# Non-Obvious Manipulability of the Rank-Minimizing Mechanism

Peter Troyan
Department of Economics
University of Virginia111Address: P.O. Box 400182, Charlottesville, VA, 22904.
Email<EMAIL_ADDRESS>

###### Abstract

In assignment problems, the average rank of the objects received is often used
as key measure of the quality of the resulting match. The rank-minimizing (RM)
mechanism optimizes directly for this objective, by returning an assignment
that minimizes the average rank. While a naturally appealing mechanism for
practical market design, one shortcoming is that this mechanism fails to be
strategyproof. In this paper, I show that, while the RM mechanism is
manipulable, it is not _obviously_ manipulable.

Keywords: matching, rank distribution, obviousness, manipulability

JEL Codes: C78, D47

##  1 Introduction

Many institutions make assignments by collecting agents’ ordinal preferences
over the possible alternatives and using them as an input to a rule that
outputs an assignment. Common examples include public school choice
(Abdulkadiroğlu and Sönmez, 2003), medical residency matching (Roth and
Peranson, 1999), teacher assignment (Combe et al., 2022), course allocation
(Budish and Cantillon, 2012; Budish and Kessler, 2017), and refugee
resettlement (Delacrétaz et al., 2016). A natural metric to measure the
success of the outcome is the rank distribution: how many students get their
first choice, how many get their second choice, and so on. Indeed, many school
districts such as those in New York City and San Francisco publicly release
statistics on the ranks as a measure of the goodness of the match.

In the context of school choice specifically, most cities rely on either some
version of Gale and Shapley’s celebrated deferred acceptance mechanism (Gale
and Shapley, 1962) or on the so-called Boston mechanism (also sometimes called
the immediate acceptance mechanism) to determine the assignment, and then
evaluate the rank distribution produced by these mechanisms.222There is
another class of mechanisms based on the top trading cycles (TTC) algorithm of
Shapley and Scarf (1974) that has been studied extensively in the theoretical
literature, but has found few adherents in practice. The only use of it in a
real-world school choice setting that I am aware of is New Orleans, where it
was used for one year before being abandoned (Abdulkadiroǧlu et al., 2020).
However, given the importance placed on the rank distribution, it is also
natural to consider mechanisms that optimize directly for this objective.
Indeed, Featherstone (2020) notes that Teach for America does exactly this,
and uses the rank distribution when _selecting_ its assignment. Featherstone
(2020) is also one of the few papers that has undertaken a serious analysis of
mechanisms based explicitly on the rank distribution (a few other papers in
this relatively small but growing literature are discussed in Section 4).

While using mechanisms that select assignments based explicitly on the rank
distribution is naturally appealing, an important consideration in any
mechanism design problem is the incentives of the agents. Indeed, one of the
most appealing properties arguing for the use of DA-based mechanisms is that
they generally give agents strong incentives to report their true preferences.
On the other hand, Proposition 10 of Featherstone (2020) shows that no ordinal
assignment mechanism is both rank-efficient (a refinement of Pareto efficiency
that he defines that takes into account the rank distribution) and
strategyproof.

However, strategyproofness is a very demanding property, and just because a
mechanism _can_ be manipulated does not mean that it is _will_ be manipulated.
As part of a recent strand of literature on “obviousness” in mechanism design,
Troyan and Morrill (2020) introduce the concept of a _(non-)obvious
manipulation_ as a way to relax strategyproofness.__ They use their definition
to taxonomize non-strategyproof mechanisms into two classes: those that are
obviously manipulable (such as the Boston mechanism) and thus are likely to be
easily manipulated in practice,333There is a plethora of evidence that the
Boston mechanism is commonly manipulated; for a detailed analysis, see, for
instance, Dur et al. (2018). and those that are non-obviously manipulable
(such as school-proposing DA, or, as they show, any mechanism that is stable-
dominating in the sense of Alva and Manjunath, 2019) which, while formally
manipulable, have manipulations that are more difficult for cognitively-
limited agents to recognize and enact successfully.

In this paper, I consider a canonical assignment model in which there is a set
of agents (such as students) to be assigned to a set of objects (such as
schools), each of which has some fixed capacity. Agents participate in a
mechanism in which they report their preferences over the objects. I consider
the _rank-minimizing (RM) mechanism,_ which takes the reported agent
preferences, and implements an assignment that minimizes the average rank of
the objects received. My main result is to show that the RM mechanism, while
manipulable, is not _obviously_ manipulable. Thus, to the extent that
minimizing the average rank is a desirable policy objective, and that the main
shortcoming of the RM mechanism is its lack of strategyproofness, my results
suggest that this shortcoming may not be so severe, and the benefits of such a
mechanism in terms of potential efficiency gains may outweigh the costs in
terms of incentives. At the very least, given their other appealing
properties, RM mechanisms are worthy of further consideration for practical
market design.444Featherstone (2020) shows that rank efficiency is
incompatible with strategyproofness, but also shows that in a low-information
environment where beliefs over other agents’ preferences satisfy a symmetry
condition as in Roth and Rothblum (1999), agents are weakly better off by
telling the truth. A main advantage of non-obvious manipulability is that it
does not rely on any assumption about agents’ beliefs. This is discussed
further in Section 3.

##  2 Model

There is a set of $N$ agents $I=\\{i_{1},\ldots,i_{N}\\}$ and a set of $M$
objects $O=\\{o_{1},\ldots,o_{M}\\}$. Each object $o_{m}$ has a capacity
$q_{m}$ which denotes the maximum number of agents who can be assigned to it.
I assume that $\sum_{m=1}^{M}q_{m}\geq N$, which is common in school choice
settings where all students must be offered a seat at a school, and is without
loss of generality if one of the objects is an “outside option” that has
enough capacity for all agents. Each agent has a strict preference relation
$\succ_{i}$ defined on the set of objects $O$, where $o\succ_{i}o^{\prime}$
denotes that agent $i$ strictly prefers object $o$ to object $o^{\prime}$. I
use $o\succsim_{i}o^{\prime}$ when either $o\succ_{i}o^{\prime}$ or
$o=o^{\prime}$. I will sometimes refer to $\succ_{i}$ as an agent’s type. I
will also write $\succ_{i}:o,o^{\prime},\ldots,$ to denote an agent who has
preferences such that her favorite object is $o$, second favorite object is
$o^{\prime}$, and the rest of her preferences can be arbitrary.

For an agent $i$ with type $\succ_{i}$, I define $r_{i}(o)=|o^{\prime}\in
O:o^{\prime}\succsim_{i}o|$ to be the rank of object $o$ according to $i$’s
preferences; in other words, $i$’s favorite object has rank 1, $i$’s second-
favorite object has rank 2, etc.555The rank function depends implicitly on
agent $i$’s preferences $\succ_{i}$, but for readability, I write this as just
$r_{i}(\cdot)$.  Notice that I use the convention that lower numbers
correspond to more preferred objects.

An allocation $\alpha:I\rightarrow O$ is a function that assigns each agent to
an object. I use $\alpha_{i}$ to denote the object assigned to agent $i$ in
allocation $\alpha$. Any allocation must satisfy $|i\in I:\alpha_{i}=o|\leq
q_{m}$ for all $o_{m}\in O$, i.e., each object cannot be assigned to more
agents than its capacity. Let $\mathcal{A}$ be the set of all possible
allocations. Let $\mathcal{P}_{i}$ be agent $i$’s preference domain, which
consists of all strict rankings over $O$, and let
$\mathcal{P}^{I}=\mathcal{P}_{1}\times\cdots\times\mathcal{P}_{N}$. I write
$\succ_{I}=(\succ_{1},\ldots,\succ_{N})\in\mathcal{P}^{I}$ to denote a profile
of preferences, one for each agent $i_{1},\ldots,i_{N}$, and sometimes write
$\succ_{I}=(\succ_{i},\succ_{-i})$ to separate $i$’s preferences $\succ_{i}$
from those of the remaining agents, $\succ_{-i}$. The average rank of any
allocation $\alpha$ is

$\bar{r}(\alpha)=\frac{1}{N}\sum_{i\in I}r_{i}(\alpha_{i}).$

As above, I suppress the dependence of $\bar{r}(\alpha)$ on the preference
profile $\succ_{I}$ for readability. A mechanism is a function
$\psi:\mathcal{P}^{I}\rightarrow 2^{\mathcal{A}}$. Notice that I define the
range of a mechanism $\psi$ to be the power set of $\mathcal{A}$. The reason
for this choice is discussed below, after introducing the main mechanism of
interest.

The focus of this paper is on one mechanism in particular, the rank-minimizing
mechanism. The rank-minimizing (RM) mechanism, $\psi^{RM}$, is given by:

$\psi^{RM}(\succ_{I})=\\{\alpha\in\mathcal{A}:\bar{r}(\alpha)\leq\bar{r}(\alpha^{\prime})\text{
for all }\alpha^{\prime}\in\mathcal{A}\\}.$

In words, $\psi^{RM}$ takes any preference profile as an input and returns
_all_ of the allocations that minimize the average rank. The reason that the
range of the function is set-valued is that there can in general be many
allocations that achieve the minimum average rank, and because the incentive
notion considered in the next section uses a worst-case analysis, it becomes
more natural to formally define a mechanism in this way.

##  3 Main Result

Before giving the main result, I first introduce the notion of non-obvious
manipulability from Troyan and Morrill (2020). They define the concept for any
direct revelation mechanism in which agents report their type. Here, I adapt
their definition to my setting and write it using the rank notation introduced
in the previous section.

Given any set of allocations $A\in\mathcal{A}$, let

$\bar{\rho}_{i}(A)=\max_{\alpha\in A}r_{i}(\alpha_{i})$

$\text{$\underline{\rho}$}_{i}(A)=\min_{\alpha\in A}r_{i}(\alpha_{i}).$

In words, $\bar{\rho}_{i}(A)$ is the rank of agent $i$’s worst assignment over
all of the allocations in the set $A$, where once again the dependence on
$\succ_{i}$ is implicit. That is, $\bar{\rho}_{i}(A)$ is the rank of $i$’s
least-preferred school among all allocations in $A$, where recall that a
higher rank corresponds to a worse school, and thus the worst-case is given by
taking the maximum. Similarly, $\text{$\underline{\rho}$}_{i}(A)$ is the rank
of agent $i$’s best assignment over all of the allocations in the set $A$.

######  Definition 1.

A mechanism $\psi$ is not obviously manipulable if, for any agent $i$ of type
$\succ_{i}$ and any $\succ_{i}^{\prime}\neq\succ_{i}$, the following are true:

  1. (i)

$\max_{\succ_{-i}}\bar{\rho}_{i}(\psi(\succ_{i},\succ_{-i}))\leq\max_{\succ_{-i}}\bar{\rho}_{i}(\psi(\succ_{i}^{\prime},\succ_{-i}))$

  2. (ii)

$\min_{\succ_{-i}}\text{$\underline{\rho}$}_{i}(\psi(\succ_{i},\succ_{-i}))\leq\min_{\succ_{-i}}\text{$\underline{\rho}$}_{i}(\psi(\succ_{i}^{\prime},\succ_{-i}))$

If either of (i) or (ii) does not hold for some agent and type, then
$\succ_{i}^{\prime}$ is an obvious manipulation for agent $i$ of type
$\succ_{i}$, and the mechanism $\psi$ is said to be obviously manipulable.

