# An extension of the proximal point algorithm beyond convexity

S.-M. Grad, F. Lara Faculty of Mathematics, University of Vienna, Oskar-
Morgenstern-Platz 1, A-1090 Vienna, Austria. E-mail: sorin-
<EMAIL_ADDRESS>ORCID-ID: 0000-0002-1139-7504, Corresponding
authorDepartamento de Matemática, Facultad de Ciencias, Universidad de
Tarapacá, Arica, Chile. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract. We introduce and investigate a new generalized convexity notion for
functions called prox-convexity. The proximity operator of such a function is
single-valued and firmly nonexpansive. We provide examples of (strongly)
quasiconvex, weakly convex, and DC (difference of convex) functions that are
prox-convex, however none of these classes fully contains the one of prox-
convex functions or is included into it. We show that the classical proximal
point algorithm remains convergent when the convexity of the proper lower
semicontinuous function to be minimized is relaxed to prox-convexity.

_Keywords._ Nonsmooth optimization, Nonconvex optimization, Proximity
operator, Proximal point algorithm, Generalized convex function

##  1 Introduction

The first motivation behind this study comes from works like [12, 19, 22, 23]
where proximal point type methods for minimizing quasiconvex functions
formulated by means of Bregman distances were proposed. On the other hand,
other extensions of the proximal point algorithm for nonconvex optimization
problems (such as the ones introduced in [10, 18, 20, 24]) cannot be employed
in such situations due to various reasons. Looking for a way to reconcile
these approaches we came across a new class of generalized convex functions
that we called prox-convex, whose properties allowed us to extend the
convergence of the classical proximal point algorithm beyond the convexity
setting into a yet unexplored direction.

In contrast to other similar generalizations, the proximity operators of the
proper prox-convex functions are single-valued (and firmly nonexpansive) on
the underlying sets. To the best of our knowledge besides the convex and prox-
convex functions only the weakly convex ones have single-valued proximity
operators (cf. [16]). This property plays a crucial role in the construction
of proximal point type algorithms as the new iterate is thus uniquely
determined and does not have to be picked from a set. Moreover, the prox-
convexity of the functions can be considered both globally or on a subset of
their domains, that can be of advantage when dealing with concrete
applications from practice. Various functions, among which several families of
(strongly) quasiconvex, weakly and DC (i.e. difference of convex) ones,
fulfill the definition of the new notion we propose. As a byproduct of our
study we also deliver new results involving (strongly) quasiconvex functions.

Different to other extensions of the proximal point algorithm, the one we
propose has a sort of a local nature, however not in the sense of properties
of a function that hold in some neighborhoods, but concerning the restriction
of the function to a (convex) set. We are not aware of very similar work in
the literature where the proximity operator of a function is taken with
respect to a given set, however in works like [6, 13] such constructions with
some employed functions not split from the corresponding sets were already
considered.

Given a proper, lower semicontinuous and convex function
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}:=\mathbb{R}\cup\\{\pm\infty\\}$,
for any $z\in\mathbb{R}^{n}$ the minimization problem

$\min_{x\in\mathbb{R}^{n}}\left[h(x)+\frac{1}{2}\lVert z-x\rVert^{2}\right]$
(1.1)

has (even in more general frameworks such as Hilbert spaces) a unique optimal
solution denoted by $\operatorname*{Prox}_{h}(z)$, that is the value of the
proximity operator of the function $h$ at the point $z$. A fundamental
property of the latter is when $z,\overline{x}\in\mathbb{R}^{n}$ (see, for
instance, [5, Proposition 12.26])

$\overline{x}={\operatorname*{Prox}}_{h}(z)~{}\Longleftrightarrow~{}z-\overline{x}\in\partial
h(\overline{x}),$ (1.2)

where $\partial h$ is the usual convex subdifferential.

These two facts (the existence of an optimal solution to (1.1) and the
characterization (1.2)) are crucial tools for proving the convergence of the
proximal point type algorithms for continuous optimization problems consisting
in minimizing (sums of) proper, lower semicontinuous and convex functions, and
even for DC programming problems (see [4] for instance). For the class of
prox-convex functions introduced in this article the first of them holds while
the second one is replaced by a weaker variant and we show that these
properties still guarantee the convergence of the sequence generated by the
proximal point algorithm towards a minimum of a prox-convex function.

The paper is constructed as follows. After some preliminaries, where we define
the framework and recall some necessary notions and results, we introduce and
investigate the new classes of prox-convex functions and strongly
G-subdifferentiable functions, showing that the proper and lower
semicontinuous elements of the latter belong to the first one, too. Finally,
we show that the classical proximal point algorithm can be extended to the
prox-convex setting without losing the convergence.

##  2 Preliminaries

By $\langle\cdot,\cdot\rangle$ we mean the inner product of $\mathbb{R}^{n}$
and by $\lVert\cdot\rVert$ the Euclidean norm on $\mathbb{R}^{n}$. Let $K$ be
a nonempty set in $\mathbb{R}^{n}$ and we denote its topological interior by
$\operatorname*{int}K$ and its boundary by $\operatorname*{bd}K$. The
indicator function of $K$ is defined by $\delta_{K}(x):=0$ if $x\in K$, and
$\delta_{K}(x):=+\infty$ elsewhere. By $\mathbb{B}(x,\delta)$ we mean the
closed ball with center at $x\in\mathbb{R}^{n}$ and radius $\delta>0$. By
$\operatorname*{Id}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ we denote the identity
mapping on $\mathbb{R}^{n}$.

Given any $x,y,z\in\mathbb{R}^{n}$, we have

$\langle x-z,y-x\rangle=\frac{1}{2}\lVert z-y\rVert^{2}-\frac{1}{2}\lVert
x-z\rVert^{2}-\frac{1}{2}\lVert y-x\rVert^{2}.$ (2.1)

For any $x,y\in\mathbb{R}^{n}$ and any $\beta\in\mathbb{R}$, we have

$\lVert\beta x+(1-\beta)y\rVert^{2}=\beta\lVert x\rVert^{2}+(1-\beta)\lVert
y\rVert^{2}-\beta(1-\beta)\lVert x-y\rVert^{2}.$ (2.2)

Given any extended-valued function
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}:=\mathbb{R}\cup\\{\pm\infty\\}$,
the effective domain of $h$ is defined by
$\operatorname*{dom}\,h:=\\{x\in\mathbb{R}^{n}:h(x)<+\infty\\}$. We say that
$h$ is proper if $\operatorname*{dom}\,h$ is nonempty and $h(x)>-\infty$ for
all $x\in\mathbb{R}^{n}$.

We denote by
$\operatorname*{epi}h:=\\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}:h(x)\leq
t\\}$ the epigraph of $h$, by
$S_{\lambda}(h):=\\{x\in\mathbb{R}^{n}:h(x)\leq\lambda\\}$ (respectively
$S^{<}_{\lambda}(h):=\\{x\in\mathbb{R}^{n}:h(x)<\lambda\\}$ the sublevel
(respectively strict sublevel) set of $h$ at the height
$\lambda\in\mathbb{R}$, and by ${\operatorname*{\arg\min}}_{\mathbb{R}^{n}}h$
the set of all minimal points of $h$. We say that a function is $L$-Lipschitz
when it is Lipschitz continuous with constant $L>0$. We adopt the usual
convention $\sup_{\emptyset}h:=-\infty$ and $\inf_{\emptyset}h:=+\infty$.

A function $h$ with a convex domain is said to be

  * $(a)$

convex if, given any $x,y\in\operatorname*{dom}\,h$, then

$h(\lambda x+(1-\lambda)y)\leq\lambda
h(x)+(1-\lambda)h(y),~{}\forall~{}\lambda\in[0,1];$ (2.3)

  * $(b)$

semistrictly quasiconvex if, given any $x,y\in\operatorname*{dom}~{}h,$ with
$h(x)\neq h(y)$, then

$h(\lambda
x+(1-\lambda)y)<\max\\{h(x),h(y)\\},~{}\forall~{}\lambda\in\,]0,1[;$ (2.4)

  * $(c)$

quasiconvex if, given any $x,y\in\operatorname*{dom}\,h$, then

$h(\lambda
x+(1-\lambda)y)\leq\max\\{h(x),h(y)\\},~{}\forall~{}\lambda\in[0,1].$ (2.5)

We say that $h$ is strictly quasiconvex if the inequality in (2.5) is strict
(see [15, page 90]).

Every convex function is quasiconvex and semistrictly quasiconvex, and every
semistrictly quasiconvex and lower semicontinuous function is quasiconvex (see
[7, Theorem 2.3.2]). The function $h:\mathbb{R}\rightarrow\mathbb{R}$, with
$h(x):=\min\\{\lvert x\rvert,1\\}$, is quasiconvex, without being semistrictly
quasiconvex.

A function $h$ is said to be neatly quasiconvex (see [3, Definition 4.1]) if
$h$ is quasiconvex and for every $x\in\mathbb{R}^{n}$ with $h(x)>\inf h$, the
sets $S_{h(x)}(h)$ and $S^{<}_{h(x)}(h)$ have the same closure (or
equivalently, the same relative interior). As a consequence, a quasiconvex
function $h$ is neatly quasiconvex if and only if every local minimum of $h$
is global minimum (see [3, Proposition 4.1]). In particular, every
semistrictly quasiconvex function is neatly quasiconvex, and every continuous
and neatly quasiconvex function is semistrictly quasiconvex by [3, Proposition
4.2]. The function in [3, Example 4.1] is neatly quasiconvex without being
semistrictly quasiconvex. Recall that

$\displaystyle h~{}{\rm is~{}convex}$ $\displaystyle\Longleftrightarrow$
$\displaystyle\operatorname*{epi}h~{}{\rm is~{}a~{}convex~{}set;}$
$\displaystyle h~{}{\rm is~{}quasiconvex}$ $\displaystyle\Longleftrightarrow$
$\displaystyle S_{\lambda}(h)~{}{\rm
is~{}a~{}convex~{}set~{}for~{}all~{}}\lambda\in\mathbb{R}.$

For algorithmic purposes, the following notions from [5, Definition 10.27]
(see also [29, 30]) are useful.

A function $h$ with a convex domain is said to be strongly convex
(respectively strongly quasiconvex), if there exists $\beta\in]0,+\infty[$
such that for all $x,y\in\operatorname*{dom}\,h$ and all $\lambda\in[0,1]$, we
have

$\displaystyle~{}~{}~{}~{}h(\lambda y+(1-\lambda)x)\leq\lambda
h(y)+(1-\lambda)h(x)-\lambda(1-\lambda)\frac{\beta}{2}\lVert x-y\rVert^{2},$
(2.6) $\displaystyle\left({\rm respectively}~{}h(\lambda
y+(1-\lambda)x)\leq\max\\{h(y),h(x)\\}-\lambda(1-\lambda)\frac{\beta}{2}\lVert
x-y\rVert^{2}.\right)$ (2.7)

For (2.7), sometimes one needs to restrict the value $\beta$ to a subset $J$
in $]0,+\infty[$ and then $h$ is said to be strongly quasiconvex for $J$.

Every strongly convex function is strongly quasiconvex, and every strongly
quasiconvex function is semistrictly quasiconvex. Furthermore, a strongly
quasiconvex function has at most one minimizer on a convex set
$K\subseteq\mathbb{R}^{n}$ that touches its domain (see [5, Proposition
11.8]).

A function $h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ is said to be

  * $(a)$

supercoercive if

$\liminf_{\lVert x\rVert\rightarrow+\infty}\frac{h(x)}{\lVert
x\rVert}=+\infty;$ (2.8)

  * $(b)$

coercive if

$\lim_{\lVert x\rVert\rightarrow+\infty}h(x)=+\infty;$ (2.9)

  * $(c)$

weakly coercive if

$\liminf_{\lVert x\rVert\rightarrow+\infty}\frac{h(x)}{\lVert x\rVert}\geq 0;$
(2.10)

  * $(d)$

$2$-weakly coercive if

$\liminf_{\lVert x\rVert\rightarrow+\infty}\frac{h(x)}{\lVert x\rVert^{2}}\geq
0.$ (2.11)

Clearly, $(a)\Rightarrow(b)\Rightarrow(c)\Rightarrow(d)$. The function
$h(x)=\sqrt{\lvert x\rvert}$ is coercive without being supercoercive; the
function $h(x)=-\sqrt{\lvert x\rvert}$ is weakly coercive without being
coercive (moreover, it is not even bounded from below). Finally, the function
$h(x)=-\lvert x\rvert$ is $2$-weakly coercive without being weakly coercive.
Recall that $h$ is coercive if and only if $S_{\lambda}(h)$ is a bounded set
for every $\lambda\in\mathbb{R}$. A survey on coercivity notions is [8].

