# Douglas–Rachford is the best projection method

Minh N. Dao, Mareike Dressler, Hongzhi Liao†, and Vera Roshchina† School of
Sciences, RMIT University, Melbourne, VIC 3000, Australia.School of
Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052,
Australia.

###### Abstract

We prove that the Douglas–Rachford method applied to two closed convex cones
in the Euclidean plane converges in finitely many steps if and only if the set
of fixed points of the Douglas–Rachford operator is nontrivial. We analyze
this special case using circle dynamics.

We also construct explicit examples for a broad family of projection methods
for which the set of fixed points of the relevant projection method operator
is nontrivial, but the convergence is not finite. This three-parametric family
is well known in the projection method literature and includes both the
Douglas–Rachford method and the classic method of alternating projections.

Even though our setting is fairly elementary, this work contributes in a new
way to the body of theoretical research justifying the superior performance of
the Douglas–Rachford method compared to other techniques. Moreover, our result
leads to a neat sufficient condition for finite convergence of the
Douglas–Rachford method in the locally polyhedral case on the plane, unifying
and expanding several special cases available in the literature.

AMS Subject Classifications: Primary: 90C25, 65K10; Secondary: 47H09, 65K05.

Keywords: Alternating projections, convex cone, Douglas–Rachford method,
finite convergence, local polyhedrality, projection methods.

##  1 Introduction

Projection methods, such as the method of alternating projections and the
Douglas–Rachford method, primarily serve the purpose of finding a point in the
intersection of two (closed convex) sets using an interative process built by
composing the identity mapping, projection and reflection operators and their
convex combinations. For instance, the method of alternating projections [19]
is an iterative application of the composition of projections onto two
(different) sets, and each step of the Douglas–Rachford method [14, 17] is the
composition of two reflections averaged with the identity, applied to the
preceding iterate. The aforementioned methods belong to a broader family of
projection methods defined in [12]. Even though there exist other popular
projection methods that do not belong to this family, such as Dykstra’s method
[5, Section 30.2] and the method of circumcentred reflections [9], for our
purposes it is reasonable to limit the scope to a specific family of methods.

Our goal is to show that the Douglas–Rachford method is “the best” method in
this family since it converges finitely under very mild assumptions, moreover,
the number of iterations needed to reach the set of fixed points has a uniform
bound that does not depend on the starting point (see Theorem 2.2). This
finite convergence property is not true for all other members of this family:
we demonstrate this by providing examples for every other case (see Theorem
2.6). We restrict our setting to the case of two closed convex cones in the
two-dimensional space. This setting is sufficient to showcase the differences
in the behavior of various projection methods, and allows for a neat proof of
the finite convergence of the Douglas–Rachford method using circle dynamics.

The finite convergence of the Douglas–Rachford method has been investigated in
[8] for the case involving an affine subspace and a locally polyhedral set, as
well as for a hyperplane and an epigraph. It is then proved in [1] for the
scenario of a finite set and a halfspace. Subsequently, some more finite-
convergence results for the Douglas–Rachford method are provided in [7].
However, most of the existing sufficient conditions for finite convergence
require the presence of Slater’s condition. Only one noteworthy exception is
when one set is a line or a halfplane, and the other is a locally polyhedral
set in the Euclidean plane, where finite convergence can still be established
without Slater’s condition. The key convergence result of this paper allows us
to obtain a general condition for the finite convergence of the
Douglas–Rachford method for locally polyhedral sets in the plane subsumes the
known results; note that in this case we may not be able to obtain a uniform
bound on the number of iterations necessary to reach the set of fixed points.
It is also worth mentioning that finite convergence of the method of
alternating projections has recently been studied in [10, 11], but only for
two non-intersecting sets.

The paper is organized as follows. In Section 2 we explain our setting, state
and briefly discuss the main results (Theorems 2.2 and 2.6). While Theorem 2.2
characterizes the finite convergence of the Douglas–Rachford method for the
case of two cones in the plane, Corollary 2.4 provides a more convenient
condition for the finite convergence, requiring the intersection of two sets
to have at least two points in their intersection. We state and prove a
general finite convergence result for locally polyhedral sets in the plane in
Corollary 2.5. In Section 3 we build the auxiliary theory needed for the proof
of Theorem 2.2, demonstrating that Douglas–Rachford converges finitely for the
case of two convex cones under natural conditions. Section 4 is dedicated to
the detailed list of examples needed to prove Theorem 2.6, that shows that
Douglas–Rachford is the only method in the broad family of projection methods
that enjoys the finite-convergence properties given by Theorem 2.2.

We work in the real finite-dimensional space, predominantly limiting ourselves
to $\mathbb{R}^{2}$. Whenever our proofs work in a more general setting than
$\mathbb{R}^{2}$, we state the results in suitable generality. In consistency
with the literature on projection methods that is often set in Hilbert spaces,
we use  $\langle x,y\rangle$ to denote the standard inner product on
$\mathbb{R}^{d}$ and
${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}\|x\|}=\sqrt{\langle
x,x\rangle}$ for the norm induced by the inner product. For any
$x,y\in\mathbb{R}^{d}$, the line segment connecting $x$ and $y$ is given by

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}[x,y]}\coloneqq\\{\lambda
x+(1-\lambda)y\ :\ \lambda\in[0,1]\\}.$

We say that $r\subseteq\mathbb{R}^{d}$ is a  _ray_ if there exists some
$x\in\mathbb{R}^{d}$ such that

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}r}=\\{\lambda
x\ :\ \lambda\in(0,+\infty)\\}.$

A set $S\subseteq\mathbb{R}^{d}$ is called a  _cone_ if for all $x\in S$ and
all $\lambda\in(0,+\infty)$, we have $\lambda x\in S$. Any cone is a union of
rays. For $S\subseteq\mathbb{R}^{d}$, its  _polar cone_ $S^{\circleddash}$ is
defined by

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}S^{\circleddash}}\coloneqq\\{y\in\mathbb{R}^{d}\
:\ \text{for all }x\in S,\quad\langle x,y\rangle\leq 0\\}$

and its  _orthogonal complement_ $S^{\perp}$ is given by

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}S^{\perp}}\coloneqq\\{y\in\mathbb{R}^{d}\
:\ \text{for all }x\in S,\quad\langle x,y\rangle=0\\}.$

We denote the  _interior_ of $S$ by  $\operatorname{int}S$ and its  _boundary_
by  $\operatorname{bdry}S$.

##  2 Generalized Projection Methods and the Main Results

Recall that for a closed convex set $C\subseteq\mathbb{R}^{d}$, by [5, Theorem
3.16], the projection
${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}P_{C}(x)}$
of $x\in\mathbb{R}^{d}$ onto $C$ is the unique point $p=P_{C}(x)\in C$ such
that, for all $u\in C$,

$\|x-p\|\leq\|x-u\|.$

Given two closed convex sets $A,B\subseteq\mathbb{R}^{d}$ and an initial point
$x_{0}\in\mathbb{R}^{d}$, the intersection of $A$ and $B$ can be approached by
alternatively projecting the current iterate onto each one of these sets. We
may consider the sequence of two projections as one iteration and treat this
method as fixed point iterations of the composition
${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}T_{AP}(x)}\coloneqq
P_{B}\circ P_{A}(x)$, called  _the method of alternating projections_. This
method, also called _von Neumann method_ after its inventor, can have very
slow convergence with rate depending on the relative geometry of the two sets
(exemplified by the case of two linear subspaces where the rate of convergence
is linear and depends on the Friedrichs angle between the subspaces; see [16,
2, 3]).

Amongst many ways to improve the convergence rate of the method of alternating
projections, one can ‘overproject’ by pushing the projection beyond the
boundary of the closed convex set. An extreme version of such an
overprojection is a reflection in the hyperplane perpendicular to the
direction of the projection.

For a closed convex set $C\subseteq\mathbb{R}^{d}$, we define the  _reflection
operator_ $R_{C}(x)$ as

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}R_{C}(x)}\coloneqq
2P_{C}(x)-x,$

that is, $R_{C}=2P_{C}-\mathrm{Id}$, with  $\mathrm{Id}$ being the  _identity
operator_.

The  _Douglas–Rachford operator_ on an ordered pair of closed convex sets
$A,B\subseteq\mathbb{R}^{d}$ is the composition of two reflections, averaged
with the identity operator,

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}T_{DR}}\coloneqq\frac{1}{2}({\rm
Id}+R_{B}\circ R_{A}).$

Analogous to overprojecting, it can be useful to underproject, and to compose
the fixed point operator from various combinations of such mappings. It is
reasonable to build such operators preserving the benign properties that help
control and track the convergence. We therefore use the following definition
of a generalized projection method from [12, 13] that allows us to work in a
fairly general setting, while ensuring converegence of our method.

######  Definition 2.1 (Generalized projection operator).

Let $A,B\subseteq\mathbb{R}^{d}$ be closed convex sets, and
$\mu,\lambda\in(0,2],\kappa\in(0,+\infty)$. Denote

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}P_{A}^{\lambda}}\coloneqq(1-\lambda)\operatorname{Id}+\lambda
P_{A}\quad{\rm
and}\quad{\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}P_{B}^{\mu}}\coloneqq(1-\mu)\operatorname{Id}+\mu
P_{B}.$

We define the  _generalized projection operator_ as

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}T_{\lambda,\mu}^{\kappa}}\coloneqq(1-\kappa)\operatorname{Id}+\kappa
P_{B}^{\mu}\circ P_{A}^{\lambda}.$

Note that $\mu=\lambda=\kappa=1$ yields the alternating projection operator,
and $\mu=\lambda=2$ and $\kappa=1/2$ generates the Douglas–Rachford operator.

Starting from $x_{0}\in\mathbb{R}^{d}$, the generalized projection method
applies the operator $T=T_{\lambda,\mu}^{\kappa}$ to the current iterate,
generating a sequence $(x_{n})_{n\in\mathbb{N}}$, with

$x_{n+1}=T(x_{n})=T^{n}(x_{0}).$

This method is said to  _converge finitely_ if for every
$x_{0}\in\mathbb{R}^{d}$, the sequence $(T^{n}(x_{0}))_{n\in\mathbb{N}}$
converges finitely, i.e., there is an $n\in\mathbb{N}$ such that
$T^{n}(x_{0})\in\operatorname{Fix}T$, where $\operatorname{Fix}T$ denotes the
_fixed point set_ of $T$, given by

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}\operatorname{Fix}T}\coloneqq\\{x\in\mathbb{R}^{d}\
:\ T(x)=x\\}.$

If we restrict the domain and codomain of $T$ to a subset $S$, we emphasize
this in the set of fixed points as
${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}\operatorname{Fix}_{S}T}$.

For the case when the two sets $A$ and $B$ are closed convex cones, we will
borrow the linear algebra notation and let

${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}\operatorname{Ker}T}\coloneqq\\{x\in\mathbb{R}^{d}\
:\ T(x)=0_{d}\\},$

where  $0_{d}$ is the origin of $\mathbb{R}^{d}$.

In the case when $A,B\subseteq\mathbb{R}^{d}$ are closed convex sets with
$A\cap B\neq\varnothing$, it is known, e.g., from [6, Corollary 3.9 and
Theorem 3.13] that

$\operatorname{Fix}T_{DR}=A\cap
B+(A-B)^{\circleddash}\text{~{}~{}and~{}~{}}P_{A}(\operatorname{Fix}T_{DR})=A\cap
B$ (1)

and for every sequence $(x_{n})_{n\in\mathbb{N}}$ generated by the
Douglas–Rachford method,

$x_{n}\to x\in\operatorname{Fix}T_{DR}\text{~{}~{}and~{}~{}}P_{A}(x_{n})\to
P_{A}(x)\in A\cap B\text{~{}as~{}}n\to+\infty.$ (2)

In the case of two linear subspaces, the Douglas–Rachford method also exhibits
linear convergence and the rate of convergence is the cosine of the Friedrichs
angle between the subspaces; see [15, 4]. For the linear convergence in a more
general setting, we refer to [18].