To understand Definition 1, first consider part (i). On the left-hand side of
the inequality, $\psi(\succ_{i},\succ_{-i})$ is the set of all possible
allocations when $i$ reports her true preferences, $\succ_{i}$, and the other
agents report $\succ_{-i}$. Next, $\bar{\rho}_{i}(\psi(\succ_{i},\succ_{-i}))$
is the rank of $i$’s worst-case outcome in this set. Finally, we take the
maximum over all of the other agent reports, $\succ_{-i}$, to give the rank of
the worst-case outcome. The RHS is the same, just replacing $\succ_{i}$ with a
misreport $\succ_{i}^{\prime}$. The inequality in (i) says that the rank of
the worst-case outcome under the misreport should be worse (i.e., higher) than
that under the truth. Part (ii) of Definition 1 is analogous, except it
compares the best-case outcomes instead, and, since lower numbers correspond
to more preferred outcomes, we replace max with min.

There are several justifications for why a designer might be concerned with
obvious manipulations. As discussed in the introduction, just because a
mechanism _can_ be manipulated does not mean that it _will_ be manipulated,
and Definition 1 is a way to separate those manipulations that are “obvious”,
and are thus likely to be identified by participants, from those that are not.
Formally, Theorem 1 of Troyan and Morrill (2020) shows that obvious
manipulations are precisely those manipulations that can be recognized by an
agent who is cognitively-limited in the sense defined by Li (2017), and is
unable to contingently reason about outcomes state-by-state. Mathematically,
such agents know the range of the function $\psi$ conditional on their own
reports, but not the full function itself, state-by-state.666This is
particularly relevant in the context of school choice. For instance, Troyan
and Morrill (2020) write: “…this could be a neighborhood parent group that
does not fully understand (or has not been told) the assignment algorithm but
has kept track of what preferences have been submitted and what the resulting
assignments were.” Such parent groups are indeed quite common; see Pathak and
Sönmez (2008). Allowing some manipulations so long as they are non-obvious
widens the space of mechanisms available to the designer, which may allow for
improvements on other dimensions, such as the average rank. Further, unlike
other relaxations of strategyproofness such as Bayesian incentive
compatibility or approximate notions such as strategyproofness-in-the-large
(SPL, Azevedo and Budish, 2019), NOM requires no assumptions on how
preferences are drawn or agent beliefs. Rather, NOM is defined using best and
worst case scenarios, which are likely to be particularly salient.777Following
Troyan and Morrill (2020), several other papers have applied non-obvious
manipulability to various settings, including Aziz and Lam (2021), Ortega and
Segal-Halevi (2019), Archbold et al. (2022), Troyan et al. (2020), and Cerrone
et al. (2022). Also similar in spirit, though technically different, is Li and
Dworczak (2021) who show that a designer can sometimes be better off using a
non-SP mechanism even when agents are unsophisticated, no matter how they
resolve their “strategic confusion”.

Given Definition 1, I can now state the main result of the paper, Theorem 1
below.

######  Theorem 1.

The rank-minimizing mechanism is not obviously manipulable.

The proof of this theorem can be found at the end of the paper, following the
Discussion and Conclusion section.

##  4 Discussion and Conclusion

The rank-minimizing mechanism is an appealing mechanism for assignment
problems, because it directly optimizes a natural objective that is desirable
to policy-makers. Thus, it is somewhat striking that there has been thus far
relatively little written about this mechanism in the literature, while there
have been hundreds of papers written about mechanisms such as DA and TTC. One
possible explanation for this gap is that the literature is overly-focused on
strategyproofness as an incentive property, which is satisfied by both DA and
TTC. While strategyproofness is a very appealing desideratum, it limits the
flexibility for a designer to optimize on other important dimensions. This
paper shows that, while not strategyproof, the rank-minimizing mechanism is at
least not obviously manipulable.

Besides Featherstone (2020), who provides a detailed analysis of rank
efficiency criteria and related mechanisms and was discussed in the
Introduction, there are only two other papers I am aware of in the economics
literature that analyze the RM mechanism. Nikzad (2022) studies the RM
mechanism in large markets, and provides an upper bound on the expected
average rank (see also Sethuraman (2022) for another proof of the same
result). He also shows that the RM mechanism is Bayesian incentive compatible
when agent preference rankings are drawn uniformly at random. Ortega and Klein
(2022) study the average rank of RM, DA, and TTC in large markets, both
theoretically and using simulations, and find that RM outperforms DA and TTC
on important dimensions such as efficiency and fairness. They also use data
from secondary school admissions in Hungary to analyze the three mechanisms in
an empirical setting, and find similar support for RM. The data is from DA, a
strategyproof mechanism, and they write:

> One potential concern is that students would submit different preferences
> when allocations are determined by RM, which is not strategy-proof. This
> concern is mitigated because the potential gains from misrepresentation are
> small…Moreover, there is evidence than [_sic_] truthful behavior can depend
> more on the complexity of the mechanism rather than actual strategy-
> proofness (Cerrone et al., 2022).

My result showing the non-obvious manipulability of RM can be seen as a
formalization of this final point of Ortega and Klein (2022). Whether the
rank-minimizing mechanism will actually be manipulated in practice is of
course ultimately an empirical question, but given my result and the other
desirable properties of the mechanism, RM is a mechanism that seems worthy of
further investigation.

### Proof of Theorem 1

I start with the following lemma.

######  Lemma 1.

Consider a preference profile $\succ_{I}^{\prime}$ such that for all $j\in I$,
agent $j$’s preferences are
$\succ_{j}^{\prime}:o_{m_{1}},o_{m_{2}},\ldots,o_{m_{M}}$, and let
$k^{*}=\min\\{k:\sum_{k^{\prime}=1}^{k^{{}^{\prime}}}q_{m_{k^{\prime}}}\geq
N\\}$. Further, let $O^{\prime}=\\{o_{m_{1}},\ldots,o_{m_{k^{*}}}\\}$ and
$A^{*}\subseteq\mathcal{A}$ be the subset of allocations that satisfy:

  1. (I)

All $o_{m_{k}}\in O^{\prime}\setminus\\{o_{m_{k^{*}}}\\}$ are assigned to
exactly $q_{m_{k}}$ agents.

  2. (II)

Object $o_{m_{k^{*}}}$ is assigned to exactly
$N-\sum_{k^{\prime}=1}^{k^{*}-1}q_{m_{k^{\prime}}}$ agents.

Then, $\psi^{RM}(\succ_{I}^{\prime})=A^{*}$. Further, for any $\tilde{o}\in
O^{\prime}=\\{o_{m_{1}},\ldots,o_{m_{k^{*}}}\\}$, there is at least one
allocation $\alpha\in\psi^{RM}(\succ_{I}^{\prime})$ such that
$\alpha_{i}=\tilde{o}.$

In words, object $o_{m_{k^{*}}}$ is the critical object, in the sense that the
set $O^{\prime}=\\{o_{m_{1}},\ldots,o_{m_{k^{*}}}\\}$ has enough total
capacity to accommodate all agents, while the set
$O^{\prime}\setminus\\{o_{m_{k^{*}}}\\}$ does not. The set $A^{*}$ is then the
set of all allocations that consist of all possible ways of assigning the $N$
agents to the objects in $O^{\prime}$ such that
$o_{m_{1}},\ldots,o_{m_{k^{*}-1}}$ are filled to capacity, and all remaining
agents are assigned to $o_{m_{k^{*}}}$. The lemma says that when all agents
have the same preferences, any rank-minimizing allocation must satisfy (I) and
(II).

_Proof of Lemma 1._ For any allocation $\alpha$, let $\alpha_{m_{k}}=\\{i\in
I:\alpha_{i}=o_{m_{k}}\\}$ be the set of agents assigned to $o_{m_{k}}$. Under
the preference profile given, the average rank of any allocation is:

$\bar{r}(\alpha)=\frac{1}{N}\left(\sum_{k=1}^{M}k|\alpha_{m_{k}}|\right).$

This is clearly minimized by an allocation that assigns $q_{m_{1}}$ students
to $o_{m_{1}}$, $q_{m_{2}}$ students to $o_{m_{2}}$, etc., until all students
are assigned. This is precisely the set of allocations $A^{*}$, and so by
construction, any allocation $\alpha\in A^{*}$ achieves the minimum average
rank, i.e., $\alpha\in\psi^{RM}(\succ_{I})$. For any allocation
$\alpha^{\prime}\notin A^{*}$, either (I) or (II) must fail. If (I) fails, let
$o_{\hat{m}}\in\\{o_{1},\ldots,o_{m_{k^{*}-1}}\\}$ be an object that is not
filled to capacity. Then, there must be some agent $j$ assigned to some object
$o_{m_{k^{*}}},\ldots,o_{m_{M}}$. Reassigning agent $j$ to object
$o_{\hat{m}}$ and leaving all other assignments the same lowers the average
rank, and so $\alpha^{\prime}\notin\psi^{RM}(\succ_{I}).$ Therefore, we have
$\psi^{RM}(\succ_{I}^{\prime})=A^{*}$. Lastly, because all agents have the
same preferences, it does not matter precisely which agents are assigned to
which schools, and so by symmetry, there exists at least one
$\alpha\in\psi^{RM}(\succ_{I}^{\prime})$ such that $\alpha_{i}=\tilde{o}$ for
all $\tilde{o}\in O^{\prime}$.

$\hfill\blacksquare$

Now, consider an agent $i$ with type $\succ_{i}$. I show that for any
$\succ_{i}^{\prime}\neq\succ_{i}$, each part of Definition 1 holds for the
rank-minimizing mechanism.

#### Part (i):
$\max_{\succ_{-i}}\bar{\rho}_{i}(\psi(\succ_{i},\succ_{-i}))\leq\max_{\succ_{-i}}\bar{\rho}_{i}(\psi(\succ_{i}^{\prime},\succ_{-i}))$.