The convex subdifferential of a proper function
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ at $x\in\mathbb{R}^{n}$ is

$\partial h(x):=\left\\{\xi\in\mathbb{R}^{n}:h(y)\geq
h(x)+\langle\xi,y-x\rangle,~{}\forall~{}y\in\mathbb{R}^{n}\right\\},$ (2.12)

when $x\in\operatorname*{dom}\,h$, and $\partial h(x)=\emptyset$ if
$x\not\in\operatorname*{dom}\,h$. But in case of nonconvex functions
(quasiconvex for instance) the convex subdifferential is too small and often
empty, other subdifferential notions (see [14, 25]) being necessary, like the
Gutiérrez subdifferential (of $h$ at $x$), defined by

$\partial^{\leq}h(x):=\left\\{\xi\in\mathbb{R}^{n}:~{}h(y)\geq
h(x)+\langle\xi,y-x\rangle,~{}\forall~{}y\in S_{h(x)}(h)\right\\},$ (2.13)

when $x\in\operatorname*{dom}\,h$, and $\partial^{\leq}h(x)=\emptyset$ if
$x\not\in\operatorname*{dom}\,h$, or the Plastria subdifferential (of $h$ at
$x$), that is

$\partial^{<}h(x):=\left\\{\xi\in\mathbb{R}^{n}:~{}h(y)\geq
h(x)+\langle\xi,y-x\rangle,~{}\forall~{}y\in S^{<}_{h(x)}(h)\right\\},$ (2.14)

when $x\in\operatorname*{dom}\,h$, and $\partial^{<}h(x)=\emptyset$ if
$x\not\in\operatorname*{dom}\,h$. Clearly, $\partial
h\subseteq\partial^{\leq}h\subseteq\partial^{<}h$. The reverse inclusions do
not hold as the function $h:\mathbb{R}\rightarrow\mathbb{R}$ given by
$h(x)=\min\\{x,\max\\{x-1,0\\}\\}$ shows (see [26, page 21]). A sufficient
condition for equality in this inclusion chain is given in [26, Proposition
10].

Note that both $\partial^{\leq}h$ and $\partial^{<}h$ are (at any point)
either empty or unbounded, and it holds (see [14, 25, 26])

$0\in\partial^{<}h(x)\Longleftrightarrow
0\in\partial^{\leq}h(x)\Longleftrightarrow
x\in\operatorname*{\arg\min}_{\mathbb{R}^{n}}\,h\Longleftrightarrow\partial^{\leq}h(x)=\mathbb{R}^{n}.$
(2.15)

We recall the following results originally given in [25, Theorem 2.3], [31,
Proposition 2.5 and Proposition 2.6] and [9, Theorem 20], respectively.

######  Lemma 2.1.

Let $h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper function.
The following results hold.

  * $(a)$

If $h$ is quasiconvex and $L$-Lipschitz, then $\partial^{<}h(x)\neq\emptyset$
for all $x\in\mathbb{R}^{n}$. Moreover, there exists $\xi\in\partial^{<}h(x)$
such that $\lVert\xi\rVert\leq L$.

  * $(b)$

If $h$ is neatly quasiconvex and $L$-Lipschitz, then
$\partial^{\leq}h(x)\neq\emptyset$ for all $x\in\mathbb{R}^{n}$. Moreover, if
$u\in\partial^{\leq}h(x)$, $u\neq 0$, then $L\frac{1}{\lVert
u\rVert}u\in\partial^{\leq}h(x)$.

For $\gamma>0$ we define the Moreau envelope of parameter ${\gamma}$ of $h$ by

${}^{\gamma}h(z)=\inf_{x\in\mathbb{R}^{n}}\left(h(x)+\frac{1}{2\gamma}\lVert
z-x\rVert^{2}\right).$ (2.16)

The proximity operator of parameter $\gamma>0$ of a function
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ at $x\in\mathbb{R}^{n}$ is
defined as

${\operatorname*{Prox}}_{\gamma
h}:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n},\
{\operatorname*{Prox}}_{\gamma
h}(x)=\operatorname*{\arg\min}\limits_{y\in\mathbb{R}}\left\\{h(y)+\frac{1}{2\gamma}\|y-x\|^{2}\right\\}.$
(2.17)

When $h$ is proper, convex and lower semicontinuous,
${\operatorname*{Prox}}_{\gamma h}$ turns out to be a single-valued operator.
By a slight abuse of notation, when ${\operatorname*{Prox}}_{\gamma h}$ is
single-valued we write in this paper ${\operatorname*{Prox}}_{\gamma h}(z)$
(for some $z\in\mathbb{R}^{n}$) to identify the unique element of the actual
set ${\operatorname*{Prox}}_{\gamma h}(z)$. Moreover, when $\gamma=1$ we write
${\operatorname*{Prox}}_{h}$ instead of ${\operatorname*{Prox}}_{1h}$.

For studying constrained optimization problems, the use of constrained notions
becomes important since they ask for weaker conditions. Indeed, for instance,
the function $h:\mathbb{R}\rightarrow\mathbb{R}$ given by $h(x)=\min\\{\lvert
x\rvert,2\\}$ is convex on $K=[-2,2]$, but is not convex on $\mathbb{R}$.

For a nonempty set $K$ in $\mathbb{R}^{n}$, by
$\partial_{K}h(x),\partial^{\leq}_{K}h(x)$ and $\partial^{<}_{K}h(x)$, we mean
the convex, Gutiérrez and Plastria subdifferentials of $h$ at $x\in K$
restricted to the set $K$, that is,

$\displaystyle\partial_{K}h(x):=\partial(h+\delta_{K})(x)=\left\\{\xi\in\mathbb{R}^{n}:~{}h(y)\geq
h(x)+\langle\xi,y-x\rangle,~{}\forall~{}y\in K\right\\},$

as well as $\partial^{\leq}_{K}h(x):=\partial^{\leq}(h+\delta_{K})(x)$ and
$\partial^{<}_{K}h(x):=\partial^{<}(h+\delta_{K})(x)$.

For $K\subseteq\mathbb{R}^{n}$, a single-valued operator
$T:K\rightarrow\mathbb{R}^{n}$ is called

  * $(a)$

monotone on $K$, if for all $x,y\in K$, we have

$\langle T(x)-T(y),x-y\rangle\geq 0;$ (2.18)

  * $(b)$

firmly nonexpansive if for every $x,y\in K$, we have

$\lVert
T(x)-T(y)\rVert^{2}+\lVert(\operatorname*{Id}-T)(x)-(\operatorname*{Id}-T)(y)\rVert^{2}\leq\lVert
x-y\rVert^{2},~{}\forall~{}x,y\in K,$ (2.19)

According to [5, Proposition 4.4], $T$ is firmly nonexpansive if and only if

$\lVert T(x)-T(y)\rVert^{2}\leq\langle
x-y,T(x)-T(y)\rangle,~{}\forall~{}x,y\in K.$ (2.20)

As a consequence, if $T$ is firmly nonexpansive, then $T$ is Lipschitz
continuous and monotone.

##  3 Prox-convex functions

In this section, we introduce and study a class of functions for which the
necessary fundamental properties presented in the introduction are satisfied.

###  3.1 Motivation, definition and basic properties

We begin with the following result, in which we provide a general sufficient
condition for the nonemptiness of the values of the proximity operator.

######  Proposition 3.1.

Let $h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper, lower
semicontinuous and $2$-weakly coercive function. Given any
$z\in\mathbb{R}^{n}$, there exists
$\overline{x}\in\operatorname*{Prox}_{h}(z)$.

###### Proof.

Given $z\in\mathbb{R}^{n}$, we consider the minimization problem:

$\min_{x\in\mathbb{R}^{n}}h_{z}(x):=h(x)+\frac{1}{2}\lVert x-z\rVert^{2}.$
(3.1)

Since $h$ is lower semicontinuous and $2$-weakly coercive, $h_{z}$ is lower
semicontinuous and coercive by [8, Theorem 2$(ii)$]. Thus, there exists
$\overline{x}\in\mathbb{R}^{n}$ such that
$\overline{x}\in{\operatorname*{\arg\min}}_{\mathbb{R}^{n}}h_{z}$, i.e.,
$\overline{x}\in\operatorname*{Prox}_{h}(z)$. ∎

One cannot weaken the assumptions of Proposition 3.1 without losing its
conclusion.

######  Remark 3.1.

  * $(i)$

Note that every convex function is $2$-weakly coercive, and every bounded from
below function is also $2$-weakly coercive. The function
$h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ given by $h(x)=-\lvert x\rvert$ is
$2$-weakly coercive, but is neither convex nor bounded from below. However,
for any $z\in\mathbb{R}^{n}$, $\operatorname*{Prox}_{h}(z)\neq\emptyset$.

  * $(ii)$

The $2$-weak coercivity assumption can not be dropped in the general case.
Indeed, the function $h:\mathbb{R}\rightarrow\mathbb{R}$ given by
$h(x)=-x^{3}$ is continuous and quasiconvex, but fails to be $2$-weakly
coercive and for any $z\in\mathbb{R}$ one has
$\operatorname*{Prox}_{h}(z)=\emptyset$.

Next we characterize the existence of solution in the definition of the
proximity operator.

######  Proposition 3.2.

Let $h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper function.
Given any $z\in\mathbb{R}^{n}$, one has

$\overline{x}\in{\operatorname*{Prox}}_{h}(z)~{}\Longleftrightarrow~{}h(\overline{x})-h(x)\leq\frac{1}{2}\langle\overline{x}+x-2z,x-\overline{x}\rangle,~{}\forall
x\in\mathbb{R}^{n}.$ (3.2)

###### Proof.

Let $z\in\mathbb{R}^{n}$. One has

$\displaystyle\overline{x}\in{\operatorname*{Prox}}_{h}(z)$
$\displaystyle\Longleftrightarrow\,h(\overline{x})+\frac{1}{2}\lVert\overline{x}-z\rVert^{2}\leq
h(x)+\frac{1}{2}\lVert x-z\rVert^{2},\ \forall\,x\in\mathbb{R}^{n}$
$\displaystyle\Longleftrightarrow\,h(\overline{x})-h(x)\leq\frac{1}{2}\lVert
x-z\rVert^{2}-\frac{1}{2}\lVert\overline{x}-z\rVert^{2}m\
\forall\,x\in\mathbb{R}^{n}$
$\displaystyle\Longleftrightarrow\,h(\overline{x})-h(x)\leq\langle\overline{x}-z,x-\overline{x}\rangle+\frac{1}{2}\lVert
x-\overline{x}\rVert^{2},\ \forall\,x\in\mathbb{R}^{n}$
$\displaystyle\Longleftrightarrow\,h(\overline{x})-h(x)\leq\frac{1}{2}\langle\overline{x}+x-2z,x-\overline{x}\rangle,\
\forall\,x\in\mathbb{R}^{n}.$

∎

Relation (3.2) is too general for providing convergence results for proximal
point type algorithms while relation (1.2) has proven to be extremely useful
in the convex case. Motivated by this, we introduce the class of prox-convex
functions below. In the following, we write

${\operatorname*{Prox}}_{h}(K,z):={\operatorname*{Prox}}_{(h+\delta_{K})}(z).$
(3.3)

Note that closed formulae for the proximity operator of a sum of functions in
terms of the proximity operators of the involved functions are known only in
the convex case and under demanding hypotheses, see, for instance, [1].
However, constructions like the one in (3.3) can be found in the literature on
proximal point methods for solving different classes of (nonconvex)
optimization problems, take for instance [6, 13].

######  Definition 3.1.

Let $K$ be a closed set in $\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper function such
that $K\cap\operatorname*{dom}\,h\neq\emptyset$. We say that $h$ is prox-
convex on $K$ if there exists $\alpha>0$ such that for every $z\in K$,
$\operatorname*{Prox}_{h}(K,z)\neq\emptyset$, and

$\overline{x}\in{\operatorname*{Prox}}_{h}(K,z)~{}\Longrightarrow~{}h(\overline{x})-h(x)\leq\alpha\langle\overline{x}-z,x-\overline{x}\rangle,~{}\forall
x\in K.$ (3.4)

The set of all prox-convex function on $K$ is denoted by $\Phi(K)$, and the
scalar $\alpha>0$ for which (3.4) holds is said to be the prox-convex value of
the function $h$ on $K$. When $K=\mathbb{R}^{n}$ we say that $h$ is prox-
convex.