Our first result (Theorem 2.2) shows that the Douglas–Rachford method
converges finitely on a pair of closed convex cones with the only exception
being the case when the set of fixed points of the Douglas–Rachford operator
is trivial. In particular, this means that these two sets intersect in a
single point, since the set of fixed points of the Douglas–Rachford operator
must contain the intersection of these sets. For an illustration of the
result, see Figure 1.

Figure 1: Left: Illustration of Theorem 2.2(i). Starting from an initial guess
$x_{0}\notin A\cup B$, the Douglas–Rachford method finds a point in $A\cap B$
in two steps.
Right: Illustration of Theorem 2.2(ii). Here, $A$ and $B$ are two straight
lines and $\operatorname{Fix}T_{DR}=A\cap B=\\{0\\}$. The sequence
$(x_{n})_{n\in\mathbb{N}}$ generated by $T_{DR}$ will converge to 0 but not
finitely.

######  Theorem 2.2.

Let $A,B\subseteq\mathbb{R}^{2}$ be two closed convex cones.

  1. (i)

If $\operatorname{Fix}T_{DR}\neq\\{0\\}$, then there exists an
$n\in\mathbb{N}$ such that
$T_{DR}^{n}(\mathbb{R}^{2})\subseteq\operatorname{Fix}T_{DR}$. That is,
starting from any point $x_{0}\in\mathbb{R}^{2}$, the Douglas–Rachford method
applied to $(A,B)$ converges in at most $n$ iterations.

  2. (ii)

If $\operatorname{Fix}T_{DR}=\\{0\\}$, then for any $n\in\mathbb{N}$,
$T_{DR}^{n}(\mathbb{R}^{2}\setminus\operatorname{Ker}T_{DR})\cap\operatorname{Fix}T_{DR}=\varnothing$.
That is, the Douglas–Rachford method applied to $(A,B)$ converges finitely if
and only if the starting point $x_{0}$ is in the kernel of $T_{DR}$.

The proof of Theorem 2.2 follows from Propositions 3.2 and 3.7 given in
Section 3. We point out that finite convergence to the set of fixed points
does not guarantee that the method actually produces a point in $A\cap B$,
since $\operatorname{Fix}T_{DR}$ may be larger than $A\cap B$. However, it is
common in the literature to consider this type of convergence results; see [7]
for example.

######  Remark 2.3.

The condition $\operatorname{Fix}T_{DR}=\\{0\\}$ in Theorem 2.2(ii) can be
formulated in several equivalent ways. Indeed, if $A,B\subseteq\mathbb{R}^{d}$
are closed convex sets with nonempty intersection, then by (1),

$\displaystyle\operatorname{Fix}T_{DR}=\\{0\\}$ $\displaystyle\iff A\cap
B+(A-B)^{\circleddash}=\\{0\\}$ $\displaystyle\iff A\cap
B=\\{0\\}=(A-B)^{\circleddash}$ $\displaystyle\iff A\cap
B=\\{0\\}\text{~{}and~{}}0\in\operatorname{int}(A-B).$

Now, assume that $A$ and $B$ are closed convex cones. Hence $A-B$ is a cone,
and so $0\in\operatorname{int}(A-B)$ if and only if $A-B=\mathbb{R}^{d}$. In
this case,

$\operatorname{Fix}T_{DR}=\\{0\\}\iff A\cap
B=\\{0\\}\text{~{}and~{}}A-B=\mathbb{R}^{d}.$

######  Corollary 2.4.

Let $A,B\subseteq\mathbb{R}^{2}$ be two closed convex cones with more than one
common point. Then there exists an $n\in\mathbb{N}$ such that for any starting
point $x_{0}\in\mathbb{R}^{2}$, the Douglas–Rachford method applied to $(A,B)$
converges in at most $n$ iterations.

  * Proof.

As $A\cap B$ contains more than one point, we have from (1) that
$\operatorname{Fix}T_{DR}\neq\\{0\\}$. The conclusion follows from Theorem
2.2(i). ∎

We can draw the following interesting consequence for two locally polyhedral
sets. Recall that a set is  _polyhedral at $x$_ if there exists a
neighbourhood of $x$ in which the set coincides with some polyhedral set.

######  Corollary 2.5.

Let $A,B\subseteq\mathbb{R}^{2}$ be two closed convex sets with more than one
common point such that $A$ and $B$ are polyhedral at every point in
$\operatorname{bdry}A\cap\operatorname{bdry}B$ (if any). Then, for any
starting point $x_{0}\in\mathbb{R}^{2}$, the Douglas–Rachford method applied
to $(A,B)$ converges in finitely many iterations to a point in
$\operatorname{Fix}T_{DR}$.

  * Proof.

As a set is polyhedral at each of its interior points, $A$ and $B$ are
actually polyhedral at every point in $A\cap B$. Let
$(x_{n})_{n\in\mathbb{N}}$ be a sequence generated by the Douglas–Rachford
method applied to $(A,B)$. According to (2), $(x_{n})_{n\in\mathbb{N}}$
converges to a point $x\in\operatorname{Fix}T_{DR}$ with $P_{A}(x)\in A\cap
B$. By the local polyhedrality of $A$ and $B$, there exists a neighbourhood of
$P_{A}(x)$ in which $A$ and $B$ coincide with some polyhedral sets. Shrinking
the neighbourhood and translating the sets if necessary, $A$ and $B$ coincide
with some closed convex cones $A_{0}$ and $B_{0}$ in a neighbourhood of
$P_{A}(x)$. Since $A\cap B$ contains more than one point, so does $A_{0}\cap
B_{0}$. In view of [7, Theorem 5.2] and Corollary 2.4,
$(x_{n})_{n\in\mathbb{N}}$ converges finitely to $x$. ∎

It can be seen that Corollary 2.5 covers the result in $\mathbb{R}^{2}$ of [7,
Theorem 4.5]. We also note that if $A$ is convex and
$A\cap\operatorname{int}B\neq\varnothing$, then $A\cap B$ contains more than
one point. Therefore, the result in $\mathbb{R}^{2}$ of [8, Theorem 3.7]
follows from Corollary 2.5.

Our second result (Theorem 2.6) establishes that the Douglas–Rachford method
is the only one in the family of projection methods having the strong finite
convergence behavior given in Theorem 2.2, hence justifying referring to it as
the best projection method.

######  Theorem 2.6.

For any allowable choice of the parameters $\mu,\lambda,\kappa$ in Definition
2.1, except for $(\mu,\lambda,\kappa)=(2,2,1/2)$, namely for all
$T=T_{\lambda,\mu}^{\kappa}\neq T_{DR}$, there exists a pair of closed convex
cones $A,B\subseteq\mathbb{R}^{2}$ and a starting point
$x_{0}\in\mathbb{R}^{2}$ such that, for any $n\in\mathbb{N}$,
$T^{n}(x_{0})\notin\operatorname{Fix}T\neq\\{0\\}$.

We prove Theorem 2.6 in Section 4 by constructing explicit examples for each
choice of the parameters.

##  3 Finite convergence of the Douglas–Rachford method

In this section, we prove Theorem 2.2. We do so by first considering the case
when the kernel of the Douglas–Rachford operator is a line (Section 3.1) and
then reducing the remaining cases to circle dynamics (Section 3.2).

###  3.1 The case when the kernel is a line

Our goal is to prove Proposition 3.2, which yields Theorem 2.2 in the case
when the kernel of the Douglas–Rachford operator is a line. Using the polar,
we deduce the following convenient characterization of the kernel of the
Douglas–Rachford operator.

######  Lemma 3.1.

Let $A,B\subseteq\mathbb{R}^{d}$ be closed convex cones. Then

$\operatorname{Ker}T_{DR}=\operatorname{cone}\,[(-B\cap
A^{\circleddash})\cup(B^{\circleddash}\cap A)].$ (3)

  * Proof.

We first observe that the set $-B\cap A^{\circleddash}$ is orthogonal to
$B^{\circleddash}\cap A$, i.e.,

$\text{for all~{}}u\in-B\cap A^{\circleddash}\text{~{}and all~{}}v\in
B^{\circleddash}\cap A,\quad\langle u,v\rangle=0.$ (4)

Indeed, take any $u\in-B\cap A^{\circleddash}$ and $v\in B^{\circleddash}\cap
A$. Since $u\in A^{\circleddash}$ and $v\in A$, we have $\langle
u,v\rangle\leq 0$. However, as $u\in-B$ and $v\in B^{\circleddash}$ we also
have $\langle u,v\rangle\geq 0$, and so (4) follows.

For convenience, denote

$Q\coloneqq\operatorname{cone}[(-B\cap
A^{\circleddash})\cup(B^{\circleddash}\cap A)].$

We show that $Q\subseteq\operatorname{Ker}T_{DR}$. Consider an arbitrary $x\in
Q$. Then $x=u+v$ for some $u\in-B\cap A^{\circleddash}$ and $v\in
B^{\circleddash}\cap A$ (by virtue of these two sets being closed convex
cones). Since $\langle u,v\rangle=0$ due to (4), we derive that, for all $w\in
A$,

$\langle x-v,w-v\rangle=\langle u,w-v\rangle=\langle u,w\rangle\leq 0,$

because $u\in A^{\circleddash}$. Since $v\in A$, by the angle characterization
of projection, we conclude that $v$ is the projection of $x$ onto $A$. That
is, $P_{A}(x)=v$, and thus, $R_{A}(x)=2v-x=2v-(u+v)=v-u$. Likewise, for all
$w\in B$,

$\langle(v-u)-(-u),w-(-u)\rangle=\langle v,w+u\rangle=\langle v,w\rangle\leq
0,$

where the final inequality is due to $v\in B^{\circleddash}$. We hence deduce
that $-u$ is the projection of $v-u$ onto $B$, i.e., $P_{B}(v-u)=-u$. This
yields

$R_{B}\circ R_{A}(x)=R_{B}(v-u)=2(-u)-(v-u)=-u-v=-x,$

and therefore,

$T_{DR}(x)=\frac{1}{2}\left(x+R_{B}\circ R_{A}(x)\right)=\frac{1}{2}(x-x)=0,$

which means $x\in\operatorname{Ker}T_{DR}$.

It remains to show that $\operatorname{Ker}T_{DR}\subseteq Q$. Consider
$x\in\operatorname{Ker}T_{DR}$. Then

$0=T_{DR}(x)=\frac{1}{2}\left(x+R_{B}\circ R_{A}(x)\right),$

and thus $R_{B}\circ R_{A}(x)=-x$. Now let $y=R_{A}(x)$. Then
$P_{A}(x)=\frac{1}{2}(x+y)$ and because $R_{B}(y)=-x$, also
$P_{B}(y)=\frac{1}{2}(y-x)$. It follows that $P_{B}(y)=-x+P_{A}(x)$ and
$P_{A}(x)=y-P_{B}(y)$.

