6.8cm Erratum, counterexample and an additional revealing poll step for a
result of “Analysis of direct searches for discontinuous functions”
NovembreNovember 2022 XX C. Audet, P.-Y. Bouchet, L. Bourdin P.-Y. Bouchet
Hamel, Benoit, Karine Hébert (2022). “Un exemple de citation”, _Journal of
Journals_ , vol. X issue Y, p. n-mhttps://www.gerad.ca/fr Internet Appendix
NovembreNovember2022

C. Audet LABEL:affil:gerad P.-Y. Bouchet LABEL:affil:gerad L. Bourdin
LABEL:affil:unilim

affil:geradGERAD, Montréal (Qc), Canada, H3T 1J4 affil:unilimXLIM Research
Institute, UMR CNRS 7252, University of Limoges, France

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract This note provides a counterexample to a theorem announced in the
last part of the paper Analysis of direct searches for discontinuous
functions, Mathematical Programming Vol. 133, pp. 299–325, 2012. The
counterexample involves an objective function $f:\mathbb{R}\to\mathbb{R}$
which satisfies all the assumptions required by the theorem but contradicts
some of its conclusions. A corollary of this theorem is also affected by this
counterexample. The main flaw revealed by the counterexample is the
possibility that a directional direct search method (dDSM) generates a
sequence of trial points $(x_{k})_{k}$ converging to a point $x_{*}$ where $f$
is discontinuous and whose objective function value $f(x_{*})$ is strictly
less than $\lim_{k\to\infty}f(x_{k})$. Moreover the dDSM generates no trial
point in one of the two branches of $f$ near $x_{*}$. This note also
investigates the proof of the theorem to highlight the inexact statements in
the original paper. Finally this work concludes with a modification of the
dDSM that allows to recover the properties broken by the counterexample.

#### Keywords:

Discontinuous optimization, direct search methods, generalized derivatives.

Work of the first author is supported by NSERC Canada Discovery Grant
2020–04448.

##  1 Introduction

Derivative-free optimization is the mathematical field that focuses on
nonsmooth optimization problems with lack of derivatives (see, e.g., [1, 3,
4]). The generic problem is

$\underset{x\in\mathbb{R}^{n}}{\operatorname*{minimize}}~{}f(x)\quad s.t.\quad
x\in\Omega,$

where $f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{+\infty\\}$ is nonsmooth and the
feasible region $\Omega$ is a nonempty subset of $\mathbb{R}^{n}$. A class of
derivative-free algorithms well studied in the literature is the so-called
directional Direct Search Methods (dDSM). When $f$ has bounded level sets, a
dDSM generates a so-called refined point $x_{*}$ and a set $D$ of so-called
refining unitary directions. In addition, when $f$ is locally Lipschitz-
continuous at $x_{*}$, it is known that the Clarke generalized derivative
$f^{\circ}_{C}$ is nonnegative at $x_{*}$ in each refining direction $d\in
D\cap H_{\Omega}(x_{*})$, where $H_{\Omega}(x_{*})$ stands for the
hypertangent cone to $\Omega$ at $x_{*}$. Relaxing this last continuity
assumption, and thus minimizing a possibly discontinuous objective function,
is challenging. The paper [5] settles several results about the behavior of a
dDSM in that context.

The main results of this paper (precisely [5, Theorem 3.1] and [5, Theorem
3.2]) show that, if a dDSM generates a refining subsequence satisfying
$f(x_{k})\to f(x_{*})$, then the Rockafellar generalized derivative
$f^{\circ}_{R}$ (applicable to discontinuous functions, as opposed to the
Clarke generalized derivative) is nonnegative at $x_{*}$ in all directions
$d\in D\cap T_{\Omega}(x_{*})$, where $T_{\Omega}(x_{*})$ stands for the
tangent cone to $\Omega$ at $x_{*}$. These important results extend the
previous analyses to discontinuous functions.

At the end of the paper, two additional results (precisely [5, Theorem 4.1]
and [5, Corollary 4.1]) attempt to derive sufficient conditions on the
refining subsequence to guarantee that $f(x_{k})\to f(x_{*})$. Unfortunately,
to the best of our knowledge, one of the statements from [5, Theorem 4.1] is
incorrect, which also affects the validity of [5, Corollary 4.1]. The present
work focuses on this inexact statement.

A dDSM iteratively repeats a two-steps process formalized in [5, Algorithm
2.1]. First, an optional Search step evaluates some points anywhere in
$\mathbb{R}^{n}$. Second, a mandatory Poll step samples $f$ around the current
incumbent solution $x_{k}$ according to some chosen step parameter
$\alpha_{k}>0$ and set $D_{k}$ of directions. For a given sequence
$(x_{k};\alpha_{k};D_{k})_{k\in\mathbb{N}}$, a so-called refining subsequence
(indexed by $K\subseteq\mathbb{N}$) to a refined point $x_{*}$ and a set $D$
of refining unitary directions mean that the three following conditions are
satisfied: (i) each iteration indexed by $k\in K$ is unsuccessful (that is,
all Search and Poll trial points fail to dominate the incumbent solution);
(ii) $(x_{k})_{k\in K}$ converges to $x_{*}$ and $(\alpha_{k})_{k\in K}$
converges to $0$; (iii) any direction in $D$ is the limit of a subsequence of
refining normalized directions $(\frac{d_{k}}{\left\lVert
d_{k}\right\rVert})_{k\in K_{1}}$ where $0\neq d_{k}\in D_{k}$ for all $k\in
K_{1}\subseteq K$. Following this terminology, [5, Theorem 4.1] and [5,
Corollary 4.1], plus [5, Assumption 2.1] and [5, Assumption 4.1] they rely on,
are stated as follows.

> [5, Assumption 2.1] The level set $L(x_{0})=\\{x\in\Omega:f(x)\leq
> f(x_{0})\\}$ is bounded. The function $f$ is bounded below on $L(x_{0})$.

> [5, Assumption 4.1] The function $f$ is such that there exists a
> neighborhood $B$ of $x_{*}$ (a limit point of a refining subsequence) which
> admits a finite partition $B=\cup_{i=1}^{n_{B}}B_{i}$ such that, for all
> $i\in\\{1,\dots,n_{B}\\}$,
>
>   1. 1.
>
> $\mathrm{int}(B_{i})\neq\emptyset$,
>
>   2. 2.
>
> $\mathrm{cl}(B_{i})$ has the exterior cone property (see [5, Definition
> 4.1]),
>
>   3. 3.
>
> $f$ is Lipschitz-continuous in $\mathrm{int}(B_{i})$ and can be continuously
> extended from $\mathrm{int}(B_{i})$ to $\partial B_{i}$.
>
>

> [5, Theorem 4.1] Consider a refining subsequence $(x_{k})_{k\in K}$
> converging to $x_{*}\in\Omega$ (and note that [5, Assumption 2.1] is
> required for the existence of such a subsequence). Assume that $f$ is lower
> semicontinuous at $x_{*}$ and satisfies [5, Assumption 4.1]. Let the sets of
> refining directions for $x_{*}$ corresponding to any infinite subsequence of
> $K$ be dense in the unit sphere.
>
> If $x_{*}$ belongs to the interior of a partition set in
> $\\{B_{1},\dots,B_{n_{B}}\\}$, then $f^{\circ}_{C}(x_{*};v)\geq 0$ for all
> refining directions $v\in T_{\Omega}(x_{*})$ (assuming here also that
> $H_{\Omega}(x_{*})$ is nonempty).
>
> Otherwise, there exists a subsequence $K^{\prime}\subset K$ and a partition
> set $B^{\prime}\in\\{B_{1},\dots,B_{n_{B}}\\}$ such that (i) $(x_{k})_{k\in
> K}\subset\mathrm{cl}(B^{\prime})$, (ii) there is an infinite number of poll
> points, corresponding to iterates in $K$, in $\mathrm{int}(B^{\prime})$, and
> (iii) there is an infinite number of poll points, corresponding to iterates
> in $K$, in $\mathbb{R}^{n}\setminus\mathrm{cl}(B^{\prime})$.

> [5, Corollary 4.1] Under [5, Assumption 2.1] and the assumptions of [5,
> Theorem 4.1] and when $n_{B}=2$, there exists a subsequence $K_{*}\subset K$
> and a partition set $B_{*}\in\\{B_{1},B_{2}\\}$ such that, when $x_{*}$ is
> in the border of the two partition sets,
>
>   1. 1.
>
> $B_{*}$ satisfies the properties stated for $B^{\prime}$ in [5, Theorem
> 4.1],
>
>   2. 2.
>
> $\lim_{k\in K_{*}}f(x_{k})=f(x_{*})$.
>
>

The present note provides in Section 2 a counterexample to [5, Theorem 4.1]
and [5, Corollary 4.1]. More precisely, Section 2.1 constructs an
unconstrained optimization problem with a single-variable real-valued
objective function $f:\mathbb{R}\to\mathbb{R}$, Section 2.2 proposes a
specific dDSM instance, and Section 2.3 shows that the counterexample
satisfies all the assumptions required in [5, Theorem 4.1] and [5, Corollary
4.1] but contradicts some of their conclusions. An analysis of the proofs of
[5, Theorem 4.1] and [5, Corollary 4.1] is provided in Section 3 to identify
the incorrect steps.

A second objective of the present work is to propose in Section 4 an
adaptation of [5, Algorithm 2.1] which recovers the properties broken by the
counterexample. Section 4.1 introduces this adaptation, relying on a strategy
proposed in [2]. It is based on the evaluation of a few additional points at
each iteration to asymptotically evaluate a dense set of points around
$x_{*}$, thus it evaluates points in each branch of $f$ near $x_{*}$ and
reveals any discontinuity. This leads to Theorem 1, stated and proved in
Section 4.2, which claims results similar to [5, Corollary 4.1] under an
assumption similar to [5, Assumption 4.1] but without requiring $n_{B}=2$.

##  2 A single-variable unconstrained counterexample

The single-variable real-valued objective function $f:\mathbb{R}\to\mathbb{R}$
in this counterexample is illustrated in the left part of Figure 1. It
possesses local minima at $x=\frac{5}{4}\times 2^{\ell}$ for each integer
$\ell\in\mathbb{Z}$. The dDSM parameters are chosen so that each incumbent
solution lies at one of these local minimizers and the corresponding step
parameters are sufficiently small so that each Poll step results in a failure.
A Search step is added only at odd-index iterations, generating a trial point
at the next local minimizer to the left of the current incumbent solution. The
poll size parameter is halved at unsuccessful iterations, and remains constant
at successful ones. Hence, by construction, all even-index iterations fail,
while all odd-index iterations succeed, and the sequence of the incumbent
solutions converges to the origin.

Figure 1: The function $f$ (left plot), and the polynomial $p$ (right plot)
used to construct $f$.

###  2.1 Construction of the objective function

This section describes the construction of the discontinuous objective
function $f:\mathbb{R}\to\mathbb{R}$ illustrated in the left plot of Figure 1.
On the closed interval $\mathbb{R}^{-}$, it is defined as the convex quadratic
function given by $f(x)=(x+1)^{2}-2$ for all $x\in\mathbb{R}^{-}$. On the open
interval $\mathbb{R}^{+}_{*}$, it is constructed from the polynomial
$p:[1,2]\to\mathbb{R}$ illustrated in the right plot of Figure 1 and given by

$p(x)=-18+60x-69x^{2}+34x^{3}-6x^{4}~{}\mbox{ for all}~{}x\in[1,2].$

It satisfies $p(1)=1$, $p(2)=2$,
$p^{\prime}(1)=p^{\prime}\left(\frac{5}{4}\right)=p^{\prime}(2)=0$,
$p^{\prime\prime}(1)<0,~{}p^{\prime\prime}(2)<0$,
$p^{\prime\prime}\left(\frac{5}{4}\right)>0$. Hence $p$ has a unique local
minimizer at $x=\frac{5}{4}$ and two local maximizers at the endpoints of the
interval $[1,2]$. The objective function $f$ is constructed on
$\mathbb{R}^{+}_{*}$ by scaling the polynomial $p$. Precisely, $f$ equals to
$2p$ on the interval $2\times[1,2[$, to $p$ on $[1,2[$, to $\frac{p}{2}$ on
$\frac{1}{2}\times[1,2[$ and, in general, to $2^{\ell}p$ on the interval
$2^{\ell}\times[1,2[$ with $\ell\in\mathbb{Z}$. Hence, on each interval
$2^{\ell}\times[1,2]$ with $\ell\in\mathbb{Z}$, $f$ has exactly three
stationary points (given by a unique local minimizer at $x=\frac{5}{4}\times
2^{\ell}$ and two local maximizers at the endpoints).

To summarize, consider the unconstrained optimization problem
($\Omega=\mathbb{R}$) given by

$\underset{x~{}\in~{}\mathbb{R}}{\operatorname*{minimize}}~{}f(x),$

where

$f(x)\ =\ \left\\{\begin{array}[]{ll}(x+1)^{2}-2&\mbox{if}~{}x\leq
0,\\\\[5.0pt] 2^{\ell}p\left(\dfrac{x}{2^{\ell}}\right)&\mbox{if}~{}x\in
2^{\ell}\times[1,2[\mbox{ with }\ell\in\mathbb{Z},\\\
\end{array}\right.~{}\mbox{ for all}~{}x\in\mathbb{R}.$

######  Proposition 1.

The function $f$ satisfies [5, Assumption 2.1] and [5, Assumption 4.1] with
$x_{*}=0$.