######  Remark 3.2.

  * $(i)$

One can immediately notice that (3.4) is equivalent to a weaker version of
(1.2), namely

$\overline{x}\in{\operatorname*{Prox}}_{h}(K,z)~{}\Longrightarrow~{}z-\overline{x}\in\partial\left(\frac{1}{\alpha}(h+\delta_{K})\right)(\overline{x}).$

  * $(ii)$

The scalar $\alpha>0$ for which (3.4) holds needs not be unique. Indeed, if
$h$ is convex, then $\alpha=1$ by Proposition 3.4. However, due to the
convexity of $h$, $\langle\overline{x}-z,x-\overline{x}\rangle\geq 0$. Hence,
$\overline{x}\in\operatorname*{Prox}_{h}(K,z)$ implies that

$\displaystyle
h(\overline{x})-h(x)\leq\langle\overline{x}-z,x-\overline{x}\rangle\leq\gamma\langle\overline{x}-z,x-\overline{x}\rangle,~{}\forall~{}\gamma\geq
1,\ ~{}\forall~{}x\in K.$

Note however that a similar result does not necessarily hold in general, as
$\langle\overline{x}-z,x-\overline{x}\rangle$ might be negative.

  * $(iii)$

Note also that, at least from the computational point of view, an exact value
of $\alpha$ needs not be known, as one can see in Section 4.

In the following statement we see that in the left-hand side of (3.4) one can
replace the element-of symbol with equality since the proximity operator of a
proper prox-convex function is single-valued and also firmly nonexpansive.

######  Proposition 3.3.

Let $K$ be a closed set in $\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ a proper prox-convex
function on $K$ such that $K\cap\operatorname*{dom}\,h\neq\emptyset$. Then the
map $z\rightarrow\operatorname*{Prox}_{h}(K,z)$ is single-valued and firmly
nonexpansive.

###### Proof.

Suppose that $h$ is a prox-convex function with prox-convex value $\alpha>0$
and assume that for some $z\in K$ one has
$\\{\overline{x}_{1},\overline{x}_{2}\\}\subseteq\operatorname*{Prox}_{h}(K,z)$.
Then

$\displaystyle
h(\overline{x}_{1})-h(x)\leq\alpha\langle\overline{x}_{1}-z,x-\overline{x}_{1}\rangle,~{}\forall
x\in K,$ (3.5) $\displaystyle
h(\overline{x}_{2})-h(x)\leq\alpha\langle\overline{x}_{2}-z,x-\overline{x}_{2}\rangle,~{}\forall
x\in K.$ (3.6)

Take $x=\overline{x}_{2}$ in (3.5) and $x=\overline{x}_{1}$ in (3.6). By
adding the resulting equations, we get

$0\leq\alpha\langle\overline{x}_{1}-\overline{x}_{2},\overline{x}_{2}-z+z-\overline{x}_{1}\rangle=-\alpha\|\overline{x}_{1}-\overline{x}_{2}\|^{2}\leq
0.$

Hence, $\overline{x}_{1}=\overline{x}_{2}$, consequently
$\operatorname*{Prox}_{h}(K,\cdot)$ is single-valued.

Let $z_{1},z_{2}\in K$ and take
$\overline{x}_{1}\in\operatorname*{Prox}_{h}(K,z_{1})$ and
$\overline{x}_{2}\in\operatorname*{Prox}_{h}(K,z_{2})$. One has

$\displaystyle
h(\overline{x}_{1})-h(x)\leq\alpha\langle\overline{x}_{1}-z_{1},x-\overline{x}_{1}\rangle,~{}\forall~{}x\in
K,$ (3.7) $\displaystyle
h(\overline{x}_{2})-h(x)\leq\alpha\langle\overline{x}_{2}-z_{2},x-\overline{x}_{2}\rangle,~{}\forall~{}x\in
K.$ (3.8)

Taking $x=\overline{x}_{2}$ in (3.7) and $x=\overline{x}_{1}$ in (3.8) and
adding them, we have

$\lVert\overline{x}_{1}-\overline{x}_{2}\rVert^{2}\leq\langle
z_{1}-z_{2},\overline{x}_{1}-\overline{x}_{2}\rangle.$

Hence, by [5, Proposition 4.4], $\operatorname*{Prox}_{h}(K,\cdot)$ is firmly
nonexpansive. ∎

Next we show that every lower semicontinuous and convex function is prox-
convex.

######  Proposition 3.4.

Let $K$ be a closed and convex set in $\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper and lower
semicontinuous function such that $K\cap\operatorname*{dom}\,h\neq\emptyset$.
If $h$ is convex on $K$, then $h\in\Phi(K)$ with $\alpha=1$.

###### Proof.

Since $h$ is convex, the function $x\mapsto h(x)+({\beta}/{2})\lVert
z-x\rVert^{2}$ is strongly convex on $K$ for all $\beta>0$ and all $z\in K$,
in particular, for $\beta=1$. Thus $\operatorname*{Prox}_{h}(K,z)$ contains
exactly one element, say $\overline{x}\in\mathbb{R}^{n}$. It follows from [5,
Proposition 12.26] that $z-\overline{x}\in\partial h(\overline{x})$, so
relation (3.4) holds for $\alpha=1$. Therefore, $h\in\Phi(K)$. ∎

Prox-convexity goes beyond convexity as shown below.

######  Example 3.1.

Let us consider $K:=[0,1]$ and the continuous and real-valued function
$h:K\rightarrow\mathbb{R}$ given by $h(x)=-x^{2}-x$. Note that

  * $(i)$

$h$ is strongly quasiconvex without being convex on $K$ (take $\beta=1$);

  * $(ii)$

For all $z\in K$,
$\operatorname*{Prox}_{h}(K,z)={\operatorname*{\arg\min}}_{K}h=\\{1\\}$;

  * $(iii)$

$\partial^{\leq}_{K}h(1)=\mathbb{R}^{n}$ since, by $(ii)$,
$S_{h(1)}(h)=\\{1\\}$, i.e., $\partial^{\leq}h(1)=\mathbb{R}^{n}$ by (2.15);

  * $(iv)$

$h$ satisfies condition (3.4) for all $\alpha>0$;

  * $(v)$

$h\in\Phi(K)$;

  * $(vi)$

more generally, $h\in\Phi([0,r])$ for all $r>0$.

In order to formulate a reverse statement of Proposition 3.4, we note that if
$h:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is a lower semicontinuous and prox-
convex function on some set $K\cap\operatorname*{dom}\,h\neq\emptyset$ which
satisfies (3.4) for $\alpha=1$, then $h$ is not necessarily convex. Indeed,
the function in Example 3.1 satisfies (3.4) for all $\alpha>0$, but it is not
convex on $K=[0,1]$.

In the following example, we show that lower semicontinuity is not a necessary
condition for prox-convexity. Note also that although the proximity operator
of the function mentioned in Remark 3.1$(ii)$ is always empty, this is no
longer the case when restricting it to an interval.

######  Example 3.2.

Take $n\geq 3$, $K_{n}:=[1,n]$ and the function
$h_{n}:K_{n}\rightarrow\mathbb{R}$ given by

$h_{n}(x)=\left\\{\begin{array}[c]{cl}1-x^{3},&\mathrm{if}~{}1\leq x\leq 2,\\\
1-x^{3}-k,&\mathrm{if}~{}k<x\leq
k+1,~{}k\in\\{2,\ldots,n-1\\}.\end{array}\right.$

Note that $h_{n}$ is neither convex nor lower semicontinuous, but it is
quasiconvex on $K_{n}$. Due to the discontinuity of $h_{n}$, the function
$f_{n}(x)=h_{n}(x)+({1}/{2})\lVert x\rVert^{2}$ is neither convex nor lower
semicontinuous on $K_{n}$, hence $h_{n}$ is not $c$-weakly convex (in the
sense of [17]) either and also its subdifferential is not hypomonotone (as
defined in [10, 18, 24]). However, for any $z\in K_{n}$,
$\operatorname*{Prox}_{h_{n}}(K_{n},z)=\\{n\\}$, and
$\partial^{\leq}_{K_{n}}h_{n}(n)=K_{n}$. Therefore, $h_{n}\in\Phi(K_{n})$.

Another example of a prox-convex function that is actually (like the one in
Example 3.1) both concave and DC follows.

######  Example 3.3.

Take $K=[1,2]$ and $h:(0,+\infty)\to\mathbb{R}$ defined by
$h(x)=5x+\ln(1+10x)$. As specified in [21], both the prox-convex function
presented in Example 3.1 and this one represent cost functions considered in
oligopolistic equilibrium problems, being thus relevant for studying also from
a practical point of view. One can show that
$\operatorname*{Prox}_{h}(K,z)={\operatorname*{\arg\min}}_{K}h=\\{1\\}$ for
all $z\in K$ and (3.4) is fulfilled for $\alpha\in(0,5)$.

######  Remark 3.3.

  * (i)

One can also construct examples of $c$-weakly convex functions (for some
$c>0$) that are not prox-convex, hence these two classes only contain some
common elements without one of them being completely contained in the other.

  * (ii)

While Examples 3.1 and 3.3 exhibit prox-convex functions that are also DC, the
prox-convex functions presented in Example 3.2 are not DC. Examples of DC
functions that are not prox-convex can be constructed as well, consequently,
like in the case of $c$-weakly convex functions, these two classes only
contain some common elements without one of them being completely contained in
the other. Note moreover that different to the literature on algorithms for DC
optimization problems (see, for instance, [2, 4]) where usually only critical
points (and not optimal solutions) of such problems are determinable, for the
DC functions that are also prox-convex proximal point methods are capable of
delivering global minima (on the considered sets).

  * (iii)

The remarkable properties of the Kurdyka-Łojasiewicz (KŁ) functions made them
a standard tool when discussing proximal point type algorithms for nonconvex
functions. As their definition requires proper closedness and the prox-convex
functions presented in Example 3.2 are not closed, one can conclude that the
class of prox-convex functions is broader in this sense than the one of KŁ
ones. Similarly one can note that prox-convexity is not directly related to
hypomonotonicity of subdifferentials (see [10, 18, 24], respectively).

  * (iv)

At least due to the similar name, a legitimate question is whether the notion
of prox-convexity is connected in any way with the prox-regularity (cf. [10,
20, 24]). While the latter asks a function to be locally lower semicontinuous
around a given point, the notion we introduce in this work does not assume any
topological properties on the involved function. Another difference with
respect to this notion can be noticed in Section 4, where we show that the
classical proximal point algorithm remains convergent towards a minimum of the
function to be minimized even if this lacks convexity, but is prox-convex. On
the other hand, the iterates of the modified versions of the proximal point
method employed for minimizing prox-regular functions converge towards
critical points of the latter. Last but not least note that, while in the
mentioned works one uses tools specific to nonsmooth analysis such as
generalized subdifferentials, in this paper we employ the convex
subdifferential and some subdifferential notions specific to quasiconvex
functions.

Necessary and sufficient hypotheses for condition (3.4) are given below.

######  Proposition 3.5.

Let $K$ be a closed set in $\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper, lower
semicontinuous and prox-convex function such that
$K\cap\operatorname*{dom}\,h\neq\emptyset$. Let $\alpha>0$ be the prox-convex
value of $h$ on $K$, and $z\in K$. Consider the following assertions

  * $(a)$

$\operatorname*{Prox}_{h}(K,z)=\\{\overline{x}\\}$;

  * $(b)$

$z-\overline{x}\in\partial_{K}\left(\frac{1}{\alpha}h\right)(\overline{x})$;

  * $(c)$

$(\frac{1}{\alpha}h)_{z}(\overline{x})-(\frac{1}{\alpha}h)_{z}(x)\leq-\frac{1}{2}\lVert
x-\overline{x}\rVert^{2}$ for all $x\in K$;

  * $(d)$

$\overline{x}\in\operatorname*{Prox}_{\frac{1}{\alpha}h}(K,z)$.

Then

$(a)~{}\Longrightarrow~{}(b)~{}\Longleftrightarrow~{}(c)~{}\Longrightarrow~{}(d).$

If $\alpha=1$, then $(d)$ implies $(a)$ and all the statements are equivalent.

###### Proof.

$(a)\Rightarrow(b)$: By definition of prox-convexity.

$(b)\Leftrightarrow(c)$: One has

$\displaystyle
z-\overline{x}\in\partial_{K}(\frac{1}{\alpha}h)(\overline{x})~{}\Longleftrightarrow~{}(\frac{1}{\alpha}h)(\overline{x})-(\frac{1}{\alpha}h)(x)\leq\langle\overline{x}-z,x-\overline{x}\rangle,~{}\forall
x\in K$
$\displaystyle\Longleftrightarrow~{}\frac{1}{\alpha}h(\overline{x})-\frac{1}{\alpha}h(x)\leq\frac{1}{2}\lVert
z-x\rVert^{2}-\frac{1}{2}\lVert z-\overline{x}\rVert^{2}-\frac{1}{2}\lVert
x-\overline{x}\rVert^{2},~{}\forall x\in K$
$\displaystyle\Longleftrightarrow~{}\frac{1}{\alpha}h(\overline{x})+\frac{1}{2}\lVert
z-\overline{x}\rVert^{2}-\frac{1}{\alpha}h(x)-\frac{1}{2}\lVert
z-x\rVert^{2}\leq-\frac{1}{2}\lVert x-\overline{x}\rVert^{2},~{}\forall x\in
K$
$\displaystyle\Longleftrightarrow~{}(\frac{1}{\alpha}h)_{z}(\overline{x})-(\frac{1}{\alpha}h)_{z}(x)\leq-\frac{1}{2}\lVert
x-\overline{x}\rVert^{2},~{}\forall x\in K.$ (3.9)

$(c)\Rightarrow(d)$: As $-({1}/{2})\lVert x-\overline{x}\rVert^{2}\leq 0$ for
all $x\in K$, (3.9) yields
$\overline{x}\in\operatorname*{Prox}_{(1/\alpha)h}(K,z)$.