Since $A$ is a cone, for all $\lambda\in(0,+\infty$, we have $\lambda
P_{A}(x)\in A$, and hence by the characterization of projection,

$0\geq\langle x-P_{A}(x),\lambda P_{A}(x)-P_{A}(x)\rangle=(\lambda-1)\langle
x-P_{A}(x),P_{A}(x)\rangle,$

and therefore $\langle x-P_{A}(x),P_{A}(x)\rangle=0$. This leads to

$\langle P_{B}(y),P_{A}(x)\rangle=-\langle x-P_{A}(x),P_{A}(x)\rangle=0.$ (5)

The angle characterization of projection yields

$\text{for all }u\in A,\quad\langle x-P_{A}(x),u-P_{A}(x)\rangle\leq 0,$ (6)

and

$\text{for all }v\in B,\quad\langle y-P_{B}(y),v-P_{B}(y)\rangle\leq 0.$ (7)

Now, for all $v\in B$, using (5), (6), and the fact that
$P_{A}(x)=y-P_{B}(y)$, we have

$\langle P_{A}(x),v\rangle=\langle P_{A}(x),v\rangle-\langle
P_{A}(x),P_{B}(y)\rangle=\langle P_{A}(x),v-P_{B}(y)\rangle=\langle
y-P_{B}(y),v-P_{B}(y)\rangle\leq 0,$

therefore $P_{A}(x)\in B^{\circleddash}$. Analogously, for all $u\in A$, using
(5), (7), and the fact that $P_{B}(y)=-x+P_{A}(x)$ leads to

$\langle P_{B}(y),u\rangle=\langle P_{B}(y),u\rangle-\langle
P_{B}(y),P_{A}(x)\rangle=\langle P_{B}(y),u-P_{A}(x)\rangle=-\langle
x-P_{A}(x),u-P_{A}(x)\rangle\geq 0,$

and thus $-P_{B}(y)\in A^{\circleddash}$.

We conclude

$x=P_{A}(x)+(-P_{B}(y)),$

where $P_{A}(x)\in A\cap B^{\circleddash}$ and $-P_{B}(y)\in-B\cap
A^{\circleddash}$, which means that $x\in Q$. ∎

Note that the proof of Lemma 3.1 shows that the two sets featuring in the
representation of the kernel are orthogonal. It may happen that one of them is
trivial, but it is perfectly possible that both of them are nontrivial, for
instance, when $A$ and $B$ are two lines orthogonal to each other. These
observations may be useful when considering the case of the kernel being a
line.

######  Proposition 3.2.

Let $T_{DR}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ be the Douglas–Rachford
operator generated by two closed convex cones $A,B\subseteq\mathbb{R}^{2}$.
Suppose that $\operatorname{Ker}T_{DR}$ is a line. Then
$\operatorname{Fix}T_{DR}\neq\\{0\\}$ and
$T_{DR}(\mathbb{R}^{2})\subseteq\operatorname{Fix}T_{DR}$.

  * Proof.

By Lemma 3.1,

$\operatorname{Ker}T_{DR}=\operatorname{cone}\,[(-B\cap
A^{\circleddash})\cup(B^{\circleddash}\cap A)].$

Since $-B\cap A^{\circleddash}$ and $B^{\circleddash}\cap A$ are orthogonal,
if $\operatorname{Ker}T_{DR}$ is a line, either $-B\cap A^{\circleddash}$ is a
line and $B^{\circleddash}\cap A=\\{0\\}$, or the other way around.

_Case 1:_ If $-B\cap A^{\circleddash}=L$, where $L$ is a line through zero,
and $B^{\circleddash}\cap A$ is $\\{0\\}$, then $A,B^{\circleddash}\subseteq
L^{\perp}$.

    * –

If $A=\\{0\\}$, then $A^{\circleddash}=\mathbb{R}^{2}$ and $B=L$;

    * –

If $B^{\circleddash}=\\{0\\}$, then $B=\mathbb{R}^{2}$ and
$A^{\circleddash}=L$;

    * –

If $A\neq\\{0\\}$ and $B^{\circleddash}\neq\\{0\\}$, we must have
$A=-B^{\circleddash}$, coinciding with a half-line orthogonal to $L$, but this
is impossible since in this case $-B\cap A^{\circleddash}$ would be a half-
space.

It follows that either $A=\\{0\\}$ and $B=L$ or $A=L^{\perp}$ and
$B=\mathbb{R}^{2}$.

_Case 2:_ If $B^{\circleddash}\cap A=L$ is a line and $-B\cap
A^{\circleddash}$ is $\\{0\\}$, analogously (or by letting $\tilde{A}=B$,
$\tilde{B}=-A$ and applying the previous case) we have

    * –

If $B=\\{0\\}$, then $B^{\circleddash}=\mathbb{R}^{2}$ and $A=L$;

    * –

If $A^{\circleddash}=\\{0\\}$, then $A=\mathbb{R}^{2}$ and
$B^{\circleddash}=L$;

    * –

If $B\neq\\{0\\}$ and $A^{\circleddash}\neq\\{0\\}$, then
$B^{\circleddash}\cap A$ is a half-space, contradicting the assumption.

Hence either $A=\mathbb{R}^{2}$ and $B=L^{\perp}$ or $B=\\{0\\}$ and $A=L$.

For all four possible cases, it is trivial to check that
$T_{DR}(\mathbb{R}^{2})\subseteq\operatorname{Fix}T_{DR}$. Moreover, since
$\operatorname{Fix}T_{DR}=(A\cap B)+(A-B)^{\circleddash}$ by (1), we deduce
that $\operatorname{Fix}T_{DR}\neq\\{0\\}$. ∎

###  3.2 Reduction to circle dynamics when the kernel is not a line

For the case when the kernel of the Douglas–Rachford operator is not a line,
the proof of Theorem 2.2 can be reduced to the study of circle dynamics of an
associated system. We first explain this identification and derive useful
results that allow us to proceed with the proof of Proposition 3.7 (and
together with Proposition 3.2 proves Theorem 2.2, as we explain at the end of
this section).

Note that if $r$ is a ray, then either it is trivial ($r=\\{0\\}$), or it has
a unique intersection with the unit circle at $x/\|x\|$. Likewise, every point
on the circle defines a unique ray. We can hence identify points on the unit
circle with nontrivial rays. Denote the  _ray generated by_
${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}x}\in\mathbb{R}^{2}$
by  $[x]$, i.e., $[x]=\\{\lambda x\ :\ \lambda\in(0,+\infty)\\}$. Now define
the mapping  $\arg$ that maps nontrivial rays to the point on the unit circle
$S^{1}$,

$\text{for all }x\neq 0,\quad\arg([x])=x/\|x\|.$

Since this mapping is bijective, its inverse is well defined:

$\text{for all }t\in S^{1},\quad\arg^{-1}(t)=[t].$

It turns out that the Douglas–Rachford operator generated by two closed convex
cones maps rays to rays, i.e., it is  _positively homogeneous_. Moreover, it
can only map rays to the trivial ray ‘once’. Thus the behaviour of the method
can be reduced to circle dynamics. We make this more precise in the following
results.

######  Lemma 3.3.

Let $T_{\lambda,\mu}^{\kappa}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the
generalized projection operator generated by two closed convex cones
$A,B\subseteq\mathbb{R}^{d}$. Then $T_{\lambda,\mu}^{\kappa}$ is positively
homogeneous, that is, for all $x\in\mathbb{R}^{d}$ and all
$\lambda\in(0,+\infty)$,

$T_{\lambda,\mu}^{\kappa}(\lambda x)=\lambda T_{\lambda,\mu}^{\kappa}(x).$

Consequently, the Douglas–Rachford operator $T_{DR}$ generated by two closed
convex cones $A,B\subseteq\mathbb{R}^{d}$ is positively homogeneous.

  * Proof.

This follows straightforwardly from the definition of the generalized
projection operator. Since the identity mapping and projections onto cones are
positively homogenenous, so are their compositions and linear combinations. ∎

Lemma 3.3 allows us to introduce a well-defined circle map induced by the
Douglas–Rachford operator. More generally, suppose that
$T\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is a positively homogeneous mapping.
Then $T$ maps rays to rays, that is, $T([x])=[T(x)]$. Therefore, we can define
the associated  _unit circle operator_
${\color[rgb]{0.2,0.2,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.75}\varphi_{T}}\colon
S^{1}\to S^{1}\cup\\{0\\}$,

$\text{for all }t\in
S^{1},\quad\varphi_{T}(t)\coloneqq\arg(T([t]))=\arg([T(t)]).$ (8)

It is evident that this mapping is well defined, moreover taking the inverses
we have

$\arg^{-1}\varphi_{T}(t)=[T(t)]=T([t])$ (9)

whenever $T(t)\neq 0$.

Having zero in the range of $\varphi_{T}$ prevents us from working exclusively
on the circle. Fortunately, we can suitably restrict this mapping to a
continuous subset of the circle and hence avoid dealing with this pesky zero.
A key observation is that the Douglas–Rachford operator never maps to nonzero
points in its kernel, and hence any starting point is either mapped to zero in
the first iteration or cannot be mapped to zero in finitely many iterations.

The next result allows us to pinpoint the specific linear pieces that comprise
the Douglas–Rachford operator. We note that this is the key result that
distinguishes the behavior of this method from the rest of the family
$T_{\lambda,\mu}^{\kappa}$, as any other combination of parameters generate
different types of linear pieces, that, in particular, prevent us from
identifying the set of fixed points of the circle map with the set of fixed
points of the generalised projection method operator.

######  Proposition 3.4.

Let $T_{DR}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the Douglas–Rachford
operator generated by two closed convex cones $A,B\subseteq\mathbb{R}^{d}$.
Then

$T_{DR}(\mathbb{R}^{d})\cap\operatorname{Ker}T_{DR}=\\{0\\}.$

  * Proof.

Suppose the statement is not true, then there exists $y\in
T_{DR}(\mathbb{R}^{d})\cap\operatorname{Ker}T_{DR}$ such that $y\neq 0$. Since
$y\in T_{DR}(\mathbb{R}^{d})$ and $y\neq 0$, there must be some
$x\in\mathbb{R}^{d}$ with $x\notin\operatorname{Ker}T_{DR}$ such that
$T_{DR}(x)=y$. Let $\lambda\in(0,+\infty)$. Since $T_{DR}$ is firmly
nonexpansive [6, Proposition 3.1], we have from [5, Proposition 4.4(i)&(iv)]
that

$\displaystyle\|T_{DR}(x)-T_{DR}(\lambda y)\|^{2}\leq\langle x-\lambda
y,T_{DR}(x)-T_{DR}(\lambda y)\rangle.$

Since $y=T_{DR}(x)\in\operatorname{Ker}T_{DR}$, by Lemma 3.3 we have
$T_{DR}(\lambda y)=\lambda T_{DR}(y)=0$. Therefore,

$\displaystyle\|T_{DR}(x)\|^{2}\leq\langle x-\lambda
T_{DR}(x),T_{DR}(x)\rangle,$

which yields

$\displaystyle(1+\lambda)\|T_{DR}(x)\|^{2}\leq\langle x,T_{DR}(x)\rangle.$

Since $T_{DR}(x)=y\neq 0$, letting $\lambda$ go to positive infinity, the left
hand side approaches positive infinity, which is impossible, since the right
hand side $\langle T_{DR}(x),x\rangle$ is a constant. We reached a
contradiction, hence our assumption is wrong and the original statement is
true. ∎

######  Proposition 3.5.

Let $T_{DR}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ be the Douglas–Rachford
operator generated by two closed convex cones $A,B\subseteq\mathbb{R}^{2}$.
Then $T_{DR}$ is a (continuous) piecewise linear mapping such that each piece
is one of the following types of a linear mapping:

  1. (i)

a composition of a rotation by some fixed angle $\theta$ and a scaling by
$|\cos\theta|$;

  2. (ii)

a projection onto some fixed line;

  3. (iii)

a projection onto zero; or

  4. (iv)

the identity mapping.

  * Proof.

Depending on the point $x\in\mathbb{R}^{2}$ the projection $P_{C}$ of $x$ onto
a closed convex cone $C\subseteq\mathbb{R}^{2}$ is either a reflection in zero
(i.e., $P_{C}(x)=-x$), a reflection in a line (that corresponds to one of the
(at most two) edges of the cone $C$), or the identity mapping. Therefore,
$P_{C}(x)$ is a piecewise linear operator with parts being one of these three
possibilities. Hence, for every point $x\in\mathbb{R}^{2}$, the
Douglas–Rachford operator is a composition of a finite number of such
elementary linear mappings, and thus is in itself a piecewise linear mapping.
Based on whether for a particular point $x$ the reflections $R_{A}(x)$ and
$R_{B}(R_{A}(x))$ are reflections in zero, in a line, or are identity
mappings, we have the nine cases listed in Table 1. Note that if both
$R_{A},R_{B}$ are reflections in lines $L_{1}$ and $L_{2}$ respectively,
suppose the angle between $L_{1}$ and $L_{2}$ is $\theta$, then $|\arg(x)-\arg
R_{B}(R_{A}(x))|=2\theta$ so that $\|T_{DR}(x)\|=|{\rm cos}\,\theta|\|x\|$.