Index agent $i$’s true type as
$\succ_{i}:o_{m_{1}},o_{m_{2}},\ldots,o_{m_{M}}$. I first show that when $i$
reports her true preferences,
$\max_{\succ_{-i}^{\prime}}\bar{\rho}_{i}(\psi^{RM}(\succ_{i},\succ_{-i}^{\prime}))=k^{*}$,
where $k^{*}$ is as defined in Lemma 1. When $\succ_{j}=\succ_{i}$ for all
$j$, there is at least one allocation $\alpha\in\psi^{RM}(\succ_{I})$ such
that $\alpha_{i}=o_{m_{k^{*}}},$ by Lemma 1. This implies
$\max_{\succ_{-i}^{\prime}}\bar{\rho}_{i}(\psi^{RM}(\succ_{i},\succ_{-i}^{\prime}))\geq
k^{*}$. To show equality, assume that
$\max_{\succ_{-i}}\bar{\rho}_{i}(\psi^{RM}(\succ_{i},\succ_{-i}))>k^{*}$.
Then, there must be some $\succ_{-i}^{\prime}$ and some
$\alpha^{\prime}\in\psi^{RM}(\succ_{i},\succ_{-i}^{\prime})$ such that
$\alpha_{i}^{\prime}=o_{m_{k^{\prime}}}$ for some $k^{\prime}>k^{*}$. But,
this implies that there is some $k^{\prime\prime}\leq k^{*}$ such that object
$o_{m_{k^{\prime\prime}}}$ is not filled to capacity. Thus, consider an
alternative allocation $\alpha^{\prime\prime}$ where agent $i$ is reassigned
to $o_{m_{k^{\prime\prime}}}$ and all other agents have the same assignment as
in $\alpha^{\prime}$. Then, we have
$\bar{r}(\alpha^{\prime\prime})<\bar{r}(\alpha^{\prime})$, i.e., this lowers
the average rank, which contradicts that
$\alpha^{\prime}\in\psi^{RM}(\succ_{i},\succ_{-i}^{\prime})$. Therefore,
$\max_{\succ_{-i}^{\prime}}\bar{\rho}_{i}(\psi^{RM}(\succ_{i},\succ_{-i}^{\prime}))=k^{*}$.

Thus, I have shown that when $i$ reports her true preferences, her worst-case
outcome is
$\max_{\succ_{-i}^{\prime}}\bar{\rho}_{-i}(\psi^{RM}(\succ_{i},\succ_{-i}^{\prime}))=k^{*}$.
What remains to show is that for any misreport
$\succ_{i}^{\prime}\neq\succ_{i},$ we have
$\max_{\succ_{-i}^{\prime}}\bar{\rho}_{i}(\psi^{RM}(\succ_{i}^{\prime},\succ_{-i}^{\prime}))\geq
k^{*}$, where of course $\bar{\rho}_{i}$ is evaluated with respect to $i$’s
true preferences. Consider a misreport $\succ_{i}^{\prime}\neq\succ_{i}$. For
notational purposes, index this preference profile as

$\succ_{i}^{\prime}:o_{r_{1}},o_{r_{2}},\ldots,o_{r_{M}}.$

Let $k^{**}=\min\\{k:\sum_{k^{\prime}=1}^{k}q_{r_{k^{\prime}}}\geq N\\}$.
Similar to Lemma 1, this is the index of the critical object in the sense that
the set $O^{\prime\prime}=\\{o_{r_{1}},\ldots,o_{r_{k^{**}}}\\}$ has enough
total capacity for all agents, but the set
$\\{o_{r_{1}},\ldots,o_{r_{k^{**}-1}}\\}$ does not. Consider a preference
profile $\succ_{I}^{\prime}$ such that $\succ_{j}^{\prime}=\succ_{i}^{\prime}$
for all $j\in I$. By Lemma 1, we have
$\psi^{RM}(\succ_{I}^{\prime})=A{}^{**}$, where $A^{**}$ is defined
analogously to $A^{*}$ above, replacing $O^{\prime}$ with the set
$O^{\prime\prime}$.

Case 1: $O^{\prime\prime}=\\{o_{1},\ldots,o_{m_{k^{*}}}\\}$.

By Lemma 1, there is at least one allocation
$\alpha\in\psi^{RM}(\succ_{I}^{\prime})$ such that $\alpha_{i}=o_{m_{k^{*}}}$.
Therefore,
$\max_{\succ_{-i}^{\prime}}\bar{\rho}_{i}(\psi^{RM}(\succ_{i}^{\prime},\succ_{-i}^{\prime})\geq
k^{*}$, as desired.

Case 2: $O^{\prime\prime}\neq\\{o_{1},\ldots,o_{m_{k^{*}}}\\}$.

In this case, there is some $\tilde{o}\in O^{\prime\prime}$ such that
$o_{m_{k^{*}}}\succ_{i}\tilde{o}$ according to $i$’s true preferences
$\succ_{i}$. By Lemma 1, there is at least one allocation
$\alpha\in\psi^{RM}(\succ_{I}^{\prime})$ such that $\alpha_{i}=\tilde{o}$.
Because $o_{m_{k^{*}}}\succ_{i}\tilde{o}$, this implies that
$\max_{\succ_{-i}^{\prime}}\bar{\rho}_{i}(\psi^{RM}(\succ_{i}^{\prime},\succ_{-i}^{\prime})>k^{*},$
as desired.

This completes the argument for part (i).

#### Part (ii):
$\min_{\succ_{-i}}\text{$\underline{\rho}$}_{i}(\psi(\succ_{i},\succ_{-i}))\leq\min_{\succ_{-i}}\text{$\underline{\rho}$}_{i}(\psi(\succ_{i}^{\prime},\succ_{-i}))$.

Without loss of generality, consider agent $i_{1}$ whose preferences are
$\succ_{1}:o_{1},\ldots,o_{M}$. As in part (i), let
$m^{*}=\min\\{m^{\prime}:\sum_{m=1}^{m^{\prime}}q_{m^{\prime}}\geq N\\}$.
Consider preference profile $\succ_{-i}$ for the other agents constructed as
follows:

  * •

Exactly $q_{1}-1$ agents have preferences such that $\succ_{j}:o_{1},\ldots.$

  * •

For all $m^{\prime}=2,\ldots,m^{*}-1$, exactly $q_{m^{\prime}}$ agents have
preferences such that $\succ_{j}:o_{m^{\prime}},\ldots.$

  * •

Exactly $N-\sum_{m^{\prime}=1}^{m^{*}-1}q_{m^{\prime}}$ agents have
preferences such that $\succ_{j}:o_{m^{*}},\ldots.$

In words, the constructed profile $\succ_{I}$ is such that each object $o_{m}$
has exactly $q_{m}$ agents who have ranked it first. This is possible by the
definition of $m^{*}$ and the assumption that $\sum_{m}q_{m}\geq N$. Now,
notice that at this profile, there is a unique rank-minimizing allocation,
$\psi^{RM}(\succ_{I})=\\{\alpha^{*}\\}$, where $\alpha^{*}$ is the allocation
such that each agent is assigned to her first-choice object. Thus,
$\text{$\underline{\rho}$}_{i}(\psi(\succ_{i},\succ_{-i}))=1$, and so
$\min_{\succ_{-i}}\text{$\underline{\rho}$}_{i}(\psi(\succ_{i},\succ_{-i}))=1$.
Since it is obvious that
$\text{$\underline{\rho}$}_{i}(\psi(\succ_{i}^{\prime},\succ_{-i}))\geq 1$ for
any $(\succ_{i}^{\prime},\succ_{-i})$, we have
$\min_{\succ_{-i}}\text{$\underline{\rho}$}_{i}(\psi(\succ_{i},\succ_{-i}))\leq\min_{\succ_{-i}}\text{$\underline{\rho}$}_{i}(\psi(\succ_{i}^{\prime},\succ_{-i}))$,
and therefore part (ii) of Definition 1 holds.

$\hfill\blacksquare$

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# Wireless Information Surveillance via STAR-RIS

Fatemeh Jafarian†, Mehrdad Ardebilipour†, Mohammadali Mohammadi‡, and Michail
Matthaiou‡
† Department of Electrical Engineering, K. N. Toosi University of Technology,
Tehran, Iran
‡Centre for Wireless Innovation (CWI), Queen’s University Belfast, U.K.
<EMAIL_ADDRESS><EMAIL_ADDRESS>{m.mohammadi<EMAIL_ADDRESS>
###### Abstract

We explore the potential of a simultaneously transmitting and reflecting
reconfigurable intelligent surface (STAR-RIS) to enhance the performance of
wireless surveillance systems. The STAR-RIS is deployed between a full-duplex
(FD) multi-antenna legitimate eavesdropper (E) and a suspicious communication
pair. It reflects the suspicious signal towards the suspicious receiver (SR),
while simultaneously transmitting the same signal to E for interception
purposes. Additionally, it enables the forwarding of a jamming signal from E
to SR, which is located on the back side of the STAR-RIS. To enhance the
eavesdropping non-outage probability, we formulate a non-convex joint
optimization problem to design the beamforming vectors at E and
reflection/transmission phase shift matrices at the STAR-RIS. We adopt the
block coordinate descent (BCD) algorithm and propose an approach, mainly based
on semi-definite relaxation (SDR) and successive convex approximation (SCA),
for solving the resulting decoupled sub-problems. Finally, we compare the
performance of the proposed design against low-complexity zero-forcing
(ZF)-based beamforming designs.

††footnotetext: The work of M. Mohammadi and M. Matthaiou was supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 101001331).

##  I Introduction

Wireless information surveillance can allow authorized parties (e.g., National
Security Agency, military) to legally supervise and identify abnormal user
behaviors in wireless networks [1]. It involves utilizing physical layer
techniques, such as jamming-assisted [1, 2] and spoofing relaying-assisted
proactive eavesdropping [3], to manipulate the suspicious link aligned with
the legitimate eavesdropping channel, i.e., the link between the suspicious
transmitter (ST) and E. This ensures that E can intercept dubious information.

The eavesdropping performance of these systems is practically limited.
Successful monitoring can only be achieved when E is in proximity to the ST or
when a direct link exists between the ST and E [4]. Therefore, wireless
surveillance systems primarily employ active FD relays to simultaneously
forward/overhear the suspicious signal and interfere with the suspicious link
[3]. To address the challenges of delay processing and energy consumption at
active relays, surveillance systems have recently adopted the emerging RIS
technology [5, 6, 7, 4, 8]. The RIS signal enhancement/ cancellation
capabilities can be leveraged to intelligently reflect the received suspicious
signal by adjusting the phase shifts of all passive elements, turning zero
delay into a reality without requiring any additional transmission power for
signal forwarding [9].

Existing research contributions [5, 6, 7, 4, 8] assume that the RIS can only
reflect the incident signals, implying that the ST and SR must be located on
the same side of the RIS. However, this geographical restriction may not
always be met in practice and limits the applicability of RISs, as users are
typically distributed on both sides of RISs. To overcome such limitations, the
novel concept of STAR-RIS has been introduced [10, 11]. Unlike conventional
RISs, each element of a STAR-RIS can simultaneously reflect and transmit the
incident signals, eliminating the need to confine their deployment to specific
geographic areas and achieving full-space coverage. Moreover, since both the
transmission and reflection coefficients can be designed, a STAR-RIS offers
new degrees-of-freedom (DoF) to enhance the performance of wireless systems.
Recently, a few initial works [12, 13] have been conducted to investigate the
potential of STAR-RISs in wireless surveillance systems. The work in [13]
considered a half-duplex E, while the authors in [12] deployed a dual-antenna
FD E, where one antenna is used to overhear the suspicious signal and the
other antenna is employed to interfere with the SR. Hence, the potential use
of multi-antenna arrays at E remains unexplored. Additionally, the design of
phase shifts was neglected in both [12, 13].

