# On some efficiency conditions for vector optimization problems with
uncertain cone constraints: a robust approach

A. Uderzoa CONTACT A. Uderzo. Email<EMAIL_ADDRESS>a Dipartimento di
Matematica e Applicazioni, Università di Milano-Bicocca, Milano, Italy

###### Abstract

In the present paper, several types of efficiency conditions are established
for vector optimization problems with cone constraints affected by
uncertainty, but with no information of stochastic nature about the uncertain
data. Following a robust optimization approach, data uncertainty is faced by
handling set-valued inclusion problems. The employment of recent results about
error bounds and tangential approximations of the solution set to the latter
enables one to achieve necessary conditions for weak efficiency via a
penalization method as well as via the modern revisitation of the Euler-
Lagrange method, with or without generalized convexity assumptions. The
presented conditions are formulated in terms of various nonsmooth analysis
constructions, expressing first-order approximations of mappings and sets,
while the metric increase property plays the role of a constraint
qualification.

###### keywords:

Vector optimization problem; data uncertainty; robust approach; weak
efficiency condition; generalized derivative; generalized convexity.

##  1 Introduction

Consider a vector optimization problem

${\rm Min}_{K}f(x)\ \hbox{\ subject to $x\in{\mathcal{R}}$},$ $None$

where ${\mathcal{R}}\subseteq\mathbb{X}$ is a decision set defining the
feasible region of the problem, $f:\mathbb{X}\longrightarrow\mathbb{Y}$
represents the criterion with respect to which decisions in ${\mathcal{R}}$
are to be optimized, and $K\subseteq\mathbb{Y}$ is a convex cone defining the
partial order, according to which the outcomes of decisions are compared in
the criteria space. Throughout the paper $(\mathbb{X},\|\cdot\|)$ and
$(\mathbb{Y},\|\cdot\|)$ denote real Banach spaces and it will be assumed that
${\rm int}\,K\neq\varnothing$. For vector optimization problems the concept of
solution is not uniquely defined but several notions, reflecting different
aspects of the issue, can be considered. Among the others, the notions of
efficient and weakly efficient solution are well recognized and largely
investigated in the literature devoted to vector optimization (see [9, 14, 15,
17, 24]). Recall that an element $\bar{x}\in{\mathcal{R}}$ is said to be a
locally weakly efficient (for short, w-eff.) solution to $({\mathcal{P}})$ if
there exists $\delta>0$ such that

$f({\mathcal{R}}\cap{\rm B}(\bar{x},\delta))\cap[f(\bar{x})-{\rm
int}\,K]=\varnothing;$

an element $\bar{x}\in{\mathcal{R}}$ is said to be a locally efficient (for
short, eff.) solution to $({\mathcal{P}})$ if there exists $\delta>0$ such
that

$f({\mathcal{R}}\cap{\rm
B}(\bar{x},\delta))\cap[f(\bar{x})-K]=\\{f(\bar{x})\\}.$

Clearly, any local eff. solution to $({\mathcal{P}})$ is also a local w-eff.
one. The present paper deals with conditions of local weak efficiency for
vector optimization problems, whose decision set ${\mathcal{R}}$ is formalized
by uncertain cone constraints, namely problems of the form

${\rm Min}_{K}f(x)\ \hbox{\ with $x\in S$ subject to }\ g(\omega,x)\in C,$
$None$

where $C\subseteq\mathbb{Z}$ is a (proper) closed, convex cone in a real
Banach space $(\mathbb{Z},\|\cdot\|)$, with $C\neq{\mathbf{0}}$,
$S\subseteq\mathbb{X}$ is a closed set expressing a geometric constraint free
from uncertainty, and $g:\Omega\times\mathbb{X}\longrightarrow\mathbb{Z}$ is a
given mapping. Here $\Omega$ represents a given uncertainty set, which allows
one to describe a decision environment characterized by a crude knowledge of
the data. This means that the constraining mapping $g$, as a structural
element of the problem, is affected by uncertainty, but this uncertainty can
not be tackled by handling probability distributions as in stochastic
optimization, because such an information is not at disposal. The only
information about the data element $\omega$ is that $\omega\in\Omega$. The
paper often credited as a first reference in undertaking an aware and
systematic study of optimization problems, whose data are affected by this
form of uncertainty, is [3] 111To be more precise, in [3] the authors do
indicate as a forerunner of their approach A.L. Soyster, who in [26]
introduced a similar point of view in dealing with uncertainly constrained
problems in mathematical programming.. There, reasons for such a crude
knowledge of the data are widely discussed. In this circumstance, situations
quite common in reality may require that the cone constraint $g(\omega,x)\in
C$ is satisfied, whatever the actual realization of $\omega\in\Omega$ is. In
other terms, the decision maker is forced to regard as feasible only those
elements of $S$ such that $g(\omega,x)\in C$ for every $\omega\in\Omega$.
Examples of such situations, emerging especially in engineering applications,
are described in [3]. On this basis the authors developed an approach hedging
the decision maker against the worst cases that may occur, called robust
approach to uncertain optimization, in analogy with robust control. This
‘pessimistic’ (or ‘ultraconservative’, in the Soyster’s words) approach to
uncertainty opened a flourishing line of research, in scalar as well as in
vector optimization, known as robust optimization (see [3, 4, 5] and
references therein).

In the case of vector optimization problems such as
$({\mathcal{P}}_{\omega})$, where the objective function is not affected by
uncertainty, this approach reduces to consider as a feasible region the set

${\mathcal{R}}=\\{x\in S:\ g(\omega,x)\in C,\ \forall\omega\in C\\}.$

Thus, by introducing the set-valued mapping
$G:\mathbb{X}\rightrightarrows\mathbb{Z}$, defined as being

$G(x)=g(\Omega,x)=\\{z=g(\omega,x)\in\mathbb{Z}:\ \omega\in\Omega\\},$ (1)

the robust counterpart of the feasible region of $({\mathcal{P}}_{\omega})$
leads naturally to consider the so-called set-valued inclusion problem: given
a (nonempty) closed set $S\subseteq\mathbb{X}$, a proper, closed and convex
cone $C\subseteq\mathbb{Z}$ and a set-valued mapping
$G:\mathbb{X}\rightrightarrows\mathbb{Z}$

$\hbox{ find $x\in S$ such that }\ G(x)\subseteq C.$ $None$

In fact, recalling that the upper inverse image of $C$ through the set-valued
mapping $G$ is the set $G^{+1}(C)=\\{x\in\mathbb{X}:\ G(x)\subseteq C\\}$, one
has

${\mathcal{R}}=S\cap G^{+1}(C).$

To the best of the author’s knowledge, problem $(\mathcal{SVI})$ began to be
investigated independently of robust optimization in [6], which focuses on
error bound estimates. Solvability and solution stability issues for
$(\mathcal{SVI})$ have been studied more recently in [27, 28]. In the light of
the role played by $(\mathcal{SVI})$ in the robust approach to optimization
problems with uncertain constraints, it seems to be natural to assess an
impact evaluation of the recent achievements about the solution set to
$(\mathcal{SVI})$ and its approximations within the theory of
optimality/efficiency conditions. Some initial results along this line of
research have been obtained in the case of scalar optimization in [27, 29].
So, the present analysis can be regarded as a development of ideas and
techniques, presented especially in [29], towards the specific context of
vector optimization, in considering problems of the form

${\rm Min}_{K}f(x)\ \hbox{\ with $x\in S$ subject to }\ G(x)\subseteq C.$
$None$

This analysis will be performed here by well-known techniques: in fact, some
first-order efficiency conditions are obtained by means of the Clarke
penalization principle, through its vector counterpart due to J.J. Ye (see
[30]). Some other first-order efficiency conditions are achieved by exploiting
tangential approximations of the solution set to $(\mathcal{SVI})$, following
a modern revisitation of the celebrated Euler-Lagrange method. In both the
cases, the main tools employed come from nonsmooth and variational analysis as
well as from generalized convexity.

Optimality conditions for vector optimization problems with uncertain
constraints are a subject intensively investigated in the last years, in
particular through the robust approach (see, among others, [7, 8, 16] and
references therein). A feature distinguishing the analysis here proposed is
the great generality kept on $\Omega$, in the very spirit of robust
optimization, owing to the introduction of the set-valued mapping $G$.

The presentation of the contents is organized according to the following
arrangement. Section 2 collects some basic technical preliminaries of large
employment in optimization and related fields. Some more specific
constructions needed in the subsequent analysis will be recalled contextually
to their use. In Section 3 first-order necessary conditions for the local weak
efficiency of solutions to $({\mathcal{P}}_{G})$ are established via a
penalization method, with and without generalized convexity assumptions. In
Section 4 different Lagrangian-type necessary conditions for local weak
efficiency are formulated in terms outer prederivatives of $G$, with or
without smoothness assumption on $f$.

The notations in use throughout the paper are mainly standard. Quite often,
capital letters in bold will denote real Banach spaces. The null vector in a
Banach space is denoted by $\mathbf{0}$. In a metric space setting, the closed
ball centered at an element $x$, with radius $r\geq 0$, is indicated with
${\rm B}(x,r)$. In particular, in a Banach space, ${\mathbb{B}}={\rm
B}(\mathbf{0},1)$. Whenever $A$ is a subset of a metric space, ${\rm B}(A,r)$
indicates the $r$-enlargement of $A$, whereas the distance of a point $x$ from
$A$ is denoted by ${\rm dist}\left(x,A\right)$. If $W$ is a subset of the same
metric space, ${\rm exc}(A,W)=\sup_{a\in A}{\rm dist}\left(a,W\right)$
indicates the excess of $A$ over $W$. Symbols ${\rm cl}\,A$ and ${\rm int}\,A$
denote the topological closure and the interior of $A$, respectively. If $A$
is a subset of a Banach space, its convex hull is denoted by ${\rm conv}\,A$
and, when $A$ is convex, its relative interior is denoted by ${\rm ri}\,A$.

By $\mathcal{L}(\mathbb{X},\mathbb{Y})$ the Banach space of all bounded linear
operators acting between $\mathbb{X}$ and $\mathbb{Y}$ is denoted, equipped
with the operator norm $\|\cdot\|_{\mathcal{L}}$. In particular,
$\mathbb{X}^{*}=\mathcal{L}(\mathbb{X},\mathbb{R})$ stands for the dual space
of $\mathbb{X}$, in which case $\|\cdot\|_{\mathcal{L}}$ is simply marked by
$\|\cdot\|$. The null vector of a dual space will be marked by
$\mathbf{0}^{*}$. The duality pairing a Banach space with its dual will be
denoted by $\langle\cdot,\cdot\rangle$. Given a function
$\varphi:\mathbb{X}\longrightarrow\mathbb{R}\cup\\{\mp\infty\\}$, by
$[\varphi\leq 0]=\varphi^{-1}([-\infty,0])$ its sublevel set is denoted,
whereas $[\varphi>0]=\varphi^{-1}((0,+\infty])$ denotes the strict superlevel
set of $\varphi$. The acronyms l.s.c., u.s.c. and p.h. stand for lower
semicontinuous, upper semicontinuous and positively homogeneous, respectively.
The symbol ${\rm dom}\,\varphi=\varphi^{-1}(\mathbb{R})$ indicates the domain
of $\varphi$, whenever $\varphi$ is a functional, whereas if
$F:\mathbb{X}\rightrightarrows\mathbb{Y}$ is a set-valued mapping, ${\rm
dom}\,F=\\{x\in\mathbb{X}:\ F(x)\neq\varnothing\\}$.