When $\alpha=1$, the implication $(d)\Rightarrow(a)$ is straightforward. ∎

######  Remark 3.4.

  * $(i)$

It follows from Proposition 3.5$(d)$ that if $h$ is prox-convex on $K$ with
prox-convex value $\alpha>0$, then the function $(1/\alpha)h$ is also prox-
convex on $K$ with prox-convex value $1$. Moreover,
$\operatorname*{Prox}_{(1/\alpha)h}=\operatorname*{Prox}_{h}$.

  * $(ii)$

Assertions $(d)$ and $(a)$ of Proposition 3.5 are not equivalent in the
general case. Indeed, let us consider $h$ and $K$ as in Example 3.1 and
$\alpha=10$. Take $z=0$. Then $((1/10)h)_{0}(x)=(2/5)x^{2}-(1/10)x$ and
$\operatorname*{Prox}_{(1/10)h}(K,0)=\\{1/8\\}$ while
$\operatorname*{Prox}_{h}(K,0)=\\{1\\}$.

If $h$ is prox-convex with prox-convex value $\alpha$, then we know that
$\operatorname*{Prox}_{(1/\alpha)h}=\operatorname*{Prox}_{h}$ is a singleton,
hence

${}^{\frac{1}{\alpha}}h(z)=\min_{x\in K}\left(h(x)+\frac{\alpha}{2}\lVert
z-x\rVert^{2}\right)=h({\operatorname*{Prox}}_{h}(z))+\frac{\alpha}{2}\lVert
z-{\operatorname*{Prox}}_{h}(z)\rVert.$ (3.10)

Consequently, ${}^{1/\alpha}h(z)\in\mathbb{R}$ for all $z\in\mathbb{R}^{n}$.
Furthermore, we have the following statements.

######  Proposition 3.6.

Let $h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be proper, lower
semicontinuous and prox-convex with prox-convex value $\alpha>0$ on a closed
set $K\subseteq\mathbb{R}^{n}$ such that
$K\cap\operatorname*{dom}\,h\neq\emptyset$. Then
${}^{1/\alpha}h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is Fréchet differentiable
everywhere and

$\nabla({{}^{\frac{1}{\alpha}}}h)=\alpha\left(\operatorname*{Id}-{\operatorname*{Prox}}_{\frac{1}{\alpha}h}\right),$
(3.11)

is $\alpha$-Lipschitz continuous.

###### Proof.

Let $x,y\in K$ with $x\neq y$. Set $\gamma=1/\alpha$,
$\overline{x}=\operatorname*{Prox}_{h}(K,x)$ and
$\overline{y}=\operatorname*{Prox}_{h}(K,y)$. As $h$ is prox-convex with prox-
convex value $\alpha$, we have

$\displaystyle h(z)-h(\overline{x})\geq\alpha\langle
x-\overline{x},z-\overline{x}\rangle\,\forall\,z\in
K\,\Longrightarrow\,h(\overline{y})-h(\overline{x})\geq\frac{1}{\gamma}\langle
x-\overline{x},\overline{y}-\overline{x}\rangle.$

From (2.17), we get

${}^{\gamma}h(y)-\,^{\gamma}h(x)$
$\displaystyle=h(\overline{y})-h(\overline{x})+\frac{1}{2\gamma}\left(\lVert
y-\overline{y}\rVert^{2}-\lVert x-\overline{x}\rVert^{2}\right)$
$\displaystyle\geq\frac{1}{2\gamma}\left(2\langle
x-\overline{x},\overline{y}-\overline{x}\rangle+\lVert
y-\overline{y}\rVert^{2}-\lVert x-\overline{x}\rVert^{2}\right)$
$\displaystyle=\frac{1}{2\gamma}\left(\lVert
y-\overline{y}-x+\overline{x}\rVert^{2}+2\langle
y-x,x-\overline{x}\rangle\right)$ $\displaystyle\geq\frac{1}{\gamma}\langle
y-x,x-\overline{x}\rangle.$ (3.12)

Exchanging above $x$ with $y$ and $\overline{x}$ with $\overline{y}$, one gets

$^{\gamma}h(x)-\,^{\gamma}h(y)\geq\frac{1}{\gamma}\langle
x-y,y-\overline{y}\rangle.$ (3.13)

It follows from equations (3.12) and (3.13) that

$\displaystyle 0$
$\displaystyle\leq\,^{\gamma}h(y)-\,^{\gamma}h(x)-\frac{1}{\gamma}\langle
y-x,x-\overline{x}\rangle$ $\displaystyle\leq-\frac{1}{\gamma}\langle
x-y,y-\overline{y}\rangle-\frac{1}{\gamma}\langle y-x,x-\overline{x}\rangle$
$\displaystyle=\frac{1}{\gamma}\lVert y-x\rVert^{2}+\frac{1}{\gamma}\langle
y-x,\overline{x}-\overline{y}\rangle.$

As $\operatorname*{Prox}_{K,h}$ is firmly nonexpansive on $K$, $\langle
y-x,\overline{y}-\overline{x}\rangle\geq\lVert\overline{y}-\overline{x}\rVert^{2}\geq
0$, then

$\displaystyle 0\leq\,^{\gamma}h(y)-\,^{\gamma}h(x)-\frac{1}{\gamma}\langle
y-x,x-\overline{x}\rangle\leq\frac{1}{\gamma}\lVert y-x\rVert^{2}$
$\displaystyle\Longrightarrow~{}\lim_{y\rightarrow
x}\frac{{}^{\gamma}h(y)-\,^{\gamma}h(x)-\frac{1}{\gamma}\langle
y-x,x-\overline{x}\rangle}{\lVert y-x\rVert}=0.$

Thus, ${}^{1/\alpha}h$ is Fréchet differentiable at every
$x\in\mathbb{R}^{n}$, and $\nabla(^{1/\alpha}h)=\alpha(\operatorname*{Id}-$
$\operatorname*{Prox}_{h})$. Since $\operatorname*{Prox}_{h}$ is firmly
nonexpansive, $\operatorname*{Id}-\operatorname*{Prox}_{h}$ is also firmly
nonexpansive, so $\nabla(^{1/\alpha}h)$ is $\alpha$-Lipschitz continuous. ∎

###  3.2 Strongly G-subdifferentiable functions

Further we introduce and study a class of quasiconvex functions whose lower
semicontinuous members are prox-convex.

######  Definition 3.2.

Let $K$ be a closed and convex set in $\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper and lower
semicontinuous function such that $K\cap\operatorname*{dom}\,h\neq\emptyset$.
We call $h$ strongly G-subdifferentiable on $K$ if

  * $(a)$

$h$ is strongly quasiconvex on $K$ for some $\beta\in[1,+\infty[$;

  * $(b)$

for each $z\in K$ there exists $\overline{x}\in\mathbb{R}^{n}$ such that
$\operatorname*{Prox}_{h}(K,z)=\\{\overline{x}\\}$ and

$\frac{1}{2}(z-\overline{x})\in\partial^{\leq}_{K}h(\overline{x}).$ (3.14)

Next we show that a lower semicontinuous and strongly G-subdifferentiable
function on $K$ is prox-convex.

######  Proposition 3.7.

Let $K$ be a closed and convex set in $\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper and lower
semicontinuous function such that $K\cap\operatorname*{dom}\,h\neq\emptyset$.
If $h$ is strongly G-subdifferentiable on $K$, then $h\in\Phi(K)$.

###### Proof.

Let $h$ be a lower semicontinuous and strongly G-subdifferentiable function.
Then for every $z\in K$, there exists $\overline{x}\in K$ with
$\overline{x}=\operatorname*{Prox}_{h}(K,z)$. Hence, given any $y\in K$, we
take $y_{\lambda}=\lambda y+(1-\lambda)\overline{x}$ with $\lambda\in[0,1]$.
Thus, by the definition of the proximity operator and the strong
quasiconvexity of $h$ on $K$ for some $\beta\geq 1$, we have

$\displaystyle h(\overline{x})$ $\displaystyle\leq h(\lambda
y+(1-\lambda)\overline{x})+\frac{1}{2}\lVert\lambda
y+(1-\lambda)\overline{x}-z\rVert^{2}-\frac{1}{2}\lVert\overline{x}-z\rVert^{2}$
$\displaystyle=h(\lambda
y+(1-\lambda)\overline{x})+\lambda\langle\overline{x}-z,y-\overline{x}\rangle+\lambda^{2}\frac{1}{2}\lVert
y-\overline{x}\rVert^{2}$
$\displaystyle\leq\max\\{h(y),h(\overline{x})\\}+\lambda\langle\overline{x}-z,y-\overline{x}\rangle+\frac{\lambda}{2}(\lambda\beta+\lambda-\beta)\lVert
y-\overline{x}\rVert^{2}.$

We have two possible cases.

  * $(i)$

If $h(y)>h(\overline{x})$, then

$h(\overline{x})-h(y)\leq\lambda\langle\overline{x}-z,y-\overline{x}\rangle+\frac{\lambda}{2}(\lambda\beta+\lambda-\beta)\lVert
y-\overline{x}\rVert^{2},\ \forall\lambda\in[0,1].$

Hence, for $\lambda=1/2$ and since $\beta\geq 1$, one has

$\displaystyle h(\overline{x})-h(y)$
$\displaystyle\leq\frac{1}{2}\langle\overline{x}-z,y-\overline{x}\rangle+\frac{1}{4}(\frac{1}{2}-\frac{\beta}{2})\lVert
y-\overline{x}\rVert^{2}$
$\displaystyle\leq\frac{1}{2}\langle\overline{x}-z,y-\overline{x}\rangle,\
\forall y\in K\backslash S_{h(\overline{x})}(h).$

  * $(ii)$

If $h(y)\leq h(\overline{x})$, then $y\in S_{h(\overline{x})}(h)$, it follows
from Definition 3.2$(b)$ that

$\displaystyle\frac{1}{2}(z-\overline{x})\in\partial^{\leq}_{K}h(\overline{x})\Longleftrightarrow
h(\overline{x})-h(y)\leq\frac{1}{2}\langle\overline{x}-z,y-\overline{x}\rangle,\
\forall y\in K\cap S_{h(\overline{x})}(h).$

Therefore, it follows that $h$ satisfies (3.4) for $\alpha={1}/{2}$, i.e.,
$h\in\Phi(K)$. ∎

######  Remark 3.5.

  * $(i)$

When $h:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is lower semicontinuous and
strongly quasiconvex, as strongly quasiconvex functions are semistrictly
quasiconvex, $h$ is quasiconvex and every local minimum of $h$ is a global
minimum, too, so $h$ is neatly quasiconvex, i.e.,
$\partial^{<}h=\partial^{\leq}h$ (see [26, Proposition 9]). Therefore, we can
replace $\partial^{\leq}_{K}h$ by $\partial^{<}_{K}h$ in condition (3.14).

  * $(ii)$

Strongly $G$-subdifferentiable functions are not necessarily convex as the
function in Example 3.1 shows.

A family of prox-convex functions that are not strongly $G$-subdifferentiable
can be found in Remark 3.6, see also Example 3.2.

Now, we study lower semicontinuous strongly quasiconvex functions for which
the Gutierréz subdifferential is nonempty. To that end, we first recall the
following definitions (adapted after [11, Definition 3.1]).

######  Definition 3.3.

Let $K$ be a nonempty set in $\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ with
$K\cap\operatorname*{dom}\,h\neq\emptyset$. We say that $h$ is

  * $(a)$

$\inf$-compact on $K$ if for all $\overline{x}\in K$,
$S_{h(\overline{x})}(h)\cap K$ is compact;

  * $(b)$

$\alpha$-quasiconvex at $\overline{x}\in K$ $(\alpha\in\mathbb{R})$, if there
exist $\rho>0$ and $e\in\mathbb{R}^{n}$, $\lVert e\rVert=1$, such that

$y\in K\cap\mathbb{B}(\overline{x},\rho)\cap
S_{h(\overline{x})}(h)\Longrightarrow~{}\langle
y-\overline{x},e\rangle\geq\alpha\lVert y-\overline{x}\rVert^{2};$ (3.15)

  * $(c)$

positively quasiconvex on $K$ if for any $\overline{x}$ there exists
$\alpha(\overline{x})>0$ such that $h$ is $\alpha(\overline{x})$-quasiconvex
on $S_{h(\overline{x})}(h)$.