We here explore a STAR-RIS-assisted wireless surveillance scenario, where a
multi-antenna FD E aims to eavesdrop on a suspicious communication between a
pair of ST-SR. The STAR-RIS adopts an energy-splitting protocol [10] and
reflects the suspicious signal to the SR, while simultaneously transmitting
the suspicious signal to the E and forwards the jamming signal from E to the
SR. Our main contributions are as follows:

  * •

We jointly optimize the passive transmission/reflection coefficients at the
STAR-RIS and the active transmit/receive beamforming vectors at the FD E.
Although the resulting problem is a complicated non-convex optimization
problem, we solve it by adopting the BCD algorithm and decompose it into
tractable sub-problems. Then, we propose an approach based on SDR and SCA for
solving the decoupled sub-problems.

  * •

As a benchmark, two low-complexity designs are proposed, where the ZF
principle is used at the beamforming design stage to effectively cancel the
self-interference (SI) at the FD E. Our numerical results reveal that the
proposed joint beamforming and phase-shift design can significantly improve
the information surveillance performance of the system compared to the
benchmarks.

_Notations:_ We use bold upper case letters to denote matrices, and lower case
letters to denote vectors. The notations $(\cdot)^{{\dagger}}$ and
$(\cdot)^{T}$ denote the Hermitian transpose and transpose, respectively;
$\|\cdot\|$ denotes the Euclidean norm of a complex vector; $|\cdot|$ denotes
the absolute value of a complex scalar; $\mathrm{tr}(\cdot)$ and
$(\cdot)^{-1}$ denote the trace and inverse operation of a matrix;
$\mathrm{diag}\\{{\bf A}\\}$ and $\mathrm{diag}\\{{\bf a}\\}$ represent a
vector whose elements are extracted from the main diagonal elements of the
matrix ${\bf A}$ and a diagonal matrix with ${\bf a}$ on its main diagonal,
respectively; ${\bf I}_{M}$ represents the $M\times M$ identity matrix;
$\bm{1}_{N}$ denotes an all-ones vector of size $N$. A zero mean circular
symmetric complex Gaussian variable having variance $\sigma^{2}$ is denoted by
$\mathcal{CN}(0,\sigma^{2})$. Finally, $\mathbb{E}\\{\cdot\\}$ denotes the
statistical expectation.

##  II System model

We consider a wireless surveillance system where FD E aims to eavesdrop on the
communication link between a single ST-SR pair, as shown in Fig. 1. However, E
is deliberately positioned far away from the suspicious system to avoid
detection, resulting in a weak eavesdropping link. To overcome this
limitation, a STAR-RIS is deployed between E and the suspicious system to
establish an effective bridge for eavesdropping and jamming. The STAR-RIS,
comprised of $N$ transmission/reflection elements, employs an energy-splitting
protocol [10]. This protocol adaptively adjusts the channel power gains of the
suspicious and legitimate eavesdropping links, thereby enhancing the
eavesdropping capabilities. The STAR-RIS assists E in overhearing the ST,
while simultaneously forwarding a jamming signal from E towards SR. E operates
in FD mode, overhearing the suspicious signal from ST via $N_{R}$ receive
antennas and sending a jamming signal via $N_{T}$ transmit antennas to
interfere with SR. ST and SR are equipped with a single antenna each.

Let $h_{SD}=\sqrt{\gamma_{SD}}\tilde{h}_{SD}$, ${{\bf
h}_{SR}}=\sqrt{\gamma_{SR}}\tilde{{\bf h}}_{SR}\in\mathbb{C}^{N\times 1}$,
${\bf h}_{RD}=\sqrt{\gamma_{RD}}\tilde{{\bf h}}_{RD}\in\mathbb{C}^{N\times
1}$, and ${\bf H}_{RE}=\sqrt{\gamma_{RE}}\tilde{{\bf
H}}_{RE}\in\mathbb{C}^{N_{R}\times N}$ represent the channel coefficients for
ST-to-SR, the ST-to-STAR-RIS, the STAR-RIS-to-SR, and the STAR-RIS-to-E link,
respectively, where $\gamma_{XY}$ denotes the large-scale fading between node
$X\in\\{S,R\\}$ and $Y\in\\{D,R,E\\}$. Moreover, $\tilde{h}_{SD}$,
$\tilde{{\bf h}}_{SR}$, $\tilde{{\bf h}}_{RD}$, and $\tilde{{\bf H}}_{RE}$
denote the small-scale fading components each having $\mathcal{CN}(0,1)$
elements. The SI link at E is denoted by ${{\bf
H}_{EE}}\in\mathbb{C}^{N_{R}\times N_{T}}$, whose elements can be modeled as
$\mathcal{CN}(0,\sigma_{\mathtt{SI}}^{2})$ random variables [14]. We express
the transmission and reflection coefficient matrices of the STAR-RIS as
$\bm{\Theta}_{t}=\mathrm{diag}\big{(}\big{[}\sqrt{\beta_{1}^{t}}e^{j\theta_{1}^{t}},\ldots,\sqrt{\beta_{N}^{t}}e^{j\theta_{N}^{t}}\big{]}\big{)}$
and
$\bm{\Theta}_{r}=\mathrm{diag}\big{(}\big{[}\sqrt{\beta_{1}^{r}}e^{j\theta_{1}^{r}},\ldots,\sqrt{\beta_{N}^{r}}e^{j\theta_{N}^{r}}\big{]}\big{)}$
respectively, where $\beta_{n}^{t}$ ($\beta_{n}^{r}$) and $\theta_{n}^{t}$
($\theta_{n}^{r}$), $\forall n\in\left\\{1,2,\ldots,N\right\\}$ denote the
transmission (reflection) energy splitting factors and phase shifts,
respectively. In addition, we note that the phase shifts
$\theta_{n}^{t},~{}\theta_{n}^{r}\in\left[0,2\pi\right]$ are generally chosen
independently of each other. In contrast, according to the law of energy
conservation, $\beta_{n}^{t},~{}\beta_{n}^{r}\in\left[0,1\right]$ are coupled
with each other, i.e., they should satisfy the constraint of
$\beta_{n}^{t}+\beta_{n}^{r}=1,\forall n\in\left\\{1,2,\ldots,N\right\\}$
[11]. Similar to [12, 13], we adopt the same energy splitting ratio for each
element of the STAR-RIS, i.e., $\beta_{1}^{t}=\ldots=\beta_{N}^{t}=\beta^{t}$,
$\beta_{1}^{r}=\ldots=\beta_{N}^{r}=\beta^{r}$ and $\beta^{t}+\beta^{r}=1$.

###  II-A Transmission Protocol

Assume that the ST transmits signal ${x}_{s}$ to SR with transmit power
${P}_{s}$. Hence, the received signal of the SR is given by

$\displaystyle~{}{y}_{D}\\!=\\!$ $\displaystyle\sqrt{P_{\mathtt{E}}}{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf H}_{\mathtt{ER}}}{\bf
w}_{t}{{x}_{j}}$ $\displaystyle\hskip
30.00005pt+\sqrt{{P_{\mathtt{s}}}}\big{(}{h_{SD}}+\\!{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf
h}_{\mathtt{SR}}\big{)}{{x}_{s}}\\!+\\!{{n}_{D}},$ (1)

where ${\bf w}_{t}\in\mathbb{C}^{N\times 1}$ is the transmit beamforming
vector, $P_{\mathtt{E}}$ denotes the jamming power of E, ${x}_{j}$ denotes the
jamming signal of E, and $n_{D}\sim\mathcal{CN}(0,\sigma_{\mathtt{D}}^{2})$.
Hence, the received signal-to-interference-plus-noise ratio (SINR) at SR is
given by

$~{}\mathrm{SINR}_{\mathtt{D}}=\frac{{{P_{\mathtt{s}}}}|{h_{SD}}+{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf H}_{\mathtt{ER}}}{\bf
w}_{t}|^{2}+\sigma_{\mathtt{D}}^{2}}.$ (2)

Let ${\bf w}_{r}\in\mathbb{C}^{N\times 1}$ denote the receive beamforming
vector at E. Then, the received signal at E is given by

$\displaystyle~{}y_{E}=$ $\displaystyle\sqrt{P}_{\mathtt{s}}{\bf
w}_{r}^{{\dagger}}{\bf H}_{\mathtt{RE}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}{x}_{s}+\sqrt{P_{\mathtt{E}}}\big{(}{\bf
w}_{r}^{{\dagger}}{\bf H}_{\mathtt{EE}}{\bf w}_{t}$ $\displaystyle+{\bf
W}_{r}^{{\dagger}}{\bf H}_{\mathtt{RE}}\bm{\Theta}_{r}{\bf
H}_{\mathtt{ER}}{\bf w}_{t}\big{)}{x}_{j}+{\bf w}_{r}^{{\dagger}}{{\bf
n}}_{E}.$ (3)

where ${\bf n}_{E}\in\mathbb{C}^{N\times 1}$ is the received noise vector such
that ${\bf n}_{E}\sim\mathcal{CN}(\bm{0},\sigma_{\mathtt{E}}^{2}{\bf I}_{N})$.
Accordingly, the received SINR at E can be expressed as

$\displaystyle~{}\mathrm{SINR}_{\mathtt{E}}$
$\displaystyle=\frac{P_{\mathtt{s}}|{\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{EE}}}{{\bf w}_{t}}+{{\bf w}_{r}^{{\dagger}}{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}{{\bf
w}_{t}}|^{2}+\sigma_{\mathtt{E}}^{2}}.$ (4)

Figure 1: Illustration of the considered STAR-RIS-assisted wireless
surveillance system.

As the performance metric for the considered surveillance system, we focus on
the eavesdropping non-outage probability, denoted as
$P_{\mathrm{NOP}}\triangleq\mathbb{E}\\{X\\}$. The indicator function $X$ in
$\mathbb{E}\\{X\\}$ denotes the event of successful eavesdropping at E, given
by

$\displaystyle X=\begin{cases}\displaystyle 1&\text{if
}\mathrm{SINR}_{\mathtt{E}}\geq\mathrm{\mathrm{SINR}_{\mathtt{D}}},\\\
\displaystyle 0&\text{otherwise}.\displaystyle\end{cases}$ (5)

where $X=1$ and $X=0$ indicate the eavesdropping non-outage and outage events,
respectively. In other words, to achieve a reliable detection at SR, ST varies
its transmission rate according to $\mathrm{SINR}_{\mathtt{D}}$. Hence, if
$\mathrm{SINR}_{\mathtt{E}}\geq\mathrm{\mathrm{SINR}_{\mathtt{D}}}$, then E
can also reliably decode the information intended to SR. On the other hand, if
$\mathrm{SINR}_{\mathtt{E}}<\mathrm{SINR}_{\mathtt{D}}$, E is unable to decode
this information without any error [15, 16]. The eavesdropping non-outage
probability is mathematically represented by

$\displaystyle P_{\mathrm{NOP}}({\bf w}_{r},{\bf
w}_{t},\bm{\Theta}_{r},\bm{\Theta}_{t})=\Pr\big{(}\mathrm{SINR}_{\mathtt{E}}>\mathrm{SINR}_{\mathtt{D}}\big{)},$
$\displaystyle\hskip 20.00003pt=\Pr\bigg{(}\frac{P_{\mathtt{s}}|{\bf
w}_{r}^{{\dagger}}{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{EE}}}{{\bf w}_{t}}+{{\bf w}_{r}^{{\dagger}}{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}{{\bf
w}_{t}}|^{2}+\sigma_{\mathtt{E}}^{2}}$ $\displaystyle\hskip
60.00009pt>\frac{{{P_{\mathtt{s}}}}|{h_{SD}}+{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf H}_{\mathtt{ER}}}{\bf
w}_{t}|^{2}+\sigma_{\mathtt{D}}^{2}}\bigg{)}.$ (6)

###  II-B Problem Formulation

The eavesdropping non-outage probability expression in (II-A) depends on the
transmit/receive beamforming vectors at E, i.e., ${\bf w}_{t}$, ${\bf w}_{r}$,
as well as transmission/reflection matrices at the STAR-RIS, i.e.,
$\bm{\Theta}_{t}$, and $\bm{\Theta}_{r}$. Therefore, we can jointly optimize
${\bf w}_{t}$, ${\bf w}_{r}$, $\bm{\Theta}_{t}$, and $\bm{\Theta}_{r}$ to
enhance the surveillance performance. To characterize the fundamental
information theoretic performance limits of the considered wireless
information surveillance system, we assume that E has perfect channel state
information (CSI) of all links [12, 13]. In practice, E can overhear the pilot
signals sent by ST and SR to acquire the CSI of ${\bf h}_{SR}$, ${\bf
h}_{SR}$, and ${\bf H}_{RE}$. To this end, the proposed channel estimation
method in [17] can be applied. On the other hand, E can obtain the CSI of
$h_{SD}$ by eavesdropping the feedback channels of the suspicious transmitter-
receiver pair [18].