##  2 Basic tools of analysis

Let $A\subseteq\mathbb{X}$ be a nonempty closed subset of a Banach space and
let $\bar{x}\in A$. Nonsmooth analysis provides a large variety of concepts
for the local, first-order conic approximation of $A$ near $\bar{x}$. For the
purposes of the present analysis, the following ones are to be mentioned:

${\rm T}(A;\bar{x})=\\{v\in\mathbb{X}:\ \exists(v_{n})_{n}\hbox{ with
}v_{n}\to v,\ \exists(t_{n})_{n}\hbox{ with }t_{n}\downarrow 0:\
\bar{x}+t_{n}v_{n}\in A,\ \forall n\in\mathbb{N}\\},$ ${\rm
I}(A;\bar{x})=\\{v\in\mathbb{X}:\ \exists\delta>0:\bar{x}+tv\in A,\ \forall
t\in(0,\delta)\\},$

and

${\rm I_{w}}(A;\bar{x})=\\{v\in\mathbb{X}:\ \forall\epsilon>0,\ \exists
t_{\epsilon}\in(0,\epsilon):\ \bar{x}+t_{\epsilon}v\in A\\},$

called the contingent (or Bouligand tangent) cone, the feasible direction cone
and the weak feasible direction cone to $A$ at $\bar{x}$, respectively. They
are known to be linked by the inclusion relation of general validity

${\rm I}(A;\bar{x})\subseteq{\rm I_{w}}(A;\bar{x})\subseteq{\rm
T}(A;\bar{x}),$

where strict inclusion may hold (see [25]). Whenever $A$ is locally convex
around $\bar{x}$, i.e. there exists $r>0$ such that $A\cap{\rm B}(\bar{x},r)$
is a convex set, the above inclusion relation collapses to

${\rm cl}\,{\rm I}(A;\bar{x})={\rm cl}\,{\rm I_{w}}(A;\bar{x})={\rm
T}(A;\bar{x})$

(see [25, Proposition 11.1.2(d)]). In such an event, ${\rm T}(A;\bar{x})$ is a
closed convex cone, while ${\rm I}(A;\bar{x})$ is a convex cone.

Let $Q\subseteq\mathbb{Y}$ be a cone. The sets

${Q}^{{}^{\oplus}}=\\{y^{*}\in\mathbb{Y}^{*}:\ \langle y^{*},y\rangle\geq
0,\quad\forall y\in Q\\}\quad\hbox{ and
}\quad{Q}^{{}^{\ominus}}=-{Q}^{{}^{\oplus}}$

are called the positive and the negative dual cone of $Q$, respectively.

######  Remark 1.

(i) Note that, whenever a set $A$ is locally convex around $\bar{x}$ (so ${\rm
T}(A;\bar{x})$ is convex) the negative dual cone operator allows one to
represent the normal cone to $A$ in the sense of convex analysis at some
element $\bar{x}\in A$ in terms of contingent cone as follows

${\rm N}(A;\bar{x})=\\{x^{*}\in\mathbb{X}^{*}:\ \langle
x^{*},x-\bar{x}\rangle\leq 0,\quad\forall x\in A\\}={{\rm
T}(A;\bar{x})}^{{}^{\ominus}}.$

(ii) The interaction of the negative dual cone operator with some set
operations is described by the following formula: given
$\Lambda\in\mathcal{L}(\mathbb{X},\mathbb{Y})$ and two closed convex cones
$Q\subseteq\mathbb{Y}$ and $P\subseteq\mathbb{X}$, it holds

${(P\cap\Lambda^{-1}(Q))}^{{}^{\ominus}}={\rm
cl}\,({P}^{{}^{\ominus}}+\Lambda^{*}({Q}^{{}^{\ominus}})),$

where $\Lambda^{*}\in\mathcal{L}(\mathbb{Y}^{*},\mathbb{X}^{*})$ denotes the
adjoint operator to $\Lambda$ (see [25, Lemma 2.4.1]). From this formula one
can derive, as a special case, the equality

${\left[\Lambda^{-1}(Q)\right]}^{{}^{\ominus}}={\rm
cl}\,\Lambda^{*}({Q}^{{}^{\ominus}}),$ (2)

and, under the qualification condition ${\rm int}\,P_{1}\cap{\rm
int}\,P_{2}\neq\varnothing$,

${(P_{1}\cap
P_{2})}^{{}^{\ominus}}={P_{1}}^{{}^{\ominus}}+{P_{2}}^{{}^{\ominus}},$ (3)

with $P_{1}$ and $P_{2}$ being closed convex cones in $\mathbb{X}$ (see [1,
Table 4.3 (5)b)]. Note that in the equality $(\ref{eq:dconeadj})$ the closure
operation can be omitted if $\Lambda^{-1}({\rm ri}\,Q)\neq\varnothing$. Such a
condition is evidently satisfied if $\Lambda\mathbb{X}\supseteq Q$ and
$Q\neq\\{\mathbf{0}\\}$ (see, for instance, [23, Corollary 16.3.2]).

Let $K\subseteq\mathbb{Y}$ be a (proper) convex cone inducing a partial order
$\leq_{{}_{K}}$ on $\mathbb{Y}$ and let
$f:\mathbb{X}\longrightarrow\mathbb{Y}$ be a mapping between Banach spaces.
Then $f$ is said to be $K$-convex on the convex set $A\subseteq\mathbb{X}$ if
the set

${\rm epi}_{K}(f)=\\{(x,y)\in\mathbb{X}\times\mathbb{Y}:\ x\in A,\
f(x)\leq_{{}_{K}}y\\}$

is convex. If, in addition, $A$ is a cone and $f$ is also positively
homogeneous, then $f$ is said to be $K$-sublinear on $A$. It is well known
that if $f$ is $K$-convex on $A$, then $f(A)+K$ is convex, while if $A$ is a
cone and $f$ is $K$-sublinear, then $f(A)+K$ is a convex cone.

Following [11, Definition 2.3], a mapping $f$ is said to be $K$-convexlike on
a set (not necessarily convex) $A$ if the set $f(A)+K$ is a convex.

######  Remark 2.

In Section 3 it will be used the fact, which is readily proved by handling the
related definitions, that if $\nu:\mathbb{X}\longrightarrow\mathbb{R}$ is a
sublinear function on $\mathbb{X}$ and $e\in K$, then the mapping $\nu
e:\mathbb{X}\longrightarrow\mathbb{Y}$, defined by $x\mapsto\nu(x)e$ is
$K$-sublinear on $\mathbb{X}$.

Generalized convexity notions apply also to set-valued mappings. Following
[6], a set-valued mapping $F:\mathbb{X}\rightrightarrows\mathbb{Z}$ between
Banach spaces is said to be $C$-concave on $\mathbb{X}$, where
$C\subseteq\mathbb{Z}$ is a (proper) convex cone, if

$F(tx_{1}+(1-t)x_{2})\subseteq tF(x_{1})+(1-t)F(x_{2})+C,\quad\forall
x_{1},\,x_{2}\in\mathbb{X}.$

Some examples of $C$-concave set-valued mappings of interest in optimization
can be found in [28]. For the purposes of the present analysis, the special
class of $C$-concave set-valued mappings known as fans is to be mentioned.
Recall that, after [13], a set-valued mapping
$H:\mathbb{X}\rightrightarrows\mathbb{Z}$ is called fan if it fulfils all the
following conditions:

(i) it is p.h.;

(ii) $\mathbf{0}\in H(\mathbf{0})$;

(iii) it is convex-valued;

(iv) $H(x_{1}+x_{2})\subseteq H(x_{1})+H(x_{2}),\quad\forall
x_{1},\,x_{2}\in\mathbb{X}$.

Fans may appear in a variety of forms. In Section 4, only fans which are
generated by bundles of linear mappings will be actually employed, i.e. fans
$H_{\mathcal{G}}:\mathbb{X}\rightrightarrows\mathbb{Z}$ that can be
represented as

$H_{\mathcal{G}}(x)=\\{\Lambda x:\ \Lambda\in\mathcal{G}\\},$

where $\mathcal{G}\subseteq\mathcal{L}(\mathbb{X},\mathbb{Z})$ is a (nonempty)
convex and weakly closed set.

######  Remark 3.

Whenever a fan $H_{\mathcal{G}}$ is generated by a bounded set $\mathcal{G}$,
it turns out to be a Lipschitz set-valued mapping, i.e. it holds

${\rm haus}(H_{\mathcal{G}}(x_{1}),H_{\mathcal{G}}(x_{2}))\leq
l\|x_{1}-x_{2}\|,\quad\forall x_{1},\,x_{2}\in\mathbb{X},$

with $l\geq\sup\\{\|\Lambda\|_{\mathcal{L}}:\ \Lambda\in\mathcal{G}\\}$, where
${\rm haus}(A,W)=\max\\{{\rm exc}(A,W),{\rm exc}(W,A)\\}$ denotes the
Hausdorff distance between two sets $A$ and $W$ (see [29, Remark 2.14(iii)]).

##  3 Weak efficiency conditions via penalization

######  Definition 3.1 ($K$-Lipschitz continuity).

Let $f:\mathbb{X}\longrightarrow\mathbb{Y}$ be a mapping between normed spaces
and let $K\subseteq\mathbb{Y}$ be a convex cone, with ${\rm
int}\,K\neq\varnothing$. $f$ is said to be $K$-Lipschitz on the set
$D\subseteq\mathbb{X}$ if there exist a constant $\ell_{f}>0$ and a vector
$e\in{\rm int}\,K\cap{\mathbb{B}}$ such that

$f(x_{1})\in f(x_{2})-\ell_{f}\|x_{1}-x_{2}\|e+K,\quad\forall x_{1},\,x_{2}\in
D.$

If $\bar{x}\in\mathbb{X}$ and $f$ is $K$-Lipschitz on a set $D={\rm
B}(\bar{x},\delta)$ for some $\delta>0$, then $f$ is said to be $K$-Lipschitz
near $\bar{x}$.

The above notion has been used in [30] as a key concept to extend the Clarke
penalization principle from the scalar case to vector optimization problems.
This is done here directly through a local error bound function, whose
definition is recalled below.

######  Definition 3.2 (Local error bound function).

Let $\bar{x}\in{\mathcal{R}}\subseteq S\subseteq\mathbb{X}$. A function
$\psi:\mathbb{X}\longrightarrow[0,+\infty]$ is said to be a local error bound
function for ${\mathcal{R}}$ near $\bar{x}$ if there exists $\delta>0$ such
that both the following conditions are satisfied:

(i) ${\rm dist}\left(x,{\mathcal{R}}\right)\leq\psi(x),\quad\forall x\in{\rm
B}(\bar{x},\delta)\cap S$;

(ii) ${\rm dist}\left(x,{\mathcal{R}}\right)=\psi(x),\quad\forall
x\in{\mathcal{R}}$.

######  Proposition 3.3.

([30, Theorem 4.2(i)]) With reference to a problem $({\mathcal{P}})$, let
$\bar{x}\in{\mathcal{R}}$ and suppose that:

(i) $f$ is $K$-Lipschitz near $\bar{x}$, with constant $\ell_{f}$ and vector
$e\in{\rm int}\,K$;

(ii) $\psi:\mathbb{X}\longrightarrow[0,+\infty]$ is an error bound function
near $\bar{x}$.

Then, for any $\ell\geq\ell_{f}$, every local $w$-eff. solution to
$({\mathcal{P}})$ is also a local w-eff. solution of the problem

${\rm Min}_{K}[f(x)+\ell\psi(x)e].$ $None$

Furthermore, if ${\mathcal{R}}$ is closed, for any $\ell>\ell_{f}$, every
local eff. solution to $({\mathcal{P}})$ is also a local eff. solution of the
problem $({\mathcal{P}}_{\ell})$.