###### Proof.

[5, Assumption 2.1] is satisfied and [5, Assumption 4.1] holds for $x_{*}=0$.
Indeed, $x_{*}=0$ has a neighborhood $B={]}-2,2[$ that can be partitioned as
$B=B_{1}\cup B_{2}$ with $B_{1}={]}-2,0]$ and $B_{2}={]}0,2[$. Then, for both
$i\in\\{1,2\\}$, $\mathrm{int}(B_{i})\neq\emptyset$, $f$ is Lipschitz-
continuous on $B_{i}$ and can be continuously extended to $\partial B_{i}$,
and $\mathrm{cl}(B_{i})$ has the exterior cone property. ∎

###  2.2 Construction of the dDSM instance

This section describes a dDSM instance to minimize $f$. The starting point is
$x_{0}=\frac{5}{4}$ and the initial step parameter is
$\alpha_{0}=\frac{1}{4}$. The Poll sets of directions are $D_{k}=\\{-1,1\\}$
for all $k\in\mathbb{N}$. The step parameter is updated as
$\alpha_{k+1}=\alpha_{k}$ if the iteration $k$ finds a better point than the
incumbent solution $x_{k}$, and $\alpha_{k+1}=\frac{\alpha_{k}}{2}$ otherwise.
The Search step generates no trial point when the iteration number $k$ is
even, and returns a single trial point $t_{k}=x_{k}-5\alpha_{k}$ when $k$ is
odd.

These parameters are chosen so that, at each even iteration $k=2q$ with
$q\in\mathbb{N}$, the incumbent solution $x_{2q}$ is the local minimum of $f$
on the interval $2^{-q}\times[1,2]$, the Poll step fails because all its trial
points lie in that interval, and the Search step running on the following odd
iteration $2q+1$ generates the trial point corresponding to the next local
minimizer on the left of $x_{2q}$ and located on the interval
$2^{-(q+1)}\times[1,2]$. Hence the sequence of incumbent solutions
$(x_{k})_{k\in\mathbb{N}}$ converges to the origin, the sequence of step
parameters $(\alpha_{k})_{k\in\mathbb{N}}$ converges to $0$, and both
directions $d_{1}=-1$ and $d_{2}=1$ are considered at each even iteration.
These claims are formalized in the next result.

######  Lemma 1.

The sequences of incumbent solutions $(x_{k})_{k}$ and of step parameters
$(\alpha_{k})_{k}$ satisfy

$\forall q\in\mathbb{N},\quad x_{2q}=\frac{5}{4}\times
2^{-q}=x_{2q+1},\quad\alpha_{2q}=\frac{1}{4}\times 2^{-q}=2\alpha_{2q+1}.$

At each even iteration $k=2q$ with $q\in\mathbb{N}$, the Poll step fails after
evaluating both trial points $x_{2q}+\alpha_{2q}d_{1}$ and
$x_{2q}+\alpha_{2q}d_{2}$.

###### Proof.

The proof is done by induction over $q\in\mathbb{N}$.

Iteration $0$ starts with $x_{0}=\frac{5}{4}$ and $\alpha_{0}=\frac{1}{4}$. It
is an even iteration and, therefore, there is no Search step. The incumbent
solution $x_{0}$ is a local minimizer on the interval
$[x_{0}-\alpha_{0},x_{0}+\alpha_{0}]=[1,\frac{3}{2}]$ and, therefore, the two
candidates $x_{0}+\alpha_{0}d_{1}$ and $x_{0}+\alpha_{0}d_{2}$ generated by
the Poll step fail to dominate $x_{0}$. Thus iteration $0$ is unsuccessful,
hence $x_{1}=x_{0}=\frac{5}{4}$ and
$\alpha_{1}=\frac{1}{2}\alpha_{0}=\frac{1}{4}\times 2^{-1}$. Then the
induction statement holds for $q=0$.

Now consider $q\in\mathbb{N}_{*}$ and assume that the induction statement
holds for $q-1$. This implies that iteration $2q-1$ starts with
$x_{2q-1}=x_{2(q-1)+1}=\frac{5}{4}\times 2^{-(q-1)}$ and
$\alpha_{2q-1}=\alpha_{2(q-1)+1}=\frac{1}{4}\times 2^{-q}$. It is an odd
iteration, so the Search step generates the trial point
$t_{2q-1}=x_{2q-1}-5\alpha_{2q-1}=\frac{5}{4}\times
2^{-(q-1)}-\frac{5}{4}\times 2^{-q}=\frac{5}{4}\times 2^{-q}$. This trial
point satisfies $f(t_{2q-1})=\frac{f(x_{2q-1})}{2}<f(x_{2q-1})$. Hence
iteration $2q-1$ is a success and iteration $2q$ starts with
$x_{2q}=t_{2q-1}=\frac{5}{4}\times 2^{-q}$ and
$\alpha_{2q}=\alpha_{2q-1}=\frac{1}{4}\times 2^{-q}$. This is an even
iteration and, therefore, there is no Search step. The incumbent solution
$x_{2q}$ is a local minimizer on the interval
$[x_{2q}-\alpha_{2q},x_{2q}+\alpha_{2q}]=2^{-q}\times[1,\frac{3}{2}]$ and
then, the two candidates $x_{2q}+\alpha_{2q}d_{1}$ and
$x_{2q}+\alpha_{2q}d_{2}$ generated by the Poll step fail to dominate
$x_{2q}$. Thus iteration $2q$ is unsuccessful, hence
$x_{2q+1}=x_{2q}=\frac{5}{4}\times 2^{-q}$ and
$\alpha_{2q+1}=\frac{1}{2}\alpha_{2q}=\frac{1}{4}\times 2^{-(q+1)}$. Then the
induction statement holds for $q\in\mathbb{N}_{*}$ if it holds for $q-1$,
which completes the proof. ∎

The sequence $(x_{k};\alpha_{k};D_{k})_{k\in\mathbb{N}}$ constructed above
satisfies the requirements of [5, Theorem 4.1] and [5, Corollary 4.1]. This
claim is formalized in the next proposition.

######  Proposition 2.

The sequence $(x_{k};\alpha_{k};D_{k})_{k\in\mathbb{N}}$ is a refining
sequence which converges to the refined point $x_{*}=0$ and the refining set
of directions $D=\\{-1,1\\}$ is dense in the unit sphere.

###### Proof.

Lemma 1 proves that the sequence of incumbent solutions
$(x_{k})_{k\in\mathbb{N}}$ converges to $x_{*}=0$ and that the sequence of
poll parameters $(\alpha_{k})_{k\in\mathbb{N}}$ converges to $0$. Furthermore,
for each even value of $k$, the Poll step evaluates $x_{k}+\alpha_{k}d$ for
all $d\in D_{k}=\\{-1,1\\}$. Thus the set of refining directions is
$D=\\{-1,1\\}$. ∎

###  2.3 Contradiction to some conclusions of [5, Theorem 4.1] and [5,
Corollary 4.1]

Propositions 1 and 2 show that the counterexample satisfies all the
requirements of [5, Theorem 4.1] and [5, Corollary 4.1]. In that context [5,
Theorem 4.1] claims that there exist a subsequence (indexed by
$K^{\prime}\subseteq\mathbb{N}$) and $i\in\\{1,2\\}$ such that the Poll step
fails at each iteration $k\in K^{\prime}$ and

$(a):\quad\forall k\in
K^{\prime},~{}\left\\{\begin{array}[]{l}x_{k}\in\mathrm{cl}(B_{i}),\\\
x_{k}+\alpha_{k}d_{2}\in\mathrm{int}(B_{i}),\end{array}\right.\qquad\mbox{and}\qquad(b):\quad\forall
k\in K^{\prime},~{}\left\\{\begin{array}[]{l}x_{k}\in\mathrm{cl}(B_{i}),\\\
x_{k}+\alpha_{k}d_{1}\in\mathbb{R}\setminus\mathrm{cl}(B_{i}).\end{array}\right.$

In addition [5, Corollary 4.1] asserts that

$(c):\quad\underset{k\in K}{\lim}~{}f(x_{k})=f(x_{*}).$

In the counterexample, the sequence of poll trial points has no element in
$B_{1}={]}-2,0]$ containing the refined point $x_{*}=0$ on its border. Thus,
the claim $(a)$ holds while the claims $(b)$ and $(c)$ are false. Indeed, by
restricting to $i=2$ (since $x_{k}\notin cl(B_{1})$ for all $k\in\mathbb{N}$)
and to the indexes $K=2\mathbb{N}$ where the Poll step fails, the claim $(a)$
holds with any $K^{\prime}\subseteq K$. However, for any $K^{\prime}\subseteq
K$, $\lim_{k\in K^{\prime}}f(x_{k})=0\neq-1=f(x_{*})$ and the trial point
$x_{k}+\alpha_{k}d_{1}=2^{-q}$ belongs to $B_{2}$ for all $k=2q\in K^{\prime}$
with $q\in\mathbb{N}$. Hence the claims $(b)$ and $(c)$ are incorrect.

##  3 Analysis of the proof leading to incorrect conclusions

The current section investigates the proof of [5, Theorem 4.1] in its general
context. In the paper [5], this proof is divided into six paragraphs that are
summarized in the following six steps. An additional seventh step leads to [5,
Corollary 4.1]. The error in the proof lies in the fifth step.

Denote $i\in\llbracket 1,n_{B}\rrbracket$ such that $x_{*}$ belongs to the
partition set $B_{i}$. First, the authors restrict the proof to the MADS
algorithm since the proof can be easily adapted to other dDSM. Second, they
address the case where $x_{*}\in\mathrm{int}(B_{i})$. Third, when
$x_{*}\in\partial B_{i}$, the authors address first the subcase where all
incumbent solutions $x_{k}$ lie in the borders of the partition sets. In the
remaining subcases, they extract a subsequence (indexed by $K_{1}\subseteq K$)
such that $x_{k}\in\mathrm{int}(B_{i})$ for all $k\in K_{1}$. Since the set of
refining directions is dense in the unit sphere, they extract a second
subsequence (indexed by $K_{2}\subseteq K_{1}$) such that, at any iteration
$k\in K_{2}$, some Poll trial points belong to $\mathrm{int}(B_{i})$. Fourth,
the authors create a function $\overline{f}$ Lipschitz-continuously extending
$f$ from $B_{i}$ to $\mathrm{cl}(B_{i})$. To the best of our knowledge, these
four steps are all valid.

Then, in the fifth paragraph, the authors aim to prove by contradiction that
some of the Poll points associated with iterations $k\in K_{2}$ do not belong
to $\mathrm{cl}(B_{i})$. Their proof involves a function $\widetilde{f}$
extending $\overline{f}$ from $B_{i}$ to $\mathbb{R}^{n}$ such that
$\widetilde{f}=\overline{f}$ on $B_{i}$, $\widetilde{f}$ is Lipschitz-
continuous on $\mathbb{R}^{n}$ and $\widetilde{f}$ is locally strictly
decreasing at any $z\in\partial B_{i}$ in all directions belonging to a cone
with nonempty interior. To the best of our knowledge, the construction of
$\widetilde{f}$ is valid. However, the authors consider a direction $d$ on
which $\widetilde{f}$ is strictly decreasing at $x_{*}$ and erroneously claim
that the Clarke derivative $\widetilde{f}^{\circ}_{C}(x_{*};d)$ is thus
strictly negative. The error in the proof lies in this incorrect claim.
Indeed, the fact that $\widetilde{f}$ is strictly decreasing at $x_{*}$ in a
direction $d$ does not imply that $\widetilde{f}^{\circ}_{C}(x_{*};d)$ is
strictly negative (for example, the single-variable function
$x\mapsto-\left\lvert{x}\right\rvert$ has a Clarke derivative equaling $1$ at
$x=0$ in both directions $d=1$ and $d=-1$). The following quote highlights the
error.

> [Erroneous claim in the fifth paragraph of the proof of [5, Theorem 4.1]]
> Let us assume that all poll points associated with the refining subsequence
> belong to $\mathrm{cl}(B_{i})$. We will see that this leads us to a
> contradiction. […] Let $\widetilde{f}$ be the extended function (and $L$ its
> Lipschitz constant). We then obtain that
>
> $\widetilde{f}^{\circ}_{C}(x_{*};v)\geq[\dots]\geq 0,$
>
> for all refining directions $v$, which is a contradiction since these
> directions are dense in the unit sphere and $\widetilde{f}$ is locally
> strictly decreasing from $x_{*}$ along all directions in a cone of nonempty
> interior.

The sixth paragraph, which considers the above claim as true, deduces that
there exists a third subsequence (indexed by $K_{3}\subseteq K_{2}$) for which
the Poll step at iterations $k\in K_{3}$ generates trial points
$t_{k}=x_{k}+\alpha_{k}d_{k}$ satisfying $f(t_{k})\geq f(x_{k})$ and
$t_{k}\notin\mathrm{cl}(B_{i})$. This concludes the proof of [5, Theorem 4.1].
Finally a seventh paragraph, also relying directly on the above claim, leads
to [5, Corollary 4.1]. In the case $n_{B}=2$, and arbitrarily denoting $i=2$,
the authors claim that some poll trial points belong to $B_{2}$ and some to
$B_{1}$ for all $k\in K_{3}$. Then, the limit of the sequence
$(f(x_{k}))_{k\in K_{3}}$ equals $f(x_{*})$ via the lower semicontinuity of
$f$ at $x_{*}$ and the failing Poll steps at iterations $k\in K_{3}$. To the
best of our knowledge, these last two steps are incorrect without assuming the
inexact claim from the fifth paragraph. Therefore their conclusions are not
valid in general.