$R_{B}$$R_{A}$ | Reflect in a line $L_{1}$ | Reflect in $0$ | Identity
---|---|---|---
Reflect in a line $L_{2}$ |  Rotate by an angle $\theta$ and scale by $|\cos\theta|$ |  Project onto $L_{2}^{\perp}$ |  Project onto $L_{2}$
Reflect in $0$ |  Project onto $L_{1}^{\perp}$ |  Identity |  Project onto 0
Identity |  Project onto $L_{1}$ |  Project onto 0 |  Identity
Table 1: Possible linear pieces of the mapping $T_{DR}$.

∎

######  Proposition 3.6.

Suppose that $I$ is a connected subset of the unit circle and $\varphi\colon
I\to I$ is a continuous piecewise function, such that each of the pieces is
either an identity, maps to some particular point, or is a fixed rotation. If
$\operatorname{Fix}\varphi\neq\varnothing$, then there exists an
$n\in\mathbb{N}$ such that $\varphi^{n}(I)\subseteq\operatorname{Fix}\varphi$.

  * Proof.

First, note that $\varphi$ is nonexpansive with respect to arc lengths.
Indeed, since $I$ is connected, for any $x,y\in I$, at least one of the two
arcs connecting them lies in $I$. By  $d(x,y)$ denote the length of the
shortest of these arcs (in case there are two). We claim that the mapping
$\varphi$ is nonexpansive with respect to this metric. Assume the contrary,
then there exist some $x,y\in I$ such that the nonexpansiveness is broken with
some $\alpha>1$, that is,

$d(\varphi(x),\varphi(y))\geq\alpha d(x,y).$

On the shortest arc connecting $x$ and $y$, choose the midpoint $z$. Then at
least one of the two resulting segments will also fail the nonexpansiveness
with the same value $\alpha$. We can continue the subdivisions until we
generate a converging sequence of nested segments $[x_{n},y_{n}]$ that
converges to some $\bar{x}$. Since $\bar{x}\in[x_{n},y_{n}]$, either
$[x_{n},\bar{x}]$ or $[\bar{x},y_{n}]$ breaks nonexpansiveness. Choosing the
expansiveness failing interval and relabeling, we have $x_{n}\to\bar{x}$, and
$\varphi$ breaking nonexpansiveness on each one of these segments with the
same $\alpha$. Because there are finitely many pieces, we can refine the
sequence further to ensure that $x_{n}$ is always mapped by the same piece. By
continuity, the value $\varphi(\bar{x})$ must be consistent. This piece is
nonexpansive, since it is either a rotation, a constant map or the identity
map, so we have a contradiction and thus, $\varphi$ is nonexpansive.

By continuity of the mapping $\varphi$, there must be a sufficiently small
neighborhood of $\operatorname{Fix}\varphi$ in $I$ where $\varphi$ is the
projection onto some point in $\operatorname{Fix}\varphi$. Indeed, if this is
not the case, then there exists a sequence $(x_{n})_{n\in\mathbb{N}}$ of
points in $I\setminus\operatorname{Fix}\varphi$ such that
$\operatorname{dist}(x_{n},\operatorname{Fix}\varphi)$ approaches zero, and
$\varphi(x)$ is identified with a piece that is not a projection onto
$\operatorname{Fix}\varphi$. Since the circle is compact, we can assume that
$(x_{n})_{n\in\mathbb{N}}$ converges to some point
$\bar{x}\in\operatorname{Fix}\varphi$, and because the number of pieces is
finite, we can choose a subsequence of $(x_{n})_{n\in\mathbb{N}}$ such that
$\varphi$ acts on $x$ using the same piece. If this piece is a fixed rotation,
then $\varphi(\bar{x})\neq\bar{x}$, which is impossible. If the piece is a
projection onto some point, then this point must be $\bar{x}$, which
contradicts our assumption. We conclude that there exists a sufficiently small
neighborhood of $\operatorname{Fix}\varphi$, say, of size $\varepsilon$ such
that this entire neighborhood is mapped to the set of fixed points. Then for
any point $x\in I$ we have either $\varphi(x)\in\operatorname{Fix}\varphi$ or
we connect $x$ to $\operatorname{Fix}\varphi$ with the shortest arc (this is
possible since the set of fixed points is closed in $I$) and take the point
$y$ on this arc at the distance $\varepsilon$ of the endpoint. Then we have

$d(\varphi(x),\operatorname{Fix}\varphi)\leq d(\varphi(x),\varphi(y))\leq
d(x,y)=d(x,\operatorname{Fix}\varphi)-\varepsilon.$

Since with each application we either bring the point to the set of fixed
points or reduce the distance to the set of fixed points by $\varepsilon$,
after at most $n=2\pi/\varepsilon$ iterations we have
$\varphi^{n}(I)\subseteq\operatorname{Fix}\varphi$. ∎

######  Proposition 3.7.

Let $T_{DR}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ be the Douglas–Rachford
operator generated by two closed convex cones $A,B\subseteq\mathbb{R}^{2}$.
Suppose that $\operatorname{Ker}T_{DR}$ is not a line. Then exactly one of the
following alternatives holds.

  1. (i)

If $\operatorname{Fix}T_{DR}\neq\\{0\\}$, then there is an $n\in\mathbb{N}$
such that $T_{DR}^{n}(\mathbb{R}^{2})\subseteq\operatorname{Fix}T_{DR}$.

  2. (ii)

If $\operatorname{Fix}T_{DR}=\\{0\\}$, then for any $n\in\mathbb{N}$,
$T_{DR}^{n}(\mathbb{R}^{2}\setminus\operatorname{Ker}T_{DR})\cap\operatorname{Fix}T_{DR}=\varnothing$.

  * Proof.

Let $W=\mathbb{R}^{2}\setminus\operatorname{Ker}T_{DR}$. Observe that $0\notin
W$ and that, by Proposition 3.4, $T_{DR}(W)\subseteq W$.

To show (ii), let $\operatorname{Fix}T_{DR}=\\{0\\}$. Then
$\operatorname{Fix}T_{DR}\cap W=\varnothing$, and given that
$T_{DR}^{n}(W)\subseteq W$, we conclude that
$T_{DR}^{n}(W)\cap\operatorname{Fix}T_{DR}=\varnothing$ for all $n$.

For (i), assume $\operatorname{Fix}T_{DR}\neq\\{0\\}$. Recall that by Lemma
3.1, $\operatorname{Ker}T_{DR}$ is a convex cone. Hence its complement $W$ is
also a cone. Moreover, since $\operatorname{Ker}T_{DR}$ is a convex cone but
not a line, its intersection with the unit circle is connected. The set
$I=W\cap S^{1}$ is therefore connected. Thus, restricting domain and codomain
to $I$, the mapping $\varphi_{T}$ in (8) is well defined. Taking into account
that $0\notin T_{DR}(W)$, Lemma 3.3 leads firstly to $T_{DR}([x])=[x]$ being
equivalent to $T_{DR}(x)=x$, that is, $T_{DR}$ maps a ray to itself if and
only if it fixes every point of the ray. This means that restricting $T_{DR}$
to $W$ and $\varphi_{T_{DR}}$ to $I$ we have

$\arg([\operatorname{Fix}_{W}T_{DR}])=\operatorname{Fix}_{I}\varphi_{T_{DR}},\;\text{
or equivalently
}[\operatorname{Fix}_{W}T_{DR}]=\arg^{-1}(\operatorname{Fix}_{I}\varphi_{T_{DR}}),$
(10)

where by  $[S]$ we denote the set of all rays in a cone $S$.

Secondly, Proposition 3.5 yields that restricted to the rays in $[W]$, the
mapping $T_{DR}([x])$ is piecewise linear with pieces being identity,
rotation, or a mapping onto a (different) fixed ray. It is easy to see that
since $T_{DR}$ is nonexpansive, and hence continuous on $W$, the associated
$\varphi_{T_{DR}}$ must be continuous on $I$. From (10) we conclude that
$\operatorname{Fix}_{I}\varphi_{T_{DR}}\neq\varnothing$. It follows from
Proposition 3.6 that there exists some $n\in\mathbb{N}$ such that
$\varphi_{T_{DR}}(I)\subseteq\operatorname{Fix}_{I}\varphi_{T_{DR}}$. Then for
any $[x]\subseteq[W]$,

$\displaystyle
T_{DR}^{n}([x])\overset{\eqref{eq:assocoperator2}}{=}T_{DR}^{n-1}(\arg^{-1}\varphi_{T_{DR}}(I))\overset{\eqref{eq:assocoperator2}}{=}T_{DR}^{n-2}(\arg^{-1}\varphi_{T_{DR}}^{2}(I))=\cdots=\arg^{-1}\varphi_{T_{DR}}^{n}(I).$

Now note that

$\displaystyle\arg^{-1}\varphi_{T_{DR}}^{n}(I)\overset{\text{Prop.}~{}\ref{prop:typesphi}}{\subseteq}\arg^{-1}\operatorname{Fix}_{I}\varphi_{T_{DR}}\overset{\eqref{eq:fixcoincide}}{=}[\operatorname{Fix}_{W}T_{DR}].$

We conclude,
$T_{DR}^{n}(W)=T_{DR}^{n}(\mathbb{R}^{2}\setminus\operatorname{Ker}T_{DR})\subseteq\operatorname{Fix}T_{DR}$
which together with
$T_{DR}(\operatorname{Ker}T_{DR})=\\{0\\}\subseteq\operatorname{Fix}T_{DR}$
(by Proposition 3.4) proves the case (i). ∎

Combining the results of Section 3 finally shows Theorem 2.2.

  * Proof of Theorem 2.2.

If $\operatorname{Ker}T_{DR}$ is not a line the statement follows from
Proposition 3.7. In the case when the kernel is a line, Proposition 3.2 shows
that the set of fixed points is nontrivial and the Douglas–Rachford method
converges in at most one step. ∎

##  4 Counterexamples

In this section, we prove Theorem 2.6 by constructing explicit examples for
each choice of the three parameters, where the method does not converge
finitely, even then the operator has a nontrivial set of fixed points. We
present six different (families of) counterexamples, each covering a subset of
the parameter space from the Definition 2.1. Note that $A\cap
B\subseteq\operatorname{Fix}T_{\lambda,\mu}^{\kappa}$ for all possible choices
of $(\lambda,\mu,\kappa)$. In the examples below, we choose $A\cap B$ to be
nontrivial so that each $\operatorname{Fix}T_{\lambda,\mu}^{\kappa}$ is
nontrivial as well. The union of these subsets together with the choice of
parameters that corresponds to the Douglas–Rachford method covers the whole
parameter space.

More specifically, in each example we demonstrate that for the prescribed
choice of parameters $(\lambda,\mu,\kappa)$ there exists a pair of sets $A$
and $B$ in the plane and a starting point $x_{0}$ such that the set of fixed
points of the corresponding operator $T_{\lambda,\mu}^{\kappa}$ is nonempty,
while the projection method based on $(\lambda,\mu,\kappa)$ and the pair
$A,B$, applied to $x_{0}$, does not converge finitely. Examples 4.1, 4.2, and
4.3 cover all parameter combinations except for the curve $(1/t,1/t,t)$,
$t\in[1/2,+\infty)$, see Figure 2.

