterminantalVariety']}. This proof can be adapted to show that Proposition \ref{['prop:ProjectiveRetractionDeterminantalVariety']} still holds if $\mathbb{R}^{m \times n}$ and $\mathbb{R}_{\le r}^{m \times n}$ are replaced with $\mathrm{S}(n) \coloneq \{X \in \mathbb{R}^{n \times n} \mid X^\top = X\}$ and $\mathrm{S}_{\le r}(n) \coloneq \{X \in \mathrm{S}(n) \mid \mathop{\mathrm{rank}}\nolimits X \le r\}$, respectively. Moreover, this result can be illustrated for $n = 2$ and $r = 1$ by taking advantage of two facts. First, the map $\varphi : \mathbb{R}^3 \to \mathrm{S}(2) : x \mapsto \frac{1}{\sqrt{2}} \left[x_3-x_1x_2 \@normalcr x_2x_3+x_1\right]$ is a bijection. Second, $\mathrm{S}_{\le 1}(2)$ is the image of the cone $C \coloneq \{x \in \mathbb{R}^3 \mid x_3^2 = x_1^2+x_2^2\}$ under $\varphi$. Given $x \in C \setminus \{0\}$, let $Z$ be the tangent vector to $\mathrm{S}_{\le 1}(2)$ at $\varphi(x)$ defined in the proof of Proposition \ref{['prop:ProjectiveRetractionDeterminantalVariety']}. Then, $Z \in \mathrm{S}(2)$. Define $z \coloneq \varphi^{-1}(Z)$. The point $x$, the half-line $\{x+tz \mid t \in [0, \infty)\}$, and $\bigcup_{t \in [0, \infty)} P_{C}(x+tz)$ are represented in the figure. The set $P_{C}(x+tz)$ is a singleton for every $t \in [0, \infty) \setminus \{t_*\}$, but $P_{C}(x+t_*z)$ contains two elements, one on the upper part of the cone, and the other on the lower part of the cone. The segments joining $x+t_*z$ to each of its two projections onto $C$ are represented in dashed line. This figure illustrates that every selection of $[0, \infty) \multimap C : t \mapsto P_{C}(x+tz)$ is discontinuous at $t_*$.

Theorems & Definitions (4)

  • Definition 1: HosseiniUschmajew2019
  • Proposition 2
  • Proposition 3
  • Proposition 4