##  4 An adapted dDSM to handle the counterexample

The counterexample introduced in Section 2 shows that a sequence of incumbent
solutions generated by a dDSM may approach a discontinuity without noticing
its existence. Section 4.1 proposes an adaptation of [5, Algorithm 2.1] (see
Algorithm 1) which avoids this drawback, as shown by the new convergence
Theorem 1 stated and proved in Section 4.2.

###  4.1 A revised algorithm with a Revealing Poll step

This section adds a second Poll step to [5, Algorithm 2.1] to overcome the
aforementioned drawback. This so-called Revealing Poll step has been
introduced in the literature as a part of the DiscoMADS algorithm [2] (which
is a variant of a dDSM called the MADS algorithm) designed for a framework
differing from the present work (including additional mechanisms to push
incumbent solutions away from discontinuities). At each iteration, the
Revealing Poll step evaluates at least one random mesh point in a ball of
constant radius centered at the current incumbent solution. Therefore, when
the sequence of incumbent solutions approaches a refined point $x_{*}$ at
which $f$ is discontinuous and when the mesh becomes sufficiently fine, the
Revealing Poll step eventually evaluates points in each branch of $f$ that has
a nonempty interior near $x_{*}$. This property allows to correct and to
strengthen the convergence analysis provided in [5], as shown in the next
Section 4.2.

Algorithm 1 below encloses the addition of a Revealing Poll step to [5,
Algorithm 2.1]. Precisely, the Search step, the Poll step and the Step size
parameter update follow [5, Algorithm 2.1]. Also, the forcing function
$\overline{\rho}$ (assumed to be continuous and nondecreasing on
$\mathbb{R}^{+}$ with $\rho(t)/t\to 0$ as $t\searrow 0$) is extracted from [5,
Algorithm 2.1]. Hence the only novelty lies in the Revealing Poll step, with
its parameter $R>0$ in the Initialization step.

Algorithm 1 Directional direct-search method (following [5, Algorithm 2.1],
except for the additional Revealing Poll step)

Initialization

Choose $x_{0}\in\Omega$ with $f(x_{0})<+\infty$, $\alpha_{0}>0$,
$0<\beta_{1}\leq\beta_{2}<1$, $\gamma\geq 1$ and a constant radius $R>0$ for
the Revealing Poll step. Let $\mathbb{D}$ be a (possibly infinite) set of
positive spanning sets.

for k = 0, 1, 2, … do

Search step

Define a finite (possibly empty) set $S_{k}\subset\mathbb{R}^{n}$. Denote
$t_{k}\in\operatorname*{argmin}\\{f(x_{k})-\overline{\rho}(\left\lVert
t-x_{k}\right\rVert):t\in S_{k}\cap\Omega\\}$. If
$f(t_{k})<f(x_{k})-\overline{\rho}(\left\lVert t-x_{k}\right\rVert)$, set
$x_{k+1}=t_{k}$ and skip the next two steps.

Revealing Poll step

Randomly define a nonempty finite set $D_{k}^{r}$ according to the uniform
independent distribution over the closed ball of radius $R$ centered at $0$.
Denote $P^{r}_{k}=\\{x_{k}+d:d\in D^{r}_{k}\\}$ and
$t_{k}\in\operatorname*{argmin}\\{f(x_{k})-\overline{\rho}(\left\lVert
t-x_{k}\right\rVert):t\in P^{r}_{k}\cap\Omega\\}$. If
$f(t_{k})<f(x_{k})-\overline{\rho}(\left\lVert t-x_{k}\right\rVert)$, set
$x_{k+1}=t_{k}$ and skip the next Poll step.

Poll step

Choose $D_{k}\in\mathbb{D}$. Denote $P_{k}=\\{x_{k}+\alpha_{k}d:d\in D_{k}\\}$
and $t_{k}\in\operatorname*{argmin}\\{f(x_{k})-\overline{\rho}(\left\lVert
t-x_{k}\right\rVert):t\in P_{k}\cap\Omega\\}$. If
$f(t_{k})<f(x_{k})-\overline{\rho}(\left\lVert t-x_{k}\right\rVert)$, then set
$x_{k+1}=t_{k}$. Otherwise, set $x_{k+1}=x_{k}$.

Step size parameter update

If $x_{k+1}\neq x_{k}$, increase the step size parameter as
$\alpha_{k+1}\in[\alpha_{k},\gamma\alpha_{k}]$. Otherwise strictly decrease
the step size parameter as
$\alpha_{k+1}\in[\beta_{1}\alpha_{k},\beta_{2}\alpha_{k}]$.

Algorithm 1 evaluates the trial points $\mathcal{P}_{k}=P^{r}_{k}\cup P_{k}$
and trial directions $\mathcal{D}_{k}=D^{r}_{k}\cup D_{k}$ at each
unsuccessful polling iteration $k\in\mathbb{N}$. Hence, the definition of a
refining sequence is adapted as follows. A refining subsequence
$(x_{k};\alpha_{k};\mathcal{D}_{k})_{k\in K}$ with corresponding refined point
$x_{*}$ and set $\mathcal{D}$ of refining unitary directions satisfy the three
following conditions: (i) $f(t)\geq f(x_{k})-\overline{\rho}(\left\lVert
t-x_{k}\right\rVert)$ for all $t\in S_{k}\cup\mathcal{P}_{k}$; (ii)
$(x_{k})_{k\in K}$ converges to $x_{*}$ and $(\alpha_{k})_{k\in K}$ converges
to $0$; (iii) any direction in $D$ is the limit of a subsequence of refining
normalized directions $(\frac{d_{k}}{\left\lVert d_{k}\right\rVert})_{k\in
K_{1}}$ where $0\neq d_{k}\in\mathcal{D}_{k}$ for all $k\in K_{1}\subseteq K$.

###  4.2 Analysis of the revised algorithm

Algorithm 1 inherits the properties of [5, Algorithm 2.1] since the Revealing
Poll step can be viewed as a specific Search step. This section proves that
Algorithm 1 has an additional asymptotic result sufficiently strong to correct
and generalize [5, Theorem 4.1] and [5, Corollary 4.1]. This result is stated
below as Theorem 1 and is based on the next Lemma 2, which is similar to [2,
Lemma 4.12], and on Assumption 1 which replaces [5, Assumption 4.1]. In the
sequel, $\mathcal{B}_{r}(x)$ denotes the closed ball of $\mathbb{R}^{n}$ of
radius $r>0$ centered at $x\in\mathbb{R}^{n}$, and $\mathbb{S}^{n}$ denotes
the unit sphere in $\mathbb{R}^{n}$.

######  Lemma 2.

Let $(x_{k};\alpha_{k};\mathcal{D}_{k})_{k\in K}$ be a refining subsequence
generated by Algorithm 1 with refined point $x_{*}$ and set $\mathcal{D}$ of
refining unitary directions. The set $\cup_{k\in K}\mathcal{P}_{k}$ is almost
surely dense in the closed ball $\mathcal{B}_{R}(x_{*})$ and $\mathcal{D}$ is
almost surely dense in $\mathbb{S}^{n}$.

###### Proof.

This result is proved in [2, Lemma 4.12] in the case of the MADS algorithm.
The current proof focuses on the general case of a dDSM. Let $\varepsilon>0$.

First, let $u\in\mathbb{S}^{n}$ and let us prove that
$\mathcal{B}_{\varepsilon}(u)$ contains a direction $d\in\mathcal{D}$. Recall
that each element of the infinite set $\cup_{k\in K}D^{r}_{k}$ is drawn from
the uniform independent distribution on $\mathcal{B}_{R}(0)$. Then, almost
surely there exists an infinite subset of $K$ denoted $L$ such that for all
$k\in L$, the random event $\\{\frac{d}{\left\lVert d\right\rVert}:d\in
D^{r}_{k}\\}\cap\mathcal{B}_{\varepsilon}(u)\neq\emptyset$ happens. Let
$(\frac{d_{k}}{\left\lVert d_{k}\right\rVert})_{k\in L}$ with $d_{k}\in
D^{r}_{k}$ and $\frac{d_{k}}{\left\lVert
d_{k}\right\rVert}\in\mathcal{B}_{\varepsilon}(u)$ for all $k\in L$. Since
$\mathcal{B}_{\varepsilon}(u)$ is a compact set, $(\frac{d_{k}}{\left\lVert
d_{k}\right\rVert})_{k\in L}$ has an accumulation unitary direction $d$
belonging to $\mathcal{D}\cap\mathcal{B}_{\varepsilon}(u)$.

Second, let $y\in\mathcal{B}_{R}(x_{*})$, and let us prove that there is a
Revealing Poll trial point $x\in\cup_{k\in K}P^{r}_{k}$ that belongs to
$\mathcal{B}_{\varepsilon}(y)$. Since $x_{*}$ is a refined point of
$(x_{k})_{k\in K}$, there exists a subsequence (indexed by $K_{1}\subseteq K$)
such that $\left\lVert x_{k}-x_{*}\right\rVert\searrow 0$ with $k\in K_{1}$.
Thus, there exist another subsequence (indexed by $K_{2}\subseteq K_{1}$) such
that the distance $d(y,\mathcal{B}_{R}(x_{k}))\searrow 0$ and an index
$k_{0}\in K_{2}$ such that
$d(y,\mathcal{B}_{R}(x_{k}))\leq\frac{\varepsilon}{2}$ for all $k\in
K_{3}=\\{k:k\in K_{2},~{}k\geq k_{0}\\}$. Then, denoting
$\Theta_{k}=\mathrm{int}(\mathcal{B}_{R}(x_{*})\cap\mathcal{B}_{\varepsilon}(y)\cap\mathcal{B}_{R}(x_{k}))$
for all $k\in K_{3}$ and $\mathrm{vol}(\cdot)$ the volume of a set, one has

$\emptyset\ \neq\ \mathrm{int}(\Theta_{k})\ =\ \Theta_{k}\ \subset\
\mathcal{B}_{R}(x_{k})\quad\mbox{and}\quad 0\ \leq\
\dfrac{\mathrm{vol}\left(\mathcal{B}_{R}(x_{k})\setminus\Theta_{k}\right)}{\mathrm{vol}\left(\mathcal{B}_{R}(x_{k})\right)}\
\leq\
\dfrac{\mathrm{vol}\left(\mathcal{B}_{R}(x_{k_{0}})\setminus\Theta_{k_{0}}\right)}{\mathrm{vol}\left(\mathcal{B}_{R}(x_{k_{0}})\right)}\
<\ 1,$

for all $k\in K_{3}$. Indeed, $\Theta_{k}$ is a nonempty open subset of
$\mathcal{B}_{R}(x_{k})$ for all $k\in K_{3}$. The second inequality follows
from
$\mathrm{vol}(\mathcal{B}_{R}(x_{k}))=\mathrm{vol}(\mathcal{B}_{R}(x_{k_{0}}))$
and
$\mathcal{B}_{R}(x_{k})\setminus\Theta_{k}=\mathcal{B}_{R}(x_{k})\setminus(A\cup\mathcal{B}_{\varepsilon}(y))$
and $\left\lVert x_{k}-x_{*}\right\rVert\searrow 0$ for all $k\in K_{3}$. The
first and third inequalities follow from
$0\leq\mathrm{vol}(\mathcal{B}_{R}(x_{k})\setminus\Theta_{k})<\mathrm{vol}(\mathcal{B}_{R}(x_{k}))$,
since $\Theta_{k}$ is an open set included in $\mathcal{B}_{R}(x_{k})$ for all
$k\in K_{3}$.

Denoting $E_{k}$ the random event $P_{k}^{r}\cap\Theta_{k}\neq\emptyset$
conditionally to $x_{k}$ for any $k\in K_{3}$, the proof is complete if at
least one of the events $(E_{k})_{k\in K_{3}}$ happens. The probability of any
of the events $E_{k}$ is

$\mathbb{P}\left(E_{k}\right)\ =\
1-\left(\dfrac{\mathrm{vol}\left(\mathcal{B}_{R}(x_{k})\setminus\Theta_{k}\right)}{\mathrm{vol}\left(\mathcal{B}_{R}(x_{k})\right)}\right)^{\\#P_{k}^{r}}\
\geq\
1-\dfrac{\mathrm{vol}\left(\mathcal{B}_{R}(x_{k_{0}})\setminus\Theta_{k_{0}}\right)}{\mathrm{vol}\left(\mathcal{B}_{R}(x_{k_{0}})\right)}\
=\ c\ >\ 0,$

where $\\#P_{k}^{r}\geq 1$ denotes the number of trial points in the Revealing
Poll step at iteration $k$. Thus, for any $k\in K_{3}$, $\mathbb{P}(E_{k})$ is
bounded below by a strictly positive constant $c$. Since moreover the events
$(E_{k})_{k\in K_{3}}$ are all independent, then almost surely at least one of
them happens. ∎

######  Assumption 1.

Any refined point has a neighborhood $B$ with a finite partition
$B=\cup_{i=1}^{n_{B}}B_{i}$ satisfying the first and third statements of [5,
Assumption 4.1] and $\mathrm{cl}(B_{i})=\mathrm{cl}(\mathrm{int}(B_{i}))$ for
each $i\in\llbracket 1,n_{B}\rrbracket$.

Assumption 1 replaces [5, Assumption 4.1], changing the requirement involving
the exterior cone property on each branch of $f$ by a requirement that no
branch has a meagre subset non-adherent to its interior. Then, with the
Revealing Poll step and Lemma 2, the following Theorem 1 holds.