Figure 2: From left to right: the values of parameters $(\lambda,\mu,\kappa)$
covered by Examples 4.1, 4.2, and 4.3.

Examples 4.4, 4.5, and 4.6 address the values $t\in(1/2,+\infty)$, and the
only remaining choice $(2,2,1/2)$ corresponds to the Douglas–Rachford method,
see Figure 3.

\begin{overpic}[width=238.49231pt]{pics/egcurve.pdf}
\put(35.0,72.0){{Example~{}\ref{Ex5}}} \put(25.0,51.0){{Example~{}\ref{Ex6}}}
\put(55.0,44.0){Example~{}\ref{Ex4}} \put(69.0,55.0){Douglas--Rachford}
\end{overpic} Figure 3: The values of parameters $(\lambda,\mu,\kappa)$
covered by Examples 4.4, 4.5, and 4.6, and the single choice corresponding to
the Douglas–Rachford method.

For convenience of describing the concrete subsets of the plane, we mostly use
polar coordinates.

Figure 4: Illustration of Example 4.1, when
$(\lambda,\mu,\kappa)\in(0,2]\times(0,2]\times(0,1/2)$.

######  Example 4.1.

Let $A=\\{(\rho,\theta)\ :\ 0\leq\theta\leq 3\pi/4\\}$, $B=\\{(\rho,\theta)\
:\ \pi/2\leq\theta\leq\pi\\}$, and set $x_{0}\in\operatorname{int}A$. Then for
any $n\in\mathbb{N}$, $T^{n}(x_{0})\notin\operatorname{Fix}T$ with
$T=T_{\lambda,\mu}^{\kappa}$ and

$(\lambda,\mu,\kappa)\in(0,2]\times(0,2]\times(0,1/2).$

  * Proof.

We know $A\cap B=\\{(\rho,\theta)\ :\ \pi/2\leq\theta\leq 3\pi/4\\}$. Let
$L\coloneqq\\{(\rho,\theta)\ :\ \theta=\pi/2\\}$. Suppose the Cartesian
coordinates of $x_{0}$ are $x_{0}=(d_{0},y)$, where $d_{0}>0,y>0$, thus
$d(x_{0},L)=d_{0}$, as illustrated in Figure 4. Since
$P_{A}^{\lambda}(x_{0})=x_{0}$, we have

$\displaystyle
P_{B}^{\mu}(P_{A}^{\lambda}(x_{0}))=P_{B}^{\mu}(x_{0})=(1-\mu)\cdot(d_{0},y)+\mu\cdot(0,y)=((1-\mu)d_{0},y),$
(11)

and therefore

$\displaystyle T(x_{0})=(1-\kappa)x_{0}+\kappa
P_{B}^{\mu}(P_{A}^{\lambda}(x_{0}))=(1-\kappa)(d_{0},y)+\kappa((1-\mu)d_{0},y)=((1-\kappa\mu)d_{0},y).$
(12)

Similarly, we can claim

$T^{n}(x_{0})=((1-\kappa\mu)^{n}d_{0},y).$

Given $(\mu,\kappa)\in(0,2]\times(0,1/2)$, we have that, for any
$n\in\mathbb{N}$, $0<(1-\kappa\mu)^{n}<1$, hence
$T^{n}(x_{0})\in\operatorname{int}A$ and $T^{n}(x_{0})\to P_{L}(x_{0})$ as
$n\to+\infty$. ∎

Figure 5: Left: Illustration of Example 4.2, when
$(\lambda,\mu,\kappa)\in(0,2]\times(0,2]\times[1/2,+\infty)\;\text{ and
}\;\kappa\mu\neq 1$.
Right: Illustration of Example 4.3, when
$(\lambda,\mu,\kappa)\in(0,2]\times(0,2]\times[1/2,+\infty)\;\text{ and
}\;\kappa\lambda\neq 1$.

######  Example 4.2.

Consider $A=\\{(\rho,\theta)\ :\ 0\leq\theta\leq\pi\\}$, $B=\\{(\rho,\theta)\
:\ \theta=\pi/2\\}$, and $x_{0}\in\operatorname{int}A$. Then for any
$n\in\mathbb{N}$, $T^{n}(x_{0})\notin\operatorname{Fix}T$ with
$T=T_{\lambda,\mu}^{\kappa}$ and

$(\lambda,\mu,\kappa)\in(0,2]\times(0,2]\times[1/2,+\infty)\quad\text{ and
}\quad\kappa\mu\neq 1.$

  * Proof.

Obviously, $A\cap B=B$. Let $x_{0}=(d_{0},y)$ be the Cartesian coordinates of
$x_{0}$, with $d_{0}\neq 0,y>0$. Then $d(x_{0},B)=|d_{0}|$, see Figure 5.
Since $P_{A}^{\lambda}(x_{0})=x_{0}$, the exact same calculations (11) and
(12) hold true and again analogously we have for any $n\in\mathbb{N}$ that
$T^{n}(x_{0})=((1-\kappa\mu)^{n}d_{0},y)$. For our choice of $(\mu,\kappa)$,
it holds that for any $n\in\mathbb{N}$, $(1-\kappa\mu)^{n}\neq 0$, leading to
$T^{n}(x_{0})\notin\operatorname{Fix}T$. ∎

######  Example 4.3.

Let $A=\\{(\rho,\theta)\ :\ \theta=\pi/2\\}$, $B=\\{(\rho,\theta)\ :\
0\leq\theta\leq\pi\\}$, and $x_{0}\in\operatorname{int}B$. Then for any
$n\in\mathbb{N}$, $T^{n}(x_{0})\notin\operatorname{Fix}T$ with
$T=T_{\lambda,\mu}^{\kappa}$ and

$(\lambda,\mu,\kappa)\in(0,2]\times(0,2]\times[1/2,+\infty)\quad\text{ and
}\quad\kappa\lambda\neq 1.$

  * Proof.

We know $A\cap B=A$. For $x_{0}=(d_{0},y)$, where $d_{0}\neq 0,y>0$, we have
$d(x_{0},A)=|d_{0}|$, as depicted in Figure 5. Here,

$P_{A}^{\lambda}(x_{0})=(1-\lambda)(d_{0},y)+\lambda(0,y)=((1-\lambda)d_{0},y)\;\in\;B.$

Hence $P_{B}^{\mu}(P_{A}^{\lambda}(x_{0}))=P_{A}^{\lambda}(x_{0})$ and

$T(x_{0})=(1-\kappa)x_{0}+\kappa
P_{B}^{\mu}(P_{A}^{\lambda}(x_{0}))=(1-\kappa)(d_{0},y)+\kappa((1-\lambda)d_{0},y)=((1-\kappa\lambda)d_{0},y).$

Similarly, for any $n\in\mathbb{N}$,
$T^{n}(x_{0})=((1-\kappa\mu)^{n}d_{0},y).$ For the specific choice of
$(\lambda,\kappa)$, we have that, for any $n\in\mathbb{N}$,
$(1-\lambda\mu)^{n}\neq 0$, thus $T^{n}(x_{0})\notin\operatorname{Fix}T$. ∎

Figure 6: Left: Illustration of Example 4.4, when
$(\lambda,\mu,\kappa)=(1/t,1/t,t),\;\text{where}\;1/2<t<1$.
Right: Illustration of Example 4.5, when
$(\lambda,\mu,\kappa)=(1/t,1/t,t),\;\text{where}\;t>1$.

######  Example 4.4.

Choose $A=\\{(\rho,\theta)\ :\ \pi/2\leq\theta\leq 3\pi/2\\}$,
$B=\\{(\rho,\theta)\ :\ -\pi/2\leq\theta\leq\pi/2\\}$, and set
$x_{0}\in\operatorname{int}B$. Then for any $n\in\mathbb{N}$,
$T^{n}(x_{0})\notin\operatorname{Fix}T$ with $T=T_{\lambda,\mu}^{\kappa}$ and

$(\lambda,\mu,\kappa)=(1/t,1/t,t),\quad\text{ where }\quad 1/2<t<1.$

  * Proof.

Here, $A\cap B=\\{(\rho,\theta)\ :\ \theta=\pi/2\;{\rm or}\;\theta=-\pi/2\\}$.
Let $L\coloneqq A\cap B$, and $x_{0}=(d_{0},y)$, where $d_{0}>0,y>0$, thus
$d(x_{0},L)=d_{0}$, see Figure 6. We have

$\displaystyle
P_{A}^{\lambda}(x_{0})=(1-1/t)\cdot(d_{0},y)+1/t\cdot(0,y)=((1-1/t)d_{0},y).$
(13)

Then

$\displaystyle
P_{B}^{\mu}(P_{A}^{\lambda}(x_{0}))=(1-1/t)\cdot((1-1/t)d_{0},y)+1/t\cdot(0,y)=((1-1/t)^{2}d_{0},y),$
(14)

and therefore

$\displaystyle T(x_{0})$ $\displaystyle=(1-\kappa)x_{0}+\kappa
P_{B}^{\mu}(P_{A}^{\lambda}(x_{0}))$
$\displaystyle=(1-t)(d_{0},y)+t(\left(1-1/t\right)^{2}d_{0},y)$ (15)
$\displaystyle=\left(\left(1/t-1\right)d_{0},y\right).$

Note, $T(x_{0})\in\operatorname{int}B$. Similarly, we obtain that, for any
$n\in\mathbb{N}$,

$T^{n}(x_{0})=((1/t-1)^{n}d_{0},y).$

Since $1/2<t<1$, we have for any $n\in\mathbb{N}$ that $0<(1/t-1)^{n}<1$.
Thus, for any $n\in\mathbb{N}$, $T^{n}(x_{0})\in\operatorname{int}B$ and
$T^{n}(x_{0})\to P_{L}(x_{0})$ as $n\to+\infty$. ∎

######  Example 4.5.

Let $A=\\{(\rho,\theta)\ :\ \theta=\pi/2\\}$ and $B=\\{(\rho,\theta)\ :\
\theta=\pi/2\;{\rm or}\;\theta=-\pi/2\\}$. Suppose the Cartesian coordinates
of $x_{0}$ are $x_{0}=(d_{0},y)$, where $d_{0}>0,y>0$. Then for any
$n\in\mathbb{N}$, $T^{n}(x_{0})\notin\operatorname{Fix}T$ with
$T=T_{\lambda,\mu}^{\kappa}$ and

$(\lambda,\mu,\kappa)=(1/t,1/t,t),\quad\text{ where }\quad t>1.$

  * Proof.

Obviously, $A\cap B=A$. Let $L\coloneqq A$, then $d(x_{0},L)=d_{0}$, as
illustrated in Figure 6. The same calculations (13)–(Proof.) can be performed
and again, we may claim for any $n\in\mathbb{N}$ that
$T^{n}(x_{0})=((1/t-1)^{n}d_{0},y)$. For $t>1$, it holds that, for any
$n\in\mathbb{N}$, $0<|1/t-1|^{n}<1$. Therefore, for any $n\in\mathbb{N}$,
$T^{n}(x_{0})\notin\operatorname{Fix}T$ and $T^{n}(x_{0})\to P_{L}(x_{0})$ as
$n\to+\infty$. ∎

Figure 7: Illustration of Example 4.6, when $(\lambda,\mu,\kappa)=(1,1,1)$.

######  Example 4.6.

Let $A=\\{(\rho,\theta)\ :\ \pi/4\leq\theta\leq 5\pi/4\\}$,
$B=\\{(\rho,\theta)\ :\ \pi\leq\theta\leq 2\pi\\}$, and $x_{0}=(x,y)$ be the
Cartesian coordinates of $x_{0}\in\mathbb{R}^{2}\setminus(A\cup B)$. Then for
any $n\in\mathbb{N}$, $T^{n}(x_{0})\notin\operatorname{Fix}T$ with
$T=T_{\lambda,\mu}^{\kappa}$ and

$(\lambda,\mu,\kappa)=(1,1,1).$

  * Proof.