Noticing that the STAR-RIS operates in energy-splitting mode, the joint
optimization problem can be formulated as

$\displaystyle\underset{{\bf w}_{r},{\bf
w}_{t},\bm{\Theta}_{r},\bm{\Theta}_{t}}{\max}\,\,\hskip 5.0pt$ $\displaystyle
P_{\mathrm{NOP}}({\bf w}_{r},{\bf w}_{t},\bm{\Theta}_{r},\bm{\Theta}_{t}),$
(7a) $\displaystyle\mathrm{s.t.}\,\,\hskip 20.00003pt$
$\displaystyle\beta^{r}+\beta^{t}=1,~{}$ (7b)
$\displaystyle~{}\|{e^{j\theta_{t}}}\|=\|{e^{j\theta_{r}}}\|=1,$ (7c)
$\displaystyle~{}\|{\bf w}_{r}\|=\|{{\bf w}_{t}}\|=1.~{}$ (7d)

##  III Joint Beamforming and Phase-shift Design

In this section, we propose an optimal design to jointly optimize the
beamforming vectors at E and transmission/reflection coefficients at STAR-RIS.
Before proceeding, by invoking (2) and (4), we first re-express the objective
function in a more tractable form. Therefore, the optimization problem (7) can
be recast as

$\displaystyle~{}\underset{{\bf w}_{r},{\bf
w}_{t},\bm{\Theta}_{r},\bm{\Theta}_{t}}{\min}\,\,\hskip 1.00006pt$
$\displaystyle\bigg{(}\frac{|{h_{SD}}+{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf H}_{\mathtt{ER}}}{\bf
w}_{t}|^{2}+\sigma_{\mathtt{D}}^{2}}$ $\displaystyle-\frac{|{\bf
w}_{r}^{{\dagger}}{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{EE}}}{{\bf w}_{t}}+{{\bf w}_{r}^{{\dagger}}{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}{{\bf
w}_{t}}|^{2}+\sigma_{\mathtt{E}}^{2}}\bigg{)},$ (8a)
$\displaystyle\mathrm{s.t.}\,\,\hskip 20.00003pt$
$\displaystyle~{}\eqref{P1:beta:const}-\eqref{P1:wrwt1}.$ (8b)

It is obvious that (8) is a non-convex problem due to its non-convex objective
and the phase shift-related constraints. To tackle this issue, we apply the
widely used classic BCD algorithm to solve (8). To this end, we partition the
optimization variables into two blocks, 1) receive/transmit semaphores at E,
i.e., ${\bf w}_{r}$ and ${\bf w}_{t}$; 2) Transmission/reflection phase shifts
at the STAR-RIS, i.e., $\bm{\Theta}_{t}$ and $\bm{\Theta}_{r}$. Then, we
minimize the objective function in (8) by iteratively optimizing each of the
above two blocks in one iteration while the other block is fixed, until
convergence is reached.

###  III-A Beamforming Design at E

By inspecting the objective function (8), we find out that only the second
term, i.e., $\frac{|{\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{EE}}}{{\bf w}_{t}}+{{\bf w}_{r}^{{\dagger}}{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}{{\bf
w}_{t}}|^{2}+\sigma_{\mathtt{E}}^{2}}$ depends on ${\bf w}_{r}$. Therefore,
for a given ${\bf w}_{t}$, the optimal ${\bf w}_{r}$ is the solution of the
following optimization problem

$\displaystyle~{}\max_{\|{\bf w}_{r}\|=1}\qquad\frac{|{\bf
w}_{r}^{{\dagger}}{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{EE}}}{{\bf w}_{t}}+{{\bf w}_{r}^{{\dagger}}{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}{{\bf
w}_{t}}|^{2}+\sigma_{\mathtt{E}}^{2}}.$ (9)

Since (9) is a generalized Rayleigh ratio problem [14], the optimal receive
beamformer can be obtained in closed-form

$\displaystyle{\bf w}_{r}^{\ast}=\frac{(\rho_{e}{\bf B}+{\bf
I}_{N_{R}})^{-1}{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}}{\|(\rho_{e}{\bf B}+{\bf I}_{N_{R}})^{-1}{\bf
H}_{\mathtt{RE}}\bm{\Theta}_{t}{\bf h}_{\mathtt{SR}}\|},$ (10)

where $\rho_{e}\\!=\\!{P_{\mathtt{E}}}/{\sigma_{\mathtt{E}}^{2}}$ is the
normalized signal-to-noise ratio (SNR) of the jamming symbols, while ${\bf
B}\triangleq({\bf H}_{\mathtt{EE}}+{\bf H}_{\mathtt{RE}}\bm{\Theta}_{r}{{\bf
H}_{\mathtt{ER}}}){\bf W}({{\bf H}_{\mathtt{EE}}^{{\dagger}}}+{{\bf
H}_{\mathtt{ER}}^{{\dagger}}}{\bm{\Theta}_{r}^{{\dagger}}}{\bf
H}_{\mathtt{RE}}^{{\dagger}})$, with ${\bf W}\triangleq{{\bf w}_{t}}{{\bf
w}_{t}^{{\dagger}}}$. By substituting ${\bf w}_{r}^{\ast}$ into (8), the
latter can be written as

$\displaystyle\underset{\|{{\bf w}_{t}}\|=1}{\min}\,\,$ $\displaystyle\hskip
20.00003pt~{}\frac{\frac{\sigma_{\mathtt{E}}^{2}}{\sigma_{\mathtt{D}}^{2}}|{h_{SD}}+{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf
h}_{\mathtt{SR}}|^{2}}{1+\frac{{P_{\mathtt{E}}}}{\sigma_{\mathtt{D}}^{2}}|{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf H}_{\mathtt{ER}}}{\bf
w}_{t}|^{2}}$ (11)
$\displaystyle+\frac{\rho_{e}|{\textbf{h}_{SR}^{{\dagger}}}{\bm{\Theta}_{t}^{{\dagger}}}{{\bf
H}_{\mathtt{RE}}^{{\dagger}}}({{\bf H}_{\mathtt{EE}}}{{{\bf w}}_{t}}+{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}{{\bf
w}_{t}})|^{2}}}{{1+\frac{{P_{\mathtt{E}}}}{\sigma_{\mathtt{D}}^{2}}{{\bf
w}_{t}^{{\dagger}}}({{\bf H}_{\mathtt{EE}}^{{\dagger}}}+{{\bf
H}_{\mathtt{ER}}^{{\dagger}}}{\bm{\Theta}_{r}^{{\dagger}}}{\bf
H}_{\mathtt{RE}}^{{\dagger}}){{({\bf H}_{\mathtt{EE}}}+{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}){{\bf w}_{t}}}}.$

The optimization problem (11) is non-convex because of the complex objective
function. To proceed, we first introduce a slack variable
$y={1+\frac{{P_{\mathtt{E}}}}{\sigma_{\mathtt{D}}^{2}}\mathrm{tr}({\bf W}{{\bf
H}_{\mathtt{ER}}^{{\dagger}}}{\bm{\Theta}_{t}^{{\dagger}}}{{\bf
H}_{\mathtt{RD}}^{{\dagger}}}{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}}{{\bf H}_{\mathtt{ER}}})$. Then, by
employing the SDR technique [19] to relax the quadratic terms of the
beamformers in the objective function and constraints, the original problem is
reformulated as follows

$\displaystyle\underset{y,{\bf W}\succeq 0}{\min}\,\,$ $\displaystyle\hskip
10.00002pt~{}\frac{\frac{\sigma_{\mathtt{E}}^{2}}{\sigma_{\mathtt{D}}^{2}}|{h_{SD}}+{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf h}_{\mathtt{SR}}|^{2}}{y}$
$\displaystyle+\frac{\rho_{e}\mathrm{tr}\big{(}{\bf
W}\big{[}{\textbf{U}^{{\dagger}}}{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{{\bf
h}_{\mathtt{SR}}}{{\bf
h}_{\mathtt{SR}}^{{\dagger}}}\bm{\Theta}_{t}^{{\dagger}}{{\bf
H}_{\mathtt{RE}}^{{\dagger}}}\textbf{U}\big{]}\big{)}}{{1+\frac{P_{\mathtt{E}}}{\sigma_{\mathtt{D}}^{2}}\mathrm{tr}({\bf
W}{\textbf{U}^{{\dagger}}}{\textbf{U}})}},$ (12a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip 5.0pt~{}\
y={1+\frac{{P_{\mathtt{E}}}}{\sigma_{\mathtt{D}}^{2}}\mathrm{tr}({\bf W}{{\bf
H}_{\mathtt{ER}}^{{\dagger}}}{\bm{\Theta}_{t}^{{\dagger}}}{{\bf
h}_{\mathtt{RD}}}{\bf h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}}{{\bf
H}_{\mathtt{ER}}}),~{}$ (12b) $\displaystyle\hskip 5.0pt~{}\ \mathrm{tr}({\bf
W})=1,$ (12c)

where $\textbf{U}\triangleq{{\bf H}_{\mathtt{EE}}}+{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}$. Problem (12) is
still non-convex in the objective function and constraints (12b) and (12c).
Note that when $y$ is fixed, problem (12) becomes a quasi-convex optimization
problem, which can be converted into a convex SDP problem after some
transformations. Hence, problem (12) can be solved by the two-stage
optimization procedure [15], where the inner stage is an SDP problem with
fixed $y$, while the outer stage is a one dimensional line search problem over
$y$. In particular, the one dimensional problem is