Proposition 3.3 enables one to free the original problem from its constraints.
Notice indeed that problem $({\mathcal{P}}_{\ell})$ is unconstrained. For
problems such as $({\mathcal{P}}_{G})$, where the feasible region is
structured as a solution set to $(\mathcal{SVI})$, one has to adequate the
local error bound function to the the constraint definition. In the present
analysis, the following merit function
$\nu_{G,C}:\mathbb{X}\longrightarrow\mathbb{R}\cup\\{\pm\infty\\}$ for
problems $(\mathcal{SVI})$ is exploited to treat the data uncertainty in the
constraints:

$\nu_{G,C}(x)=\sup_{z\in G(x)}{\rm dist}\left(z,C\right)={\rm exc}(G(x),C).$

Henceforth, as a standing assumption it is assumed that ${\rm
dom}\,G=\mathbb{X}$. As a consequence, one has
$\nu_{G,C}:\mathbb{X}\longrightarrow[0,+\infty]$ and therefore the following
characterization of the feasible region of $({\mathcal{P}}_{G})$:

${\mathcal{R}}=S\cap[\nu_{G,C}\leq 0].$

The next lemma singles out a constraint qualification, under which the merit
function $\nu_{G,C}$ is shown to actually work as a local error bound
function. In order to formulate it, let us recall that, after [10], given a
function $\varphi:X\longrightarrow\mathbb{R}\cup\\{\pm\infty\\}$ defined on a
metric space $(X,d)$ and $\bar{x}\in\varphi^{-1}(\mathbb{R})$, the strong
slope of $\varphi$ at $\bar{x}$ is defined as the quantity

$\displaystyle|\nabla\varphi|(\bar{x})=\left\\{\begin{array}[]{ll}0,&\hbox{ if
$\bar{x}$ is a local minimizer of $\varphi$},\\\
\displaystyle\limsup_{x\to\bar{x}}{\varphi(\bar{x})-\varphi(x)\over
d(x,\bar{x})},&\hbox{ otherwise.}\end{array}\right.$

In view of the formulation of the next lemma, it is useful to observe that, if
as a metric space $X$ one takes a closed subset $S\subseteq\mathbb{X}$
containing $\bar{x}$ and as a distance $d$ one takes the distance induced by
$\|\cdot\|$, the above definition becomes

$\displaystyle|\nabla_{S}\varphi|(\bar{x})=\left\\{\begin{array}[]{ll}0,&\hbox{
if $\bar{x}$ is a local minimizer}\\\ &\hbox{ of $\varphi$ over $S$},\\\
\displaystyle\inf_{r>0}\sup_{x\in{\rm B}(\bar{x},r)\cap
S\backslash\\{\bar{x}\\}}{\varphi(\bar{x})-\varphi(x)\over\|x-\bar{x}\|},&\hbox{
otherwise.}\end{array}\right.$

######  Lemma 3.4.

Let $G:\mathbb{X}\rightrightarrows\mathbb{Z}$, $S$ and $C$ as in problem
$(\mathcal{SVI})$, and let $\bar{x}\in{\mathcal{R}}$. Suppose that:

(i) $G$ is l.s.c. in a neighbourhood of $\bar{x}$;

(ii) there exist positive $\sigma$ and $r$ such that

$|\nabla_{S}\nu_{G,C}|(x)\geq\sigma,\quad\forall x\in{\rm B}(\bar{x},r)\cap
S\cap[\nu_{G,C}>0].$ $None$

Then function $\psi=\sigma^{-1}\nu_{G,C}$ is a local error bound function for
${\mathcal{R}}$.

###### Proof.

From [27, Lemma 2.3(i)] it is known that the lower semicontinuity of $G$ (in
the sense of set-valued mappings) implies the lower semicontinuity for the
functional $\nu_{G,C}$. Thus, by hypothesis (i), for some $\delta_{0}>0$ it is
true that $\nu_{G,C}$ is l.s.c. on ${\rm B}(\bar{x},\delta_{0})\cap S$. Notice
that, as a closed subset of a Banach space, ${\rm B}(\bar{x},\delta_{0})\cap
S$ is a complete metric space, if equipped with the induced metric. Besides,
without any loss of generality, it is possible to assume that, if $r>0$ is as
in hypothesis (ii), it is $r<\delta_{0}$. Notice that the case ${\rm
B}(\bar{x},r)\cap S\cap[\nu_{G,C}>0]=\varnothing$ means ${\rm
B}(\bar{x},r)\cap S\subseteq G^{+1}(C)\cap S$, so it holds ${\rm
dist}\left(x,{\mathcal{R}}\right)=0\leq\psi(x)$ for every $x\in{\rm
B}(\bar{x},r)\cap S$ and any $\psi:\mathbb{X}\longrightarrow[0,+\infty]$.
Otherwise, it is possible to apply [2, Corollary 3.1] with $X={\rm
B}(\bar{x},r)\cap S$, according to which

${\rm dist}\left(x,{\mathcal{R}}\right)={\rm dist}\left(x,S\cap[\nu_{G,C}\leq
0]\right)\leq\frac{\nu_{G,C}(x)}{\sigma},\quad\forall x\in{\rm
B}(\bar{x},r/2)\cap S.$

Thus, setting $\delta=r/2$ and $\psi(x)=\sigma^{-1}\nu_{G,C}$, the condition
(i) in Definition 3.2 is fulfilled. Since under the above assumptions
${\mathcal{R}}$ is closed, one has

${\rm dist}\left(x,{\mathcal{R}}\right)=0=\psi(x),\quad\forall
x\in{\mathcal{R}},$

so also the condition (ii) in Definition 3.2 is readily satisfied. This
completes the proof. ∎

With the specialization of $\psi$ above introduced, upon the constraint
qualification $(\mathcal{CQ})$, the penalization principle for vector
optimization takes the following form.

######  Proposition 3.5.

With reference to a problem $({\mathcal{P}}_{G})$, let
$\bar{x}\in{\mathcal{R}}=S\cap G^{+1}(C)$. Suppose that:

(i) $f$ is $K$-Lipschitz near $\bar{x}$, with constant $\ell_{f}$ and
$e\in{\rm int}\,K$;

(ii) $G$ is l.s.c. in a neighbourhood of $\bar{x}$ and condition
$(\mathcal{CQ})$ is satisfied.

Then, for any $\ell\geq\ell_{f}$, every local w-eff. solution to
$({\mathcal{P}}_{G})$ is also a local w-eff. solution to problem

${\rm Min}_{K}[f(x)+\ell\sigma^{-1}\nu_{G,C}(x)e]\quad\hbox{ subject to $x\in
S$ }.$ $None$

For any $\ell>\ell_{f}$, every local eff. solution to $({\mathcal{P}}_{G})$ is
a local eff. solution to $({\mathcal{P}}_{G,\ell})$.

###### Proof.

Since $f$ is $K$-Lipschitz near $\bar{x}$, ${\rm int}\,K$ is open and, under
the above assumptions, according to Lemma 3.4, $\sigma^{-1}\nu_{G,C}$ is a
local error bound function for ${\mathcal{R}}$, then the first assertion in
Proposition 3.3 can be invoked. This yields that $\bar{x}$ is a local w-eff.
solution to problem $({\mathcal{P}}_{G,\ell})$, for any $\ell\geq\ell_{f}$.

Since ${\mathcal{R}}$ is closed, in the case $\bar{x}\in{\mathcal{R}}$ is a
local eff. solution to $({\mathcal{P}}_{G})$, it suffices to apply the second
assertion in Proposition 3.3, in order to conclude that $\bar{x}$ is a local
eff. solution to problem $({\mathcal{P}}_{G,\ell})$, for any $\ell>\ell_{f}$.
∎

The constraint qualification $(\mathcal{CQ})$ is expressed in terms of
$\nu_{G,C}$. This function can be built by means of the problem data.
Nevertheless, it would be useful to formulate conditions ensuring the validity
of $(\mathcal{CQ})$ directly on $G$. This can be done by exploiting the metric
increase property, as introduced in [27].

######  Definition 3.6 (Metrically $C$-increasing mapping).

Let $S\subseteq\mathbb{X}$ be a (nonempty) closed set and let
$C\subseteq\mathbb{Z}$ be a closed, convex cone, with $C\neq\\{\mathbf{0}\\}$.
A set-valued mapping $F:\mathbb{X}\rightrightarrows\mathbb{Z}$ between Banach
spaces is said to be metrically $C$-increasing around $\bar{x}\in{\rm
dom}\,G$, relative to $S$, if there exist $\delta>0$ and $\alpha>1$ such that

$\forall x\in{\rm B}(\bar{x},\delta)\cap S,\ \forall
r\in(0,\delta)\quad\exists z\in{\rm B}(x,r)\cap S:\ {\rm B}(F(z),\alpha
r)\subseteq{\rm B}(F(x)+C,r).$ (6)

The quantity

${\rm inc}_{C}(F;S;\bar{x})=\sup\\{\alpha>1:\ \exists\delta>0\hbox{ for which
(\ref{in:mincrSx}) holds}\\}$

is called exact exact bound of metric $C$-increase of $F$ around $\bar{x}$,
relative to $S$.

Several examples of metrically increasing mappings, along with an
infinitesimal criterion for detecting the occurrence of this property, are
provided in [27]. The next proposition enlightens the role of the metric
increase property as a constraint qualification condition.

######  Proposition 3.7.

Let $G:\mathbb{X}\rightrightarrows\mathbb{Z}$, $S$ and $C$ as in problem
$(\mathcal{SVI})$, and let $\bar{x}\in{\mathcal{R}}$. Suppose that:

(i) $G$ is l.s.c. in a neighbourhood of $\bar{x}$;

(ii) $G$ is metrically $C$-increasing around $\bar{x}$, relative to $S$.

Then condition $(\mathcal{CQ})$ holds true with $\sigma=\alpha-1$ and
$r=\delta$, for any $\alpha\in(1,{\rm inc}_{C}(G;S;\bar{x}))$ and $\delta$ as
in $(\ref{in:mincrSx})$.

###### Proof.

As already seen, by hypothesis (i) the function $\nu_{G,C}$ is l.s.c. in ${\rm
B}(\bar{x},\delta_{0})$, for some $\delta_{0}>0$. According to hypothesis
(ii), fixed $\alpha\in(1,{\rm inc}_{C}(G;S;\bar{x})))$, there exists
$\delta_{\alpha}>0$ such that $(\ref{in:mincrSx})$ holds. Observe that the
nature of the metric $C$-increase property around $\bar{x}$ allows one to
assume without loss of generality that $\delta_{\alpha}<\delta_{0}$.

Now, let us take an arbitrary $x\in{\rm B}(\bar{x},\delta_{\alpha})\cap
S\cap[\nu_{G,C}>0]$. Since, under the current assumptions, $\nu_{G,C}$ is
l.s.c. at $x\in{\rm B}(\bar{x},\delta_{0})$, there exists $\delta_{x}>0$ such
that $\nu_{G,C}(z)>0$ for every $z\in{\rm B}(x,\delta_{x})$. Take any $r>0$
such that $r<\min\\{\delta_{\alpha},\delta_{x}\\}$. According to
$(\ref{in:mincrSx})$, there exists $z_{r}\in{\rm B}(x,r)\cap S$ such that

${\rm B}(G(z_{r}),\alpha r)\subseteq{\rm B}(G(x)+C,r).$ (7)

Notice that it must be $z_{r}\neq x$. Indeed, since it is $\nu_{G,C}(x)>0$
(namely, it is ${\rm exc}(G(x),C)>0$), if it were $z_{r}=x$, then by inclusion
$(\ref{in:mincrSzr})$ and [27, Lemma 2.2], one would find

$\displaystyle\nu_{G,C}(x)+\alpha r$ $\displaystyle=$ $\displaystyle{\rm
exc}({\rm B}(G(x),\alpha r),C)={\rm exc}({\rm B}(G(z_{r}),\alpha r),C)$
$\displaystyle\leq$ $\displaystyle{\rm exc}({\rm
B}(G(x)+C,r),C)=\nu_{G,C}(x)+r,$

wherefrom $\alpha\leq 1$, in contrast with the fact that $\alpha>1$.
Furthermore, by applying once again inclusion $(\ref{in:mincrSzr})$ and taking
into account that $\nu_{G,C}(z_{r})>0$, so [27, Lemma 2.2] still works, one
obtains

$\displaystyle\nu_{G,C}(z_{r})$ $\displaystyle=$ $\displaystyle{\rm exc}({\rm
B}(G(z_{r}),\alpha r),C)-\alpha r\leq{\rm exc}({\rm B}(G(x)+C,r),C)-\alpha r$
$\displaystyle=$ $\displaystyle{\rm exc}(G(x)+C,C)+r-\alpha
r=\nu_{G,C}(x)+(1-\alpha)r.$

As it is $z_{r}\in{\rm B}(x,r)\cap S$, from the last inequality chain it
follows

$\nu_{G,C}(x)-\nu_{G,C}(z_{r})\geq(\alpha-1)r\geq(\alpha-1)\|z_{r}-x\|.$

This inequality says that $x$ fails to be a local minimizer for $\nu_{G,C}$
and therefore allows one to get the following estimate

$|\nabla_{S}\nu_{G,C}|(x)=\limsup_{z\xrightarrow{S}x}\frac{\nu_{G,C}(x)-\nu_{G,C}(z)}{\|x-z\|}\geq\lim_{r\downarrow
0}\frac{\nu_{G,C}(x)-\nu_{G,C}(z_{r})}{\|x-z_{r}\|}\geq\alpha-1.$

By arbitrariness of $x\in{\rm B}(\bar{x},\delta_{\alpha})\cap
S\cap[\nu_{G,C}>0]$, the last inequalities show that condition
$(\mathcal{CQ})$ is satisfied with $\sigma=\alpha-1$ and $r=\delta_{\alpha}$,
thereby completing the proof. ∎

On the base of the constraint system analysis exposed above, one is now in a
position to formulate necessary weak efficiency condition for problems
$({\mathcal{P}}_{G})$. To this aim, it remains to recall some further element
of nonsmooth analysis.