The following result presents a connection between strongly quasiconvex
functions and positively quasiconvex ones.

######  Proposition 3.8.

Let $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a strongly quasiconvex
function, $\overline{x}\in\mathbb{R}^{n}$ and $\alpha>0$. Then the following
assertions hold

  * $(a)$

If $\xi\in\partial\left(({1}/{\alpha})h\right)(\overline{x})$, then

$\langle\xi,y-\overline{x}\rangle\leq-\frac{\beta}{2\alpha}\lVert
y-\overline{x}\rVert^{2},~{}\forall y\in S_{h(\overline{x})}(h).$ (3.16)

  * $(b)$

If $\xi\in\partial^{\leq}h(\overline{x})$, then

$\langle\xi,y-\overline{x}\rangle\leq-\frac{\beta}{2}\lVert
y-\overline{x}\rVert^{2},\ \forall y\in S_{h(\overline{x})}(h).$ (3.17)

As a consequence, in both cases, $h$ is positively quasiconvex on
$\mathbb{R}^{n}$.

###### Proof.

The proofs are similar, so we only show $(a)$. Take
$\overline{x}\in\mathbb{R}^{n}$ and
$\xi\in\partial\left(({1}/{\alpha})h\right)(\overline{x})$. Then,

$\alpha\langle\xi,z-\overline{x}\rangle\leq h(z)-h(\overline{x}),\ \forall
z\in\mathbb{R}^{n}.$

Take $y\in S_{h(\overline{x})}(h)$ and $z=\lambda y+(1-\lambda)\overline{x}$
with $\lambda\in[0,1]$. Then

$\displaystyle\lambda\alpha\langle\xi,y-\overline{x}\rangle$
$\displaystyle\leq h(\lambda y+(1-\lambda)\overline{x})-h(\overline{x})$
$\displaystyle\leq\max\\{h(y),h(\overline{x})\\}-\lambda(1-\lambda)\frac{\beta}{2}\lVert
y-\overline{x}\rVert^{2}-h(\overline{x})$
$\displaystyle=-\lambda(1-\lambda)\frac{\beta}{2}\lVert
y-\overline{x}\rVert^{2}.$

Then, for every $y\in S_{h(\overline{x})}(h)$, by dividing by $\lambda>0$ and
taking the limit when $\lambda$ descends towards $0$, we have

$\displaystyle\langle\xi,y-\overline{x}\rangle\leq\lim_{\lambda\downarrow
0}\left(-(1-\lambda)\frac{\beta}{2\alpha}\lVert
y-\overline{x}\rVert^{2}\right)=-\frac{\beta}{2\alpha}\lVert
y-\overline{x}\rVert^{2}.$

Now, since $h$ is strongly quasiconvex,
${\operatorname*{\arg\min}}_{\mathbb{R}^{n}}h$ has at most one point. If
$\overline{x}\in{\operatorname*{\arg\min}}_{\mathbb{R}^{n}}h$, then condition
(3.15) holds immediately. If
$\overline{x}\not\in{\operatorname*{\arg\min}}_{\mathbb{R}^{n}}h$, then
$\xi\neq 0$, i.e., condition (3.15) holds for
${\beta}/{(2\alpha\lVert\xi\rVert)}>0$.

Therefore, $h$ is positively quasiconvex on $\mathbb{R}^{n}$. ∎

As a consequence, we have the following result.

######  Corollary 3.1.

Let $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a lower semicontinuous and
strongly quasiconvex function with $\beta=1$, let $z\in\mathbb{R}^{n}$ and
$\overline{x}\in\operatorname*{Prox}_{h}(z)$. If there exists
$\xi\in\partial^{\leq}h(\overline{x})$ such that

$h_{z}(\overline{x})-h_{z}(x)\leq\langle\xi,y-\overline{x}\rangle,\ \forall
y\in S_{h(\overline{x})}(h),$ (3.18)

then $h$ is prox-convex on its sublevel set at the height $h(\overline{x})$,
i.e., $h\in\Phi(S_{h(\overline{x})}(h))$.

###### Proof.

If $\xi\in\partial^{\leq}h(\overline{x})$, and since $h$ is lower
semicontinuous and strongly quasiconvex with $\beta=1$, then by Proposition
3.8$(b)$, we have

$\displaystyle
h_{z}(\overline{x})-h_{z}(x)\leq\langle\xi,y-\overline{x}\rangle\leq-\frac{1}{2}\lVert
y-\overline{x}\rVert^{2},\ \forall y\in S_{h(\overline{x})}(h),$
$\displaystyle\Longrightarrow~{}h(\overline{x})-h(x)\leq\frac{1}{2}\lVert
z-y\rVert^{2}-\frac{1}{2}\lVert z-\overline{x}\rVert^{2}-\frac{1}{2}\lVert
y-\overline{x}\rVert^{2},\ \forall y\in S_{h(\overline{x})}(h)$
$\displaystyle\Longleftrightarrow~{}h(\overline{x})-h(x)\leq\langle\overline{x}-z,x-\overline{x}\rangle,\
\forall y\in S_{h(\overline{x})}(h).$

Therefore, $h\in\Phi(S_{h(\overline{x})}(h))$. ∎

Another consequence is the following sufficient condition for
$\inf$-compactness under an $L$-Lipschitz assumption, which revisits [29,
Corollary 1].

######  Corollary 3.2.

Let $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be an $L$-Lipschitz and strongly
quasiconvex function. Then $h$ is $\inf$-compact on $\mathbb{R}^{n}$.

###### Proof.

If $h$ is strongly quasiconvex, then $h$ is neatly quasiconvex, and since $h$
is $L$-Lipschitz, $\partial^{\leq}h(x)\neq\emptyset$ for all
$x\in\mathbb{R}^{n}$ by Lemma 2.1$(b)$. Now, by Proposition 3.8$(b)$, it
follows that $h$ is positively quasiconvex on $\mathbb{R}^{n}$. Finally, $h$
is $\inf$-compact on $\mathbb{R}^{n}$ by [11, Corollary 3.6]. ∎

We finish this section with the following observation.

######  Remark 3.6.

There are (classes of) prox-convex functions which are neither convex nor
strongly quasiconvex. Indeed, for all $n\in\mathbb{N}$, we take
$K_{n}:=[-n,+\infty[$ and the continuous quasiconvex functions
$h_{n}:K_{n}\rightarrow\mathbb{R}$ given by $h_{n}(x)=x^{3}$. Clearly, $h_{n}$
is neither convex nor strongly quasiconvex on $K_{n}$ hence also not strongly
G-subdifferentiable either.

Take $n\in\mathbb{N}$. Then for all $z\in K_{n}$,
${\operatorname*{\arg\min}}_{K_{n}}h_{n}=\operatorname*{Prox}_{h_{n}}(z)=\\{-n\\}$,
thus $S_{h_{n}(\overline{x})}(h_{n})=\\{\overline{x}\\}$, i.e.,
$\partial^{\leq}_{K_{n}}h_{n}(\overline{x})=\mathbb{R}^{n}$. Therefore,
$h_{n}\in\Phi(K_{n})$ for all $n\in\mathbb{N}$. Taking also into consideration
Corollary 3.1 one can conclude that the classes of strongly quasiconvex and
prox-convex functions intersect without being included in one another.

######  Remark 3.7.

All the prox-convex functions we have identified so far are semistrictly
quasiconvex, too, while there are semistrictly quasiconvex functions that are
not prox-convex (for instance $h:\mathbb{R}\to\mathbb{R}$ defined by $h(x)=1$
if $x=0$ and $h(x)=0$ if $x\neq 0$), hence the connection between the classes
of prox-convex and semistrictly quasiconvex functions remains an open problem.

For a further study on strong quasiconvexity, positive quasiconvexity and
$\inf$-compactness we refer to [11, 29, 30].

##  4 Proximal point type algorithms for nonconvex problems

In this section we show that the proximal point type algorithm remains
convergent when the function to be minimized is proper, lower semicontinuous
and prox-convex (on a given closed convex set), but not necessarily convex.
Although the algorithm considered below is the simplest and most basic version
available and some of the advances achieved in the convex case, such as
accelerations and additional flexibility by employing additional parameters,
are at the moment still open in the prox-convex setting, our investigations
show that the proximal point type methods can be successfully extended towards
other classes of nonconvex optimization problems for which they could not be
employed so far due to lack of a theoretical fundament.

######  Theorem 4.1.

Let $K$ be a closed and convex set in $\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ be a proper, lower
semicontinuous and prox-convex on $K$ function such that
${\operatorname*{\arg\min}}_{K}h\neq\emptyset$ and
$K\cap\operatorname*{dom}\,h\neq\emptyset$. Then for any $k\in\mathbb{N}$, we
set

$x^{k+1}={\operatorname*{Prox}}_{h}(K,x^{k})$ (4.1)

Then $\\{x^{k}\\}_{k}$ is a minimizing sequence of $h$ over $K$, i.e.,
$h(x^{k})\rightarrow\min_{x\in K}h(x)$ when $k\to+\infty$.

###### Proof.

Since $h$ is prox-convex on $K$, denote its prox-convex value by $\alpha>0$
and for all $k\in\mathbb{N}$ one has

$\displaystyle
x^{k+1}={\operatorname*{Prox}}_{h}(K,x^{k})~{}\Longrightarrow~{}x^{k}-x^{k+1}\in\partial\left(\frac{1}{\alpha}h+\delta_{K}\right)(x^{k+1})$
$\displaystyle\Longleftrightarrow\alpha\langle
x^{k}-x^{k+1},x-x^{k+1}\rangle\leq h(x)-h(x^{k+1}),\ \forall x\in K.$ (4.2)

Take $x=x^{k}\in K$, and since $\alpha>0$, we have

$0\leq\langle
x^{k}-x^{k+1},x^{k}-x^{k+1}\rangle\leq\frac{1}{\alpha}(h(x^{k})-h(x^{k+1}))\Longrightarrow~{}h(x^{k+1})\leq
h(x^{k}),~{}\forall k\in\mathbb{N}.$

On the other hand, take $\overline{x}\in{\operatorname*{\arg\min}}_{K}h$.
Then, for any $k\in\mathbb{N}$, by taking $x=\overline{x}$ in equation (4.2),
we have

$\displaystyle\lVert x^{k+1}-\overline{x}\rVert^{2}$ $\displaystyle=\lVert
x^{k+1}-x^{k}+x^{k}-\overline{x}\rVert^{2}$ $\displaystyle=\lVert
x^{k+1}-x^{k}\rVert^{2}+\lVert x^{k}-\overline{x}\rVert^{2}+2\langle
x^{k+1}-x^{k},x^{k}-\overline{x}\rangle$ $\displaystyle=-\lVert
x^{k+1}-x^{k}\rVert^{2}+\lVert x^{k}-\overline{x}\rVert^{2}+2\langle
x^{k+1}-x^{k},x^{k+1}-\overline{x}\rangle$ $\displaystyle\leq\lVert
x^{k}-\overline{x}\rVert^{2}+\frac{2}{\alpha}(h(\overline{x})-h(x^{k+1}))\leq\lVert
x^{k}-\overline{x}\rVert^{2},$ (4.3)

where we used that $h(\overline{x})\leq h(x^{k+1})$. Thus,
$\\{x^{k}-\overline{x}\\}_{k}$ is bounded. Then, and passing to a subsequence
if needed, $x^{k}\rightarrow\overline{x}$ when $k\rightarrow+\infty$. Finally,
since $h$ is lower semicontinuous and $K$ is closed, we have
$\liminf_{k\rightarrow+\infty}h(x^{k})=\min_{x\in K}h(x)$. ∎

######  Remark 4.1.

From (4.3) one can deduce straightforwardly that the known
${\mathcal{O}}(1/n)$ rate of convergence of the proximal point algorithm holds
in the prox-convex case, too.

######  Remark 4.2.

Although the function to be minimized in Theorem 4.1 by means of the proximal
point algorithm is assumed to be prox-convex, its prox-convex value $\alpha>0$
needs not be known, even if it plays a role in the proof.

######  Remark 4.3.

One can modify the proximal point algorithm by replacing in (4.1) the proximal
step by ${\operatorname*{Prox}}_{h}(S_{h(x^{k})}(h),x^{k})$ without affecting
the convergence of the generated sequence. Note also that taking
$K=\mathbb{R}^{n}$ in Theorem 4.1 one obtains the classical proximal point
algorithm adapted for prox-convex functions and not for a restriction of such
a function to a given closed convex set $K\subseteq\mathbb{R}^{n}$.