$\displaystyle\underset{y}{\min}\,\,$ $\displaystyle\hskip
10.00002pt~{}f(y)+\frac{\frac{\sigma_{\mathtt{E}}^{2}}{\sigma_{\mathtt{D}}^{2}}|{h_{SD}}+{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf h}_{\mathtt{SR}}|^{2}}{y},$
(13a) $\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
20.00003pt~{}1<y<{1+\frac{{P_{\mathtt{E}}}}{\sigma_{\mathtt{D}}^{2}}|{\bf
h}_{\mathtt{RD}}^{\dagger}\Theta_{t}{{\bf H}_{\mathtt{ER}}}{\bf
w}_{t}|^{2}},~{}$ (13b)

where $f(y)$ is the optimal value of the inner optimization problem presented
below

$\displaystyle\underset{s>0,\textbf{Z}\succeq{\bf 0}}{\min}\,\,$
$\displaystyle\hskip
10.00002pt~{}\rho_{e}\mathrm{tr}\big{(}\textbf{Z}\big{[}{\textbf{U}^{{\dagger}}}{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{t}{{\bf h}_{\mathtt{SR}}}{{\bf
h}_{\mathtt{SR}}^{{\dagger}}}\bm{\Theta}_{t}^{{\dagger}}{{\bf
H}_{\mathtt{RE}}^{{\dagger}}}\textbf{U}\big{]}\big{)},$ (14a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip 5.0pt~{}\
\mathrm{tr}(\textbf{Z})=s,$ (14b) $\displaystyle\hskip 5.0pt~{}\
{{s+\frac{{P_{\mathtt{E}}}}{{\sigma_{\mathtt{D}}^{2}}}\mathrm{tr}(\textbf{Z}{\textbf{U}^{{\dagger}}}{\textbf{U}})}}=1,$
(14c) $\displaystyle\hskip 5.0pt~{}\
{s}{(y\\!-1)}\\!=\\!{\frac{{P_{\mathtt{E}}}}{\sigma_{\mathtt{D}}^{2}}\mathrm{tr}(\textbf{Z}{{\bf
H}_{\mathtt{ER}}^{{\dagger}}}{\bm{\Theta}_{t}^{{\dagger}}}{{\bf
h}_{\mathtt{RD}}}{\bf h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}}{{\bf
H}_{\mathtt{ER}}}),$ (14d)

where we have introduced ${\textbf{Z}=s{\bf W}}$, while ${s>0}$ satisfies
${{{s+\frac{{P_{\mathtt{E}}}}{{\sigma_{\mathtt{E}}^{2}}}\mathrm{tr}(s{\bf
W}{\textbf{U}^{{\dagger}}}{\textbf{U}})}}=1}$. Problem (14) consists of a
linear objective function with a set of linear constraints, hence it is a
convex SDP problem that can be efficiently solved. Thus, optimization problems
(13) and (14) are iteratively solved to provide the optimal transmit
beamformer, ${\bf w}_{t}^{\star}$.

###  III-B STAR-RIS Transmission/Reflecting Phase Design

In this subsection, we propose a joint design of $\bm{\Theta}_{r}$ and
$\bm{\Theta}_{t}$ at the STAR-RIS for given ${\bf w}_{t}$ and ${\bf w}_{r}$ at
E. To this end, the optimization problem (8) is formulated as

$\displaystyle\underset{\bm{\Theta}_{r},\bm{\Theta}_{t}}{\min}\,\,$
$\displaystyle\Big{(}\frac{|{h_{SD}}+{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf H}_{\mathtt{ER}}}{\bf
w}_{t}|^{2}+\sigma_{\mathtt{D}}^{2}}$ $\displaystyle-\frac{|{\bf
w}_{r}^{{\dagger}}{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{EE}}}{{\bf w}_{t}}+{{\bf w}_{r}^{{\dagger}}{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}{{\bf
w}_{t}}|^{2}+\sigma_{\mathtt{E}}^{2}}\Big{)},$ (15a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
20.00003pt~{}\eqref{P1:beta:const},\eqref{P1:phase1}.$ (15b)

By defining
${\widetilde{\bm{\theta}}_{r}}=\mathrm{diag}(\bm{\Theta}_{r})^{{\dagger}}$,
${\widetilde{\bm{\theta}}_{t}}=\mathrm{diag}(\bm{\Theta}_{t})^{{\dagger}}$,
${\bf a}_{k_{1}}=\mathrm{diag}({\bf h}_{\mathtt{RD}}^{\dagger}){{\bf
h}_{\mathtt{SR}}}$, ${\overline{\bf a}_{k_{1}}}=[{\bf
a}_{k_{1}}^{H},h_{SD}^{H}]^{H}$, ${\bf A}_{k_{1}}={\overline{\bf
a}_{k_{1}}}{\overline{\bf a}_{k_{1}}^{H}}$, ${\bf
a}_{k_{2}}=\mathrm{diag}({\bf h}_{\mathtt{RD}}^{\dagger}){{\bf
H}_{\mathtt{ER}}}{{\bf w}_{t}}$, ${\overline{\bf a}_{k_{2}}}=[{\bf
a}_{k_{2}}^{H},0]^{H}$, ${\bf A}_{k_{2}}={\overline{\bf
a}_{k_{2}}}{\overline{\bf a}_{k_{2}}^{H}}$, $a_{k_{3}}=\mathrm{diag}({\bf
w}_{r}^{{\dagger}}{{\bf H}_{\mathtt{RE}}}){{\bf h}_{\mathtt{SR}}}$,
${\overline{\bf a}_{k_{3}}}=[{\bf a}_{k_{3}}^{H},0]^{H}$, ${\bf
A}_{k_{3}}={\overline{\bf a}_{k_{3}}}{\overline{\bf a}_{k_{3}}^{H}}$,
$a_{k_{4}}=\mathrm{diag}({\bf w}_{r}^{{\dagger}}{{\bf H}_{\mathtt{RE}}}){{\bf
H}_{\mathtt{ER}}}{{\bf w}_{t}}$, $b={\bf w}_{r}^{{\dagger}}{{\bf
H}_{\mathtt{EE}}}{{\bf w}_{t}}$, ${\overline{\bf a}_{k_{4}}}=[{\bf
a}_{k_{4}}^{H},b]^{H}$, ${\bf A}_{k_{4}}=\overline{\bf
a}_{k_{4}}{\overline{\bf a}_{k_{4}}^{H}}$,
$\overline{\bm{\theta}}_{r}=[\widetilde{\bm{\theta}}_{r},1]$,
${\overline{\bm{\theta}}_{t}}=[\widetilde{\bm{\theta}}_{t},1]$, ${\bf
Q}_{r}=\overline{\bm{\theta}}_{r}^{H}\overline{\bm{\theta}}_{r}$, and ${\bf
Q}_{t}=\overline{\bm{\theta}}_{t}^{H}\overline{\bm{\theta}}_{t}$, the
optimization problem (15) can be reformulated as

$\displaystyle\underset{{\bf Q}_{t},{\bf Q}_{r}\succeq 0}{\min}\,\,$
$\displaystyle\hskip 10.00002pt~{}\frac{\mathrm{tr}({\bf A}_{k_{3}}{\bf
Q}_{t})\mathrm{tr}({\bf A}_{k_{2}}{\bf Q}_{t})}{\mathrm{tr}({\bf
A}_{k_{4}}{\bf Q}_{r})\mathrm{tr}({\bf A}_{k_{1}}{\bf Q}_{r})},~{}$ (16a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
20.00003pt\mathrm{diag}\\{{\bf Q}_{r}\\}+\mathrm{diag}\\{{\bf
Q}_{t}\\}=\bm{1}_{N},~{}$ (16b) $\displaystyle\hskip
20.00003pt\mathrm{Rank}({\bf Q}_{r})=1,~{}~{}$ (16c) $\displaystyle\hskip
20.00003pt\mathrm{Rank}({\bf Q}_{t})=1.~{}$ (16d)

The optimization problem (16) is still non-convex due to non-convex objective
function and constraints (16d). To deal with the non-convex objective
function, we introduce two positive slack variables
$\frac{1}{I_{k}}=\mathrm{tr}({\bf A}_{k_{3}}{\bf Q}_{t})\mathrm{tr}({\bf
A}_{k_{2}}{\bf Q}_{t})$ and $S_{k}=\mathrm{tr}({\bf A}_{k_{4}}{\bf
Q}_{r})\mathrm{tr}({\bf A}_{k_{1}}{\bf Q}_{r})$. As a result, the problem (16)
is recast as

$\displaystyle\underset{{\bf Q}_{t},{\bf Q}_{r},I_{k},S_{k}}{\min}\,\,\hskip
10.00002pt~{}\frac{1}{I_{k}S_{k}},$ (17a) $\displaystyle\hskip
25.00003pt\mathrm{s.t.}\,\,\hskip
30.00005pt\frac{1}{I_{k}}\leq\mathrm{tr}({\bf A}_{k_{3}}{\bf
Q}_{t})\mathrm{tr}({\bf A}_{k_{2}}{\bf Q}_{t}),~{}$ (17b) $\displaystyle\hskip
72.00008ptS_{k}\geq\mathrm{tr}({\bf A}_{k_{4}}{\bf Q}_{r})\mathrm{tr}({\bf
A}_{k_{1}}{\bf Q}_{r}),~{}$ (17c) $\displaystyle\hskip
72.00008pt~{}\eqref{eq:p77:d}-\eqref{eq:p77:j}.$ (17d)

Problem (17) is still non-convex due to the first two constraints and the non-
convex rank-one constraints. To deal with the non-convex constraints (17b) and
(17c), we apply the following lower bounds

$\displaystyle
4xy\\!\leq\\![(x\\!+\\!y)^{2}\\!-\\!2(x^{(n)}\\!-\\!y^{(n)})(x\\!-\\!y)\\!+\\!(x^{(n)}\\!-\\!y^{(n)})^{2}],$
(18)
$\displaystyle~{}-4xy\\!\leq\\![(x\\!-\\!y)^{2}\\!-\\!2(x^{(n)}\\!+\\!y^{(n)})(x\\!+\\!y)\\!+\\!(x^{(n)}\\!+\\!y^{(n)})^{2}],$
(19)

where $x\geq 0,y\geq 0$. Then, by relaxing the rank-one constraints for ${\bf
Q}_{t}$ and ${\bf Q}_{r}$, we can solve the following SDP problem