Let $\varphi:\mathbb{X}\longrightarrow\mathbb{R}\cup\\{\pm\infty\\}$ be a
function which is finite around $\bar{x}\in\varphi^{-1}(\mathbb{R})$.
Following [19, Section 1.3.2], the set

$\widehat{\partial}^{+}\varphi(\bar{x})=\left\\{x^{*}\in\mathbb{X}^{*}:\
\limsup_{x\to\bar{x}}{\varphi(x)-\varphi(\bar{x})-\langle
x^{*},x-\bar{x}\rangle\over\|x-\bar{x}\|}\leq 0\right\\}$

is called the Fréchet upper subdifferential of $\varphi$ at $\bar{x}$. It is
readily seen that, whenever $\varphi$ is (Fréchet) differentiable at
$\bar{x}$, then $\widehat{\partial}^{+}\varphi(\bar{x})=\\{{\rm
D}\varphi(\bar{x})\\}$, whereas whenever
$\varphi:\mathbb{X}\longrightarrow\mathbb{R}$ is concave, the set
$\widehat{\partial}^{+}\varphi(\bar{x})$ reduces to the superdifferential of
$\varphi$ at $\bar{x}$, in the sense of convex analysis.

######  Remark 4.

The following variational description of the Fréchet upper subdifferential of
$\varphi$ at $\bar{x}$ will be exploited in the sequel: for every
$x^{*}\in\widehat{\partial}^{+}\varphi(\bar{x})$ there exists a function
$\varsigma:\mathbb{X}\longrightarrow\mathbb{R}$, Fréchet differentiable at
$\bar{x}$ and with $\varphi(\bar{x})=\varsigma(\bar{x})$, such that
$\varphi(x)\leq\varsigma(x)$ for every $x\in\mathbb{X}$ and ${\rm
D}\varsigma(\bar{x})=x^{*}$ (to get it, it suffices to remember that
$\widehat{\partial}^{+}\varphi(\bar{x})=-\widehat{\partial}(-\varphi)(\bar{x})$,
where $\widehat{\partial}$ denotes the Fréchet subdifferential, and then apply
[19, Theorem 1.88(i)]).

Given a mapping $f:\mathbb{X}\longrightarrow\mathbb{Y}$ between Banach spaces
and $\bar{x}\in\mathbb{X}$ $f^{\prime}(\bar{x};v)$ indicates the directional
derivative of $f$ at $\bar{x}$, in the direction $v\in\mathbb{X}$. If its
directional derivative exists for every $v\in\mathbb{X}$, $f$ is said to be
directionally differentiable at $\bar{x}$.

A first-order necessary condition for weak efficiency of solutions to
$({\mathcal{P}}_{G})$ can be stated as follows.

######  Theorem 3.8 (Weak efficiency condition via penalization).

With reference to a problem $({\mathcal{P}}_{G})$, let
$\bar{x}\in{\mathcal{R}}=S\cap G^{+1}(C)$ be a local w-eff. solution to
$({\mathcal{P}}_{G})$. Suppose that:

(i) $f$ is $K$-Lipschitz near $\bar{x}$, with constant $\ell_{f}$ and
$e\in{\rm int}\,K$, and is directionally differentiable at $\bar{x}$;

(ii) $G$ is l.s.c. in a neighbourhood of $\bar{x}$ and metrically
$C$-increasing around $\bar{x}$, relative to $S$;

(iii) $\widehat{\partial}^{+}\nu_{G,C}(\bar{x})\neq\varnothing$.

Then for any $\alpha\in(1,{\rm inc}_{C}(G;S;\bar{x}))$, $\ell\geq\ell_{f}$ and
$x^{*}\in\widehat{\partial}^{+}\nu_{G,C}(\bar{x})$ it must be

$f^{\prime}(\bar{x};v)+\frac{\ell}{\alpha-1}\langle x^{*},v\rangle
e\not\in{\rm int}\,K,\quad\forall v\in{\rm I}(S;\bar{x}).$ (8)

###### Proof.

Under the assumptions made, in the light of Proposition 3.7 the condition
$(\mathcal{CQ})$ is satisfied. Thus, it is possible to invoke Proposition 3.5,
according to which $\bar{x}$ turns out to be a w-eff. solution of problem
$({\mathcal{P}}_{G,\ell})$, for any $\alpha\in(1,{\rm inc}_{C}(G;S;\bar{x}))$
and $\ell\geq\ell_{f}$. This means that there exists $\delta>0$ such that

$\left(f+\frac{\ell}{\alpha-1}\nu_{G,C}e\right)({\rm B}(\bar{x},\delta)\cap
S)\cap[f(\bar{x})-{\rm int}\,K]=\varnothing.$ (9)

Take an arbitrary $v\in{\rm I}(S;\bar{x})\cap{\mathbb{B}}$. By reducing the
value of $\delta>0$ in $(\ref{eq:weffVOPGlempty})$ if needed, one can assume
that $\bar{x}+tv\in S$, for all $t\in(0,\delta)$. Therefore, from the relation
in $(\ref{eq:weffVOPGlempty})$ it follows

$\frac{f(\bar{x}+tv)-f(\bar{x})}{t}+\frac{\ell\nu_{G,C}(\bar{x}+tv)e}{(\alpha-1)t}\in\mathbb{Y}\backslash(-{\rm
int}\,K),\quad\forall t\in(0,\delta).$ (10)

Let $x^{*}$ be an arbitrary element of
$\widehat{\partial}^{+}\nu_{G,C}(\bar{x})$. According to the characterization
of upper Fréchet subgradients mentioned in Remark 4, there exists a Fréchet
differentiable function $\varsigma:\mathbb{X}\longrightarrow\mathbb{R}$ such
that $\varsigma(\bar{x})=\nu_{G,C}(\bar{x})=0$, $\varsigma(x)\geq\nu_{G,C}(x)$
for every $x\in\mathbb{X}$, and ${\rm D}\varsigma(\bar{x})=x^{*}$. Therefore,
one has

$\varsigma(\bar{x}+tv)-\nu_{G,C}(\bar{x}+tv)\geq 0,\quad\forall
t\in(0,+\infty),$

whence

$\frac{\ell[\varsigma(\bar{x}+tv)-\nu_{G,C}(\bar{x}+tv)]e}{(\alpha-1)t}\in
K,\quad\forall t\in(0,+\infty).$ (11)

By combining $(\ref{eq:weffpenal})$ and $(\ref{in:sigmameritposK})$ and
observing that, for every $y\in\mathbb{Y}\backslash(-{\rm int}\,K)$ it holds
$y+K\subseteq\mathbb{Y}\backslash(-{\rm int}\,K)$, one obtains

$\frac{f(\bar{x}+tv)-f(\bar{x})}{t}+\frac{\ell\varsigma(\bar{x}+tv)e}{(\alpha-1)t}\in\mathbb{Y}\backslash(-{\rm
int}\,K),\quad\forall t\in(0,\delta).$

By passing to the limit as $t\downarrow 0$ in the last inclusion, while taking
into account that the cone $\mathbb{Y}\backslash(-{\rm int}\,K)$ is closed and
that $f$ is directionally differentiable at $\bar{x}$, one achieves the
relation in $(\ref{notin:weffconpenal})$ for any $v\in{\rm
I}(S;\bar{x})\cap{\mathbb{B}}$. Since the mapping $v\mapsto
f^{\prime}(\bar{x};v)+\frac{\ell}{\alpha-1}\langle x^{*},v\rangle e$ is
positively homogeneous and $\mathbb{Y}\backslash(-{\rm int}\,K)$ is a cone,
the validity of relation in $(\ref{notin:weffconpenal})$ can be extended to
the whole set ${\rm I}(S;\bar{x})$. By arbitrariness of $x^{*}$, this
reasoning completes the proof. ∎

Among the hypotheses of Theorem 3.8, the most involved is (iii), so its
deserves some comment. In the next remark, some elements for discussion are
provided in order to clarify the meaning of such an assumption.

######  Remark 5.

According to its definition, the merit function $\nu_{G,C}$ is nonnegative
and, since it is $\bar{x}\in G^{+1}(C)$ one has $\nu_{G,C}(\bar{x})=0$, so the
hypothesis (iii) in Theorem 3.8 is about the nontriviality of the Fréchet
upper subdifferential at a (global) minimizer. A systematic study of this tool
of nonsmooth analysis (actually, not so often employed as its lower
counterpart) and related optimality conditions for constrained minimization
problems can be found in [18, 20]. In particular, it was shown that, for given
a function $\varphi:\mathbb{X}\longrightarrow\\{\pm\infty\\}$, which is
defined on an Asplund space and locally Lipschitz around $\bar{x}$, the
nonemptiness of $\widehat{\partial}^{+}\varphi(\bar{x})$ is automatic if
$\varphi$ is upper regular at $\bar{x}$, i.e.
$\widehat{\partial}^{+}\varphi(\bar{x})=\partial^{+}\varphi(\bar{x})$, where
$\partial^{+}\varphi(\bar{x})$ denotes the limiting upper subdifferential of
$\varphi$ at $\bar{x}$, defined through the basic normals to the hypergraph of
$\varphi$ (see [19, Definition 1.78]). In such a circumstance, it holds

$\partial_{\rm Cl}\varphi(\bar{x})={\rm
cl}^{*}\,\widehat{\partial}^{+}\varphi(\bar{x}),$

where $\partial_{\rm Cl}\varphi(\bar{x})$ denotes the Clarke generalized
gradient of $\varphi$ at $\bar{x}$ and ${\rm cl}^{*}\,A$ marks the closure of
a set $A$ with respect to the weak${}^{*}\,$ topology (see, for more details,
[20, Remark 4.5] and [20, Section 5.5.4]). Note that, as it is possible to
check at once, $\nu_{G,C}$ is locally Lipschitz around $\bar{x}$ whenever $G$
is Lipschitz continuous around $\bar{x}$.

By introducing proper convexity/concavity assumptions on the problem data $S$,
$f$ and $G$, it is possible to establish a first-order necessary weak
efficiency condition in a scalarized form. To this aim, the next remark will
be useful.

######  Remark 6.

(i) It is readily seen that, whenever
$G:\mathbb{X}\rightrightarrows\mathbb{Z}$ is $C$-bounded around a point
$\bar{x}\in\mathbb{X}$, i.e. there exists $\delta>0$ such that $G(x)\backslash
C$ is bounded for every $x\in{\rm B}(\bar{x},\delta)$, then $\bar{x}\in{\rm
int}\,({\rm dom}\,\nu_{G,C})$.

(ii) whenever $G:\mathbb{X}\rightrightarrows\mathbb{Z}$ is $C$-concave on
$\mathbb{X}$ the function $\nu_{G,C}$ is convex on $\mathbb{X}$ (see, for
instance, [28, Remark 4.14]).

######  Theorem 3.9 (Weak efficiency condition via penalization under
convexity).

With reference to a problem $({\mathcal{P}}_{G})$, let
$\bar{x}\in{\mathcal{R}}=S\cap G^{+1}(C)$ be a local w-eff. solution to
$({\mathcal{P}}_{G})$. Suppose that:

(i) $S$ is locally convex around $\bar{x}$;

(ii) $f$ is $K$-Lipschitz near $\bar{x}$, with constant $\ell_{f}$ and
$e\in{\rm int}\,K$, and is directionally differentiable at $\bar{x}$, with
$f^{\prime}(\bar{x};\cdot):\mathbb{X}\longrightarrow\mathbb{Y}$ being
$K$-sublinear;

(iii) $G$ is l.s.c. in a neighbourhood of $\bar{x}$ and metrically
$C$-increasing around $\bar{x}$, relative to $S$;

(iv) $G$ is $C$-bounded around $\bar{x}$ and Hausdorff u.s.c. at $\bar{x}$;

(v) $G$ is $C$-concave in $\mathbb{X}$.