######  Example 4.1.

Let $K=[0,2]\times\mathbb{R}$ and consider the function
$h:K\rightarrow\mathbb{R}$ given by
$h(x_{1},x_{2})=x^{2}_{2}-x^{2}_{1}-x_{1}$. Observe that $h$ is continuous,
strongly quasiconvex in the second argument, convex and strongly quasiconvex
in the first argument, hence $h$ is strongly quasiconvex without being convex
on $K$. Furthermore, by Example 3.1 $h$ is prox-convex on $K$. The global
minimum of $h$ over $K$ is $(2,0)^{\top}$ and it can be found by applying
Theorem 4.1, i.e. via the proximal point algorithm, although the function $h$
is not convex. First one determines the proximity operator

${\operatorname*{Prox}}_{h}(K,(z_{1},z_{2})^{\top})=\left(\left\\{\begin{array}[]{cc}0,&\mbox{if
}z_{1}\leq-2\\\ 2,&\mbox{if
}z_{1}>-2\end{array},\right.\frac{z_{2}}{3}\right)^{\top},\
z_{1},z_{2}\in\mathbb{R}.$

Taking into consideration the way $K$ is defined, it follows that the proximal
step in Theorem 4.1 delivers $x^{k+1}=(2,x^{k}_{2}/3)^{\top}$, where
$x^{k}=(x^{k}_{1},x^{k}_{2})^{\top}$. Whatever feasible starting point
$x^{1}\in K$ of the algorithm is chosen, it delivers the global minimum of $h$
over $K$ because $x^{k}_{1}=2$ and $x^{k}_{2}=x^{1}_{2}/(3^{k-1})$ for all
$k\in\mathbb{N}$.

##  5 Conclusions and future work

We contribute to the discussion on the convergence of proximal point
algorithms beyond convexity by introducing a new generalized convexity notion
called prox-convexity. We identify some classes of quasiconvex, weakly convex
and DC functions (and not only) that satisfy the new definition and different
useful properties of these functions are proven. Then we show that the
classical proximal point algorithm remains convergent when the convexity of
the proper lower semicontinuous function to be minimized is relaxed to prox-
convexity (on a certain subset of the domain of the function).

In a future work, we aim to uncover more properties and develop calculus rules
for prox-convex functions as well as to extend our investigation to nonconvex
equilibrium problems and nonconvex mixed variational inequalities, to Hilbert
spaces and to splitting methods, also employing Bregman distances instead of
the classical one where possible.

##  6 Declarations

###  6.1 Funding

This research was partially supported by FWF (Austrian Science Fund), project
M-2045, and by DFG (German Research Foundation), project GR 3367/4-1 (S.-M.
Grad) and Conicyt–Chile under project Fondecyt Iniciación 11180320 (F. Lara).

###  6.2 Conflicts of interest/Competing interests

There are no conflicts of interest or competing interests related to this
manuscript.

###  6.3 Availability of data and material

Not applicable.

###  6.4 Code availability

Not applicable.

###  6.5 Authors’ contributions

Both authors contributed equally to the study conception and design.

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Cylindrical drift chamber sec:dch The positron momentum and position vectors
are measured with CDCH. In this section we review the main advantages of CDCH
over the MEG drift chamber system, its construction, commissioning,
reconstruction, alignment and performance. Concept and design sec:dchconc The
MEG drift chamber system consisted of sixteen individual radially aligned
chambers. With this detector configuration, relevant efficiency loss and
resolution degradation were caused by the material of the chamber’s mechanical
support, electronic cards and cables, which were frequently crossed by the
positron trajectories. The CDCH, extensively discussed in baldini_2018,
Chiappini:2023egy, overcomes these limitations by replacing the segmented
structure of the MEG chambers with a single volume one.

The CDCH is a 1.93 long low-mass cylindrical volume, filled with a
helium–isobutane gas mixture (the exact composition is specified in
sec:dchgeo) and equipped with nine concentric layers of 192 gold-plated
tungsten sense wires each, 1728 in total, arranged in a stereo configuration
with two views and ∼10000 silver-plated aluminium cathode and guard wires. The
sense wires collect the signals from the drift electrons, while the cathode
and guard wires form nearly square drift cells and define the electric field
within and at the boundaries of the sensitive volume; the cell dimensions
range from 5.87.5 at center and from 6.78.7 at the end-plates. The sense wires
within the experimental acceptance, about 1200, are read out by the data
acquisition system.

Construction sec:dchcons Construction of the CDCH, which began in late 2016
and was completed in spring 2018, was performed modularly by soldering wires
in groups of sixteen onto printed circuit boards (PCBs), which were then
mounted between radial spokes at the ends of the chamber. The mechanical
structure of the detector achieves its final form by integrating a carbon
fibre component securely fastened to both end-plates, effectively enclosing
the sensitive volume.

The chamber was delivered to PSI for the commissioning phase in summer 2018
and integrated into the MEG II detector in winter 2018. After installation,
the CDCH was operated at PSI with engineering runs in $2019$ and $2020$ and
with physics data taking runs in 2021 and 2022.

[width=1angle=0] cdchopen_pisa.png

Figure : Picture of the open CDCH equipped with all the wires.

###  DAQ and services

High voltage (HV) is supplied with a commercial system by ISEG
Spezialelektronik GmbH, made of mod. EHS F230 and EHS F430 boards in a WIENER
crate with MPOD controller, for a total of 144 channels. HV channels are split
between two cables by custom-built distribution boards, and eight wires are
powered through each cable. Ionisation signals are read at both ends of the
chamber wires by 216 front-end (FE) boards with a bandwidth of ∼400, which are
connected to the wire PCBs. The signals are digitised at 1.2GSPS by the
WaveDREAM boards of the integrated trigger and DAQ system WaveDAQ, described
in sec:rtdaq. The low voltage (LV, 5 V) is supplied to the FE boards via a
dedicated distribution system through the WaveDREAM boards. As the FE boards
have a total power consumption of ≈350, they have to be cooled with water and
glycol by a cooling system built into the board holders and purged with dry
air to avoid condensation of water vapour. The gas is supplied by a dedicated
gas system [Baldini:2018ing] that mixes helium with isobutane and oxygen in
the required proportions. Isopropyl alcohol is added by passing a fraction of
the helium flow through a thermostated, alcohol-filled bubbler.

##  Final geometry and operation

The final geometry of the CDCH differs in some details from the original
design. The chamber was designed to have ten layers of sense wires, but during
the wiring phase there were delays in establishing the environmental
conditions and procedures needed to eliminate the risk of wire breakage. Then,
the outermost layer was not installed to fit into the schedule of the CDCH
operation. We used Monte Carlo (MC) simulations to verify that the degradation
of tracking efficiency and performance due to the missing layer is ¡1. First,
using Garfield++ simulations [garfield++], the HV operating point was set in
the range [range-units = single]14001480, with the innermost layer at the
highest voltage, to achieve a gas gain of $\sim$5e5. This configuration gives
a reasonable sensitivity to individual ionisation clusters, as confirmed by
the distance of closest approach (DOCA) measurements presented in
sec:dchhitrec. The HV can be set individually for each cell at 10 steps to
account for difference in size of cell due to their radial position within the
chamber. The mechanical tension of the wires, which is required to ensure
electrostatic stability and to avoid short circuits between the anode and
cathode wires, was also calculated using Garfield++ simulations. Several tests
were carried out, both on a full-size prototype and on the CDCH itself. The
required tension value was determined by stretching the wires by +5.2 with
respect to the nominal length (65 of the the elastic limit), based on
calculations. The gas mixture was also optimised in $2019$ and $2020$ to avoid
corona discharges and current spikes and to restore normal operation after a
sustained short circuit between an anode and a cathode wire. The original
He–isobutane gas mixture $\left(90:10\right)$ was modified by the addition of
oxygen and isopropyl alcohol (1.5). The addition of oxygen is a particularly
delicate operation due to its electronegativity, as it can capture drift
electrons. The oxygen level was initially set to 2 to dampen anomalous
currents of ∼400 and then gradually reduced to a stable value of 0.5 after
normal current levels were restored. In dch:current we show typical currents
on six HV channels during normal MEG II beam operation; each HV channel powers
16 sense wires.

[width=1angle=0] L3Current.pdf

Figure : Typical currents drawn by six CDCH HV channels when the beam is on.
Currents fall abruptly to zero when the beam goes off.

###  Issues and problems

The main problem encountered during the construction and commissioning of CDCH
was the breakage of 107 cathode wires. The wire breaks were investigated in
detail by microscopic inspections, chromatography, SEM/EDX analyses and
immersion tests in water, and were proved to be due to galvanic corrosion of
the aluminium core, caused by air humidity penetrating through small cracks in
the silver coating. A phenomenological model [wireanalysis] was developed to
predict the number of breaks as a function of the time the wire is exposed to
the ambient moisture and the elongation of the wire. The problem was solved by
keeping the CDCH in a controlled dry atmosphere during maintenance and
operation. When these precautions were taken, the rate of wire breakage
dropped to 0 so far. The impact of ∼100 missing cathodes (¡ 1 of the total)
was assessed by using MC simulations and found to be negligible. Nevertheless,
the presence of broken wire fragments within the CDCH is dangerous as they can
cause short circuits affecting several sectors of the chamber. All fragments
were carefully removed with a dedicated tool, but one of them caused the
strong short circuit cited in sec:dchgeo, which increased the chamber current
up to ∼400. A dedicated optimisation procedure for the gas mixture was
necessary to restore normal current values. During the standard data taking it
is very important to keep the isopropyl alcohol level stable, to avoid current
spikes, by continuously bubbling it within the chamber; an example of the
current spikes occurring in case of a shortage of isopropyl alcohol is shown
in dch:anomaly.

[width=1angle=0] L1Spike.pdf

Figure : Currents in four sectors of layer 1. Spikes are due to shortage of
isopropyl alcohol.

An additional problem is the noise caused by occasional short circuits on FE
boards. Because the FE boards are densely mounted on the back of the chamber
end plates, small movements of a board can cause short circuits between the
capacitors on the board and the aluminium support structure of the board
itself; such short circuits cause oscillations in the amplifier circuits that
can affect the behaviour of large parts of the chamber. These oscillations can
only be stopped by switching off the LV of the boards, but since the LV is
distributed in groups of eight channels, a single noisy channel will cause
eight channels to fail. This problem was solved by inserting suitable
insulating plastic blocks to ensure the proper separation between the board
elements and the metallic parts of the support structure.

##  Reconstruction algorithms

The purpose of the CDCH analysis is to identify and reconstruct the positron
tracks using the information on the sense wires provided by the ionisation
clusters. The CDCH operates in a high-rate environment; the hit rate per cell
can be ¿1 at the innermost wires for $R_{\muon}=\SI{5e7}{\per\second}$,
corresponding to a cell occupancy of 25 in the time window of maximum drift
time. Moreover, the geometrical characteristics, with only nine layers and
small stereo angles, makes the track finding complicated. A more detailed
description on the reconstruction algorithms is available elsewhere
[CDCHPerfor].

###  Waveform processing and hit reconstruction

The first step is to identify the signals induced by drift electrons in the
waveforms of the cells traversed by positrons; such signals are called “hits”.
dch:waveform shows typical waveforms due to a hit, which consists of several
temporally separated pulses from different ionisation clusters that are
stretched by the slow drift time of the electrons. For good hit reconstruction
efficient signal-to-noise discrimination and pile-up identification are
required. Two waveform processing algorithms have been developed to detect
hits.

[width=1angle=0] CDCHwfm.pdf

Figure : Example of typical waveforms in a drift cell crossed by a track after
the noise reduction applied. The top plot is read by the upstream electronics,
the bottom by the downstream.