$\displaystyle\underset{{\bf Q}_{t},{\bf Q}_{r},I_{k},S_{k}}{\min}\,\,\hskip
1.00006pt\frac{1}{I_{k}S_{k}},$ (20a) $\displaystyle\hskip
20.00003pt\mathrm{s.t.}\,\,\hskip 15.00002pt\\!\big{(}\mathrm{tr}({\bf
A}_{k_{3}}{\bf Q}_{t})\\!-\\!\mathrm{tr}({\bf A}_{k_{2}}{\bf
Q}_{t})\big{)}^{\\!2}\\!\\!\\!-\\!2\big{(}\mathrm{tr}({\bf A}_{k_{3}}{\bf
Q}_{t}^{\\!(n)})$ $\displaystyle\hskip 50.00008pt+\mathrm{tr}({\bf
A}_{k_{2}}{\bf Q}_{t}^{(n)})\big{)}\\!\big{(}\mathrm{tr}({\bf A}_{k_{3}}{\bf
Q}_{t})\\!+\\!\mathrm{tr}({\bf A}_{k_{2}}{\bf Q}_{t})\big{)}$
$\displaystyle\hskip 50.00008pt+\\!\big{(}\mathrm{tr}({\bf A}_{k_{3}}{\bf
Q}_{t}^{(n)})+\mathrm{tr}({\bf A}_{k_{2}}{\bf
Q}_{t}^{(n)})\big{)}^{\\!2}+\frac{4}{I_{k}}\\!\leq\\!0,~{}$ (20b)
$\displaystyle\hskip 50.00008pt\big{(}\mathrm{tr}({\bf A}_{k_{4}}{\bf
Q}_{r})+\mathrm{tr}({\bf A}_{k_{1}}{\bf
Q}_{r})\big{)}^{2}-2\big{(}\mathrm{tr}({\bf A}_{k_{4}}{\bf Q}_{r}^{(n)})$
$\displaystyle\hskip 50.00008pt-\mathrm{tr}({\bf A}_{k_{1}}{\bf
Q}_{r}^{(n)})\big{)}\big{(}\mathrm{tr}({\bf A}_{k_{4}}{\bf
Q}_{r})\\!-\\!\mathrm{tr}({\bf A}_{k_{1}}{\bf Q}_{r})\big{)}\\!$
$\displaystyle\hskip 48.00009pt+\\!\big{(}\mathrm{tr}({\bf A}_{k_{4}}{\bf
Q}_{r}^{\\!(n)})\\!-\\!\mathrm{tr}({\bf A}_{k_{1}}{\bf
Q}_{r}^{\\!(n)})\big{)}^{\\!2}\\!\\!-\\!4S_{k}\\!\leq\\!0,~{}$ (20c)
$\displaystyle\hskip 50.00008pt~{}\eqref{eq:p77:d},\eqref{eq:p77:j}.$ (20d)

1:Initialize feasible points $\left\\{{{\bf Q}}^{\left(0\right)}_{t},{{\bf
Q}}^{\left(0\right)}_{r},{{\bf w}}^{\left(0\right)}_{t}\right\\}$.

2:Set iteration index $n=0$.

3:Repeat

4:For given ${\bf Q}^{\left(0\right)}_{t}$ and ${{\bf
Q}}^{\left(0\right)}_{r}$, solve the relaxed problem (14) and return ${\bf
w}^{\left(n\right)}_{t}$.

5:For given ${\bf w}^{\left(n\right)}_{t}$, solve the relaxed problem (20) and
return ${{\bf Q}}^{\left(n\right)}_{t}$ and ${{\bf Q}}^{\left(n\right)}_{r}$.

6:Update $n=n+1$.

7:Until the fractional decrease of the objective function value is below a
predefined threshold $\epsilon_{1}>0$ or the maximum number of inner
iterations $n_{max}$ is reached.

8:Return ${\bf Q}^{\star}_{t}$, ${\bf Q}^{\star}_{r}$, and ${\bf
w}^{\star}_{t}$ with the current solutions ${\bf Q}^{\left(n\right)}_{t}$,
${\bf Q}^{\left(n\right)}_{r}$, and ${\bf w}^{\left(n\right)}_{t}$.

Algorithm 1 Proposed Double-Layer Algorithm for Solving Problem (16)

This SDP problem can be solved efficiently via CVX [20]. The optimal solution
of ${\bf Q}_{t}$ and ${\bf Q}_{r}$, however, are not generally rank-one
matrices. Therefore, after obtaining the optimal ${\bf Q}_{t}$ and ${\bf
Q}_{r}$, we need to find a rank-one solution by using the Gaussian
randomization procedure [19].

###  III-C Overall Algorithm and Complexity Analysis

Putting together the solution for beamforming vectors and
transmission/reflection matrices presented respectively in Sections III-A and
III-B, our proposed algorithm for maximizing the eavesdropping non-outage
probability is summarized in Algorithm 1. Since at each iteration $n$, the
proposed algorithm decreases the value of the non-outage probability,
convergence of the algorithm is guaranteed.

The complexity of Algorithm 1 is determined by the complexity of iteratively
solving the SDP problem (14) and (20). An SDP problem with an $a\times a$
semidefinite matrix and $b$ SDP constraints is solved with complexity
$\mathcal{O}\left(\sqrt{a}\left(a^{3}b+a^{2}b^{2}+b^{3}\right)\right)$ by
interior-point methods [19]. For problem (14), we have $a=N_{T}$ and $b=3$,
while for problem (20) we have $a=N$ and $b=3$.

##  IV Low-Complexity Designs

The optimal design necessitates an SDP approach for beamforming design at E,
which entails high computational complexity. In light of this, we propose low-
complexity suboptimal beamforming designs using the ZF principle and linear
processing, incorporating maximum ratio transmission (MRT) and maximum ratio
combining (MRC).

###  IV-A Suboptimal Designs and Problem Formulation

####  IV-A1 ZF/MRT Beamforming Design

With ZF/MRT beamforming design, also known as RZF, ZF beamforming is employed
on the receiving side to design ${\bf w}_{r}$, while MRT beamforming is
utilized on the transmitting side to design ${\bf w}_{t}$. To ensure
feasibility, we need to deploy at least two receive antennas at E, i.e.,
${N_{R}}>1$. Accordingly, we set ${\bf w}_{t}^{\mathtt{MRT}}=\frac{{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf H}_{\mathtt{ER}}}}{\|{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf H}_{\mathtt{ER}}}\|}$, and the
optimal ${\bf w}_{r}$, which maximizes the eavesdropping non-outage
probability, is the solution of

$\displaystyle\underset{\left\|{{\bf w}_{r}}\right\|=1}{\max}\,\,$
$\displaystyle\hskip 10.00002pt~{}{|{{\bf w}_{r}^{{\dagger}}}{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf h}_{\mathtt{SR}}|^{2}},$ (21a)
$\displaystyle\mathrm{s.t.}$ $\displaystyle\hskip 10.00002pt~{}{{{{\bf
w}_{r}^{{\dagger}}}}\big{(}{\bf H}_{\mathtt{EE}}+{\bf
H}_{\mathtt{RE}}\bm{\Theta}_{r}{\bf H}_{\mathtt{ER}}\big{)}{{\bf
w}_{t}^{\mathtt{MRT}}}}=0.$ (21b)

By using projection matrix theory [14], the receive beamformer, which
satisfies the condition in (21a), is given by ${{\bf
w}_{r}^{\mathtt{ZF}}}=\frac{\bm{\Xi}^{\bot}{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}}{\left\|\bm{\Xi}^{\bot}{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf h}_{\mathtt{SR}}\right\|}$, where
$\bm{\Xi}^{\bot}={\bf I}_{N_{R}}-\frac{\tilde{{\bf H}}_{\mathtt{EE}}{\bf
w}_{t}^{\mathtt{MRT}}{({\bf w}_{t}^{\mathtt{MRT}})^{{\dagger}}}\tilde{{\bf
H}}_{\mathtt{EE}}^{{\dagger}}}{\|\tilde{{\bf H}}_{\mathtt{EE}}{{\bf
w}_{t}^{\mathtt{MRT}}}\|^{2}}$ is the projection idempotent matrix, with
$\tilde{{\bf H}}_{\mathtt{EE}}={\bf H}_{\mathtt{EE}}+{\bf
H}_{\mathtt{RE}}\bm{\Theta}_{r}{\bf H}_{\mathtt{ER}}$.

Accordingly, by substituting ${{\bf w}_{r}^{\mathtt{ZF}}}$ and ${{\bf
w}_{t}^{\mathtt{MRT}}}$ into (8) that completely mitigates SI, the
optimization problem (7) is reduced to

$\displaystyle\underset{\bm{\Theta}_{r},\bm{\Theta}_{t}}{\min}\,\,$
$\displaystyle\hskip 1.00006pt~{}\frac{|{h_{SD}}\\!+\\!{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}\|{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{t}{{\bf
H}_{\mathtt{ER}}}\|^{2}\\!+\\!\sigma_{\mathtt{D}}^{2}}\\!-\\!\frac{|({{\bf
w}_{r}^{\mathtt{ZF}}})^{{\dagger}}{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}|^{2}}{\sigma_{\mathtt{E}}^{2}},$ (22a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
20.00003pt~{}\eqref{P1:beta:const},\eqref{P1:phase1}.$ (22b)

####  IV-A2 MRC/ZF Beamforming Design

With MRC/ZF beamforming design, which is also abbreviated as TZF, ${\bf
w}_{t}$ is designed based on the ZF principle to cancel SI, while ${\bf
w}_{r}$ is set according to the MRC principle to maximize the received SINR at
the E, ${\text{SINR}_{E}}$. Therefore, ${{\bf w}_{r}^{\mathtt{MRC}}}$
=$\frac{{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{{\bf
h}_{\mathtt{SR}}}}{\|{{{\bf H}_{\mathtt{RE}}}\bm{\Theta}_{t}{{\bf
h}_{\mathtt{SR}}}}\|}$, and the optimal ${\bf w}_{t}$, which maximizes the
eavesdropping non-outage probability, is the solution of

$\displaystyle\underset{\left\|{{\bf w}_{t}}\right\|=1}{\max}\,\,$
$\displaystyle\hskip 10.00002pt~{}|{\bf h}_{\mathtt{RD}}\bm{\Theta}_{t}{\bf
H}_{\mathtt{ER}}{\bf w}_{t}|^{2},$ (23a) $\displaystyle\mathrm{s.t.}$
$\displaystyle\hskip 10.00002pt~{}{{({\bf
w}_{r}^{\mathtt{MRC}})^{{\dagger}}}\big{(}{{\bf H}_{\mathtt{EE}}}+{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{r}{{\bf H}_{\mathtt{ER}}}\big{)}{{\bf
w}_{t}}}=0.$ (23b)

Using projection matrix theory, the receive beamformer, which satisfies the
condition in (23a), is given by ${{\bf
w}_{t}^{\mathtt{ZF}}}=\frac{\bm{\Upsilon}^{\bot}{{\bf
h}_{\mathtt{RD}}\bm{\Theta}_{t}{\bf
H}_{\mathtt{ER}}}}{\left\|\bm{\Upsilon}^{\bot}{{\bf
h}_{\mathtt{RD}}\bm{\Theta}_{t}{\bf H}_{\mathtt{ER}}}\right\|}$, where
$\bm{\Upsilon}^{\bot}={\bf I}_{N_{T}}-\frac{\tilde{{\bf
H}}_{\mathtt{EE}}^{{\dagger}}{\bf w}_{r}^{\mathtt{MRC}}{({\bf
w}_{r}^{\mathtt{MRC}})^{{\dagger}}}\tilde{{\bf
H}}_{\mathtt{EE}}}{\|\tilde{{\bf H}}_{\mathtt{EE}}{{\bf
w}_{r}^{\mathtt{MRC}}}\|^{2}}$ with $\tilde{{\bf H}}_{\mathtt{EE}}={\bf
H}_{\mathtt{EE}}+{\bf H}_{\mathtt{RE}}\bm{\Theta}_{r}{\bf H}_{\mathtt{ER}}$ is
the projection idempotent.