######  Theorem 1.

Under [5, Assumption 2.1] and Assumption 1, let $x_{*}\in\Omega$ be the
refined point and $\mathcal{D}$ be the set of refining unitary directions of a
refining subsequence $(x_{k};\alpha_{k};\mathcal{D}_{k})_{k\in K}$ generated
by Algorithm 1. Then $\mathcal{D}$ is almost surely dense in $\mathbb{S}^{n}$.
If $f$ is lower semicontinuous at $x_{*}$, then the partition set
$B_{*}\in\\{B_{1},\dots,B_{n_{B}}\\}$ for which $x_{*}\in B_{*}$ satisfies

  1. 1.

if $x_{*}\in\mathrm{int}(B_{*})$, then $\lim_{k\in K}f(x_{k})=f(x_{*})$ and
$f^{\circ}_{C}(x_{*};d)\geq 0$ for all $d\in\mathcal{D}\cap
T_{\Omega}(x_{*})$;

  2. 2.

otherwise $x_{*}\in\partial B_{*}$ and there exists a subsequence (indexed by
$K_{*}\subseteq K$) such that $x_{k}\in\mathrm{cl}(B_{*})$ for all $k\in
K_{*}$, $\lim_{k\in K_{*}}f(x_{k})=f(x_{*})$, and $\cup_{k\in
K_{*}}\mathcal{P}_{k}$ is almost surely dense in
$B\cap\mathcal{B}_{R}(x_{*})$.

###### Proof.

The first three steps mentioned in Section 3 remain valid in the present
framework. The remaining steps are not required. Thanks to these three steps,
the proof is restricted to $\overline{\rho}\equiv 0$, the case where
$x_{*}\in\mathrm{int}(B_{*})$ is proved and, when $x_{*}\in\partial B_{*}$,
there exists a subsequence (indexed by $K_{1}\subseteq K$) such that
$x_{k}\in\mathrm{cl}(B_{*})$ for all $k\in K_{1}$. Recall that, in contrary to
the proof of [5, Theorem 4.1] and [5, Corollary 4.1], the set of trial points
at any iteration $k\in\mathbb{N}$ is $\mathcal{P}_{k}=P^{r}_{k}\cup P_{k}$
instead of $P_{k}$. Set $K_{*}=K_{1}$. Lemma 2 ensures that $\mathcal{D}$ is
almost surely dense in the unit sphere. Hence, it suffices to show that
$\cup_{k\in K_{*}}\mathcal{P}_{k}$ is dense in $B\cap\mathcal{B}_{R}(x_{*})$,
and that $f_{*}=\lim_{k\in K_{*}}f(x_{k})$ equals $f(x_{*})$.

First, let us prove that $\cup_{k\in K_{*}}\mathcal{P}_{k}$ is almost surely
dense in $B\cap\mathcal{B}_{R}(x_{*})$ and in
$B_{*}\cap\mathcal{B}_{R}(x_{*})$. Recall that, for any sets $S_{1}$ and
$S_{2}$, $\mathrm{cl}(S_{1}\cap
S_{2})=\mathrm{cl}(S_{1})\cap\mathrm{cl}(S_{2})$ when $\partial
S_{1}\cap\partial S_{2}\subseteq\partial(S_{1}\cap S_{2})$, and note that
$\partial(\mathrm{int}(B))\cap\partial(\mathrm{int}(\mathcal{B}_{R}(x_{*})))\subseteq\partial(\mathrm{int}(B)\cap\mathrm{int}(\mathcal{B}_{R}(x_{*})))$
and $\partial
B\cap\partial\mathcal{B}_{R}(x_{*})\subseteq\partial(B\cap\mathcal{B}_{R}(x_{*}))$.
Note also that Assumption 1 leads to
$\mathrm{cl}(\mathrm{int}(B))=\mathrm{cl}(B)$. Then, Lemma 2 guarantees that
$\cup_{k\in K_{*}}\mathcal{P}_{k}$ is dense in
$\mathrm{cl}(\mathrm{int}(B\cap\mathcal{B}_{R}(x_{*})))=\mathrm{cl}(\mathrm{int}(B)\cap\mathrm{int}(\mathcal{B}_{R}(x_{*}))=\mathrm{cl}(\mathrm{int}(B))\cap\mathrm{cl}(\mathrm{int}(\mathcal{B}_{R}(x_{*})))=\mathrm{cl}(B)\cap\mathrm{cl}(\mathcal{B}_{R}(x_{*}))=\mathrm{cl}(B\cap\mathcal{B}_{R}(x_{*}))$.
Then $\cup_{k\in K_{*}}\mathcal{P}_{k}$ is dense in
$B\cap\mathcal{B}_{R}(x_{*})$. The second claim holds by a similar argument.

Second, let us prove that $\lim_{k\in K_{*}}f(x_{k})=f(x_{*})$. The sequence
$(f(x_{k}))_{k\in K_{*}}$ is decreasing and bounded below, thus it converges,
and $f_{*}\geq f(x_{*})$ by lower semicontinuity of $f$ at $x_{*}$. Moreover,
assuming that $f_{*}>f(x_{*})$ raises a contradiction. Indeed it implies that
$f(x_{k})\geq f(x_{*})+\varepsilon$ for some $\varepsilon>0$ and all $k\in
K_{*}$. Then $\inf\\{f(x):x\in\cup_{k\in K_{*}}\mathcal{P}_{k}\\}\geq
f(x_{*})+\varepsilon$ since $f(x)\geq f(x_{k})$ for all $k\in K_{*}$ and
$x\in\mathcal{P}_{k}$. Thus $\inf\\{f(x):x\in(\cup_{k\in
K_{*}}\mathcal{P}_{k})\cap B_{*}\\}\geq f(x_{*})+\varepsilon$. This
contradicts the continuity at $x_{*}$ of the restriction of $f$ to $B_{*}$,
since $\cup_{k\in K_{*}}\mathcal{P}_{k}$ is almost surely dense in
$B_{*}\cap\mathcal{B}_{R}(x_{*})$. ∎

Following [5, Theorem 4.1], its corollary [5, Corollary 4.1] attempts to
provide sufficient conditions to guarantee that a dDSM generates a refining
sequence $(x_{k})_{k\in K}$ with a refined point $x_{*}$ satisfying
$\lim_{k\in K}f(x_{k})=f(x_{*})$. Theorem 1 recovers this property. It also
guarantees that, for any value of $n_{B}$, an infinite subset of each branch
of $f$ near $x_{*}$ is evaluated, while [5, Corollary 4.1] claims this
property under the restriction $n_{B}=2$. Note that Theorem 1 is established
under Assumption 1 which replaces [5, Assumption 4.1] (precisely the exterior
cone property is replaced by the assumption that
$\mathrm{cl}(Bi)=\mathrm{cl}(\mathrm{int}(B_{i}))$ for each $i$). Finally
Theorem 1 also ensures the density of the set of refining directions, which is
required in [5, Theorem 3.2] but not ensured by [5, Corollary 4.1].
Consequently [5, Theorems 3.1 and 3.2] remain usable.

###  4.3 Algorithm 1 handles the counterexample

This section shows how Algorithm 1 handles the counterexample constructed in
Section 2, thanks to the Revealing Poll step. Consider the dDSM constructed in
Section 2.2 with the addition of a Revealing Poll step which generates a
single trial point per iteration with the polling radius $R>0$ chosen as $R=2$
for the ease of presentation. This instance of Algorithm 1 satisfies the next
Proposition 3, proving that the counterexample constructed in Section 2 is
addressed. The proof of this proposition can be adapted to remain valid with
any polling radius $R>0$.

######  Proposition 3.

The aforementioned instance of Algorithm 1 almost surely generates a refining
subsequence of $(x_{k};\alpha_{k};\mathcal{D}_{k})_{k\in\mathbb{N}}$ with the
refined point $x_{*}=-1$ (the global minimizer of $f$) and
$\mathcal{D}=\\{-1,1\\}$ as the set of refining unitary directions.

###### Proof.

Denote $I=[-\sqrt{2}-1,0]$ and note that $f(x)\leq 0$ if and only if $x\in I$.
If there exists $k_{0}\in\mathbb{N}$ such that $x_{k_{0}}\in I$, then all
incumbent solutions $x_{k}$ with $k\geq k_{0}$ belong to $I$ as well, and thus
Algorithm 1 necessarily converges to $x_{*}=-1$ since it is the only point
satisfying some optimality conditions on $I$. Hence it suffices to prove that
the index $k_{0}$ almost surely exists.

At any even iteration $k\in 2\mathbb{N}$, the Revealing Poll step is executed
and the incumbent solution satisfies $x_{k}\leq x_{0}=\frac{5}{4}$. Then, if
$x_{k}>0$, the Revealing Poll step has a probability
$\frac{2-x_{k}}{4}>\frac{2-x_{0}}{4}=\frac{3}{16}$ to generate a trial point
$t_{k}\in[x_{k}-2,0]\subset[-2,0]\subset I$ which would imply that
$x_{k+1}=t_{k}\in I$. Thus there almost surely exists $k_{0}\in\mathbb{N}$
such that $x_{k}\in I$, which concludes the proof. ∎

##  5 Discussion

This work proposes a counterexample to a result announced in [5] on direct
search methods applied to discontinuous functions. It shows in the one-
dimensional case that a dDSM may converge to a point of discontinuity without
noticing it, since it evaluates points in only one of the branches of the
function near the refined point. This counterexample may be generalized to any
dimension. The paper also identifies the incorrect step in the proof and
proposes an algorithmic modification that recovers the validity of the results
presented in [5]. Future works could improve this modified algorithm in order
to save some evaluations of the function. Another perspective for future works
could consist in analyzing additional hypotheses on the objective function and
on the domain to preserve the validity of the results without modifying the
algorithm.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] S. Alarie, C. Audet, A.E. Gheribi, M. Kokkolaras, and S. Le Digabel.  Two decades of blackbox optimization applications.  EURO Journal on Computational Optimization, 9:100011, 2021.
  * [2] C. Audet, A. Batailly, and S. Kojtych.  Escaping unknown discontinuous regions in blackbox optimization.  SIAM Journal on Optimization, 32(3):1843–1870, 2022.
  * [3] C. Audet and W. Hare.  Derivative-Free and Blackbox Optimization.  Springer Series in Operations Research and Financial Engineering. Springer, Cham, Switzerland, 2017.
  * [4] A.R. Conn, K. Scheinberg, and L.N. Vicente.  Introduction to Derivative-Free Optimization.  MOS-SIAM Series on Optimization. SIAM, Philadelphia, 2009.
  * [5] L.N. Vicente and A.L. Custódio.  Analysis of direct searches for discontinuous functions.  Mathematical Programming, 133(1-2):299–325, 2012.


11institutetext:  Rensselaer Polytechnic Institute, Troy, NY, USA
11email: {charis, senevo, gruend2}<EMAIL_ADDRESS>
22institutetext: IBM Research, Cambrdige, MA, USA, 02142
22email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Explanation Ontology in Action: A Clinical Use-Case ††thanks: Copyright
©2020 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

Shruthi Chari 11 0000-0003-2946-7870 Oshani Seneviratne 11 0000-0001-8518-917X
Daniel M. Gruen 11 0000-0003-0155-3777 Morgan A. Foreman 22
0000-0002-2739-5853 Amar K. Das 22 0000-0003-3556-0844 Deborah L. McGuinness
11 0000-0001-7037-4567

###### Abstract

We addressed the problem of a lack of semantic representation for user-centric
explanations and different explanation types in our Explanation Ontology
(https://purl.org/heals/eo). Such a representation is increasingly necessary
as explainability has become an important problem in Artificial Intelligence
with the emergence of complex methods and an uptake in high-precision and
user-facing settings. In this submission, we provide step-by-step guidance for
system designers to utilize our ontology, introduced in our resource track
paper, to plan and model for explanations during the design of their
Artificial Intelligence systems. We also provide a detailed example with our
utilization of this guidance in a clinical setting.
Resource: https://tetherless-world.github.io/explanation-ontology

###### Keywords:

Modeling of Explanations and Explanation Types Supporting Explanation Types in
Clinical Reasoning Tutorial for Explanation Ontology Usage

##  1 Introduction

Explainable Artificial Intelligence (AI) has been gaining traction due to
increasing adoption of AI techniques in high-precision settings. Consensus is
lacking amongst AI developers on the type of explainability approaches and we
observe a lack of infrastructure for user-centric explanations. User-centric
explanations address a range of users’ questions, have different foci, and
such variety provides an opportunity for end-users to interact with AI systems
beyond just understanding why system decisions were made. In our resource
paper [2], we describe an Explanation Ontology (EO), which we believe is a
step towards semantic encoding of the components necessary to support user-
centric explanations. This companion poster focuses on describing the usage
steps (Section 2) that would serve as a guide for system developers hoping to
use our ontology. We demonstrate the usage of the protocol (Section 3) as a
means to support and encode explanations in a guideline-based clinical
decision support setting.

##  2 Usage Directions for the Explanation Ontology

Protocol 1 Usage of Explanation Ontology at System Design Time
---
Inputs: A list of user questions, knowledge sources and AI methods
Goal: Model explanations that need to be supported by a system based on inputs
from user studies
The protocol: 1. Gathering requirements (a) Conduct a user study to gather the
user’s requirements of the system (b) Identify and list user questions to be
addressed 2. Modeling (a) Align user questions to explanation types (b)
Finalize explanations to be included in the system (c) Identify components to
be filled in for each explanation type (d) Plan to populate slots for each
explanation type desired (e) Use the structure of sufficiency conditions to
encode the desired set of explanations

System designers can follow the usage directions for EO at design time when
planning for the capabilities of an AI system. The guidance aims to ensure
that end-user requirements are translated into user-centric explanations. This
protocol guidance is supported by resources made available on our website.
These resources include queries to competency questions to retrieve sample
user questions addressed by each explanation type111https://tetherless-
world.github.io/explanation-ontology/competencyquestions/##question2 and the
components to be filled for each explanation type.222https://tetherless-
world.github.io/explanation-ontology/competencyquestions/##question3
Additionally, the sufficiency conditions that serve as a means for structuring
content to fit the desired explanation type can be browsed via our explanation
type details page.333https://tetherless-world.github.io/explanation-
ontology/modeling/##modelingexplanations

##  3 Clinical Use Case

We applied the protocol to understand the need for explanations in a
guideline-based care setting and identify which explanation types would be
most relevant for this clinical use case. [2] Further, we utilized the EO to
model some of the explanations we pre-populated into the system prototype.
Hereafter, we describe our application of the EO usage guidelines and
highlight how we used a user study to guide system design. As a part of the
user study, we first held an expert panel interview to understand the
clinicians’ needs when working with guideline-based care. We utilized the
expert panel input to design a cognitive walkthrough of a clinical decision
support system (CDSS) with some explanations pre-populated at design time to
address questions that clinicians would want answered.