In this case, $T$ is the alternating projection operator, i.e., $T=P_{B}\circ
P_{A}$. We see that

$P_{A}(x_{0})=\left(\frac{x+y}{2},\frac{x+y}{2}\right)\quad\mbox{and}\quad
P_{B}(P_{A}(x_{0}))=\left(\frac{x+y}{2},0\right).$

Hence, for any $n\in\mathbb{N}$,
$T^{n}(x_{0})=\left((x+y)/2^{n},0\right)\notin\operatorname{Fix}T$ and
$T^{n}(x_{0})\to 0\in\operatorname{Fix}T$ as $n\to+\infty$. ∎

We conclude Section 4 with a formal proof of Theorem 2.6.

  * Proof of Theorem 2.6.

As described earlier in this section, the proof follows from the explicit
constructions in Examples 4.1–4.6. Notice that union of the sets of the
parameters values $(\lambda,\mu,\kappa)$ used in these examples together with
the Douglas–Rachford selection $(2,2,1/2)$ comprises all possibilities given
in the Definition 2.1. Since each demonstrates the existence of a pair of sets
$A,B\subseteq\mathbb{R}^{2}$ and $x_{0}\in\mathbb{R}^{2}$ for which the method
defined by the specific choice of the parameters does not converge finitely
(while the set of fixed points of the corresponding operator is nonempty),
these examples together show the desired result. ∎

##  5 Conclusion and Outlook

We have considered a reasonably large family of projection methods in a toy
setting of two convex cones in the plane. We have shown that within this
simple model, the Douglas–Rachford method outperforms all other methods from
this class in terms of finite convergence.

Even though this result appears new, and can be extended to the case of two
convex polytopes in the plane, it is unclear if similar superiority in terms
of finite convergence is typical for this method in other settings, in
particular, for higher-dimensional and non-polyhedral problems. We note here
that in the absence of Slater’s condition, the Douglas–Rachford method may
fail to have finite convergence in high dimensions even for polyhedral sets;
see [7, Example 8.8].

It is an exciting direction for future work to obtain a geometric
characterization for the finite convergence of the Douglas–Rachford method for
the nonpolyhedral cases, and for higher dimensions.

### Acknowledgements

Part of this work was done during MND’s visit to the University of New South
Wales, Sydney. We thank UNSW Sydney for its hospitality. We are also grateful
to the Australian Research Council for continuing support. Specifically, this
research was partially supported by Discovery Projects DP200100124 and
DP230101749.

## References

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# Linearized Ginzburg-Landau equation on a curved surface and curvature-
induced superconductivity enhancement

Long Du  Guangxi University of Science and Technology, China  Minsi Li
Guangxi University of Science and Technology, China  Jiahong Gu  Guangxi
University of Science and Technology, China  Lijuan Zhou  Guangxi University
of Science and Technology, China  Huiqing Tang  Guangxi University of Science
and Technology, China  Qinghua Chen<EMAIL_ADDRESS>Guangxi
University of Science and Technology, China

###### Abstract

We derive the linearized Ginzburg-Landau (GL) equation on a spatially curved
surface in the presence of a magnetic field. By introducing a novel transverse
order parameter varying very slowly along the surface, we decouple the
linearized GL equation into a surface part and a transverse part, and obtain
the superconducting geometric potential (GP) under the superconducting/vacuum
boundary condition. Further, we numerically study the nucleation of a
rectangle thin superconducting film bent around the surface of a cylinder. Our
results show that, the geometric curvature can weaken the effect of the
magnetic field, and the competition between the applied magnetic field and the
superconducting GP will lead to an abnormal $T_{c}(B)$ dependence in the GP
dominant region and a $T_{c}$ enhancement in the magnetic field dominant
region.

Ginzburg-Landau equation, thin-layer quantization, superconducting phase
transtition, geometric effect

††preprint: APS/123-QED

With the progress in micro- and nanofabrication techniques, quantization
effects associated with the confinement of superconducting condensate at
length scales of the order of the superconducting coherence length, $\xi(T)$,
have been intensively studied for more than four decades Little and Parks
(1962); Saint-James (1965); Fink and Presson (1966). In these mesoscopic
superconducting systems, both the superconducting critical parameters and the
vortex matter change substantially with respect to the reference of the bulk
materials Moshchalkov  _et al._ (1995, 2000); Bonča and Kabanov (2001). Good
examples here are the normal-superconducting phase boundary $T_{c}(H)$ of a
submicron cylinder that shows a cusplike behavior superimposed with a linear
background Little and Parks (1962) and the phase transitions accompanied by
bistable region Geim  _et al._ (1997). Close to $T_{c}$, the vortices do not
order themselves in a triangular vortex lattice. Instead, they form vortex
patterns consistent with the geometry of the sample, so that superconductivity
nucleates in form of a giant vortex at the center of mesoscopic disks Saint-
James (1965); Fink and Presson (1966) and vortex-antivortex pairs can be
spontaneously generated in structures with discrete symmetry Chibotaru  _et
al._ (2000); Bonča and Kabanov (2001); Mel’nikov  _et al._ (2002); Chibotaru
_et al._ (2001); Misko  _et al._ (2003). These symmetry-induced effects will
disappear and the triangular vortex pattern will be recovered by decreasing
the temperature Chen  _et al._ (2009).

In addition to the effects caused by the scale confinement, the geometric
curvatures of quasi-two-dimensional (quasi-2D) mesoscopic superconductors can
also act on the Cooper-pairs inside. Generally, the quasi-2D structures with
radius of curvature much larger than $\xi$, can be treated as fully 2D Szameit
_et al._ (2010). While the studies for systems with radii of curvature
comparable to the coherence length become interesting but still at its infancy
stage. A much rigorous and physically sound candidate for studying low-
dimensional curved structures embedded in three-dimensional (3D) Euclidean
space is the thin-layer quantization scheme (TLQS), which was proposed by
Jensen, Koppe, and da Costa Jensen and Koppe (1971); da Costa (1981), and has
been constantly developing until now Schuster and Jaffe (2003); Ferrari and
Cuoghi (2008); Ortix  _et al._ (2011); Wang  _et al._ (2014). In this
approach, the extrinsic curvature leads to an alteration of the kinetic term
in the tangential Hamiltonian with an additional geometric potential (GP)
Spittel  _et al._ (2015). This geometric effect is not only a theoretical
result, but has been observed through the Tomonaga-Luttinger liquid exponent
in a one-dimensional metallic peanut-shaped $\mathrm{C}_{60}$ polymer Onoe
_et al._ (2012), and experimentally realized by an optical analogue in a
photonic topological crystal Szameit  _et al._ (2010). In addition, such GP
can lead to a topological band structure for electrons confined to periodic
minimal surfaces Aoki  _et al._ (2001), an oscillatory behavior of the
transmission coefficient of carriers bound to bent cylindrical surface Marchi
_et al._ (2005), and atomic-like bound states in rolled-up nanotubes Ortix and
Van Den Brink (2010).

Inspired by the TLQS, in this letter, we derive the linearized GL equation on
an ultra-thin superconducting film with curvature, and decouple it into a
surface part and a transverse part, which demonstrates that decoupling the
linearized GL equation under the superconducting/vacuum (S/V) boundary
condition is feasible. To show the effect of geometric curvature acted on
superconductivity, we numerically study the nucleation of a rectangle thin
superconducting film bent around the surface of a cylinder, and discuss the
phase boundary modified by the superconducting GP. Finally, the conclusions
are given.

We start the derivation with writing the linearized GL equation, which can
well describe the nucleation of the superconducting state close to the phase
transitions. In the presence of a magnetic field $\bm{B}=\nabla\times\bm{A}$,
with $\bm{A}$ representing the vector potential, the stationary linearized GL
equation is defined as Tinkham (1996); Baelus  _et al._ (2006)

$\frac{1}{2\mu}\left(-\mathrm{i}\hbar\nabla-\frac{Q}{c}\bm{A}\right)^{2}\psi=-\alpha\psi,$
(1)

where $\mu$ and $Q$ are the mass and charge of a Cooper-pair, respectively,
$\psi$ is the superconducting order parameter with its modulus squared
proportional to the Cooper-pair density, and the GL parameter
$-\alpha=\hbar^{2}/\left(2\mu\xi_{0}^{2}\right)\left(1-T/T_{\mathrm{c}}\right)$
where $\xi_{0}$ is the zero-temperature coherent length, $T$ is the actual
temperature of the superconductor, and $T_{\mathrm{c}}$ represents the bulk
critical temperature at zero magnetic field. Eq. (1) is invariant under the
following gauge transformations Fleckinger-Pellé and Kaper (1995):

$\left(\psi,\bm{A}\right)\mapsto\left(\psi^{\prime},\bm{A}^{\prime}\right)=\big{(}\psi\,\mathrm{exp}\left(\mathrm{i}{Q}\gamma/\hbar\right),\bm{A}+\nabla\gamma\big{)},$
(2)

with $\gamma$ being an arbitrary scalar function.

Let us consider a curved thin superconducting film with uniform thickness $d$,
sketched in Fig. 1, immersed in an insulating medium or vacuum. The space
occupied by the superconductor, denoted by $\Omega$, is bounded by two
parallel surfaces, $S_{1}$ and $S_{2}$, and the side surfaces, denoted by
$S_{0}$, around. For clarity, we introduce the central surface $S$ which is
equidistant from both $S_{1}$ and $S_{2}$, and suppose that $S_{0}$ is
perpendicular to $S$ at their intersections. The boundary condition for this
sample is
$\left[\bm{n}\cdot\left(\mathrm{i}\hbar\nabla+(Q/c)\bm{A}\right)\psi\right]\big{|}_{\partial\Omega}=0$,
where $\partial\Omega\equiv S_{0}\cup S_{1}\cup S_{2}$, and $\bm{n}$ indicates
the outward unit vector normal to $\partial\Omega$.

Figure 1: (Color online) Schematic diagram of the curved superconducting film,
denoted by $\Omega$, with uniform thickness $d$ bounded by surfaces $S_{1},\
S_{2}$ and $S_{0}$. The point on the surface $S$, equidistant from both
$S_{1}$ and $S_{2}$, is described by $\bm{r}\left(q^{1},q^{2}\right)$ with
$q^{1},\ q^{2}$ the surface coordinates on $S$. $\bm{e}_{1}$ and $\bm{e}_{2}$
are the base vectors for the surface coordinates $q^{1}$ and $q^{2}$,
respectively. $\bm{e}_{3}$ is the normal base vector perpendicular to $S$.