Then there exists $y^{*}\in{K}^{{}^{\oplus}}\backslash\\{\mathbf{0}^{*}\\}$
such that

$\langle y^{*},f^{\prime}(\bar{x};v)\rangle+\frac{\ell}{\alpha-1}\langle
y^{*},\nu_{G,C}^{\prime}(\bar{x};v)e\rangle\geq 0,\quad\forall v\in{\rm
I}(S;\bar{x}).$ (12)

###### Proof.

Let us start with observing that, by virtue of the hypotheses of
$C$-concavity, $\nu_{G,C}$ is convex on $\mathbb{X}$. Moreover, by hypothesis
(iv), it is $\bar{x}\in{\rm int}\,({\rm dom}\,\nu_{G,C})$. Moreover, since $G$
is Hausdorff u.s.c. at $\bar{x}$, function $\nu_{G,C}$ is also u.s.c. at
$\bar{x}$ (see [27, Lemma 2.3(ii)]). So, remembering that it is also l.s.c.
around $\bar{x}$ on the account of hypothesis (iii), $\nu_{G,C}$ turns out to
be continuous and hence directionally differentiable at $\bar{x}$ (remember
[31, Theorem 2.4.9]). From Remark 2 it follows that the mapping
$v\mapsto\nu_{G,C}(v)e$ is $K$-sublinear on $\mathbb{X}$. As a sum of two
$K$-sublinear mappings,
$f^{\prime}(\bar{x};\cdot)+\frac{\ell}{\alpha-1}\nu_{G,C}^{\prime}(\bar{x};\cdot)e$
is $K$-sublinear on $\mathbb{X}$. On the other hand, since according to
hypothesis (i) $S$ is locally convex near $\bar{x}$, the cone ${\rm
I}(S;\bar{x})$ is convex. It follows that the subset of $\mathbb{Y}$, given by

$f^{\prime}(\bar{x};{\rm
I}(S;\bar{x}))+\frac{\ell}{\alpha-1}\nu_{G,C}^{\prime}(\bar{x};{\rm
I}(S;\bar{x}))e+K,$

is a convex cone as an image of a convex cone through a $K$-sublinear mapping.

Since $\bar{x}$ is a local w-eff. solution of $({\mathcal{P}}_{G})$, by
arguing as in the proof of Theorem 3.8, it is possible to show that

$\left[f^{\prime}(\bar{x};{\rm
I}(S;\bar{x}))+\frac{\ell}{\alpha-1}\nu_{G,C}^{\prime}(\bar{x};{\rm
I}(S;\bar{x}))e\right]\cap(-{\rm int}\,K)=\varnothing.$

This entails that it holds also

$\left[f^{\prime}(\bar{x};{\rm
I}(S;\bar{x}))+\frac{\ell}{\alpha-1}\nu_{G,C}^{\prime}(\bar{x};{\rm
I}(S;\bar{x}))e+K\right]\cap(-{\rm int}\,K)=\varnothing.$

In such a circumstance one can invoke the Eidelheit theorem (see, for
instance, [31, Theorem 1.1.3]). It ensures the existence of
$y^{*}\in\mathbb{Y}^{*}\backslash\\{\mathbf{0}^{*}\\}$ and
$\gamma\in\mathbb{R}$, such that

$\langle
y^{*},f^{\prime}(\bar{x};v)+\frac{\ell}{\alpha-1}\nu_{G,C}^{\prime}(\bar{x};v)e\rangle\geq\gamma\geq\langle
y^{*},y\rangle,$ (13) $\forall v\in{\rm I}(S;\bar{x}),\ \forall y\in{\rm
cl}\,(-{\rm int}\,K)=-K.$

Since, in particular, it holds

$\langle
y^{*},f^{\prime}(\bar{x};\mathbf{0})+\frac{\ell}{\alpha-1}\nu_{G,C}^{\prime}(\bar{x};\mathbf{0})e\rangle=0\geq\gamma\geq
0=\langle y^{*},\mathbf{0}\rangle,$

it follows that $\gamma$ must be $0$ and, by consequence, the second
inequality in $(\ref{in:linsepdderfmer})$ gives $y^{*}\in{K}^{{}^{\oplus}}$.
This completes the proof. ∎

##  4 Weak efficiency conditions via tangential approximations

Throughout this section, as a first-order approximation of set-valued mappings
the notion of outer prederivative, introduced in [13], will be employed.

######  Definition 4.1 (Outer prederivative).

Let $F:\mathbb{X}\rightrightarrows\mathbb{Z}$ be a set-valued mapping between
Banach spaces and let $\bar{x}\in{\rm dom}\,F$. A p.h. set-valued mapping
$H_{F}(\bar{x};\cdot):\mathbb{X}\rightrightarrows\mathbb{Z}$ is said to be an
outer prederivative of $F$ at $\bar{x}$ if for every $\epsilon>0$ there exists
$\delta>0$ such that

$F(x)\subseteq
F(\bar{x})+H_{F}(\bar{x};x-\bar{x})+\epsilon\|x-\bar{x}\|{\mathbb{B}},\quad\forall
x\in{\rm B}(\bar{x},\delta).$

Extended discussions about this generalized differentiation concept can be
found, for instance, in [13, 12, 21].

For subsequent considerations, it is worth observing that the notion of outer
prederivative collapses to the notion of Bouligand-derivative (or
B-derivative), when both $F$ and $H_{F}(\bar{x};\cdot)$ are single-valued and
$H_{F}(\bar{x};\cdot)$ is continuous. More precisely, following [22], a
mapping $f:\mathbb{X}\longrightarrow\mathbb{Z}$ between Banach spaces is said
to be $B$-differentiable at $\bar{x}\in\mathbb{X}$ if there exists a p.h. and
continuous mapping ${\rm
D}_{B}f(\bar{x};\cdot):\mathbb{X}\longrightarrow\mathbb{Z}$, called the
B-derivative of $f$ at $\bar{x}$, such that

$\lim_{x\to\bar{x}}{f(x)-f(\bar{x})-{\rm
D}_{B}f(\bar{x};x-\bar{x})\over\|x-\bar{x}\|}=0.$

By exploiting as a constraint qualification the metric $C$-increase property
of $G$, the following inner tangential approximation of ${\mathcal{R}}$ has
been established in [29], which is expressed in terms of outer prederivatives
and tangent cones. Its proof was provided in a finite-dimensional setting, but
a perusal of the involved arguments reveals that it can be extended without
any modification to a Banach space setting.

######  Proposition 4.2.

([29, Theorem 3.1]) Let $G:\mathbb{X}\rightrightarrows\mathbb{Z}$, $S$ and $C$
as in problem $(\mathcal{SVI})$, and let $\bar{x}\in{\mathcal{R}}=S\cap
G^{+1}(C)$. Suppose that:

(i) $G$ is l.s.c. in a neighbourhood of $\bar{x}$;

(ii) $G$ is metrically $C$-increasing around $\bar{x}$, relative to $S$;

(iii) $G$ admits $H_{G}(\bar{x};\cdot):\mathbb{X}\rightrightarrows\mathbb{Z}$
as an outer prederivative at $\bar{x}$.

Then it holds

$H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm I_{w}}(S;\bar{x})\subseteq{\rm
T}({\mathcal{R}};\bar{x}).$ (14)

If, in addition,

(iv) $H_{G}(\bar{x};\cdot)$ is Lipschitz,

then the stronger inclusion holds

$H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x})\subseteq{\rm
T}({\mathcal{R}};\bar{x}).$ (15)

Following the well-known Euler-Lagrange scheme for deriving necessary
optimality conditions in the presence of constraints, from the above
tangential approximation of the feasible region of $({\mathcal{P}}_{G})$, it
is possible to obtain the below first-order weak efficiency condition.

######  Theorem 4.3.

With reference to a problem $({\mathcal{P}}_{G})$, let
$\bar{x}\in{\mathcal{R}}$ be a local w-eff. solution. Suppose that:

(i) $f$ is $B$-differentiable at $\bar{x}$;

(ii) $G$ is l.s.c. in a neighbourhood of $\bar{x}$ and is metrically
$C$-increasing around $\bar{x}$, relative to $S$;

(iii) $G$ admits $H_{G}(\bar{x};\cdot):\mathbb{X}\rightrightarrows\mathbb{Z}$
as an outer prederivative at $\bar{x}$.

Then,

${\rm D}_{B}f(\bar{x};v)\not\in-{\rm int}\,K,\quad\forall v\in
H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm I_{w}}(S;\bar{x}).$ (16)

If, in addition,

(iv) ${\rm D}_{B}f(\bar{x};\cdot):\mathbb{X}\longrightarrow\mathbb{Y}$ is
$K$-convexlike on the set $H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm
T}(S;\bar{x})$;

(v) $H_{G}(\bar{x};\mathbf{0})\subseteq C$;

(vi) $H_{G}(\bar{x};\cdot)$ is Lipschitz,

there exists $y^{*}\in{K}^{{}^{\oplus}}\backslash\\{\mathbf{0}^{*}\\}$ such
that

$y^{*}\circ{\rm D}_{B}f(\bar{x};v)\geq 0,\quad\forall v\in
H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x}).$ (17)

In particular, whenever $f$ is Fréchet differentiable at $\bar{x}$, it results
in

$-{\rm D}f(\bar{x})^{*}y^{*}\in{[H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm
T}(S;\bar{x})]}^{{}^{\ominus}}.$ (18)

###### Proof.

Upon hypotheses (ii) and (iii), the inner tangential approximation given by
$(\ref{in:intanappIang})$ can be employed. So, take an arbitrary $v\in
H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm I_{w}}(S;\bar{x})$. As it is also
$v\in{\rm T}({\mathcal{R}};\bar{x})$, there exist sequences $(v_{n})$, with
$v_{n}\to v$, and $(t_{n})$, with $t_{n}\downarrow 0$, as $n\to\infty$, such
that $\bar{x}+t_{n}v_{n}\in{\mathcal{R}}$. Since $\bar{x}$ is a local w-eff.
solution to $({\mathcal{P}}_{G})$, by recalling hypothesis (i), one obtains

${\rm
D}_{B}f(\bar{x};v_{n})+\frac{o(\bar{x};t_{n}v_{n})}{t_{n}}=\frac{f(\bar{x}+t_{n}v_{n})-f(\bar{x})}{t_{n}}\in\mathbb{Y}\backslash(-{\rm
int}\,K).$

By passing to the limit as $n\to\infty$, taking into account that
$\mathbb{Y}\backslash(-{\rm int}\,K)$ is a closed set and the mapping ${\rm
D}_{B}f(\bar{x};\cdot)$ is continuous, one achieves the inequality in
$(\ref{in:nweffcond1})$.

Upon the hypothesis (iv), the set ${\rm
D}_{B}f(\bar{x};H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x}))+K$ is a
convex subset of $\mathbb{Y}$. By arguing as in the first part of the proof,
one can show that

${\rm D}_{B}f(\bar{x};v)\not\in-{\rm int}\,K,\quad\forall v\in
H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x}),$

which amounts to say

$\left[{\rm D}_{B}f(\bar{x};H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm
T}(S;\bar{x}))\right]\cap(-{\rm int}\,K)=\varnothing.$

Notice that this implies

$[{\rm D}_{B}f(\bar{x};H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm
T}(S;\bar{x}))+K]\cap(-{\rm int}\,K)=\varnothing.$

By the Eidelheit theorem there exists
$y^{*}\in\mathbb{Y}^{*}\backslash\\{\mathbf{0}^{*}\\}$ and
$\gamma\in\mathbb{R}$ such that

$\langle y^{*},y\rangle\leq\gamma\leq\langle y^{*},{\rm
D}_{B}f(\bar{x};v)\rangle,$ (19) $\quad\forall y\in{\rm cl}\,(-{\rm
int}\,K)=-K,\quad\forall v\in H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm
T}(S;\bar{x}).$

Since owing to hypothesis (v) it is $\mathbf{0}\in-K\cap
H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x})$, according to the
inequalities in $(\ref{in:Eidsepar})$ it must be $\gamma=0$. Consequently, the
first inequality in $(\ref{in:Eidsepar})$ gives
$y^{*}\in{\mathbb{Y}}^{{}^{\oplus}}$, whereas the second one yields
$(\ref{in:thmscalariz})$. In the case of Fréchet differentiability of $f$ at
$\bar{x}$, inclusion $(\ref{in:thminclus})$ is a direct consequence of
inequality $(\ref{in:thmscalariz})$. The proof is complete. ∎

######  Remark 7.

The property of a mapping to be $K$-convexlike on a set $D$ depends
essentially on the set $D$. Notice that, if $D_{1}\subseteq D$, a mapping
$K$-convexlike on $D$ may fail to be $K$-convexlike on $D_{1}$. Thus,
hypothesis (iv) links crucially the behaviour of ${\rm D}_{B}f(\bar{x};\cdot)$
with the geometry of the set $H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm
T}(S;\bar{x})$. On the other hand, the $K$-sublinearity property is stable
under convex restrictions, in the sense that if
$h:\mathbb{X}\longrightarrow\mathbb{Y}$ is $K$-sublinear on a set
$D\subseteq\mathbb{X}$, it still remains so on each convex subset
$D_{1}\subseteq D$. This fact makes it convenient to consider the following
replacement of hypothesis (iv), with separate (but stricter) requirements on
the involved problem data:

(iv’) ${\rm D}_{B}f(\bar{x};\cdot)$ is $K$-sublinear on $\mathbb{X}$,
$H_{G}(\bar{x};\cdot)$ is $C$-superlinear on $\mathbb{X}$, and $S$ is locally
convex near $\bar{x}$.