Figure 1: An overview of the user interface affordances we designed into the
prototype to accommodate the various explanation types.

The prototype of a CDSS we designed for the walkthrough, included allowances
for explanations on different screens of the system (Fig. 1), in all, allowing
clinicians to inspect a complicated type-2 diabetes patient case. Some
examples of the pre-populated explanations are: contrastive explanations to
help clinicians decide between drugs, trace based explanations to expose the
guideline reasoning behind why a drug was suggested, and counterfactual
explanations that were generated based on a combination of patient factors.
Other examples were provided to us by clinicians during the walkthrough.

Below, we step-through how the protocol guidance (Section 2) can be used to
model an example of a counterfactual explanation from this clinical use case.
A counterfactual explanation can be generated by modifying a factor on the
treatment planning screen of the CDSS prototype. For illustration’s sake, let
us suppose that this explanation is needed to address a question, “What if the
patient had an ASCVD risk factor?" where the “ASCVD risk factor" is an
alternate set of inputs the system had not previously considered. From our
definition of a counterfactual explanation (https://tetherless-
world.github.io/explanation-ontology/modeling/#counterfactual), in response to
the modification a system would need to generate a system recommendation based
on the consideration of the new input. More specifically, in our use case, a
system would need to consider the alternate input of the ASCVD risk factor in
conjunction with the patient context and consult evidence from the American
Diabetes Association (ADA) guidelines [5] to arrive at a new suitable drug
recommendation for the patient case. With the alternate set of inputs and the
corresponding recommendation in place, the counterfactual explanation
components can now be populated as slots based on the sufficiency condition
for this explanation type. A Turtle snippet of this counterfactual explanation
example can be viewed in Fig. 2. Additionally, there have been some promising
machine learning (ML) model efforts in the explainability space [1, 6] that
could be used to generate system recommendations to populate specific
explanation types. We are investigating how to integrate some of these AI
methods with the semantic encoding of EO.

Figure 2: An annotated turtle representation of a counterfactual explanation
that the CDSS would output in response to an additional factor the system had
not considered.

##  4 Related Work

In the explainable AI space, several research projects [4, 7, 3] have begun to
explore how to support end-user specific needs such as ours. Wang et al. [7]
present a framework for aligning explainable AI practices with human reasoning
approaches, and they test this framework in a co-design exercise using a
prototype CDSS. However, their framework isn’t in a machine-readable format
and hence is difficult to reuse for supporting explanations in a system. The
findings from their co-design process that clinicians seek different
explanations in different scenarios corroborate with those from our cognitive
walkthrough, and EO can help support system designers in this endeavor.
Similarly, Dragoni et al. [3] propose a rule-based explanation system in the
behavior change space capable of providing trace based explanations to end-
users to encourage better lifestyle and nutrition habits. While this
explanation system was tested with real target users and subject matter
experts, it is limited in scope by the types of explanations it provides. Vera
et al. [4] have released a question bank of explanation question types that
can be used to drive implementations such as ours.

##  5 Conclusion

We have presented guidelines for using our Explanation Ontology in user-facing
settings and have demonstrated the utility of this guidance in a clinical
setting. This guidance describes an end-to-end process to support the
translation of end-user requirements into explanations supported by AI
systems. We are taking a two-pronged approach to pursue future work in
operationalizing the use of our explanation ontology for clinical decision
support systems. To this end, we are working towards building an explanation
as a service that would leverage our explanation ontology and connect with AI
methods to support the generation of components necessary to populate
explanation types in different use cases. Further, we are implementing the
clinical prototype as a functional UI with affordances for explanations. We
expect the guidance along with open-sourced resources on our website will be
useful to system designers looking to utilize our ontology to model and plan
for explanations to include in their systems.

## Acknowledgments

This work is done as part of the HEALS project, and is partially supported by
IBM Research AI through the AI Horizons Network.

## References

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# Higgs Alignment from Multicritical-Point Principle
in Two Higgs Doublet Models

Hikaru Kawai<EMAIL_ADDRESS>Department of Physics and Center for
Theoretical Physics, National Taiwan University, Taipei 106, Taiwan  Physics
Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
Kiyoharu Kawana<EMAIL_ADDRESS>School of Physics, KIAS, Seoul 02455,
Korea  Kin-ya Oda<EMAIL_ADDRESS>Department of Information and
Mathematical Sciences, Tokyo Woman’s Christian University, Tokyo 167-8585,
Japan  Kei Yagyu<EMAIL_ADDRESS>Department of Physics,
Osaka University, Toyonaka, Osaka 560-0043, Japan

###### Abstract

In models with non-minimal Higgs sectors, enforcing (near) Higgs alignment,
necessary to prevent significant deviations in the Higgs boson coupling from
the standard model prediction, causes a serious fine-tuning problem. We
demonstrate that the Higgs alignment is naturally deduced from the
multicritical point principle (MPP) in the general two Higgs doublet model.
Furthermore, we discuss the possibility of realizing the Yukawa alignment from
the MPP, which is necessary to prevent flavor-changing neutral currents
mediated by Higgs bosons at tree level.

††preprint: OU-HET-1229

###### Contents

  1. I Introduction
  2. II Fine-tuning problem caused by the Alignments
    1. II.1 Higgs alignment
    2. II.2 Yukawa alignment
  3. III Higgs alignment via MPP
    1. III.1 $\lambda_{6}=0$ basis
    2. III.2 Taking unitary gauge
    3. III.3 Charged Higgs VEV
    4. III.4 Neutral Higgs VEV
    5. III.5 Vacuum structure when $\mu_{3}=0$
    6. III.6 $\mu_{3}=0$ is extremum of vacuum energy
  4. IV Towards Yukawa alignment via MPP
  5. V Conclusion
  6. A Canonical form of the kinetic term

##  I Introduction

Current LHC data reveal two crucial insights: the absence of clear evidence
for particles beyond the Standard Model (SM) and the consistent nature of the
observed Higgs boson’s properties within experimental uncertainties [1, 2].
This situation is economically explained by the SM whose Higgs sector is a
minimal form composed of a single isospin doublet field. However, new physics
beyond the SM must exist because the SM cannot explain the established
phenomena such as neutrino oscillations, existence of dark matter and baryon
asymmetry of the Universe. This leads to the perplexing question of why signs
of new physics have not been observed despite their necessity.

One simple explanation is that the scale of new physics is much greater than
the electroweak scale, leading to a decoupling of its effects from the SM. The
canonical seesaw scenario [3] is one of the well known examples of such new
physics, in which super heavy right-handed neutrinos are introduced to explain
neutrino masses and oscillations [4, 5, 6] as well as baryon asymmetry via the
leptogenesis [7]. Such scenarios, if realized, are generally challenging to
test experimentally.

On the other hand, there are possibilities of realizing new physics at the
electroweak and/or TeV scales. In these scenarios, the Higgs sector is often
extended from the minimal form. In such models with non-minimal Higgs sectors,
the couplings of the SM-like Higgs boson, which can be identified with one of
the neutral Higgs bosons, can deviate from the SM predictions due to scalar
mixings. Therefore, the so-called Higgs alignment [8] is required, that is the
assumption that the mass eigenstate of the Higgs boson identified with the
observed one does not mix with the other neutral states.

The Higgs alignment can naturally be realized by imposing an unbroken
$\mathbb{Z}_{2}$ symmetry such as the inert doublet model [9], but it spoils
the CP-violating phases which are required for the successful electroweak
baryogenesis, see e.g., Refs. [10, 11, 12] as earlier works in two Higgs
doublet models (2HDMs). We thus encounter a fine-tuning problem when we need
to impose the Higgs alinment in the non-minimal Higgs sectors.

The Multicritical-Point Principle (MPP) [13, 14, 15] offers a natural solution
to such fine-tuning issues. The MPP can be described in such a way that
coupling constants in a theory are naturally tuned to one of the critical
points at which quantum phase transitions occur. We briefly outline the
essence of this mechanism; see Ref. [16] for a more detailed account.

Consider a parameter $\lambda$, such as a mass or a coupling constant, in the
action $S_{\lambda}[\phi]$. In conventional quantum field theory (QFT), a
canonical partition function is used:

$\displaystyle Z_{\rm can}(\lambda)=\int{\cal
D}\phi\exp\left(i\left\\{S_{0}[\phi]+\lambda
S_{\lambda}[\phi]\right\\}\right),$ (1)

where $S_{0}[\phi]$ is the other parts irrelevant to $\lambda$. Here,
$\lambda$ is a free parameter with no inherent principle for specific values,
leading to the fine-tuning problem. Alternatively, a micro-canonical
formulation fixes $S_{\lambda}[\phi]$ to a certain value $A$:

$\displaystyle Z_{\rm mic}(A)=\int{\cal D}\phi
e^{iS_{0}}\delta\left(S_{\lambda}[\phi]-A\right)=\int_{-\infty}^{\infty}\frac{d\lambda}{2\pi}e^{i\lambda
A}Z_{\rm can}(\lambda).$ (2)

If the integration over $\lambda$ is dominated by a point
$\lambda=\lambda_{*}(A)$ as $V\rightarrow\infty$, the micro-canonical
formulation effectively becomes a canonical one with $\lambda$ fixed at
$\lambda_{*}(A)$. A simple example is a saddle point where

$\displaystyle e^{i\lambda A}Z_{\rm can}(\lambda)\propto
e^{iV\left[(\lambda-\lambda_{*}(A))^{2}+\cdots\right]},$ (3)

leading to

$\displaystyle\log Z_{\rm mic}(A)=\log Z_{\rm
can}(\lambda)\bigg{|}_{\lambda=\lambda_{*}(A)}\left(1+{\cal O}(V^{-1})\right)$
(4)

for $V\rightarrow\infty$. Refer to Refs. [17, 18, 16] for recent developments
in this area.

In this paper, we show how the MPP naturally leads to the Higgs alignment in
the most general 2HDM without arbitrary $\mathbb{Z}_{2}$ symmetries. This
approach can facilitate successful scenarios of electroweak baryogenesis [19,
20, 21, 22, 23, 24, 25, 26] due to sufficient CP-violating phases in Yukawa
interactions under the constraints by electric dipole moments [27, 28]. We
also discuss the possibility of realizing the Yukawa alignment [29] which is
required to avoid flavor-changing neutral currents (FCNCs) at tree level,
through the MPP.

The paper is organized as follows. Section II reviews the general 2HDM,
highlighting fine-tuning issues associated with the Higgs and Yukawa
alignments. Section III demonstrates the natural derivation of the Higgs
alignment via the MPP. In Section IV, we discuss the possibility to realize
the Yukawa alignment from the MPP. Conclusions are given in Section V. In
Appendix. A, we present how we can obtain the canonical kinetic terms for the
physical Higgs boson in the unitary gauge employed in this study.

##  II Fine-tuning problem caused by the Alignments

We address the fine-tuning problem arising from the Higgs alignment and the
Yukawa alignment in the 2HDM. We consider the general model without imposing
any additional symmetries such as a $\mathbb{Z}_{2}$ symmetry.

The Higgs sector of the 2HDM is composed of two isospin doublet fields
$\tilde{\Phi}_{1}$ and $\tilde{\Phi}_{2}$. Since these two fields have the
same charge and representation, we have degrees of freedom to rotate them as

$\begin{pmatrix}\tilde{\Phi}_{1}\\\
\tilde{\Phi}_{2}\end{pmatrix}=U\begin{pmatrix}\Phi_{1}\\\
\Phi_{2}\end{pmatrix},\quad U\in U(2).$ (5)

Among the various choices for $(\Phi_{1},\Phi_{2})$, the Higgs basis [30] is
frequently utilized due to its clarity in conveying the physical properties of
the scalar fields, as will be demonstrated. In order to distinguish the Higgs
basis and other bases, we particularly represent the former as
$H_{1}\equiv\Phi_{1}$ and $H_{2}\equiv\Phi_{2}$, and they can be parameterized
as

$\displaystyle H_{1}=\begin{pmatrix}G^{+}\\\
\frac{1}{\sqrt{2}}(h_{1}^{\prime}+v+iG^{0})\end{pmatrix},\quad
H_{2}=\begin{pmatrix}H^{+}\\\
\frac{1}{\sqrt{2}}(h_{2}^{\prime}+ih_{3}^{\prime})\end{pmatrix},$ (6)

where $G^{\pm}$ and $G^{0}$ are the Nambu-Goldstone (NG) bosons which are
absorbed into the longitudinal components of $W^{\pm}$ and $Z$ bosons,
respectively, while $H^{\pm}$ $(h_{1,2,3}^{\prime})$ are the physical singly-
charged (neutral) Higgs bosons. In this basis, only $H_{1}$ obtains the non-
zero vacuume expectaion value (VEV) $v$ defined as
$v\equiv(\sqrt{2}G_{F})^{-1}$ with $G_{F}$ being the Fermi constant, so that
$H_{1}$ behaves as the Higgs field. This means that the neutral component
$h_{1}^{\prime}$ plays a role of the Higgs boson in the SM, because the gauge
interaction and the Yukawa interaction for $h_{1}^{\prime}$ are the same as
those of the SM Higgs boson at tree level. We here have implicitly assumed
that the configuration of the VEVs in Eq. (6), i.e., $\langle
H_{1}\rangle=(0,v/\sqrt{2})^{T}$ and $\langle H_{2}\rangle=0$, corresponds to
the global minimum of the potential.