When $d$ goes to zero, the Cooper-pairs in $\Omega$ are constrained in the
immediate neighborhood of the surface $S$. Following the parametrization of da
Costa da Costa (1981), the surface $S$ can be parametrized by the position
vector $\bm{r}\left(q^{1},q^{2}\right)$ with $q^{1},\ q^{2}$ the surface
coordinates on $S$, and the portion of $\Omega$ can be parametrized as

$\bm{R}\left(q^{1},q^{2},q^{3}\right)=\bm{r}\left(q^{1},q^{2}\right)+q^{3}\bm{e}_{3}\left(q^{1},q^{2}\right),$
(3)

where $\bm{e}_{3}\left(q^{1},q^{2}\right)$ is the normal basis vector
depending only on $q^{1}$ and $q^{2}$, and $\left|q^{3}\right|$ gives the
distance from the surface $S$. In what follows, the indices $i,j=1,2,3$ stand
the spatial coordinates and the spatial components of vector fields in
$\Omega$, whereas the indices $a,b,c=1,2$ numerate the surface coordinates and
the surface components for vector field on $S$. Einstein summation is also
used in the following. The base vectors for surface coordinates $q^{a}$ are
$\bm{e}_{a}=\partial_{a}\bm{r}$ with $\partial_{a}=\partial/{\partial q^{a}}$,
and $\displaystyle\bm{e}_{3}=\bm{\eta}/\left|\bm{\eta}\right|$ with
$\bm{\eta}=\bm{e}_{1}\times\bm{e}_{2}$. The derivatives of
$\bm{e}_{3}\left(q^{1},q^{2}\right)$ lie in the tangent plane and satisfy
$\partial_{a}\bm{e}_{3}=w_{a}^{b}\partial_{b}\bm{r}$ with $w_{a}^{b}$ the
components of the Weingarten curvature matrix on $S$. The covariant components
of the metric tensor in $\Omega$ and on $S$ are, respectively, defined by
$G_{ij}=\partial_{i}\bm{R}\cdot\partial_{j}\bm{R}$ and
$g_{ab}=\partial_{a}\bm{r}\cdot\partial_{b}\bm{r}$. From Eq.(3) we obtain

$\begin{split}&G_{ab}=g_{ab}+\left(\hat{w}\hat{g}+\hat{g}^{T}\hat{w}^{T}\right)_{ab}q^{3}+\left(\hat{w}\hat{g}\hat{w}^{T}\right)_{ab}\left(q^{3}\right)^{2},\\\
&G_{a3}=G_{3a}=0,\ G_{33}=1,\end{split}$ (4)

where $\hat{w}$ and $\hat{g}$ are the matrices of $w_{a}^{b}$ and $g_{ab}$,
respectively, and the superscript $T$ denotes the matrix transpose. From Eq.
(4), we have $G=f^{2}g$, where $G$ and $g$ are the determinants with respect
to the metric tensors $G_{ij}$ and $g_{ab}$, respectively, and
$f=1+\mathrm{Tr}(\hat{w}){q^{3}}+\det(\hat{w})\left(q^{3}\right)^{2}$.

Therefore, in da Costa’s 3D curvilinear coordinate, the linearized GL equation
and the boundary conditions can be rewritten as

$\begin{split}&{1\over
2\mu}\left[{-\hbar^{2}\over\sqrt{G}}\partial_{i}\left(\sqrt{G}G^{ij}\partial_{j}\psi\right)+{2\mathrm{i}\hbar
Q\over c}G^{ij}A_{i}\partial_{j}\psi\right.\\\ &\left.+{\mathrm{i}\hbar Q\over
c\sqrt{G}}\partial_{i}\left(\sqrt{G}G^{ij}A_{j}\right)\psi+{Q^{2}\over
c^{2}}G^{ij}A_{i}A_{j}\psi\right]=-\alpha\psi\end{split}$ (5)

and

$\left.\left\\{\bm{n}\cdot\left[G^{ij}\left(\mathrm{i}\hbar\,\partial_{i}+(Q/c)A_{i}\right)\psi\bm{e}_{j}\right]\right\\}\right|_{\partial\Omega}=0,$
(6)

where $G^{ij}$ are the reciprocals of $G_{ij}$, and $A_{j}$ denote the
covariant components of the vector potential $\bm{A}$. The structure of
$G_{ij}$ implies that the linearized GL equation can be separated into surface
and transverse parts. Meanwhile, the order parameter $\psi$ from Eq. (5) can
take the factorizable form $\psi=f^{-1/2}\chi$, where $f^{-1/2}$ is employed
for the conservation of the Cooper-pair density da Costa (1981); H. Liu
(2013). Then, we take into account the compressing procedure: in the limit of
confinement, the order parameter $\psi$ is constrained on $S$ with its value
being different from zero only for an extremely small range of
$\left|q^{3}\right|\leq d/2$. By performing the limit $d\to 0$, i.e. $q^{3}\to
0$, in both Eqs. (5) and (6), we have the short hand form of the linearized GL
equation

$\begin{split}&{1\over
2\mu}\left[\left(-\mathrm{i}\hbar\nabla_{\mathrm{s}}-{Q\over
c}\bm{A}_{\mathrm{s}}\right)^{2}\chi+\left(-\mathrm{i}\hbar\partial_{3}-{Q\over
c}A_{3}\right)^{2}\chi\right]\\\
&\quad+V_{\mathrm{g}}\chi=-\alpha\chi,\end{split}$ (7)

where $\nabla_{\mathrm{s}}=g^{ab}\bm{e}_{a}\partial_{b}$ and
$\bm{A}_{\mathrm{s}}=g^{ab}\bm{e}_{a}A_{b}$, and the boundary conditions (6)
separated into the surface part and the transverse part as follows,

$\displaystyle\left.\left\\{\bm{n}\cdot\left[g^{ab}\left(\mathrm{i}\hbar\partial_{a}\chi+(Q/c)A_{a}\chi\right)\bm{e}_{b}\right]\right\\}\right|_{S_{0}}=0,$
(8)
$\displaystyle\left[\mathrm{i}\hbar\left(\partial_{3}-M\right)\chi+(Q/c)A_{3}\chi\right]\big{|}_{S_{1}\bigcup
S_{2}}=0,$ (9)

where $V_{\mathrm{g}}=-\hbar^{2}/(2\mu)(M^{2}-K)$ is the well-known GP da
Costa (1981); Ferrari and Cuoghi (2008); Burgess and Jensen (1993), with
$M=\mathrm{Tr}(\hat{w})/2$ and $K=\det(\hat{w})$ being the mean curvature and
the Gaussian curvature, respectively, $A_{j}$ and its derivative are
calculated at $q^{3}=0$. In Eq. (7), the term $A_{3}\partial_{3}\chi$ couples
the vector potential along $q^{3}$ with the order parameter on $S$. Applying
the gauge transformations (2), we can cancel the coupling term by choosing

$\gamma\left(q^{1},q^{2},q^{3}\right)=-\int_{0}^{q^{3}}A_{3}\left(q^{1},q^{2},z\right)\,\mathrm{d}z$
(10)

Ferrari and Cuoghi (2008). In the limit $q^{3}\to 0$, we have
$A^{\prime}_{3}=0$ and $\partial_{3}A_{3}^{\prime}=0$ while $A_{1}$ and
$A_{2}$ unchanged. In the conventional TLQS, due to the restriction that
transverse fluctuations depend only on $q^{3}$ da Costa (1981), _a transverse
Neumann-type constraint generally makes Schödinger-type equations of motion on
surfaces with non-constant mean curvature nonseparable_ Jensen and Dandoloff
(2009). Here, to obtain a surface linearized GL equation accommodating the
transverse part of the boundary conditions (9), we introduce a new order
parameter

$\chi=\chi_{\mathrm{s}}(q^{1},q^{2})\,\chi_{\mathrm{t}}(q^{1},q^{2},q^{3}),$
(11)

with $\chi_{\mathrm{s}}$ and $\chi_{\mathrm{t}}$ the surface part and the
transverse part of the order parameter, respectively. It is easy to verify
that the separability requirement in TLQS is still met when
$\chi_{\mathrm{t}}$ is such a slowly varying function along $S$ that
$\lim_{q^{3}\to 0}\partial_{a}\chi_{\mathrm{t}}=0$.

After substituting the gauge transformation (10) into Eq. (7) and assuming
$\alpha=\alpha_{\mathrm{s}}+\alpha_{\mathrm{t}},$ with $\alpha_{\mathrm{s}}$
and $\alpha_{\mathrm{t}}$ the eigenvalues corresponding to the surface
eigenstate and the transverse eigenstate, respectively, Eq. (7) is separated
into two independent equations:

$\displaystyle-{\hbar^{2}\over
2\mu}\partial_{3}^{2}\chi_{\mathrm{t}}=-\alpha_{\mathrm{t}}\chi_{\mathrm{t}},$
(12) $\displaystyle{1\over
2\mu}\left(-\mathrm{i}\hbar\nabla_{\mathrm{s}}-{Q\over
c}\bm{A}_{\mathrm{s}}\right)^{2}\chi_{\mathrm{s}}+V_{\mathrm{g}}\chi_{\mathrm{s}}=-\alpha_{\mathrm{s}}\chi_{\mathrm{s}}.$
(13)

The expression (12) is the 1D linearized GL equation along $q^{3}$, while
expression (13) describes the order parameter attached to the surface $S$ in
the magnetic field. The transverse part of the boundary condition from Eq. (9)
reads

$\left[\left(\partial_{3}-M\right)\chi_{\mathrm{t}}\right]\big{|}_{S_{1}\bigcup
S_{2}}=0.$ (14)

From Eq. (12), the available eigenstate subjected to the boundary condition
(14) is $\chi_{\mathrm{t}}\sim\exp\left(Mq^{3}\right)$, and the corresponding
eigenvalue $-\alpha_{\mathrm{t}}=-\hbar^{2}/(2\mu)M^{2}$. Under the surface
part of the boundary condition (8), $-\alpha$ can be determined by calculating
the lowest eigenvalue of the following equation:

${1\over 2\mu}\left(-\mathrm{i}\hbar\nabla_{\mathrm{s}}-{Q\over
c}\bm{A}_{\mathrm{s}}\right)^{2}\chi_{\mathrm{s}}+V_{\mathrm{g}}^{\mathrm{s}}\chi_{\mathrm{s}}=-\alpha\chi_{\mathrm{s}},$
(15)

where $V_{\mathrm{g}}^{\mathrm{s}}=-\hbar^{2}/(2\mu)\left(2M^{2}-K\right)$ is
the GP for the thin superconducting film, and is always negative on curved
surfaces. At this point, we can state one of the most fundamental conclusions
in this letter: _By introducing a novel transverse order parameter, depending
both on the surface and transverse coordinates, that varies very slowly along
the surface_ , the linearized GL equation can be analytically decoupled into
the transverse part and the surface part under the S/V boundary condition (8),
and the nucleation of the superconducting state can be described by Eq. (15).

To present the geometric effects on the normal-superconducting phase
transition, in the following, we numerically study the nucleation of a
rectangular thin superconducting film bent around the surface of a cylinder.
First of all, we introduce the dimensionless variables for convenience,

$\displaystyle\xi=\xi_{0}\xi^{\prime},\ q^{a}=\xi_{0}{q^{\prime}}^{a},\
\partial_{a}=\xi_{0}^{-1}\partial^{\prime}_{a},\ M=M^{\prime}/\xi_{0},$
$\displaystyle K=K^{\prime}/\xi_{0}^{2},\ \bm{A}=\frac{\hbar
c}{\xi_{0}Q}\bm{A}^{\prime},\
\alpha=\frac{\hbar^{2}}{2\mu\xi_{0}^{2}}\alpha^{\prime},$ (16)

with $\partial^{\prime}_{a}=\partial/\partial{q^{\prime}}^{a}$ and $\xi_{0}$
the coherent length at zero temperature. Using the algebraic transform (16)
and omitting the primes in the dimensionless variables, we have the scaled
Eqs. (15) and (8) in the following form:

$\begin{split}&{-1\over\sqrt{g}}\partial_{a}\left(\sqrt{g}g^{ab}\partial_{b}\chi_{\mathrm{s}}\right)+{\mathrm{i}\over\sqrt{g}}\partial_{a}\left(\sqrt{g}g^{ab}A_{b}\right)\chi_{\mathrm{s}}\\\
&+2\mathrm{i}g^{ab}A_{a}\partial_{b}\chi_{\mathrm{s}}+g^{ab}A_{a}A_{b}\chi_{\mathrm{s}}+V_{\mathrm{g}}^{\mathrm{s}}\chi_{\mathrm{s}}=-\alpha\chi_{\mathrm{s}},\end{split}$
(17)
$\left.\left\\{\bm{n}\cdot\big{[}g^{ab}(\mathrm{i}\partial_{a}+A_{a})\chi_{\mathrm{s}}\bm{e}_{b}\big{]}\right\\}\right|_{S_{0}}=0,$
(18)

where the scaled superconducting GP $V_{\mathrm{g}}^{\mathrm{s}}=-(2M^{2}-K)$.
The scaled temperature-dependent coherent length becomes
$\xi(T)=\left(1-T/T_{\mathrm{c0}}\right)^{-1/2}$, and the scaled GL parameter
$-\alpha=1-T/T_{\mathrm{c0}}$, with $T_{\mathrm{c0}}$ the zero field critical
temperature. As shown in Fig. 2, we consider a rectangular superconducting
film, with width $U$ and height $V$, bent around the surface of a cylinder of
radius $R$. This superconducting film can be parameterized by two
dimensionless surface coordinates $(u,v)$ where $u$ and $v$ are the
coordinates along the direction of width and the direction of height,
respectively. The applied magnetic field $\bm{B}=B\bm{e}_{3}$ is perpendicular
to the surface with uniform intensity over the surface. The most suitable
vector potential on the surface, determined by Eq. (10), is
$\left(A_{u},A_{v}\right)=\left(0,Bu\right)$, which corresponds to
$\nabla\times\bm{A}=\bm{B}$. From Eq. (17), the linearized GL equation on the
surface reads

$-\partial_{u}^{2}\chi_{\mathrm{s}}-\left(\partial_{v}-\mathrm{i}Bu\right)^{2}\chi_{\mathrm{s}}-\frac{1}{2R^{2}}\chi_{\mathrm{s}}=-\alpha\chi_{\mathrm{s}},$
(19)

where $-1/(2R^{2})$ is the dimensionless superconducting GP. With the origin
of the coordinates $(u,v)$ chosen at the center of the surface, the boundary
conditions (8) for Eq. (19) become

$\partial_{u}\chi_{\mathrm{s}}\big{|}_{u=\pm U/2}=0,\
\left(\partial_{v}-\mathrm{i}Bu\right)\chi_{\mathrm{s}}\big{|}_{v=\pm V/2}=0.$
(20)

\begin{overpic}[tics=10,width=260.17464pt]{cylindricalFilm.png}
\put(29.0,17.0){\rotatebox{20.0}{\hbox{\pagecolor{white}$R$}}}
\put(40.0,30.0){\rotatebox{-25.0}{\hbox{\pagecolor{white}$U$}}}
\put(13.0,57.0){\rotatebox{60.0}{\hbox{\pagecolor{white}$V$}}}
\put(51.0,75.0){\color[rgb]{1,0,0}$\bm{B}$}
\put(62.0,48.0){\rotatebox{-30.0}{\color[rgb]{0,0,1}$u$}}
\put(62.0,63.0){\rotatebox{-30.0}{\color[rgb]{0,0,1}$v$}} \end{overpic} Figure
2: (Color online) A rectangular thin superconducting film, with width $U$ and
height $V$, bent around the surface of a cylinder with radius $R$ along the
direction of width. This bending film is parameterized by two orthogonal
surface coordinates $(u,v)$ with $u$ and $v$ the coordinates along the
direction of width and the direction of height, respectively. An external
magnetic field $\bm{B}$, marked by red arrows, is applied perpendicularly to
the surface with uniform intensity over the surface.

Based on the commercial software COMSOL Multiphysics, we perform finite
element simulations to compute the critical temperature $T_{\mathrm{c}}$,
corresponding to the lowest positive eigenvalue of $-\alpha$, for the model.
In panel $(a)$ of Fig. 3, the phase diagram of the critical temperature
$T_{\mathrm{c}}$ (measured in units of $T_{\mathrm{c0}}$), with $U=8\xi_{0}$
and $V=6\xi_{0}$, as a function of the mean curvature $M$ (measured in units
of $1/\xi_{0}$) and the intensity of magnetic field $B$ (measured in units of
$\hbar c/(\xi_{0}^{2}Q)$) are given. The white dashed curve, connecting the
points at which $T_{\mathrm{c}}/T_{\mathrm{c0}}=1$, divides the $M\textendash
B$ plane into the GP dominant region and the magnetic field dominant region
(denoted by Roman numerals I and II, respectively), and forms the boundary
between them. In panel $(b)$ of Fig. 3, the slices of $T_{\mathrm{c}}$ at
different intensities of magnetic field as a function of the mean curvature
are presented. Clearly, the geometric effect can weaken the action of the
magnetic field, which causes $T_{\mathrm{c}}$ to rise with the mean curvature
both in regions I and II. However, at the boundary between regions I and II,
we have the $T_{\mathrm{c}}$ equals to $T_{\mathrm{c0}}$, which means that the
corresponding lowest positive eigenvalue $-\alpha$ reducing to $0$. Further
increasing the mean curvature, i.e., crossing the boundary, the original
$-\alpha$ becomes negative while the subminimum positive eigenvalue becomes
the minimum one, giving rise to a sudden drop in $T_{\mathrm{c}}(M)$ curves at
the boundary. Fig. 3$(c)$ shows the slices of $T_{\mathrm{c}}$ at different
mean curvatures as a function of the intensity of magnetic field. The GP
dominates when the magnetic field is weak. Increasing the intensity of
magnetic field can gradually diminish the effect of GP, which leads to an
increase in $T_{\mathrm{c}}$. When the magnetic field completely compensates
for the geometric effect at the boundary, the $T_{\mathrm{c}}(B)$ curve jumps
to $T_{\mathrm{c0}}$. In region II, i.e. the magnetic field dominates, the
critical temperature $T_{\mathrm{c}}$, starting at $T_{\mathrm{c0}}$,
decreases monotonously with increasing the intensity of the magnetic field and
shows Little-Parks behavior with each of cusps corresponding to a change in
fluxoid quantum number $L$ by one Moshchalkov  _et al._ (1995). Connecting the
corresponding cusps on all $T_{\mathrm{c}}(B)$ curves at different mean
curvatures in Fig. 3$(a)$ (black dashed line), we can subdivided region II
into areas with different $L$. Compared to the model without curvature, all
$T_{\mathrm{c}}(B)$ curves for finite $M$ have positive shifts, increasing
with $M$, along the critical temperature. This means the geometric curvature
can effectively enhance the critical temperature of superconducting films.

At last, we would like to explain the reason why the critical temperature for
the curved thin superconducting film cannot exceed that for the
correspondingly planar one. We expand the GL free energy density and keep the
terms up to the quadratic order in $\left|\psi\right|$ and obtain Tinkham
(1996)

$f=f_{\mathrm{n}0}+\alpha\left|\psi\right|^{2}+\frac{1}{2\mu}\left|\left(-\mathrm{i}\hbar\nabla-\frac{Q}{c}\bm{A}\right)\psi\right|^{2}+\cdots,$
(21)

where $f_{\mathrm{n0}}$ is the free energy density of the normal state, and
$\alpha<0$ corresponds to the superconducting state. The superconducting GP,
coming from the third term (kinetic energy) in the right side of Eq. (21),
competes with and partly compensates for the action of the magnetic field
governed by $\bm{A}$ Burgess and Jensen (1993), which leads to a modification
of the kinetic term. However, changing the curvature can only modify the
position of the minimum of the kinetic energy rather than equivalently reduce
$\alpha$. Therefore, the critical temperature of the superconducting film with
curvature cannot exceed that of the correspondingly planar one.

\begin{overpic}[width=433.62pt,tics=10]{T_M_H.png}
\put(14.0,10.0){\color[rgb]{1,1,1}$(a)$}
\put(22.0,15.0){\hbox{\pagecolor{white}\footnotesize$L=0$}}
\put(44.0,15.0){\hbox{\pagecolor{white}\footnotesize$L=1$}}
\put(62.0,15.0){\hbox{\pagecolor{white}\footnotesize$L=2$}}
\put(78.0,15.0){\hbox{\pagecolor{white}\footnotesize$L=3$}}
\put(14.0,25.0){\rotatebox{90.0}{I \scriptsize(GP dominant)}}
\put(45.0,30.0){\shortstack{II \scriptsize(Magnetic\\\ \scriptsize field
dominant)}} \end{overpic}

\begin{overpic}[width=433.62pt,tics=10]{T_M.pdf} \put(14.0,10.0){$(b)$}
\end{overpic}\begin{overpic}[width=433.62pt,tics=10]{T_H.pdf}
\put(14.0,10.0){$(c)$} \end{overpic}

Figure 3: (Color online) $(a)$ Phase diagram of the critical temperature
$T_{\mathrm{c}}$ (measured in units of $T_{\mathrm{c0}}$) for the model
sketched in Fig. 2 with $U=8\xi_{0}$ and $V=6\xi_{0}$ as a function of the
mean curvature $M$ (measured in units of $1/\xi_{0}$) and the intensity of the
magnetic field $B$ (measured in units of $\hbar c/\left(\xi_{0}^{2}Q\right)$).
The leftmost white dashed curve indicates the boundary between regions I (GP
dominant region) and II (magnetic field dominant region), and three black
dashed curves denotes the boundary between the regions with different flux
vortices $L$. $(b)$ The slices of critical temperature $T_{\mathrm{c}}$ at
$B=0.0,\ 0.06,\ 0.12,\ 0.30,\ 0.60\ \text{and}\ 0.90$ as a function of the
mean curvature $M$. $T_{\mathrm{c}}$ rises with increasing $M$. $(c)$ The
slices of critical temperature $T_{\mathrm{c}}$ at $M=0,\ 0.05,\ 0.08\
\text{and}\ 0.12$ as a function of the intensity of magnetic field $B$. This
panel shows the competition between the curvature and the applied magnetic
field.

In conclusion, we have developed a general linearized GL equation valid for
any curved thin superconducting film surrounded by an insulating medium or
vacuum in the presence of a magnetic field. By introducing the novel order
parameter $\chi$, given in Eq. (11), whose transverse component varies
extremely slowly along the surface, and taking into account Ferrari’s gauge
transformation to eliminate $A_{3}$ and $\partial_{3}A_{3}$, we have decoupled
the linearized GL equation into the surface part and the transverse part. In
addition, we have provided an analytical solution for $\chi_{\mathrm{t}}$, and
have obtained the superconducting GP,
$V_{\mathrm{g}}^{\mathrm{s}}=-\hbar^{2}/\left(2\mu\right)\left(2M^{2}-K^{2}\right)$,
under the S/V boundary condition. To present the effects of geometric
curvature on the superconductivity, we have studied the normal-superconducting
phase transition of a rectangular thin superconducting film bent around the
surface of a cylinder. With applying the magnetic field perpendicular to the
film with uniform intensity over the film, we have numerically computed the
critical temperature $T_{\mathrm{c}}$ depending both on the intensity of
magnetic field $B$ and the mean curvature $M$, and have charted the phase
diagram. In this phase diagram, the $M\textendash B$ plane can be divided into
the GP dominant region and the magnetic field dominant region by the
equilibrium boundary at which $T_{\mathrm{c}}/T_{\mathrm{c0}}=1$. Due to the
competition between the applied magnetic field and the superconducting GP, in
the GP dominant region, the critical temperature abnormally raises with the
intensity of the magnetic field and suddenly jumps to $T_{\mathrm{c0}}$ at the
boundary. In the magnetic field dominant region, the critical temperature
decreases with increasing the intensity of the magnetic field and shows
Little-Parks behavior. Since the geometric effect partly compensates for the
action of the magnetic field, the $T_{\mathrm{c}}\textendash B$ phase boundary
for the model with curvature has positive shift along the critical temperature
compared to the planar one. By illustrating the provenance of the
superconducting GP in the GL free energy and its modification of the kinetic
energy term, we have shown that the critical temperature of the
superconducting film with curvature cannot exceed that of the correspondingly
planar one. The analytic method in this work for decoupling and dimensionality
reduction can be extended to many low-dimensional curved systems with Neumann-
type constraints.

This work is supported by the Science and Technology Program of Guangxi, China
(Grant No. 2018AD19310) and the National Natural Science Foundation of
China(under Grants No. 11564003 and No. 11865005).

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