In such a circumstance, ${\rm T}(S;\bar{x})$ is a convex cone as well as
$H_{G}(\bar{x};\cdot)^{+1}(C)$. Since the set ${\rm
D}_{B}f(\bar{x};H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x}))+K$ is
convex, ${\rm D}_{B}f(\bar{x};\cdot)$ turns out to be $K$-convexlike on the
set $H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x})$.

The next result provides a refinement of Theorem 4.3, which can be
established, under proper qualification conditions, by replacing general
first-order approximations of the data with the local convexity of $S$ and
linear approximations of $f$ and $G$.

######  Theorem 4.4 (Multiplier rule via fans).

Let $\bar{x}\in{\mathcal{R}}$ be a local w-eff. solution to problem
$({\mathcal{P}}_{G})$. Suppose that hypotheses (i)-(iii) are satisfied and, in
addition, that:

(iv) $S$ is locally convex near $\bar{x}$;

(v) $f$ is Fréchet differentiable at $\bar{x}$;

(vi) $H_{G}(\bar{x};\cdot)$ is a fan finitely generated by ${\mathcal{G}}={\rm
conv}\,\\{\Lambda_{1},\dots,\Lambda_{p}\\}$;

(vii) the further qualification condition holds

$\left(\bigcap_{i=1}^{p}{\rm int}\,\Lambda_{i}^{-1}(C)\right)\cap{\rm
int}\,{\rm T}(S;\bar{x})\neq\varnothing.$ (20)

Then there exist $y^{*}\in{K}^{{}^{\oplus}}\backslash\\{\mathbf{0}^{*}\\}$
and, for each $i=1,\dots,p$, $x^{*}_{i}\in\mathbb{X}^{*}$ and sequences
$(z^{*}_{i,n})_{n}$ in $\mathbb{Z}^{*}$, with
$z^{*}_{i,n}\in{C}^{{}^{\ominus}}$ and $\Lambda_{i}^{*}z^{*}_{i,n}\to
x_{i}^{*}$, such that

$\mathbf{0}^{*}\in{\rm D}f(\bar{x})^{*}y^{*}+\sum_{i=1}^{p}x_{i}^{*}+{\rm
N}(S;\bar{x}).$ (21)

###### Proof.

Observe that by hypothesis (v), it is
$H_{G}(\bar{x};\mathbf{0})=\\{\mathbf{0}\\}\subseteq C$, so hypothesis (v) of
Theorem 4.3 is fulfilled. As recalled in Remark 3, since the bundle generating
$H_{G}(\bar{x};\cdot)$ is bounded according to hypothesis (vi),
$H_{G}(\bar{x};\cdot)$ is Lipschitz. Moreover, it is readily seen that if
$H_{G}(\bar{x};\cdot)$ is generated by ${\mathcal{G}}={\rm
conv}\,\\{\Lambda_{1},\dots,\Lambda_{p}\\}$, it holds

$H_{G}(\bar{x};\cdot)^{+1}(C)=\bigcap_{i=1}^{p}\Lambda_{i}^{-1}(C).$

Since each element $\Lambda_{i}^{-1}(C)$ in the above intersection is a convex
cone as well as ${\rm T}(S;\bar{x})$ by hypothesis (iv), the set
$H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x})$ turns out to be a convex
cone. By hypothesis (v) it is ${\rm D}_{B}f(\bar{x};\cdot)={\rm D}f(\bar{x})$,
so, as a linear mapping it is $K$-convexlike on
$H_{G}(\bar{x};\cdot)^{+1}(C)\cap{\rm T}(S;\bar{x})$. One is therefore in a
position to apply Theorem 4.3. Thus, there exists
$y^{*}\in{K}^{{}^{\oplus}}\backslash\\{\mathbf{0}^{*}\\}$ such that

$-{\rm
D}f(\bar{x})^{*}y^{*}\in{\left[\bigcap_{i=1}^{p}\Lambda_{i}^{-1}(C)\cap{\rm
T}(S;\bar{x})\right]}^{{}^{\ominus}}.$

By virtue of the qualification condition in hypothesis (vii), on account of
the relations discussed in Remark 1, the last inclusion implies

$-{\rm
D}f(\bar{x})^{*}y^{*}\in\sum_{i=1}^{p}{\left(\Lambda_{i}^{-1}(C)\right)}^{{}^{\ominus}}+{{\rm
T}(S;\bar{x})}^{{}^{\ominus}}=\sum_{i=1}^{p}{\rm
cl}\,\Lambda_{i}^{*}({C}^{{}^{\ominus}})+{\rm N}(S;\bar{x}).$

This means that there must exist $x_{i}^{*}\in{\rm
cl}\,\Lambda_{i}^{*}({C}^{{}^{\ominus}})$, for every $i=1,\dots,p$, such that

$-{\rm D}f(\bar{x})^{*}y^{*}\in\sum_{i=1}^{p}x_{i}^{*}+{\rm N}(S;\bar{x}),$

which immediately entails the existence of such sequences $(z^{*}_{i,n})_{n}$
in $\mathbb{Z}^{*}$ as in asserted in the thesis, thereby completing the
proof. ∎

######  Remark 8.

It is worth noting that, whenever ${\rm int}\,C\neq\varnothing$ and
$\bar{x}\in{\rm int}\,S$, the qualification condition in $(\ref{ne:conecq})$
is satisfied provided that the following Slater-type condition holds:

$\exists x_{0}\in\mathbb{X}:\ \Lambda_{i}x_{0}\in{\rm int}\,C,\quad\forall
i=1,\dots,p.$ (22)

As one expects, the formulation of the multiplier rule expressed by
$(\ref{in:multrule})$ simplifies if specialized to a finite-dimensional space
setting. This is done in the next result, where the adjoint operation (which
can be viewed as a matrix transposition) is now denoted by the symbol $\top$.

######  Corollary 4.5 (Weak Pareto efficiency condition in finite-dimensional
spaces).

Let $\bar{x}\in{\mathcal{R}}$ be a local w-eff. solution to problem
$({\mathcal{P}}_{G})$, with $\mathbb{X}=\mathbb{R}^{n}$,
$\mathbb{Y}=\mathbb{R}^{m}$, $\mathbb{Z}=\mathbb{R}^{p}$,
$K=\mathbb{R}^{m}_{+}$ and ${\rm int}\,C\neq\varnothing$. Suppose that
hypotheses (i)-(vi) are satisfied, whereas (vii) is replaced by condition
$(\ref{in:Slater})$. Then there exist
$v\in\mathbb{R}^{m}_{+}\backslash\\{\mathbf{0}\\}$ and
$c_{i}\in{C}^{{}^{\ominus}}$, $i=1,\dots,p$, such that

$\mathbf{0}\in{\rm
D}f(\bar{x})^{\top}v+\sum_{i=1}^{p}\Lambda_{i}^{\top}c_{i}+{\rm
N}(S;\bar{x}).$ (23)

If, in particular, $\bar{x}\in{\rm int}\,S$, it results in

$\mathbf{0}={\rm D}f(\bar{x})^{\top}v+\sum_{i=1}^{p}\Lambda_{i}^{\top}c_{i}.$

###### Proof.

Under condition $(\ref{in:Slater})$, also the hypothesis (vii) of Theorem 4.4
is fulfilled. So, in applying this result, it suffices to observe that
$\Lambda_{i}x_{0}\in{\rm int}\,C$ implies $x_{0}\in\Lambda_{i}^{-1}({\rm
int}\,C)=\Lambda_{i}^{-1}({\rm ri}\,C)\neq\varnothing$. Thus, by taking into
account what noted in Remark 1(ii), it is true that

${[\Lambda^{-1}_{i}(C)]}^{{}^{\ominus}}=\Lambda_{i}^{\top}({C}^{{}^{\ominus}}),\quad\forall
i=1,\dots,p.$

This completes the proof. ∎

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# Towards more scientific meta-analyses

Lily H. Zhang1 Menelaos Konstantinidis2 Marie-Abèle Bind3,4 Donald B.
Rubin5,6,7 1Center for Data Science, New York University  2Li Ka Shing
Knowledge Institute, St. Michael’s Hospital, Unity Health Toronto, ON, Canada
3Biostatistics Center, Massachusetts General Hospital, Boston, MA 02114
4Department of Medicine, Harvard Medical School, Boston, MA 02115  5Professor
Emeritus, Department of Statistics, Harvard University,
Cambridge MA, US  6Professor, Yau Center for Mathematical Sciences, Tsinghua
University, Beijing China  7Murray Shusterman Senior Research Fellow,
Department of Statistical Science, Fox School of Business, Temple University,
Philadelphia, PA

###### Abstract

Meta-analysis can be a critical part of the research process, often serving as
the primary analysis on which the practitioners, policymakers, and individuals
base their decisions. However, current literature synthesis approaches to
meta-analysis typically estimate a different quantity than what is implicitly
intended; concretely, standard approaches estimate the average effect of a
treatment for a population of imperfect studies, rather than the true
scientific effect that would be measured in a population of hypothetical
perfect studies. We advocate for an alternative method, called response-
surface meta-analysis, which models the relationship between the quality of
the study design as predictor variables and its reported estimated effect size
as the outcome variable in order to estimate the effect size obtained by the
hypothetical ideal study. The idea was first introduced by Rubin several
decades ago Rubin , and here we provide a practical implementation. First, we
reintroduce the idea of response-surface meta-analysis, highlighting its focus
on a scientifically-motivated estimand while proposing a straightforward
implementation. Then we compare the approach to traditional meta-analysis
techniques used in practice. We then implement response-surface meta-analysis
and contrast its results with existing literature-synthesis approaches on both
simulated data and a real-world example published by the Cochrane
Collaboration. We conclude by detailing the primary challenges in the
implementation of response-surface meta-analysis and offer some suggestions to
tackle these challenges.

##  1 Background

Gene Glass first coined the term “meta-analysis” in 1976 to mean “the
statistical analysis of a large collection of individual studies” Glass . Many
statistical methods have emerged to address various challenges in conducting a
meta-analysis, from various estimators for between-study heterogeneity
Veroniki  to methods for detecting and possibly correcting publication bias
Thornton . Yet the central methodology in a meta analysis typically involves
combining studies via some form of averaging, such as is the case when
estimating the relevant mean under a fixed-effect(s) or random-effects model
Deeks . Consequently, there is an implicit assumption that the researcher’s
target is a summary of the existing literature.

With current meta-analysis methodologies, which we will refer to as
“literature synthesis meta-analysis,” we are addressing the question, “What is
the typical effect in the underlying population of studies from which the
published studies can be viewed as a random sample?” Although this is a
reasonable question, it is usually not the actual question researchers set out
to address when they conduct a meta-analysis. Instead, the hope is usually to
use the collection of existing research studies to better estimate the true
scientific effect if it could be measured perfectly. The latter is the focus
of response-surface meta-analysis, which has been praised by the inventor of
the concept of meta-analysis himself Glass review .

In this work, we formulate response-surface meta-analysis in Section 2 and
compare it with existing approaches in Section 3. Then, we show how response-
surface meta-analysis can yield different conclusions from traditional
approaches via a simulation study where the true effect is known (Section 4)
as well as via a real-world example published by the Cochrane Collaboration
(Section 5).

##  2 Formulating Response-surface Extrapolation

Here, we review the central parts of the response-surface approach first
proposed in Rubin . Let us suppose that the goal is to estimate the true
effect of a treatment relative to a control on an outcome. We denote this
effect as $\tau$. Individual studies report average effect sizes $\psi$, which
can be influenced by factors $(X,Z)$; here, $X$ denotes scientific factors, or
variables that have scientific interest, such as sex, age, or education of
study participants; Z denotes design factors that influence the quality of
studies, such as sample size and the experimental design itself (e.g.,
randomized vs. observational). The response-surface is a probabilistic model
for $\psi$ as a function of $X$ and $Z$, denoting how both scientific and
design factors can influence the study effect sizes reported by individual
studies. To estimate the true effect size $\tau_{x}$ for given scientific
factors $X=x$, we need to estimate this response-surface where $Z$ is
extrapolated to the perfect study, which we denote as $z_{\infty}$. The reason
for this focus is that scientifically, we are only interested in the effect as
a function of scientific factors $X$ in the collection of perfect studies.