We note that the $h_{1}^{\prime}$ state generally does not correspond to a
mass eigenstate: It may mix with the other neutral states, $h_{2}^{\prime}$
and $h_{3}^{\prime}$, as will be discussed in the following subsection. Such
mixing can lead to deviations in the couplings of $h_{1}^{\prime}$ from those
predicted for the SM Higgs boson. The Higgs alignment limit is thus the one in
which the $h_{1}^{\prime}$ state does not mix with the $h_{2}^{\prime}$ and
$h_{3}^{\prime}$ states. In this limit, the couplings of $h_{1}^{\prime}$
align with the SM predictions at the tree level.

###  II.1 Higgs alignment

The most general form of the Higgs potential is written in the general basis
($\Phi_{1}$,$\Phi_{2}$) as

$\displaystyle V({\Phi}_{1},{\Phi}_{2})=$
$\displaystyle{\mu}_{1}({\Phi}_{1}^{\dagger}{\Phi}_{1})+{\mu}_{2}({\Phi}_{2}^{\dagger}{\Phi}_{2})+({\mu}_{3}{\Phi}^{\dagger}_{1}{\Phi}_{2}+\text{h.c.})$
$\displaystyle+\frac{{\lambda}_{1}}{2}({\Phi}_{1}^{\dagger}{\Phi}_{1})^{2}+\frac{{\lambda}_{2}}{2}({\Phi}_{2}^{\dagger}{\Phi}_{2})^{2}+{\lambda}_{3}({\Phi}_{1}^{\dagger}{\Phi}_{1})({\Phi}_{2}^{\dagger}{\Phi}_{2})+{\lambda}_{4}({\Phi}_{1}^{\dagger}{\Phi}_{2})({\Phi}_{2}^{\dagger}{\Phi}_{1})$
$\displaystyle+\left[\frac{{\lambda}_{5}}{2}({\Phi}_{1}^{\dagger}{\Phi}_{2})^{2}+{\lambda}_{6}({\Phi}_{1}^{\dagger}{\Phi}_{1})({\Phi}_{1}^{\dagger}{\Phi}_{2})+{\lambda}_{7}({\Phi}_{2}^{\dagger}{\Phi}_{2})({\Phi}_{1}^{\dagger}{\Phi}_{2})+\text{h.c.}\right],$
(7)

where ${\mu}_{3}$ and ${\lambda}_{5,6,7}$ are complex parameters, and the
others are real. Here we consider the Higgs basis as introduced in Eq. (6)
i.e., $\Phi_{1,2}=H_{1,2}$. To clearly indicate the use of the Higgs basis, we
re-label the parameters in the Higgs potential from $\mu_{i}$ to $M_{i}$ and
from $\lambda_{j}$ to $\Lambda_{j}$.

The stationary conditions,

$\displaystyle\frac{\partial V}{\partial
h_{i}^{\prime}}\Bigg{|}_{0}=0~{}~{}(i=1,\dots,3),$ (8)

yield the following relations:

$\displaystyle M_{1}$ $\displaystyle=-\frac{v^{2}}{2}\Lambda_{1},$
$\displaystyle M_{3}$ $\displaystyle=-\frac{v^{2}}{2}\Lambda_{6},$ (9)

where $|_{0}$ denotes that all fields are set to zero.

By applying Eq. (9), we eliminate the parameters $M_{1}$ and $M_{3}$. The mass
of the charged Higgs boson is then given by

$\displaystyle m_{H^{\pm}}^{2}=M_{2}+\frac{\Lambda_{3}}{2}v^{2},$ (10)

and the mass matrix for the neutral Higgs bosons in the
$(h_{1}^{\prime},h_{2}^{\prime},h_{3}^{\prime})$ basis is

$\displaystyle\mathcal{M}_{\rm
neut}=v^{2}\begin{pmatrix}\Lambda_{1}&\text{Re}\Lambda_{6}&-\text{Im}\Lambda_{6}\\\
\text{Re}\Lambda_{6}&\frac{M_{2}}{v^{2}}+\frac{1}{2}(\Lambda_{3}+\Lambda_{4}+\text{Re}\Lambda_{5})&-\frac{1}{2}\text{Im}\Lambda_{5}\\\
-\text{Im}\Lambda_{6}&-\frac{1}{2}\text{Im}\Lambda_{5}&\frac{M_{2}}{v^{2}}+\frac{1}{2}(\Lambda_{3}+\Lambda_{4}-\text{Re}\Lambda_{5})\end{pmatrix}.$
(11)

It is evident that the Higgs alignment limit can be expressed by the
conditions:

$\displaystyle({\cal M}_{\rm neut})_{12}$ $\displaystyle\to 0,$
$\displaystyle({\cal M}_{\rm neut})_{13}$ $\displaystyle\to 0.$ (12)

This can be realized by $\Lambda_{6}=0$, which is equivalent to the following
condition according to Eq. (9):

$\displaystyle M_{3}=0.$ (13)

It is noteworthy that the Higgs alignment naturally occurs in the decoupling
limit as $M_{2}\to\infty$ because the mixings are suppressed by
$v^{2}/M_{2}\to 0$ in this limit. However, this scenario is less interesting
from a phenomenological standpoint since all predictions converge to those of
the SM, diminishing the various motivations for considering the 2HDM.

One of the simplest ways to realize the Higgs alignment is by imposing an
unbroken $\mathbb{Z}_{2}$ symmetry in Eq. (7), such that
$\Phi_{1}\to+\Phi_{1}$ and $\Phi_{2}\to-\Phi_{2}$. This symmetry explicitly
forbids the terms $\mu_{3}$ and $\lambda_{6,7}$. Consequently, the Higgs
alignment is achieved through this symmetry. This approach leads to the so-
called inert doublet model [9], which is a compelling model for explaining the
existence of dark matter. However, it does not support electroweak
baryogenesis due to the absence of new CP-violating phases.

As we have seen above, symmetries tend to impose too strict constraints on the
parameter space. On the other hand, the MPP naturally realizes fine-tuning of
parameters without such an over-constraint. In the next section, we will show
that the MPP can indeed realize the conditions for the Higgs alignment without
the extra constraints, contrary to the case with additional symmetries. In
order to do that, it would be useful to restart from the general basis (7),
instead of the Higgs basis. In particular, we can choose the basis with
$\lambda_{6}=0$ (see the discussion in Sec. III), in which the necessary and
sufficient condition of the Higgs alignment turn out to be

$\displaystyle\mu_{3}$ $\displaystyle=0,$
$\displaystyle\langle\Phi_{2}\rangle$ $\displaystyle=0.$ (14)

###  II.2 Yukawa alignment

The most general form of the Yukawa interactions are described as follows:

$\displaystyle{\cal L}_{Y}$
$\displaystyle=-\sum_{a=1}^{2}\sum_{i,j=1}^{3}\left[\overline{{Q}_{{\text{L}}i}}({Y}_{d}^{(a)})^{ij}{\Phi}_{a}{d}_{{\text{R}}j}+\overline{{Q}_{{\text{L}}i}}({Y}_{u}^{(a)})^{ij}{\Phi}^{c}_{a}{u}_{{\text{R}}j}+\overline{{L}_{{\text{L}}i}}({Y}_{e}^{(a)})^{ij}{\Phi}_{a}{e}_{{\text{R}}j}+\text{h.c.}\right],$
(15)

where ${Y}_{f}^{(a)}$ ($f=u,d$ and $e$) are $3\times 3$ complex matrices, and
$\Phi^{c}_{a}\equiv i\tau_{2}\Phi_{a}^{*}$. Here, ${Q}_{{\text{L}}i}$ and
${L}_{{\text{L}}i}$ represent the left-handed quark and lepton doublets of the
$i$th generation, respectively, while ${u}_{{\text{R}}i}$,
${d}_{{\text{R}}i}$, and ${e}_{{\text{R}}i}$ denote the right-handed up-type
quarks, down-type quarks, and charged leptons. In the following, we suppress
the flavor indices for simplicity.

In the Higgs basis, we identify $\Phi_{1}$ to $H_{1}$ and $\Phi_{2}$ to
$H_{2}$. Utilizing a bi-unitary transformation, $f_{\text{L}}\to
V_{f}f_{\text{L}}$ and $f_{\text{R}}\to U_{f}f_{\text{R}}$ with
$V_{f},U_{f}\in U(3)$ and $f_{\text{L}}$ and $f_{\text{R}}$ representing left-
handed and right-handed fermions, respectively,111 We use the same symbols
$f_{{\text{L}},{\text{R}}}$ for fermions before and after the transformation
for simplicity.  one of the Yukawa matrices can be diagonalized as

$\displaystyle Y_{f}^{(1)}\to
V_{f}^{\dagger}\,Y_{f}^{(1)}\,U_{f}=:\frac{\sqrt{2}}{v}M_{f},$ (16)
$\displaystyle Y_{f}^{(2)}\to V_{f}^{\dagger}\,Y_{f}^{(2)}\,U_{f}=:\rho_{f},$
(17)

where $M_{f}$ denotes a diagonal mass matrix with real-positive eigenvalues,
and $\rho_{f}$ is a general complex $3\times 3$ matrix. Consequently, Eq. (15)
becomes

$\displaystyle\mathcal{L}_{Y}=-\Bigg{[}$
$\displaystyle\overline{Q_{\text{L}}}\left(\frac{\sqrt{2}}{v}M_{d}H_{1}+\rho_{d}H_{2}\right)d_{\text{R}}+\overline{Q_{\text{L}}}V_{\rm
CKM}^{\dagger}\left(\frac{\sqrt{2}}{v}M_{u}H_{1}^{c}+\rho_{u}H_{2}^{c}\right)u_{\text{R}}$
$\displaystyle+\overline{L_{\text{L}}}\left(\frac{\sqrt{2}}{v}M_{e}H_{1}+\rho_{e}H_{2}\right)e_{\text{R}}+\text{h.c.}\Bigg{]},$
(18)

where $V_{\rm CKM}=V_{u}^{\dagger}V_{d}$ is the Cabibbo-Kobayashi-Maskawa
(CKM) matrix, and

$\displaystyle{Q}_{\text{L}}=\begin{pmatrix}V_{\rm
CKM}^{\dagger}{u}_{\text{L}}\\\ d_{\text{L}}\end{pmatrix}.$ (19)

It is apparent that off-diagonal elements of $\rho_{f}$ can lead to FCNCs
mediated by the $h_{2,3}^{\prime}$ and/or $H^{\pm}$ fields. These FCNCs occur
even in the Higgs alignment limit, where the Yukawa couplings for $h_{2,3}$
provide flavor-violating interactions.

The Yukawa alignment, originally proposed in Ref. [29], is defined as

$\displaystyle Y_{f}^{(2)}\propto Y_{f}^{(1)}.$ (20)

In the Higgs basis, this alignment can be expressed as

$\displaystyle\rho_{f}=\zeta_{f}\frac{\sqrt{2}M_{f}}{v},$ (21)

where $\zeta_{f}$ represents a complex, non-matrix parameter. However, from a
theoretical view point, there is no inherent reason to assume such a Yukawa
alignment as defined in (20). This lack of theoretical background further
motivates the exploration of the realization of the Yukawa alignment through
the MPP, which will be discussed in Sec. IV.

##  III Higgs alignment via MPP

We now demonstrate how the Higgs alignment condition, as shown in Eq. (14),
can be achieved using the MPP.

###  III.1 $\lambda_{6}=0$ basis

In analyzing the vacuum structure of the 2HDM, it is useful to apply the
$U(2)$ transformation (5) to simplify the potential (7) by eliminating some
coupling constants.

An $SU(2)$ component of this transformation,

$\displaystyle U$
$\displaystyle=e^{ic_{1}\tau_{1}+ic_{2}\tau_{2}}=\left(\begin{matrix}\cos|c|&\frac{c}{|c|}\sin|c|\\\
-\frac{c^{*}}{|c|}\sin|c|&\cos|c|\end{matrix}\right),$ $\displaystyle c$
$\displaystyle=c_{2}+ic_{1}\quad(c_{1,2}\in\mathbb{R})$ (22)

mixes the two Higgs fields. By using this transformation, one can eliminate
one of the complex coupling constants $\mu_{3}$, $\lambda_{5},\lambda_{6}$ and
$\lambda_{7}$ in a finite region of the parameter space. In the following, we
take222 In fact, one can see that
$(\tilde{\Phi}_{1}^{\dagger}\tilde{\Phi}_{1})^{2}$ term gives $\sim
c(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{1}^{\dagger}\Phi_{2}+{\rm h.c})$ in the
new basis, which can be used to cancel $\lambda_{6}$ unless it is too large
compared with the other coupling constants. This is sufficient because the
philosophy of MPP is to realize automatic fine-tuning within a finite region
of parameter space.

$\displaystyle\lambda_{6}=0.$ (23)

Furthermore, we can take $\mu_{3}\in\mathbb{R}$ by using the remaining phase
transformations, i.e. $U={\rm diag}(e^{i\theta_{1}},e^{i\theta_{2}})$.
Finally, we note that this basis does not necessarily coincide with the Higgs
basis.