A coherent low frequency noise was observed mainly over adjacent 16 wires.
Therefore, the first algorithm starts with coherent noise subtraction using
the averaged waveform for adjacent channels excluding the region with signal
pulses. This reduces the noise level (FWHM) from 23 at 13. Significant
incoherent high frequency noise is observed for ¿200, while the signal power
in this frequency range is negligible. Therefore, we apply high-frequency cut-
off at 225 using a discrete Fourier transform technique. These noise reduction
algorithms have been optimised by maximising the number of hits per track and
the tracking efficiency, and minimising the chi-square of track fitting. Hit
detection is based on a fixed voltage threshold for two adjacent sampling
points and a fixed threshold for the integration over 20 from the two initial
points. After the hit is detected, the thresholds are lowered to look for a
low-amplitude cluster pulse before the detected pulse. The second method uses
a deep-learning algorithm based on a convolutional neural network (CNN). The
network model accepts waveforms from eight neighbouring cells as input to
learn the pattern of the coherent noise as well as that of the signal and
outputs the probability of the first cluster arrival time of a hit at each
sampling point. It was trained with samples of simulated waveform data, where
hits are added randomly at the expected rate for
$R_{\muon}=\SI{3e7}{\per\second}$, overlaid with real noise data taken without
beam. Combining the results by the two methods results in a higher hit
efficiency but also a higher fake hit rate than the first method. To make the
best use of the results of the two methods, the following reconstruction is
repeated twice, with only the hits found with the first method and with the
hits found with the two methods after combining them. The results are combined
after the reconstruction is completed. If the reconstruction is successful
with both methods, the higher quality tracks are selected. This approach
improves the final tracking efficiency (see sec:dchperformeff), by a factor of
1.26 compared to applying only the first method, but at the cost of 4 times
the computational time for the track finding process described in
sec:track_rec due to the higher number of hits. The difference in the arrival
times of the signals and charges collected at the two ends of a wire provides
information about the $z$-coordinate of the hit along the wire. These
measurements are made by fitting one end of the waveform to the other. This
approach requires the detection of hits at only one of the two ends, and
hence, the efficiency of hit reconstruction is higher than with an approach
measuring the timings and charges at the two ends separately. The resolution
of the $z$-coordinate is several centimetres. Although this is moderate, it
helps to ensure that the track finding process is efficient and robust against
pile-up. The $z$-coordinate resolution is improved by exploiting the stereo
configuration of the wires in the tracking stage. Another piece of information
provided by the digitised waveforms is the arrival time of the ionisation
clusters on the sense wires. Since multiple clusters are frequently generated
on one waveform, the time of the first cluster must be identified to correctly
reconstruct the drift circle. This is measured from the summed waveform of the
two ends after adjusting the relative timing of the two. The arrival time of
cluster is the sum of the common track time ($T_{0}$) and the drift time of
the cluster, where $T_{0}$ is measured by pTC measurement (see sec:pTC). The
drift time of the first cluster is converted to the DOCA using the time-
distance relationship (TXY tables) described in sec:track_rec.

###  Track finding and fitting

After identification and reconstruction, the hits are fed into a pattern
recognition algorithm (track finder), followed by a track-fitter algorithm,
both based on Kalman filters. The former combines hits belonging to the same
positron track into a track candidate with a preliminary estimate of the
positron’s kinematics. The latter reconstructs the complete trajectory of the
positron and provides the best estimate of the kinematics at the target: the
positron energy $\epositron$, the emission angles ($\thetae,\phie$) and the
coordinates of the intersection of the track with the target,
$(\xpos,\ypos,\zpos)$. The track-finding algorithm is based on a track
following method that starts from hit pairs in outer layers, where occupancy
is lower. All compatible combinations of two pairs in different layers form a
set of track seeds. Each seed is propagated backward to the adjacent layers,
with checking the consistency between the track and hits and updating the
track parameters using the Kalman filter algorithm, until the innermost layer,
and then propagated forward with attempting to find additional compatible hits
to form full single turn track candidates. A track candidate is required to
have at least seven hits. The track fitter uses an extension of the Kalman
filter, namely the deterministic annealing filter (DAF) [daf] with iterative
weighting and annealing process, implemented in the GENFIT package [genfit1,
genfit2] with a proper treatment of the material effect. The left/right
ambiguity is also resolved by DAF. The fitter first fits individual track
candidates from the track finder and then merge the fitted segments to form
full multi-turn tracks inside CDCH. Then, the tracks are propagated forward to
the pTC and backward to the target. Once the track matches a pTC cluster, the
track is re-fitted with updated DOCA using the best estimated $T_{0}$, with a
correction of the time of flight from each hit to the pTC, and the best method
of DOCA estimation discussed bellow. During this re-fitting, missing hits that
the track finder was unable to associate to the track are searched for.
Frequently, hits in the final half turn are missed by the track finder but can
be added in this process, resulting in improved momentum resolution. The
deviation of the target from planarity is modelled by a triangle mesh in the
fitter. The track is first propagated to a virtual plane, a few mm in front of
the real target, and then to the nearest triangle of the mesh, where the
positron’s kinematics at the emission is reconstructed. The target crossing
point ($\xpos,\ypos,\zpos$) is used as a candidate of muon decay point to
correlate the track direction with the first interaction point of the -ray
observed in the LXe detector. The length of the trajectory from the target to
the matched pTC counter is converted to the time of flight and subtracted from
the pTC hit time to determine the positron emission time at the target
$\tpositron$. During the tracking process, the DOCA of each hit is iteratively
refined. The DOCA value is estimated at the beginning using angle-averaged TXY
tables extracted using Garfield++ simulations, which determine the drift
velocity and lines taking into account the electric and magnetic fields, the
ionisation pattern, the diffusion process, etc. The Garfield++ simulations are
performed in two-dimensions in $(r,\phi)$ sampling at slices at fixed
$z$-coordinates for each layer and then are interpolated to take into account
the longitudinal change in cell size and shape as well as the effect of the
magnetic field. Once the track is reconstructed, the DOCA is recalculated
taking into account the cell crossing angle of the track. This DOCA estimation
is biassed by the low cluster density; the small number of clusters within the
cell overestimates the hit distance from the wire. Another DOCA estimate less
prone to intrinsic biases is obtained by neural network approaches [DocaUCI].
The networks take as input various hit properties (wire and plane number,
charge, timing, hit coordinates, track angle, $T_{0}$ etc.) and waveforms (for
the CNN model) and train on the fitted track DOCA (made using the Garfield++
TXY tables) as an estimator of the true DOCA to create a “data-driven” TXY
table. This TXY optimally accounts for ionisation statistics biases, removes
errors from the simulated TXY, and uses information from all ionisation
clusters. The distribution of DOCA residuals for the conventional and two
neural network based approaches, a dense neural network (DNN) and a CNN, are
compared in dch:docaresiduals. The best result is obtained with the CNN that
processes the waveforms from all ionisation clusters. The main improvement
from the neural network is the suppression of the positive right tail
presumably from the suppression of the ionisation statistics bias. This
improves the positron kinematics by ∼10 as shown in sec:dchperformreso.
Therefore, the DOCA is finally updated with the CNN method in the re-fit
process.

[width=1angle=0] DocaOnly.pdf

Figure : The distribution of the DOCA residuals (hit DOCA $-$ track DOCA)
estimated with the conventional (NML) and the two (DNN and CNN) neural network
based approaches. The curves are fitted double Gaussian functions with a core
sigma of 114 (119) and a tail sigma of 236 (259) with a core fraction of 0.59
(0.70) for the neural network (conventional) approaches.

##  Alignment

###  Wire alignment

The global position of CDCH was measured in both $2021$ and $2022$ with an
optical survey. The information on the relative wire-by-wire alignment was
extracted from several measurements during the construction of the chamber.
However, after the complete reconstruction of the track, the distributions of
the difference between the nominal position of the wires and the wire position
calculated by the tracking algorithm using the measured DOCA show systematic
deviations of the order of ∼100, which worsen the tracking resolutions. The
deviations were resolved by implementing a track-based alignment. Two types of
tracks can be used to improve relative alignment: Michel positrons and cosmic
rays. The former have the advantage of being collected and reconstructed
during normal data taking, but also the disadvantage of requiring a more
complex reconstruction and being more affected by possible uncertainties in
the knowledge of the magnetic field. The latter have the advantage of being
straight tracks, but the disadvantage of requiring dedicated data taking
periods and a different coverage of the tracking volume. Currently, alignment
to cosmic rays is being investigated; in the following we only present the
alignment based on Michel positrons. The alignment procedure is an iterative
adjustment of the wire coordinates driven by the mean residual. The residual
$r$ is fitted, wire by wire, as a function of the longitudinal position along
the wire ($z$-coordinate) with a parabolic shape: r ( z ) = p_0 + p_1 z + p_2
[ ( 2zL )^2 - 1 ]. The parameter $p_{0}$ corresponds to a global wire
displacement, the linear term results from the inclination of the wire to the
chamber axis and the quadratic term takes into account the wire sagitta, due
to the electrostatic and gravitational forces acting on the wire. $p_{2}$ is
the (absolute) maximum value of the sagitta, which can reach ∼100. The number
of events required for high quality alignment of a single wire depends
strongly on the position of the wire in the chamber due to the trigger
criteria to select candidate events. Furthermore, the alignment algorithm is
almost insensitive to alignment errors when the track direction is parallel to
the misalignment vector. This is because the DOCA estimate does not change
when the wire position is shifted along the track direction. Therefore,
chamber sectors crossed by a large variety of angles are aligned more
efficiently than those crossed with a small scatter of angles. The alignment
procedure was based on 17 million hits, with an average 40 hits per track. We
required ¿5000 hits for each wire to be aligned; 152 wires were excluded from
this alignment and the survey results are used. They are at the edges of each
layer or with electronic problems. With larger data statistics, the alignment
for the edge region can be improved. One way to increase the statistics for
the region is to use tracks outside the time window that do not cross the pTC
but cross more frequently the CDCH regions outside or at the boundaries of the
MEG II geometric and trigger acceptance. These tracks are collected
accidentally when another event that meets the trigger conditions opens the
WaveDREAM time window. dch:xalign shows the final mean residuals in the $x$\-
(top plot) and $y$-coordinates (bottom plot) compared to the same residuals
obtained using the survey-based alignment.

[width=1angle=0] WireMean.pdf

Figure : The mean residuals in the $x$\- (top plot) and in the $y$-coordinates
(bottom plot). The black curves are obtained after 14 alignment iterations,
the green curves using the alignment based on the survey. The curves are the
results of Gaussian fits.

The spreads of the Gaussian cores of the two distributions are
$\sigma^{\rm{sur}}_{x}=\SI{22}{\micro\meter}$ and
$\sigma_{y}^{\rm{sur}}=\SI{35}{\micro\meter}$ when the survey-based alignment
is used and decrease to $\sigma_{x,y}^{\rm{trk}}<\SI{5}{\micro\meter}$ for
both coordinates after the alignment procedure. The fit parameters $p_{0}$
(global displacement) and $p_{2}$ (sagitta) are strongly correlated, as
expected. The remaining sagitta is $\sigma_{p_{2}}\sim\SI{13}{\micro\meter}$
and the wires with the largest sagittas are concentrated in the peripheral
chamber sectors, crossed by fewer tracks than central sectors. The
distributions of the displacements of the wire centres with respect to those
in the survey have $\sigma_{x}\sim\SI{100}{\micro\meter}$,
$\sigma_{y}\sim\SI{80}{\micro\meter}$ and
$\sigma_{z}\sim\SI{30}{\micro\meter}$. The distributions of the differences of
the wire angles $\Theta$ and $\Phi$ with respect to the survey have
$\sigma_{\Theta}\sim\SI{0.1}{\milli\radian}$ and
$\sigma_{\Phi}\sim\SI{1.1}{\milli\radian}$, respectively. For the $\Phi$ angle
the mean value is 1.05, which means that the alignment requires a global
azimuthal rotation, while the required polar rotation is minimal, i.e. ¡0.1.
The remaining error in wire-alignment is estimated to be between 2 at the
centre and 15 at the end plates. Some correlations between the wire centre
translation or the wire rotation and the wire number layer-by-layer have been
observed and are currently being investigated. The effects of the alignment
procedure on the resolutions of the kinematic variables are evaluated using
the double-turn method; see sec:dchperformreso for the details. The
distributions of $\zpos$ and $\phie$ are shown in dch:DTalign.

[width=1angle=0] DTSurveyUpdated.pdf

Figure : The double-turn analysis results for $\zpos$ and $\phie$. The blue
curves are obtained using the survey-based alignment, the black and the red
curves using the alignment algorithm results after $5$ and $12$ steps,
respectively.

The advantage of the alignment procedure is highlighted by the strong
reduction of the systematic biases and by the narrowing of the distributions.

###  Relative alignment between magnetic field and CDCH

Due to the gradient magnetic field, misalignment between the CDCH and the
magnet results in a non-uniformity of the energy scale with respect to the
positron emission angle. In the track reconstruction, we use a 3D map of the
magnetic field calculated using a finite element method based on the measured
coil dimensions and taking into account the thermal shrinkage of the coil. The
angular dependence is minimised by shifting the calculated magnetic field by
$(\SI{100}{\micro\meter},\SI{700}{\micro\meter},\SI{300}{\micro\meter})$ from
the nominal position, with an estimated alignment accuracy of [range-
phrase=–,range-units = single]100200.

[width=]BFieldAlignment.pdf

Figure : The angular dependence of the positron energy scale versus the
angular kinematic variables before and after alignment. The offset on the
$y$-axis is the difference of the measured value with the expected value of
the Michel edge. The three superimposed plots show the effects of shifting the
magnetic field by 1 in $x$, $y$, and $z$.