The primary benefit for response-surface approach is its theoretical clarity
and focus on the correct estimand. For further discussion, see Rubin 1990
Rubin .

In this work, we implement response-surface meta-analysis using fixed- and
random-effects meta-regression HigginsMR  with study design as a covariate.
Then, we report the predicted mean effect under a prespecified ideal study
design. This approach allows us to take advantage of the statistical advances
in meta-analysis research made since response-surface meta-analysis was first
proposed Tipton .

##  3 Comparing Response-Surface and Traditional Meta-Analysis

The primary difference between literature synthesis and response-surface meta-
analysis is the estimand of interest. Traditionally, the conceptual framework
for meta-analysis assumes that the study-specific effect sizes are distributed
around some mean treatment effect. However, estimating the mean treatment
effect for an underlying population of existing imperfect studies is not the
same as estimating the mean treatment effect that would be observed in a
population of perfect studies. In other words, current literature-synthesis
meta-analysis methods focus on the estimands at each value of X=x,
$\tau_{\text{synthesis},x}=\mathbb{E}_{\psi}(\psi|X=x)=\mathbb{E}_{\psi}\mathbb{E}_{Z}(\psi|X=x,Z)$,
where the distribution of $Z$ is that of the population of study designs of
which published studies are a random sample. In contrast, response-surface
meta-analysis targets the estimand
$\tau_{\infty,x}=\mathbb{E}_{\psi}(\psi|X=x,Z=z_{\infty})$, which we argue is
more conceptually aligned with the scientific purpose of meta-analysis.

Literature synthesis meta-analysis takes into account study design but in a
different way than response-surface meta-analysis does. Namely, literature
synthesis methods will often bias the distribution of study design quality
towards higher quality studies by excluding certain “unworthy” studies in the
analysis. Examples include considering only randomized control trials or peer-
reviewed articles in an analysis. However, averaging over such studies is
still distinct from extrapolating to the ideal study; by excluding studies,
one has changed the distribution of studies being considered, but the included
studies themselves do not typically reflect a sample of perfect studies. In
addition, literature-synthesis methods choose a hard threshold for study
inclusion and exclusion whereas response-surface meta-analysis takes advantage
of all studies to fit the response-surface and estimate the effect under
perfect conditions as well as possible.

Our implementation of response-surface meta-analysis is closely related to
meta-regression, the method of identifying sources of heterogeneity in a meta-
analysis. A meta-regression regresses study effects on covariates such as
study design or scientific factors to estimate whether such factors
significantly affect the measured effect. In this paper, we implement
response-surface meta-analysis by first performing a meta-regression using
design quality as the independent variable and then using this regression
model to estimate the predicted mean effect under optimal study design
conditions. How the resulting model is used for analysis is where meta-
regression and our implementation of response-surface meta-analysis differ:
meta-regression focuses on interpreting the coefficients in the regression
(e.g., testing significance), whereas response-surface meta-analysis focuses
on the predicted mean effect conditioned on perfect study quality. Response-
surface meta-analysis is not concerned with the significance (or lack thereof)
of an associated coefficient; instead, any uncertainty associated with the
coefficient estimates, whether they are significant or not, is reflected in
the resulting standard errors of the estimate of the predicted mean at $X=x$.

##  4 Case Study with Simulated Data

Here, we highlight the differences between response-surface and literature-
synthesis meta-analysis in a simulated data setup. Consider a situation where,
as the studies get better, the treatment effect gets more pronounced. Examples
include effects that can only be estimated as scientific instrumentation and
measurement improve, or interventions which only exhibit an effect once the
necessary measures to ensure increased compliance are put in place.

###  4.1 Setup

In our simulation, we let $Z$ be a univariate measure of design quality, where
$z_{\infty}=10$ defines the best design quality possible. We assume that
existing studies have design qualities $Z$ that follow a scaled Beta
distribution: $Z\sim 10\;\text{Beta}(\alpha=5,\beta=2)$. Studies can therefore
range in quality between 0 and 10, and most studies have reasonably high
design quality (i.e., mode at 8). See Figure 1 (a) for the sampling
distribution for design quality.

(a) Design quality.

(b) Individual studies.

Figure 1: Simulation study setup. Study design qualities are sampled from the
distribution $\text{Beta}(5,2)$ in (a), and individual study estimated mean
effects $\psi$ and standard errors $s$ are sampled subsequently, resulting in
(b).

Next, we sample $n=20$ studies in the following way. First, we sample a design
quality score $Z_{i}$ from the aforementioned sampling distribution. Then, we
sample a sample mean and standard error reported by each study as a function
of the design quality of the study. For a given study, we draw a sample mean
effect $\psi_{i}$ assuming that individual studies report sample means that
are on average closer to the true effect $\tau$ when the design quality is
higher: $\psi_{i}\sim\mathcal{N}(\tau-\gamma(Z_{\infty}-Z_{i}),1)$, where
$\gamma=2$. Then, we draw individual standard errors $s_{i}$ for each study
assuming that standard errors of a study’s estimate shrink with better design
quality: $s_{i}=\max\Big{(}\delta(Z_{\infty}-Z)+\epsilon_{i},0.1\Big{)}$,
where $\delta=0.2$ and $\epsilon_{i}\sim\mathcal{N}(0,0.01)$.

###  4.2 Methods

We now compare estimated effect sizes using literature-synthesis and response-
surface meta-analysis approaches. For our literature-synthesis meta-analysis,
we provide results under both fixed-effects and random-effects models, the
latter using the Dersimonian-Laird method. Following convention, we report the
estimated mean treatment effect and its 95% confidence interval ($\pm
1.96\;\text{SE}$). For response-surface meta-analysis, we implement both a
fixed- and random-effects-based response-surface. For the former, we fit a
fixed-effects meta-regression (i.e., weighted least squares regression with
inverse study variances as weights) with design quality as a covariate. Then,
we report the predicted mean treatment effect conditioned on ideal design
quality (i.e., $Z=10$). We do the same for the random-effects-based response-
surface, except that we fit a random effects meta-regression using the
Dersimonian-Laird method to estimate the between-study variance. We report the
95% confidence interval of both estimates.

Figure 2: A comparison of literature synthesis (red, purple) and response-
surface (green) meta-analysis on the simulated setup in Figure 2. The dashed
line indicates the true scientific effect, which only the response-surface
approaches (RSF, RSR) capture in their 95% confidence interval. Fixed effects
(FE) and random effects (RE) meta-analysis underestimate the true scientific
effect even when the analysis only includes studies with high design qualities
(e.g., FE8 and RE8 only include studies with $Z>8$).

###  4.3 Results

Figure 2 presents the differences in results (estimate and 95% confidence
interval) between the literature-synthesis and response-surface approaches. In
comparison to the true scientific effect size $\tau=20$, the fixed-effects
meta-analysis (FE), random-effects meta-analysis (RE), and response-surface
models (RSF for fixed-effects version and RSR for random-effects version)
yield estimates of 17.9 (95% CI: 17.8, 18.0), 14.8 (95% CI: 13.9, 15.6), 19.6
(95% CI: 18.4, 20.7), and 20.0 (95% CI: 17.9, 22.0), respectively. In other
words, the literature synthesis approaches produce estimates far from the true
effect size $\tau$, whereas the response-surface estimates are close to the
true $\tau$ (within one standard error).

When only considering high quality studies (i.e., those with design qualities
of 9 and above), the fixed-effects and random-effects models yield estimates
that are closer but still do not cover the true effect size: the fixed-effects
estimate is 18.5 (95% CI: 18.4, 18.6), and the random-effects estimate is 18.5
(95% CI: 17.8, 19.3). Including only high-quality studies brings the
literature synthesis results closer to the true scientific effect, but these
methods still yield biased estimates of the scientific estimand of interest
since neither appropriately address the relationship between design quality
and effect size. In contrast, given an accurate model of the relationship
between design quality and study results, response-surface meta-analysis can
appropriately estimate the average effect size conditioned on ideal design
quality.

It is worth noting that the response-surface estimates exhibit wider
confidence intervals than the corresponding literature synthesis approaches.
This occurs because the latter assume a fixed relationship between design
quality and reported effect (i.e., none) whereas the former relaxes this
assumption and instead estimates this relationship, the estimate of which
incorporates additional uncertainty into the estimate for $\tau$. Thus, when
there is indeed no relationship between design quality and individual study
estimates, literature-synthesis approaches should provide accurate estimates
with tighter confidence intervals than response-surface approaches. However,
the absence of a relationship is unlikely to be true in practice, as various
study design decisions (e.g., sample size, blinding) generally affect the bias
and variance of an individual study’s effect estimate.

To highlight the differences between literature synthesis and response-surface
approaches further, we run a simulation studying the same setup described
above while varying the parameters of the data generating process. We allow
the following parameters to vary: 1) distribution of design quality $Z$ via
the parameter $\alpha$; 2) $\gamma$, the unit change in the bias of an
observed study estimate relative to the true estimate $\tau$ for every unit
change in design quality; and 3) $\delta$, the unit change in the standard
error of the observed study estimate for every unit change in design quality.
We consider four $\alpha$ values (i.e., $\alpha$=2, 5, 8, 11), where
increasing $\alpha$ results in a distribution where a larger proportion of
studies are of high-quality: $\alpha=2$ yields a symmetric Beta distribution
with mode at 5 whereas $\alpha=11$ yields a distribution where over 65% of the
studies have a quality score about 9. We choose four $\gamma$ values (i.e.,
$\gamma$=0, 1, 2, 4), where 0 suggests that the center of the distribution of
study effect means is the same regardless of study design. As the value of
$\gamma$ increases, lower-quality studies on average exhibit effects $\psi$
that deviate farther from the true scientific effect $\tau$. Finally, we
choose $\delta$ values (i.e., $\delta$=0, 0.1, 0.2, 0.4, 0.8), where 0
suggests that on average, all studies report the same standard errors
regardless of design quality. As the value of $\delta$ increases, so do the
standard errors associated with lower-quality studies. We vary all factors
fully factorially and run 100 repetitions for each setting.

In the following plots, we compare methods based on their median bias and
coverage (i.e., proportion of runs where the 95% CI covers the true effect
$\tau$). We compare response-surface meta-analysis (i.e., RSF, RSR), described
above, with literature synthesis-based fixed-effects (FE) and random-effects
(RE) meta-analysis, as well as fixed-effects and random-effects meta-analysis
using only studies with design quality higher than 9 (i.e., FE9, RE9). We see
that results are quite consistent across the values of sample size $N$ we
examined (i.e., N=10, 20, 40, 80) and report results at $N=20$ to stay
consistent with the above case study.

Figure 3: Median bias of literature-synthesis approaches (i.e., FE, FE9, RE,
and RE9) can be quite large for various data generating processes, while the
median bias for response-surface approaches (i.e., RSF and RSR) stays near-
zero for all simulation settings. As the true effect $\tau$ is fixed at 20 for
all settings, results from this simulation suggest that literature-synthesis
approaches can yield biases as large as magnitude of the true effect itself.
Figure 4: The coverage (i.e., percentage of 95% confidence intervals that
cover the true effect) stays high for response-surface approaches (i.e., RSF,
RSR) but varies widely for literature-synthesis approaches (i.e., FE, FE9, RE,
and RE9).

Figure 3 shows that the median bias of literature synthesis-based estimates
(i.e., FE, FE9, RE, and RE9) increases as $\gamma$ increases (i.e., larger
deviations in individual reported effects), where the bias is larger in FE and
RE as $\alpha$ decreases (i.e., more lower-quality studies). In contrast,
response-surface approaches (i.e., RSF and RSR) maintain a near-zero median
bias regardless of the values of $\alpha,\gamma,\delta$. Figure 4 shows that
response-surface meta-analysis almost always has the highest coverage,
especially when comparing the response-surface approach with the analogous
literature-synthesis approach (i.e., RSF vs. FE, RSR vs. RE). The only time
literature-synthesis approaches exhibit similar or better coverage than their
analogous response-surface approaches occurs for the lowest values of
$\alpha,\gamma,\delta$ (i.e. top-left corner of Figure 4). In other words,
literature synthesis meta-analysis is advantageous only when there truly is no
relationship between design quality and reported study effect. Otherwise,
literature synthesis approaches can yield significant bias and sometimes as
low as near-zero coverage.

##  5 A Real Data Example

We now present an application of the response-surface approach to real data
from a meta-analysis published in the Cochrane Database of Systematic Reviews.