###  III.2 Taking unitary gauge

Next, by employing the usual $SU(2)$ gauge invariance, we can choose the
unitary gauge as

$\displaystyle\Phi_{1}$ $\displaystyle=\left(\begin{matrix}0\\\
\frac{\eta}{\sqrt{2}}\end{matrix}\right),$ $\displaystyle\Phi_{2}$
$\displaystyle=\left(\begin{matrix}\varphi^{+}\\\
\varphi^{0}\end{matrix}\right),$ $\displaystyle\eta$
$\displaystyle\in\mathbb{R},\quad\varphi^{\pm},\varphi^{0}\in\mathbb{C}.$ (24)

Note that when $\langle\varphi^{\pm}\rangle=\langle\varphi^{0}\rangle=0$, the
fields set to zero in Eq. (24) correspond precisely to the unphysical NG
bosons. Conversely, if there are non-zero VEVs for $\Phi_{2}$, the
$\varphi^{\pm}$ and $\varphi^{0}$ fields can mix with the longitudinal
components of the weak gauge bosons via the kinetic term (see Appendix A for
details ).

The potential, when rewritten in the new basis and in the unitary gauge, is
expressed as

$\displaystyle V(\eta,\varphi^{\pm},\varphi^{0})$
$\displaystyle=\frac{\mu_{1}}{2}\eta^{2}+\mu_{2}\left(\left|\varphi^{0}\right|^{2}+\left|\varphi^{+}\right|^{2}\right)+\frac{\mu_{3}}{\sqrt{2}}\eta\left(\varphi^{0}+\varphi^{0*}\right)$
$\displaystyle+\frac{\lambda_{1}}{8}\eta^{4}+\frac{\lambda_{2}}{2}\left(\left|\varphi^{0}\right|^{2}+\left|\varphi^{+}\right|^{2}\right)^{2}+\frac{\lambda_{3}}{2}\eta^{2}\left(\left|\varphi^{0}\right|^{2}+\left|\varphi^{+}\right|^{2}\right)+\frac{\lambda_{4}}{2}\eta^{2}\left|\varphi^{0}\right|^{2}$
$\displaystyle+\frac{\lambda_{5}}{4}\eta^{2}\left(\varphi^{0}\right)^{2}+\frac{\lambda_{7}}{\sqrt{2}}\eta\varphi^{0}\left(\left|\varphi^{0}\right|^{2}+\left|\varphi^{+}\right|^{2}\right)+\text{h.c.}$
(25)

###  III.3 Charged Higgs VEV

Let us first examine the VEV for the $\varphi^{\pm}$ fields. Since
$\varphi^{\pm}$ appear only via the $\left|\varphi^{+}\right|^{2}$ form in the
potential, we can determine whether $\langle\varphi^{\pm}\rangle=0$ or
$\langle\varphi^{\pm}\rangle\neq 0$ by looking at the sign of the mass term
given as

$\displaystyle
m_{\varphi^{\pm}}^{2}\equiv\frac{\partial^{2}V}{\partial\varphi^{+}\partial\varphi^{-}}\bigg{|}_{\varphi^{\pm}=0}=\mu_{2}+\lambda_{2}\left|\varphi^{0}\right|^{2}+\frac{\lambda_{3}}{2}\eta^{2}+\sqrt{2}\eta\,\text{Re}\left(\lambda_{7}\varphi^{0}\right).$
(26)

To prevent a charge-breaking vacuum $\langle\varphi^{\pm}\rangle\neq 0$, we
ensure $m_{\varphi^{\pm}}^{2}>0$, imposing an inequality among the coupling
constants.

###  III.4 Neutral Higgs VEV

Next, we examine the VEV for the neutral component $\varphi^{0}$. This
analysis is conducted by setting $\varphi^{\pm}=0$ in Eq. (25), leading to the
following expression for the potential:

$\displaystyle V(\eta,\varphi^{0})$
$\displaystyle=\frac{\mu_{1}}{2}\eta^{2}+\frac{\lambda_{1}}{8}\eta^{4}+\mu_{2}\left|\varphi^{0}\right|^{2}+\frac{\lambda_{2}}{2}|\varphi^{0}|^{4}$
$\displaystyle\quad+\sqrt{2}\mu_{3}\eta\,\text{Re}\left(\varphi^{0}\right)+\frac{1}{2}\left(\lambda_{3}+\lambda_{4}\right)\eta^{2}\left|\varphi^{0}\right|^{2}+\left[\frac{\lambda_{5}}{4}\eta^{2}\left(\varphi^{0}\right)^{2}+\frac{\lambda_{7}}{\sqrt{2}}\eta\varphi^{0}\left|\varphi^{0}\right|^{2}+\text{h.c.}\right].$
(27)

For a given value of $\langle\eta\rangle$, we can express the potential in
terms of the real and imaginary parts,
$\varphi^{0}=(\varphi_{\text{R}}+i\varphi_{\text{I}})/\sqrt{2}$:

$\displaystyle V(\varphi_{\text{R}},\varphi_{\text{I}})$
$\displaystyle=\mu_{3}\langle\eta\rangle\varphi_{\text{R}}+\frac{1}{2}\begin{pmatrix}\varphi_{\text{R}}&\varphi_{\text{I}}\end{pmatrix}{\cal
M}\begin{pmatrix}\varphi_{\text{R}}\\\
\varphi_{\text{I}}\end{pmatrix}+(\text{cubic and quartic terms of
}\varphi_{{\text{R}},{\text{I}}}),$ (28)

where ${\cal M}$ is a $2\times 2$ matrix defined by

$\displaystyle{\cal
M}\equiv\begin{bmatrix}\frac{\partial^{2}V}{\partial\varphi_{\text{R}}^{2}}&\frac{\partial^{2}V}{\partial\varphi_{\text{R}}\partial\varphi_{\text{I}}}\\\
\frac{\partial^{2}V}{\partial\varphi_{\text{R}}\partial\varphi_{\text{I}}}&\frac{\partial^{2}V}{\partial\varphi_{\text{I}}^{2}}\end{bmatrix}_{\varphi_{\text{R}}=\varphi_{\text{I}}=0}=\begin{pmatrix}\mu_{2}+\frac{\langle\eta\rangle^{2}}{2}\left(\lambda_{3}+\lambda_{4}+\text{Re}\lambda_{5}\right)&-\frac{\langle\eta\rangle^{2}}{2}\text{Im}\lambda_{5}\\\
-\frac{\langle\eta\rangle^{2}}{2}\text{Im}\lambda_{5}&\mu_{2}+\frac{\langle\eta\rangle^{2}}{2}\left(\lambda_{3}+\lambda_{4}-\text{Re}\lambda_{5}\right)\end{pmatrix}.$
(29)

###  III.5 Vacuum structure when $\mu_{3}=0$

Here and in the next subsection, we show that, when regarding the vacuum
energy as a function of $\mu_{3}$, the vacuum energy becomes extremum at the
point $\mu_{3}=0$. In this sense, $\mu_{3}=0$ is a critical point that can be
chosen by the MPP.

First, we discuss the conditions under which
$\varphi_{\text{R}}=\varphi_{\text{I}}=0$ is a true vacuum when $\mu_{3}=0$.
It is clear that in the limit of $\mu_{3}\to 0$, the potential (28) has a
local minimum at $\varphi_{\text{R}}=\varphi_{\text{I}}=0$ if the following
conditions are satisfied:

$\displaystyle|{\cal M}|$ $\displaystyle>0,$ $\displaystyle{\cal M}_{22}$
$\displaystyle>0,$ (30)

Even for $\mu_{3}=0$, there can be the other local minima at
$\varphi_{\text{R}}\neq 0$ and/or $\varphi_{\text{I}}\neq 0$ due to the
existence of the cubic terms of $\varphi_{\text{R}}^{3}$ and
$\varphi_{\text{I}}^{3}$. In order to examine such a minimum, it would be
convenient to move to the new basis of ($\varphi_{\text{R}}$,
$\varphi_{\text{I}}$) by which the matrix ${\cal M}$ is diagonalized. Namely,

$\displaystyle\begin{pmatrix}\varphi_{\text{R}}\\\
\varphi_{\text{I}}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}\varphi_{\text{R}}^{\prime}\\\
\varphi_{\text{I}}^{\prime}\end{pmatrix},\qquad\text{with}\quad\tan
2\theta=-\frac{\text{Im}\lambda_{5}}{\text{Re}\lambda_{5}}.$ (31)

In this new basis, the potential (28) with $\mu_{3}=0$ becomes

$\displaystyle
V\\!\left(\varphi_{\text{R}}^{\prime},\varphi_{\text{I}}^{\prime}\right)$
$\displaystyle=\frac{m_{\text{R}}^{2}}{2}\varphi_{\text{R}}^{\prime
2}+\frac{m_{\text{I}}^{2}}{2}\varphi_{\text{I}}^{\prime
2}+\left(\kappa_{\text{R}}\varphi_{\text{R}}^{\prime}+\kappa_{\text{I}}\varphi_{\text{I}}^{\prime}\right)\left(\varphi_{\text{R}}^{\prime
2}+\varphi_{\text{I}}^{\prime
2}\right)+\frac{\lambda_{2}}{8}\left(\varphi_{\text{R}}^{\prime
2}+\varphi_{\text{I}}^{\prime 2}\right)^{2},$ (32)

where

$\displaystyle m_{\text{R}}^{2}$
$\displaystyle=\mu_{2}+\frac{\langle\eta\rangle^{2}}{2}\left(\lambda_{3}+\lambda_{4}+\frac{\text{Re}\lambda_{5}}{\cos
2\theta}\right),$ (33) $\displaystyle m_{\text{I}}^{2}$
$\displaystyle=\mu_{2}+\frac{\langle\eta\rangle^{2}}{2}\left(\lambda_{3}+\lambda_{4}-\frac{\text{Re}\lambda_{5}}{\cos
2\theta}\right),$ (34) $\displaystyle\kappa_{\text{R}}$
$\displaystyle=\frac{\langle\eta\rangle}{2}\left(\text{Re}\lambda_{7}\cos\theta-\text{Im}\lambda_{7}\sin\theta\right),$
(35) $\displaystyle\kappa_{\text{I}}$
$\displaystyle=-\frac{\langle\eta\rangle}{2}\left(\text{Re}\lambda_{7}\sin\theta+\text{Im}\lambda_{7}\cos\theta\right).$
(36)

The condition (30) is then rewritten as

$\displaystyle m_{\text{R}}^{2}$ $\displaystyle>0,$ $\displaystyle
m_{\text{I}}^{2}$ $\displaystyle>0.$ (37)

Now the absence of other local minima is guaranteed when

$\displaystyle\frac{\partial
V(\varphi_{\text{R}}^{\prime},\varphi_{\text{I}}^{\prime})}{\partial\varphi_{\text{R}}^{\prime}}$
$\displaystyle\neq 0,$ $\displaystyle\frac{\partial
V(\varphi_{\text{R}}^{\prime},\varphi_{\text{I}}^{\prime})}{\partial\varphi_{\text{I}}^{\prime}}$
$\displaystyle\neq 0,$
$\displaystyle\text{for}\quad\varphi_{{\text{R}},{\text{I}}}^{\prime}$
$\displaystyle\neq 0.$ (38)

We find that the condition (38) is satisfied when

$\displaystyle\frac{\kappa_{\text{R}}^{2}}{m_{\text{R}}^{2}}+\frac{\kappa_{\text{I}}^{2}}{m_{\text{I}}^{2}}<\frac{2}{9}\lambda_{2}.$
(39)

One can numerically check that the inequality (39) is the sufficient condition
to realize the condition (38). Note that even if this condition is not
satisfied, the minimum at the origin could still be the global minimum, since
other potential minima away from the origin may have higher values.

In any case, over a wide range of the parameter space, $\varphi^{0}=0$ is the
global minimum of the potential when $\mu_{3}=0$.

###  III.6 $\mu_{3}=0$ is extremum of vacuum energy

Next, we discuss how the vacuum energy varies as a function of $\mu_{3}$. When
a small value of $\mu_{3}$ is turned on, non-zero VEVs
$\langle\varphi_{\text{R}}\rangle$ and $\langle\varphi_{\text{I}}\rangle$ are
induced as 333 It is equivalent to
$\displaystyle\langle\varphi_{\text{R}}^{\prime}\rangle$
$\displaystyle=-\frac{\langle\eta\rangle\cos\theta}{m_{\text{R}}^{2}}\mu_{3}+{\cal
O}\\!\left(\mu_{3}^{2}\right),$
$\displaystyle\langle\varphi_{\text{I}}^{\prime}\rangle$
$\displaystyle=\frac{\langle\eta\rangle\sin\theta}{m_{\text{I}}^{2}}\mu_{3}+{\cal
O}\\!\left(\mu_{3}^{2}\right).$

$\displaystyle\langle\varphi_{\text{R}}\rangle=-\frac{{\cal M}_{22}}{|{\cal
M}|}\langle\eta\rangle\mu_{3}+{\cal
O}\\!\left(\mu_{3}^{2}\right),\quad\langle\varphi_{\text{I}}\rangle=\frac{{\cal
M}_{12}}{|{\cal M}|}\langle\eta\rangle\mu_{3}+{\cal
O}\\!\left(\mu_{3}^{2}\right).$ (40)

At the same time, the VEV of $\eta$ is slightly shifted to the
$\varphi_{\text{R}}$ direction as

$\displaystyle\langle\eta\rangle^{2}=v_{0}^{2}+\sqrt{\frac{2}{-\lambda_{1}\mu_{1}}}\langle\varphi_{\text{R}}\rangle\mu_{3}+{\cal
O}(\mu_{3}^{3}),$ (41)

where

$\displaystyle
v_{0}^{2}\equiv\langle\eta\rangle^{2}\Big{|}_{\mu_{3}=0}=-\frac{2\mu_{1}}{\lambda_{1}}.$
(42)

The vacuum energy ${\cal E}$ with a small $\mu_{3}$ is then expressed as

$\displaystyle{\cal E}={\cal E}_{0}+\delta{\cal E}_{0},$ (43)

where ${\cal E}_{0}$ is the vacuum energy for $\mu_{3}=0$, i.e., ${\cal
E}_{0}=\mu_{1}/2\lambda_{1}$, and from Eq. (28) we have

$\displaystyle\delta{\cal E}_{0}=C\,\mu_{3}^{2}+{\cal O}(\mu_{3}^{3}),$ (44)

in which $C$ is a coefficient determined by other coupling constants.