After the shift, there is no more bias as shown in fig:BFieldAlignment, and
the scatter of the order of ∼10 is negligible compared to the energy
resolution.

###  Relative alignment between target and CDCH

The precise alignment of the muon stopping target with respect to the
spectrometer is crucial in determining the positron emission angle and
position as discussed in detail in sec:target. We use the holes on the target
to align the target with respect to the CDCH. The six holes are visible in the
reconstructed track distribution in fig:PositronTGTDist2D, from which the
$y$\- and $z$-coordinates of the holes (thus those of the target) can be
easily estimated.

[width=]PositronDistributionTGT.pdf

Figure :  Distribution of the reconstructed positron origin ($\ypos$,$\zpos$)
on the target.

The $x$-alignment exploits the dependence of the estimated $y$-position on
$\phie$, as shown in fig:TGTXYAlignment.

[width=]TGTXYAlignment.pdf

Figure : Estimated hole $y$ position as a function of $\phie$. The slope
parameter in the fit gives the residual $x$ misalignment while the offset
parameter gives the $y$ offset of the target.

Combined with the photogrammetric method using the photo-camera system, which
corrects for temporal variation in the target positions, the hole analysis
yields the hole-by-hole residual misalignment. The precision of the hole-by-
hole position estimation is limited both by the statistical uncertainty of the
positron tracks and by the systematic uncertainties of the method. The
statistical uncertainty is [range-phrase=–,range-units = single]100200 in each
coordinate. All the systematic uncertainties originate from the non-uniform
vertex position distribution due to the beam profile, which biases the $(y,z)$
distribution produced by the holes to be closer to the beam centre. This
effect is corrected in the hole position estimation, but the uncertainties in
the correction factor dominates the hole-by-hole systematic uncertainties of
$(\SI{50}{\micro\meter},\SI{100}{\micro\meter},\SI{200}{\micro\meter})$. The
global target geometry, both the rotational and translational parameters, is
finally fitted to the residuals of the holes, yielding a translation of
$(\SI{73}{\micro\metre},\SI{799}{\micro\metre},\SI{437}{\micro\meter})$ and a
rotation of 10mrad mainly along the minor axis of the target from the results
of the photogrammetric method. The accuracy, including systematic
uncertainties, is 100 for the translation in each axis and 1.4mrad (6mrad) for
the rotation along the minor (major) axis.

###  Relative alignment between LXe detector and CDCH

The alignment procedure described in sec:dchcalib is insensitive to a global
misalignment between CDCH and other subdetectors. Among them, the LXe detector
is especially important since it provides information about the -ray
direction. Events that result with hits in both the CDCH and the LXe detector
are used to compare information from both. The appropriate event category is
the cosmic rays, which can release energy in the LXe detector and then (or
vice versa, depending on the trajectory) pass through the CDCH, producing hits
on multiple wires. The cosmic ray events are collected with a specific
triggering scheme and with the magnetic field turned off, to obtain straight
tracks that are easier to reconstruct and fit. The idea is to select cosmic
ray tracks that cross the LXe detector inner face almost perpendicularly and
to determine the coordinates of this crossing point using the LXe detector and
the CDCH information independently. On the CDCH side, a method based on the
Legendre transform [fruhwirth2021pattern, Primor_2007] is used for finding and
fitting the cosmic ray tracks. The fitted tracks are then extrapolated to the
inner face of the LXe detector. On the LXe detector side, the usual position
reconstruction algorithm (see subsubsec:LXec-position) is applied. The
selection of cosmic ray tracks almost perpendicular to the detector entrance
face is motivated by the fact that the continuous stream of energy release by
such cosmic rays is seen from the photosensors on the inner face as a point-
like energy deposit, whose position is more reliably reconstructed.
fig:dchxecali shows the LXe–CDCH difference in the reconstructed
$z$-coordinate $\mathrm{\Delta}z=z_{\mathrm{LXe}}-z_{\mathrm{CDCH}}$.

[width=1angle=0] zcdch-zLXe_fit.pdf

Figure : Difference in the reconstructed $z$-coordinate on the entrance face
of the LXe detector for cosmic rays performed by using the LXe detector and
CDCH information independently. The red curve is a Gaussian fit.

The distribution has a Gaussian shape, centred at
$\mathrm{\Delta}z=\SI{-1.0\pm 0.8}{\milli\meter}$. This result shows that a
small global shift can be applied to better align the two detectors even if it
is compatible with zero. We do not observe any significant deviation in the
difference in $\phi$ within its uncertainty, which is more subject to
systematic errors in the measurement because of non-uniform distribution of
the cosmic rays with respect to $\phi$. The relative alignment between the LXe
detector and the CDCH, combined with the relative alignment between the CDCH,
the magnetic field and the target, is sufficient to determine the relative
positron–photon angle, which is independent of the absolute position of the
detectors. For practical reasons, we need to define an absolute alignment.
Reference marks on the COBRA cryostat are used to define the absolute MEG II
reference frame. The position of the CDCH in this reference frame is
determined by an optical survey with an estimated resolution of a few hundred
micrometres, while the absolute position of the LXe detector is measured with
uncertainties of ∼600 as discussed in subsubsec:LXec-position. On this basis,
we decided to use the CDCH position as a reference and apply global shifts to
the LXe detector, the magnetic field and the target, according to the results
of the data analysis for the relative alignments. The shifts of the magnetic
field and target, reported in sec:bfield-cdch and sec:target-cdch, are quite
similar in size and direction, which could indicate a bias in the CDCH survey.
The shift in the $z$ direction is also confirmed by the relative alignment
between the CDCH and the LXe detector, but with a large uncertainty. For these
reasons, the shift to be applied to the LXe detector was determined by
combining all available information on a possible bias in the CDCH survey. We
obtained
$(\SI{90}{\micro\metre},\SI{800}{\micro\metre},\SI{630}{\micro\metre})$, with
a systematic uncertainty of 1 assigned in all directions.

##  Performance

In this section, we discuss the performance of the CDCH in terms of resolution
and tracking efficiency. A more extensive discussion is given in a dedicated
paper [CDCHPerfor]. We recall that the main motivation for building the CDCH
was the unsatisfactory performance of the MEG drift chamber system in terms of
angular and energy resolutions and tracking efficiency due to its segmented
structure.

###  Resolutions

One technique for evaluating the resolutions of the kinematic variables is the
“double-turn” method, which was already used in the MEG
experiment[baldini_2016]. In the MEG II experiment, ∼15 of the positron tracks
cross the chamber volume five times, passing through 9 x 5 sense wire layers.
In these tracks, one can identify two separate track segments (“turns”), the
first of which corresponds to two chamber crossings and the second to three
chamber crossings. Both track segments are independently fitted and propagated
to a plane parallel to the target between the two turns and the distributions
of the differences between the kinematic variables reconstructed by the two
segments are compared. dch:doubleturn shows the double-turn distributions for
$\zpos$; the shapes of the $\ypos$, $\phie$ and $\thetae$ distributions are
similar.

[width=1angle=0] ZDifferenceNN.pdf

Figure : The distribution of the double-turn difference for $\zpos$ obtained
using the conventional (NML) and neural network (DNN and CNN) approaches to
DOCA reconstruction.

The three distributions of dch:doubleturn are based on the conventional and
the neural network based DOCA reconstructions described in sec:track_rec. The
neural network approaches improve the width of the distribution by [range-
phrase=–,range-units = single]111. These distributions do not directly
represent the effective resolutions of the CDCH, but a combination in
quadrature of the resolutions of the first and second turn. The following
corrections are required to convert the double-turn results into reliable
estimates of the real resolutions. One is to correct for the bias in the
double-turn evaluation. Another is to correct for the difference between
Michel positrons and the signal positrons. Both correction factors were
evaluated via MC simulations. Good similarity was obtained between the double-
turn results on data and those on the Michel MC simulation, showing that we
have a solid knowledge of the response of the CDCH. In addition, we can
correct for the correlation effect between the variables for the detection of
events as discussed in sec:sensitivity. The correlation parameters are also
evaluated with the double turn analysis. The effective resolutions after the
corrections are summarized in dch:resolutions.

Table : Effective resolutions (core $\sigma$s) for experimental data obtained
by combining the double-turn results on data with the MC correction factors.

$\yposerr\left(\unit{mm}\right)$ | $\zposerr\left(\unit{mm}\right)$ | $\phieerr\left(\unit{\milli\radian}\right)$ | $\thetaeerr\left(\unit{\milli\radian}\right)$
---|---|---|---
$0.74$ | $2.0$ | $4.1$ | $7.2$

The energy resolution for experimental data can be measured by fitting the
convolution of the theoretical Michel spectrum multiplied by an efficiency
function that takes into account the high-energy selection of the spectrum by
spectrometer acceptance, and a resolution function formed by the sum of three
Gaussian functions to the measured Michel positron spectrum. The result is
shown in dch:michel in both logarithmic (a) and linear (b) scales.

[width=1angle=0] MichelEdge.pdf

Figure : Fit of the Michel positron spectrum in logarithmic (a) and linear (b)
scales. The black histogram is the experimental distribution, the blue curve
is the sum of three Gaussian functions describing the resolution around the
signal region and the red curve is the fitted function, obtained by convolving
the theoretical spectrum with the resolution and acceptance curves. (c) The
acceptance curve of the spectrometer modelled with an error function.

The $\sigma$ of the core Gaussian function, which accounts for ∼67 of the
integral of the resolution curve, is $\epositronerr=\SI{91}{\keV}$, better by
40 of the value quoted in the MEG II proposal. The corresponding value for the
MEG experiment was $\epositronerr=\SI{320}{\keV}$.

###  Efficiency

dch:cdcheff shows the CDCH tracking efficiency for signal positrons
$\varepsilon_{\positron,{\rm CDCH}}$ versus $R_{\muon}$. The efficiency is
defined as the ratio of the number of reconstructed positrons in the signal
energy region to the number of emitted positrons in the direction opposite to
the LXe acceptance region and detected by the pTC. Since the efficiency
depends on the positron energy, it is measured for energies just below the
signal energy and its value is extrapolated to the signal energy. The sample
used for the calculation is taken through a minimum bias trigger that requires
only one hit on the pTC.

[width=1angle=0] CDCHEff.pdf

Figure : CDCH tracking efficiency as a function of $R_{\muon}$ for signal
positrons. The blue dotted line is the design value.

It is expected that the efficiency decreases with $R_{\muon}$, because as
$R_{\muon}$ increases, the probability of pile-up also increases, making the
track-finder algorithm less effective at identifying hits belonging to
individual tracks. Nevertheless, the reduction in efficiency from the lowest
value of $R_{\muon}=\SI{2e7}{\per\second}$ to the highest
$R_{\muon}=\SI{5e7}{\per\second}$ is moderate ¡ 15. The blue dotted line
represents the design value, which is almost reached at the smallest value of
$R_{\muon}$ and is not far from being reached also at the highest. As a
reference we measure $\varepsilon_{\positron,{\rm CDCH}}\sim\SI{74}{\percent}$
at $R_{\muon}=\SI{3e7}{\per\second}$. The CDCH is highly transparent towards
pTC and allows a high signal detection efficiency for pTC,
$\varepsilon_{\positron,\mathrm{pTC}}=\SI{91(2)}{\percent}$ (discussed in
sec:ptc_performance), which is twice higher than with the MEG drift chamber.
The MEG experiment used $R_{\muon}=\SI{3e7}{\per\second}$, with a positron
efficiency of $\varepsilon_{\positron}\sim\SI{30}{\percent}$. At the same
$R_{\muon}$, the positron efficiency is
$\varepsilon_{\positron}=\varepsilon_{\positron,\mathrm{pTC}}\times\varepsilon_{\positron,{\rm
CDCH}}\sim\SI{67}{\percent}$.

##  Back-up chamber

The detailed analysis of the problems of wire breakage stimulated a long R&D
effort to explore possible alternatives to the silver-coated aluminium wires,
which were found to be fragile and easily damaged by ambient humidity, with
the aim of designing and building a new chamber (CDCH2) without the above
problems. Several possible types of wire were explored, with different coating
materials (gold and nickel), without coating etc. They were studied from
aspects: sensitivity to corrosion processes, mechanical strength, soldering
technique on PCBs, etc. We developed a system for measuring wire tension
[tensionsystem], based on the resonant frequency method, as well as with
wiring and assembly stations [chiarello_wiring]. The final choice was a wire
made of pure aluminium with a diameter of 50, which is almost insensitive to
corrosion and can be efficiently and solidly fixed on PCBs by a combination of
soldering and gluing with suitable chemical products. The CDCH2 is scheduled
for completion by the end of $2023$ and delivery to PSI in spring 2024. At
that time, the collaboration will decide whether and when to replace the
current chamber.