Accordingly, by substituting ${{\bf w}_{r}^{\mathtt{MRC}}}$ and ${{\bf
w}_{t}^{\mathtt{ZF}}}$ into (8) that completely mitigates SI, the optimization
problem (7) can be written as

$\displaystyle\underset{\bm{\Theta}_{r},\bm{\Theta}_{t}}{\min}\,\,$
$\displaystyle\hskip 10.00002pt~{}\frac{|{h_{SD}}+{\bf
h}_{\mathtt{RD}}^{\dagger}\bm{\Theta}_{r}{\bf
h}_{\mathtt{SR}}|^{2}}{{P_{\mathtt{E}}}|{\bf
h}_{\mathtt{RD}}^{\dagger}{\bm{\Theta}_{t}}{{\bf H}_{\mathtt{ER}}}{{{{\bf
w}_{t}^{\mathtt{ZF}}}}}|^{2}+\sigma_{\mathtt{D}}^{2}}-\frac{\|{{\bf
H}_{\mathtt{RE}}}\bm{\Theta}_{t}{\bf
h}_{\mathtt{SR}}\|^{2}}{\sigma_{\mathtt{E}}^{2}},$ (24a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
20.00003pt~{}\eqref{P1:beta:const},\eqref{P1:phase1}.$ (24b)

###  IV-B Solution

In this subsection, we present the solution of the optimization problems (22)
and (24). Before proceeding, we define ${\bf
a}_{k_{2}}^{\mathtt{bs}}=\mathrm{diag}({\bf h}_{\mathtt{RD}}^{\dagger}){{\bf
H}_{\mathtt{ER}}}\bm{{\bf f}}^{\mathtt{bs}}$, where the superscript
“$\mathtt{bs}$” refers to the “beamforming scheme” with
$\mathtt{bs}=\\{\mathtt{RZF},\mathtt{TZF}\\}$, and $\bm{{\bf
f}}^{\mathtt{TZF}}={{{\bf w}_{t}^{\mathtt{ZF}}}}$, while $\bm{{\bf
f}}^{\mathtt{RZF}}=\bm{1}_{N}$. Moreover, let $\overline{\bf
a}_{k_{2}}^{\mathtt{bs}}=[({\bf a}_{k_{2}}^{\mathtt{bs}})^{H},0]^{H}$, ${\bf
A}_{k_{2}}^{\mathtt{bs}}=\overline{\bf a}_{k_{2}}^{\mathtt{bs}}(\overline{\bf
a}_{k_{2}}^{\mathtt{bs}})^{H}$, ${\bf
a}_{k_{3}}^{\mathtt{bs}}=\mathrm{diag}({{\bf g}^{\mathtt{bs}}{\bf
H}_{\mathtt{RE}}}){{\bf h}_{\mathtt{SR}}}$ where ${\bf
g}^{\mathtt{RZF}}={({{\bf w}_{r}^{\mathtt{ZF}}})^{{}^{{\dagger}}}}$ and ${\bf
g}^{\mathtt{TZF}}=\bm{1}_{N}$, ${\overline{\bf a}_{k_{3}}^{\mathtt{bs}}}=[{\bf
a}_{k_{3}}^{\mathtt{bs}},0]^{H}$, ${\bf A}_{k_{3}}^{\mathtt{bs}}=\overline{\bf
a}_{k_{3}}^{\mathtt{bs}}(\overline{\bf a}_{k_{3}}^{\mathtt{bs}})^{H}$.
Therefore, the optimization problems (22) and (24) can be reformulated as

$\displaystyle\underset{{\bf Q}_{t},{\bf Q}_{r}\succeq 0}{\min}\,\,$
$\displaystyle\hskip 10.00002pt~{}\frac{\mathrm{tr}({\bf
A}_{k_{3}}^{\mathtt{bs}}{\bf Q}_{t})\mathrm{tr}({\bf
A}_{k_{2}}^{\mathtt{bs}}{\bf Q}_{t})}{\mathrm{tr}({\bf A}_{k_{1}}{\bf
Q}_{r})},$ (25a) $\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
20.00003pt\mathrm{diag}\\{{\bf Q}_{r}\\}+\mathrm{diag}\\{{\bf
Q}_{t}\\}=\bm{1}_{N},~{}$ (25b) $\displaystyle\hskip
20.00003pt\mathrm{Rank}({\bf Q}_{r})=1,~{}\mathrm{Rank}({\bf Q}_{t})=1.~{}$
(25c)

The optimization problem (25) is non-convex due to (16a)-(16d). To tackle this
issue, we first introduce two slack variables
$\frac{1}{\bar{I}_{k}}=\mathrm{tr}({\bf A}_{k_{3}}^{\mathtt{bs}}{\bf
Q}_{t})(\mathrm{tr}({\bf A}_{k_{2}}^{\mathtt{bs}}{\bf Q}_{t}))$ and
$\bar{S}_{k}=\mathrm{tr}({\bf A}_{k_{1}}{\bf Q}_{r})$.

Now, following a similar approach as in the optimal case and relaxing the
rank-one constraints, optimization problem (25) is recast as

$\displaystyle\underset{{\bf Q}_{t},{\bf
Q}_{r},\bar{I}_{k},\bar{S}_{k}}{\min}\,\,$ $\displaystyle\hskip
1.99997pt~{}\frac{1}{\bar{I}_{k}\bar{S}_{k}},$ (26a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
1.99997pt\big{(}\mathrm{tr}({\bf A}_{k_{3}}^{\mathtt{bs}}{\bf
Q}_{t})\\!-\\!\mathrm{tr}({\bf A}_{k_{2}}{\bf
Q}_{t})\big{)}^{\\!2}\\!-\\!2\big{(}\mathrm{tr}({\bf
A}_{k_{3}}^{\mathtt{bs}}{\bf Q}_{t}^{(n)})$ $\displaystyle\hskip
1.99997pt+\mathrm{tr}({\bf A}_{k_{2}}{\bf
Q}_{t}^{(n)})\big{)}\big{(}\mathrm{tr}({\bf A}_{k_{3}}^{\mathtt{bs}}{\bf
Q}_{t})\\!+\\!\mathrm{tr}(\\!{\bf A}_{k_{2}}{\bf Q}_{t})\big{)}\\!$
$\displaystyle\hskip 1.99997pt+\\!\big{(}\mathrm{tr}({\bf
A}_{k_{3}}^{\mathtt{bs}}{\bf Q}_{t}^{(n)})+\mathrm{tr}({\bf A}_{k_{2}}{\bf
Q}_{t}^{(n)})\big{)}^{\\!\\!2}+\frac{4}{\bar{I}_{k}}\leq\\!0,~{}$ (26b)
$\displaystyle\hskip 10.00002pt\\!\mathrm{tr}({\bf A}_{k_{1}}{\bf
Q}_{r})-\bar{S}_{k}\\!\leq\\!0,~{}$ (26c) $\displaystyle\hskip
5.0pt~{}\eqref{eq:p91:d}.~{}$ (26d)

This SDP problem can be solved efficiently via CVX [20], in an iterative way.
After obtaining the optimal ${\bf Q}_{t}$ and ${\bf Q}_{r}$, we need to find a
rank-one solution by using the Gaussian randomization procedure [19], if the
results are not rank-one.

Figure 2: Eavesdropping non-outage probability for the proposed optimal and
suboptimal designs versus $\rho_{e}$ ($N_{R}=N_{T}=4$).

##  V Numerical Results

In this section, numerical results are presented to evaluate the performance
of the proposed proactive eavesdropping schemes. The optimal scheme refers to
the case where the beamforming vectors at E and transmission/reflection
coefficients at STAR-RIS are jointly optimized. Unless otherwise stated, in
all the simulations, we set
$\sigma_{\mathtt{E}}^{2}\\!=\\!\sigma_{\mathtt{D}}^{2}\\!=\\!1$, and
$\epsilon_{1}=10^{-3}$. We model the large-scale fading as
$\gamma_{XY}=C_{0}(d_{XY}/D_{0})^{-\mu}$, where $d_{XY}$ is the individual
link distance, $C_{0}=-30$ dB is the reference channel gain at a distance of
$D_{0}=1$ m, and $\mu$ denotes the path loss exponent of the individual link
and set $\mu\\!=\\!3.6$ [4]. Moreover, the normalized SNR of the suspicious
link, i.e., $\rho_{s}\\!\\!\triangleq\\!\\!P_{s}/\sigma_{\mathtt{D}}^{2}$, is
set to $10$ dB. As a benchmark, we also include the results for MRC/MRT
beamforming design with optimized trans- mission and reflection coefficient
matrices at STAR-RIS.

Figure 2 shows the eavesdropping non-outage probability versus $\rho_{e}$ for
the proposed optimal and suboptimal schemes. We observe that the optimal
scheme yields the best performance, reflecting the impact of joint beamforming
and transmission/reflection phase shift design. Moreover, by increasing
$\rho_{e}$, the RZF outperforms all suboptimal designs, while MRC/MRT
outperforms the RZF and TZF schemes in the low $\rho_{e}$ regime. The superior
performance of RZF over TZF at higher values of $\rho_{e}$ can be attributed
to the transmission of stronger jamming signals (RZF sacrifices one DoF at the
transmit side to cancel SI), leading to an increased dominance of the jamming
phase.

Figure 3 shows the eavesdropping non-outage probability versus the SI strength
for the proposed schemes. We observe that the optimal design can effectively
cancel SI. However, for a given number of antennas at E, increasing the number
of receiving antennas can improve performance. As expected, SI does not affect
the ZF-based suboptimal schemes, while the eavesdropping non-outage
probability of the MRC/MRT scheme significantly decreases when the SI strength
increases. Moreover, with a decrease in the number of receiving antennas,
$N_{R}$, the MRC/MRT scheme outperforms the ZF/MRT beamforming scheme at lower
$\sigma_{\mathtt{SI}}^{2}$ values. This is due to the fact that decreasing
$N_{R}$ results in weak SI, which is beneficial for the MRC/MRT design.

Figure 3: Eavesdropping non-outage probability for the proposed optimal and
suboptimal designs versus $\sigma_{\mathtt{SI}}^{2}$ ($\rho_{e}=10$ dB).

##  VI Conclusion

We proposed a STAR-RIS-assisted proactive eavesdropping system, where the
STAR-RIS is deployed to assist a FD multi-antenna E in overhearing a ST and
interfering with a SR simultaneously. We formulated a non-convex joint
optimization problem to design the beamforming vectors at $E$ and
reflection/transmission phase shift matrices at the STAR-RIS. Moreover, low-
complexity ZF-based beamforming designs were proposed that can balance between
performance and system complexity. Our results suggest that the optimal design
can effectively cancel the SI. Moreover, by increasing the number of receiving
antennas at FD E, the surveillance performance of the optimal design is
improved.

In the future, our work will involve monitoring multiple untrusted
communication links. Moreover, we can explore other operating protocols for
the STAR-RIS operation in wireless surveillance systems, namely, mode
switching and time switching.

## References

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