It is now clear that $\mu_{3}=0$ corresponds to the extremum of ${\cal E}$,
and such a point is naturally realized by the MPP in the large volume limit.
(See the discussion of the MPP in the Introduction.) Note that we have not
assumed any fine-tuning for other couplings, and our conclusions hold for a
wide range of parameter space satisfying inequalities such as (39).

##  IV Towards Yukawa alignment via MPP

Figure 1: Vacuum bubble diagrams with quark loops.

We discuss a possible mechanism that may realize the Yukawa alignment in the
2HDM. We impose the Higgs alignment which can naturally be realized by
considering the MPP as we have seen in the previous section. In this case,
flavor-changing Yukawa couplings arise from those for $\varphi^{\pm}$ and
$\varphi^{0}$, see Eq. (18).

Let us consider the vacuum bubble diagrams shown in Fig. 1, focusing on the
quark loops as an example. The amplitude is calculated as

$\displaystyle{\cal E}_{\rm bubble}$
$\displaystyle=-\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}\frac{1}{(k_{1}-k_{2})^{2}-m_{\varphi}^{2}}$
$\displaystyle\qquad\times\Bigg{\\{}\left(2k_{1}\cdot
k_{2}\right)\text{tr}\Bigg{[}\sum_{q=u,d}\left(\frac{1}{k_{1}^{2}-M_{q}^{2}}\rho_{q}\frac{1}{k_{2}^{2}-M_{q}^{2}}\rho_{q}^{\dagger}\right)$
$\displaystyle\phantom{\qquad\times\Bigg{\\{}\left(2k_{1}\cdot
k_{2}\right)\text{tr}\Bigg{[}}+\frac{1}{k_{1}^{2}-M_{u}^{2}}V_{\rm
CKM}\rho_{d}\frac{1}{k_{2}^{2}-M_{d}^{2}}\rho_{d}^{\dagger}V_{\rm
CKM}^{\dagger}$ $\displaystyle\phantom{\qquad\times\Bigg{\\{}\left(2k_{1}\cdot
k_{2}\right)\text{tr}\Bigg{[}}+\frac{1}{k_{1}^{2}-M_{u}^{2}}\rho_{u}^{\dagger}V_{\rm
CKM}\frac{1}{k_{2}^{2}-M_{d}^{2}}V_{\rm CKM}^{\dagger}\rho_{u}\Bigg{]}$
$\displaystyle\phantom{\times\Bigg{\\{}\left(2k_{1}\cdot
k_{2}\right)\text{tr}\Bigg{[}}-\text{tr}\Bigg{[}\frac{M_{u}}{k_{1}^{2}-M_{u}^{2}}V_{\rm
CKM}\rho_{d}\frac{M_{d}}{k_{2}^{2}-M_{d}^{2}}V_{\rm CKM}^{\dagger}\rho_{u}$
$\displaystyle\phantom{\qquad\times\Bigg{\\{}\left(2k_{1}\cdot
k_{2}\right)\text{tr}\Bigg{[}}+\frac{M_{u}}{k_{1}^{2}-M_{u}^{2}}\rho_{u}^{\dagger}V_{\rm
CKM}\frac{M_{d}}{k_{2}^{2}-M_{d}^{2}}\rho_{d}^{\dagger}V_{\rm
CKM}^{\dagger}\Bigg{]}\Bigg{\\}},$ (45)

where terms proportional to $(2k_{1}\cdot k_{2})$ arise from the left and
center diagrams, while the contribution in the last two lines originate from
the right diagram in Fig. 1. For simplicity, we assume the masses of neutral
and charged scalars to be identical, denoted as $m_{\varphi}$.

Let us consider the contribution from the exchange of the neutral Higgs boson
$\varphi^{0}$ (left diagram in Fig. 1), corresponding to the first term of the
second line in Eq. (45). After taking the trace in the flavor space, we can
express this contribution by terms proportional to $|\rho_{q}^{\rm diag}|^{2}$
and $|\delta\rho_{q}|^{2}$, where $\rho_{q}^{\rm diag}$ and $\delta\rho_{q}$
represent the diagonal and off-diagonal elements of $\rho_{q}$, respectively.
Consequently, ${\cal E}_{\rm bubble}$ reaches an extremum at
$\delta\rho_{q}=0$ in the parameter space, and the diagonal form of $\rho_{q}$
is naturally realized by the MPP:

$\displaystyle\frac{\partial{\cal E}_{\rm
bubble}(\rho_{q})}{\partial\rho_{q}^{ij}}=0\quad(i\neq
j)\qquad\text{at}\quad\rho_{q}=\rho_{q}^{\rm diag}.$ (46)

However, this description is modified by the contribution from the exchange of
charged Higgs $\varphi^{\pm}$, as $\rho_{q}$ is coupled with the CKM matrix
$V_{\rm CKM}$. Therefore, the extrema discussed above are shifted by the off-
diagonal elements of $V_{\rm CKM}$.

We would like to comment on the lepton sector. In this context, the quark
states $q^{i}$ in the left diagram are replaced by the charged lepton states
$e^{i}$, and the pairs $(d^{i},u^{j})$ in the center and right diagrams are
replaced by $(e^{i},\nu^{j})$. By neglecting neutrino masses, the vacuum
energy ${\cal E}_{\rm bubble}$ can be calculated by substituting
$(\rho_{d},\rho_{u},M_{d},M_{u},V_{\rm CKM})$ with
$(\rho_{e},0,M_{e},0,I_{3\times 3})$ in Eq. (45).444When we introduce three
right-handed neutrinos, another Yukawa matrix $\rho_{\nu}$ appears in Eq.
(18), similar to $\rho_{u}$. Even in this case, the MPP predicts a diagonal
form of $\rho_{\nu}$ as long as we take the massless limit of the neutrinos.
Consequently, Eq. (45) is satisfied, and the diagonal form of $\rho_{e}$ is
naturally realized by the MPP.

##  V Conclusion

The general two Higgs doublet model (2HDM) encounters fine-tuning problems to
cope with current experimental data, specifically regarding the realization of
the Higgs and Yukawa alignments. In this work, we have implemented the
multicritical point principle (MPP) to solve these problems.

We have shown in the basis with $\lambda_{6}=0$ (a dimensionless parameter in
the Higgs potential) that setting the dimensionful mixing parameter $\mu_{3}$
between the two doublets to zero leads to the vanishing vacuum expectation
value (VEV) of the second Higgs doublet in a finite region of the parameter
space, bounded by several inequalities. Moreover, we have demonstrated that
the point $\mu_{3}\rightarrow 0$ coincides with a critical point—specifically
an extremum of the vacuum energy. Thus, the Higgs alignment is naturally
deduced through the MPP.

For the Yukawa alignment, our analysis shows that the MPP effectively predicts
the Yukawa alignment in the quark sector assuming negligible off-diagonal
elements of the Cabibbo-Kobayashi-Maskawa matrix. The predicted Yukawa
matrices $\rho_{f}$ for the additional Higgs bosons present a more general
structure unlike the one given by the conventional Yukawa alignment, i.e.,
these matrices can have three different values in their diagonal complex
elements. The Yukawa alignment for the lepton sector is naturally deduced by
the MPP assuming massless neutrinos.

This study’s outcomes not only deepen our understanding of the Higgs mechanism
but also pave the way for experimental tests of the MPP in 2HDM settings.
While predicated on certain assumptions, the scope of this research
potentially reaches into broader realms, such as cosmology and the underlying
principles of particle physics. Future advancements in technology and cross-
disciplinary research could further elucidate these findings and their broader
impact.

###### Acknowledgements.

H.K. thanks Prof. Shin-Nan Yang and his family for their kind support through
the Chin-Yu chair professorship. H.K. is partially supported by JSPS (Grants-
in-Aid for Scientific Research Grants No. 20K03970), by the Ministry of
Science and Technology, R.O.C. (MOST 111-2811-M-002-016), and by National
Taiwan University. The work of K.K. is supported by KIAS Individual Grants,
Grant No. 090901, and that of K.O. is in part supported by the JSPS Kakenhi
Grant No. 21H01107.

##  Appendix A Canonical form of the kinetic term

The basis (24) results in a non-canonical kinetic term for massive gauge
bosons, owing to the mixing between the gauge boson and the corresponding
Nambu-Goldstone boson. We will detail the process of achieving a canonical
kinetic term in this context. To simplify our discussion, we assume that
$\langle\varphi^{\pm}\rangle=0$.

It is convenient to decompose the massive gauge bosons as

$\displaystyle
W_{\mu}^{\pm}=\tilde{W}_{\mu}^{\pm}+\frac{1}{m_{W}}\partial_{\mu}\phi_{W}^{\pm},\quad
Z_{\mu}=\tilde{Z}_{\mu}+\frac{1}{m_{Z}}\partial_{\mu}\phi_{Z},$ (47)

where $\tilde{W}_{\mu}^{\pm}$ ($\tilde{Z}_{\mu}$) and $\phi_{W}^{\pm}$
($\phi_{Z}$) are respectively the transverse and longitudinal modes of the
$W_{\mu}^{\pm}$ ($Z_{\mu})$ boson. We note that
$\partial^{\mu}\tilde{W}_{\mu}^{\pm}=\partial^{\mu}\tilde{Z}_{\mu}=0$ are
satisfied by definition. The kinetic term is then expressed as

$\displaystyle{\cal L}_{\rm kin}$
$\displaystyle=(\partial_{\mu}\phi_{W}^{-},\partial_{\mu}\varphi^{-})\begin{pmatrix}1&c^{*}\\\
c&1\end{pmatrix}\begin{pmatrix}\partial^{\mu}\phi_{W}^{+}\\\
\partial^{\mu}\varphi^{+}\end{pmatrix}$
$\displaystyle\quad+\frac{1}{2}(\partial_{\mu}\phi_{Z},\partial_{\mu}\varphi_{\text{I}},\partial_{\mu}\varphi_{\text{R}})\begin{pmatrix}1&\frac{v_{\text{R}}}{v}&-\frac{v_{\text{I}}}{v}\\\
\frac{v_{\text{R}}}{v}&1&0\\\
-\frac{v_{\text{I}}}{v}&0&1\end{pmatrix}\begin{pmatrix}\partial_{\mu}\phi_{Z}\\\
\partial_{\mu}\varphi_{\text{I}}\\\
\partial_{\mu}\varphi_{\text{R}}\end{pmatrix},\quad
c=-i\frac{v_{\text{R}}+iv_{\text{I}}}{v},$ (48)

where $v\equiv\sqrt{v_{\eta}^{2}+v_{\text{R}}^{2}+v_{\text{I}}^{2}}$ with
$v_{\eta}=\langle\eta\rangle$, $v_{\text{R}}=\langle\varphi_{\text{R}}\rangle$
and $v_{\text{I}}=\langle\varphi_{\text{I}}\rangle$. The kinetic terms for the
transverse modes $\tilde{W}_{\mu}^{\pm}$ and $\tilde{Z}_{\mu}$ already take
the canonical form. By introducing the non-unitary transformation

$\displaystyle\begin{pmatrix}\phi_{W}^{+}\\\
\varphi^{+}\end{pmatrix}=\begin{pmatrix}\frac{c^{*}}{|c|}&-\frac{c^{*}}{\sqrt{1-|c|^{2}}}\\\
0&\frac{1}{\sqrt{1-|c|^{2}}}\end{pmatrix}\begin{pmatrix}\phi_{W}^{+\prime}\\\
\varphi^{+\prime}\end{pmatrix},~{}~{}\begin{pmatrix}\phi_{Z}\\\
\varphi_{\text{I}}\\\
\varphi_{\text{R}}\end{pmatrix}=\begin{pmatrix}1&-\frac{\rho}{\sqrt{1-\rho^{2}}}&0\\\
0&\frac{\rho_{\text{R}}}{\rho\sqrt{1-\rho^{2}}}&\frac{\rho_{\text{I}}}{\rho}\\\
0&-\frac{\rho_{\text{I}}}{\rho\sqrt{1-\rho^{2}}}&\frac{\rho_{\text{R}}}{\rho}\end{pmatrix}\begin{pmatrix}\phi_{Z}^{\prime}\\\
\varphi_{\text{I}}^{\prime}\\\ \varphi_{\text{R}}^{\prime}\end{pmatrix},$ (49)

$\phi_{W}^{\pm\prime}$ and $\varphi^{\pm\prime}$ can take the canonical form
of the kinetic term, and we can identify the former (latter) with the
unphysical NG boson (physical charged Higgs boson), where
$\rho=\sqrt{v_{\text{R}}^{2}+v_{\text{I}}^{2}}/v$ and
$\rho_{{\text{R}},{\text{I}}}=v_{{\text{R}},{\text{I}}}/v$.

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