###  5.1 Setup

We re-analyze a meta-analysis comparing the joint effect of diet and physical
activity on the z-score of BMI (zBMI) among children aged 6 to 12 years old
Brown19 . The original meta-analysis consists of 25 studies, all randomized
control trials.

Although the definition of the perfect study has not been established in
general, to motivate the use of the response-surface methodology, we assign
design quality using the Cochrane Risk of Bias (RoB2) tool HigginsBias , which
provides an ordinal rating (i.e., low, unclear, and high risk of bias) for
each of the following six categories: (i) bias due to random sequence
generation, which occurs when the process used to generate a random allocation
sequence for assigning participants to different groups in a study is flawed
or inadequate, leading to an unequal distribution of known or unknown
confounding factors between the groups; (ii) bias due to allocation
concealment, or the possibility that knowledge of the allocation sequence
could influence the allocation of participants to different groups in a study,
leading to a biased distribution of known or unknown confounding factors
between the groups; (iii) bias due to blinding, or the possibility that
knowledge of the treatment or intervention being received by participants or
researchers could influence the measurement or reporting of outcomes; (iv)
bias due to incomplete outcome data, which can occur when participants drop
out of a study or are lost in follow-up; (v) bias due to selective reporting,
which occurs when the study results and analyses are reported selectively
based on the direction or statistical significance of the findings; and (vi)
other bias. Based on these ratings, individual studies are assigned either a
low, unclear, or high risk of bias overall.

We define the ideal study quality (i.e., $z_{\infty}$) to be a study with low
risk of bias overall, which only three of the 25 studies satisfy. Thirteen
studies have an unclear risk of bias, and nine have a high risk of bias. To
visualize the additional information present when considering study results in
relation to design quality, see Figure 5 (b) for an visualization of the study
effect estimates categorized by overall risk of bias, as compared to a forest
plot of the studies in Figure 5 (a).

###  5.2 Methods

To perform response-surface meta-analysis, we model the average study effect
with a random effects meta-regression that includes risk of bias as an ordinal
covariate. To estimate $\tau$, we report both the mean of the prediction at a
low risk of bias, along with its 95% confidence interval. We compare our
result to that of Brown19 , who utilize a random effects model without taking
risk of bias into account.

(a) Forest plot of studies measuring effect of diet and exercise on zBMI for
children ages 6-12.

(b) Study effect sizes categorized by risk of bias.

Figure 5: (a): A forest plot of the results only takes the study estimates and
their 95% CIs into account. The corresponding random effects model estimate
favors treatment with estimate -0.05 (CI: -0.10, -0.01). (b): Plotting the
study estimates and CIs against overall risk of bias reveals that studies with
higher risk of bias tend to favor the treatment (negative mean difference) as
well as exhibit wider CIs. The response-surface estimate concludes not enough
evidence of an effect, with estimate 0.01 (CI: -0.17, 0.19).

###  5.3 Results

In contrast to the negative effect estimate published by Brown et al. of -0.05
(95% CI: -0.10,-0.01) favoring the intervention, the response-surface approach
yields a non-negative estimate of 0.01 (95% CI: -0.17, 0.19). Thus, by
adjusting for risk of bias and estimating the effect under ideal study
conditions rather than averaged over existing published studies, the inference
regarding the treatment effect changes from favoring the intervention to
concluding there is not enough information to reject the null hypothesis.

The forest plot and mean difference vs. risk of bias plots in Figure 5 help
build intuition about this result: the forest plot shows that there exist
several studies in the set of published studies that show a significant effect
favoring the treatment, such that performing a weighted averaging of all
studies via the assumptions of a random effects model yields an overall
negative effect. However, the corresponding mean difference vs. risk of bias
plot shows that the studies that report the most negative effects also have
high or unclear risk of bias, while the few low risk of bias studies yield
precise effect estimates close to zero mean difference. Rather than allowing
the higher risk of bias studies to bias the overall average towards a negative
effect, the response-surface approach takes into account risk of bias and its
relationship to reported effect size to obtain a conservative and non-negative
effect estimate.

In line with the response-surface result, the significant effect estimate in
Brown et al. Brown19  is accompanied by a low certainty GRADE rating (grade,
), in part due to the many high risk of bias studies, along with the caveat
that “the true effect may be substantially different from the estimate of the
effect.” In other words, the authors note in the analysis that the reported
effect estimate is questionable. However, reporting a result with a confidence
interval while conceding that it may be far from the true effect, although
good, is not the same as conducting a response-surface meta-analysis, which
seeks to directly estimate this true scientific effect $\tau$ and report an
accurate reflection of the uncertainty around the current estimate of this
effect.

##  6 Challenges in Response-Surface Meta-Analysis

The simulated and real-world examples presented in the previous sections
illustrate why the results of a response-surface meta-analysis can differ from
those of literature synthesis meta-analysis. This difference is due to (i) the
targeted estimand and (ii) the assumption of how study design quality affects
the study-specific estimates. For instance, if an effect becomes more
pronounced as the studies get better, then traditional meta-analysis might
underestimate the true effect whereas response-surface meta-analysis should
yield more accurate estimates by incorporating the relationship between design
quality and the effect estimate. Our synthetic setup provided an example of
this situation. On the other hand, literature synthesis meta-analysis might
claim an effect due the results of lower-quality studies whereas response-
surface meta-analysis is more conservative by incorporating the uncertainty
that arises when extrapolating to the perfect study. Our real-world case study
reflects this.

Although there is conceptual clarity in the response-surface formulation of
meta-analysis, there are practical challenges in its implementation. In
particular, response-surface meta-analysis can be sensitive to choices about
how design quality is scored and how the response-surface is modeled. We
believe that the lack of standard guidance along these axes is the key reason
why this approach to meta-analysis has not been adopted. Thus, to encourage
adoption, we offer initial solutions and suggestions for standardizing the
response-surface approach to meta-analysis.

###  6.1 Determining Design Quality

The results of a response-surface meta-analysis depend highly on how design
quality scores are assigned to existing studies and tothe idealized study.
Without a standard procedure to assign design quality, it is easy to dismiss
the results of a response-surface meta-analysis simply by disagreeing with the
design quality scale used. For instance, in the above real-data example, we
used a coarse rating based on the Cochrane RoB2 such that three studies out of
25 were assigned the same quality as an idealized study. Some may argue that
it is inappropriate to assign any existing study a quality score of
$z_{\infty}$, while others may argue against such a coarse assignment. Yet
others may advocate for an entirely different scale. In each of these cases, a
disagreement with the quality assignment can lead to distrust in the results
of the analysis. Thus, for the results of a response-surface meta-analysis to
be widely accepted, it is important to establish standard guidelines for
assigning design quality.

Finding a standard rating that is generally accepted will require community
discussion and engagement. This issue is not specific to the response-surface
formulation of meta-analysis, however. For instance, one could just as easily
disagree with the inclusion/exclusion criteria used in a literature synthesis
meta-analysis; existing meta-analyses address potential concerns by defining
eligibility criteria in advance and justifying the criteria McKenzie , and
similar strategies can help legitimize the results of a response-surface meta-
analysis.

Tools such as the Cochrane RoB2 tool for randomized control trials HigginsBias
and ROBINS-I risk of bias tool for observational studies Sterne  already
capture some degree of study quality. However, such tools imply that
randomized control trials and observational studies should be treated
separately and cannot be ranked using a common set of criteria. A response-
surface meta-analysis, on the other hand, may wish to take advantage of all
studies in estimating a response-surface, which would require a way to rate
different studies relative to each other.

The difficulty in assigning quality scores has been discussed in the past
Greenaland2001 ; Juni . However, this difficulty should not deter the
scientific community from pursuing the estimation of the true estimand of
interest. Strategies for incorporating quality scores in a scientific manner
include setting community standards for scoring design quality, assigning
quality using study design alone without knowledge of reported effect, and
conducting sensitivity analyses based on a range of plausible assignments.

###  6.2 Modeling the response-surface

The results of a response-surface meta-analysis are also generally sensitive
to how the relationship between design quality and effect size is modeled. Put
another way, it is not possible to draw inferences about perfect studies using
imperfect studies without assumptions on how such study results relate to each
other.

One strategy to help enable accurate response surfaces is to build and
incorporate external information about the role of study design and effect
size. For instance, information from historical patterns or field-specific
knowledge could help suggest the types of relationships that are plausible.
Additionally, the relationships estimated on widely-studied questions could
help inform the modeling assumptions of a response surface on newer questions,
given a plausible justification for why one would expect these relationships
to be similar. To carry this idea out on a more systematic scale, individual
fields of study could also invest in estimating and understanding the causal
relationship between study design factors and reported effect size. Such
estimates could then benefit a wide array of downstream meta-analyses which
could incorporate such prior knowledge into response surface modeling efforts.

Another way to increase confidence in the conclusions of a response-surface
meta-analysis is to conduct a sensitivity analysis based on various modeling
choices. For instance, in the above real-world example, we reported an
estimate of 0.01 (95% CI: -0.17, 0.19) using a random-effects response-surface
when encoding risk of bias as an ordinal variable. If we instead encoded risk
of bias as a numeric variable, representing the assumption of a linear
relationship between equally spaced risk of bias scores, the resulting
prediction mean at $z_{\infty}$ is also 0.01 (95% CI: -0.07, 0.09), with a
tighter confidence interval due to stronger assumptions on how the different
qualities relate. Alternatively, ignoring the high and unclear risk of bias
studies all together and simply fitting a random-effects model on the low risk
of bias studies also yields a non-negative estimate of 0.00 (95% CI: -0.04,
0.05). This result has an even tighter confidence band because incorporating
high and unclear risk of bias studies introduces additional unexplained
heterogeneity in the aforementioned models. Though the uncertainties around
these different response-surface variants differ, all the resulting estimates
are non-negative and the overall conclusion is the same, that there is not
enough evidence to favor the intervention. Reporting multiple results obtained
from different modeling choices can help increase confidence that the
conclusions of a meta-analysis are not simply the results of modeling-specific
choices. Other strategies to build confidence in a response-surface meta-
analysis include reporting several model summaries and evaluations.

##  7 Conclusion

In this work, we reintroduce response-surface meta-analysis, which reframes
meta analysis as an endeavor to estimate true scientific effects as would be
measured under perfect studies, rather than as a way to summarize the existing
population of imperfect studies. Much of the statistical machinery developed
for meta-regression can be used to fit a response-surface of the individual
studies, thereby enabling an estimate under the perfect study via a predicted
mean. We illustrate how the results of a response-surface meta-analysis can
differ from traditional approaches. Finally, we discuss two main challenges to
implementing response-surface meta-analyses. The first is determining a rating
system for design quality, which requires consensus building within research
communities. The second is modeling and extrapolating from the response-
surface, which would benefit from well-formulated assumptions and
comprehensive sensitivity analyses. These challenges may seem daunting, but
they reflect the effort required to make inferences about true scientific
effects in the presence of imperfect data.

Our attitude toward an endeavor such as a scientific meta-analysis is that it
must start with a clear statement of the objective of the investigation:
namely, the causal estimand of interest. It is important to realize that the
underlying estimand targeted by current meta-analysis methods is the typical
result among the collection of studies currently conducted by researchers in a
particular field. Although a valid question, it is often distinct from the
question of the real scientific effect. In some ways, the current literature
synthesis approach is analogous to the intention-to-treat (ITT) approach often
used in human experiments when there is noncompliance. “Intention-to-treat”
estimates the causal effects of the assignment to take a drug versus a
placebo, ignoring whether a person actually takes the drug or not. As
discussed previously Sheiner , this estimation process mixes two different
scientific effects: the medical effect of the drug itself, which is the
typical objective when conducting the study, and the combined effect of the
drug and the participant’s willingness to comply with the assignment in the
trial setting, a context which differs from the context of your doctor
treating you for a disease. In a similar way, literature synthesis confuses
two scientific questions: the true effect, which would be estimated by a
perfect study, and the average effect in the choice of studies conducted by
current researchers. Both questions are legitimate, but we believe current
practice in meta-analysis usually addresses the second while reporting results
as if the first had been addressed. Understanding the differences can lead to
a more appropriate interpretation of the results of any meta-analysis. We hope
that this article helps to encourage more clarity in research and educational
training about meta-analysis, while inspiring future work towards a more
complete and direct formulation.

##  8 Acknowledgements

Research reported in this publication was supported by the John Harvard
Distinguished Science Fellows Program within the FAS Division of Science of
Harvard University; by the Office of the Director, National Institutes of
Health under Award Numbers DP5OD021412 and R01AI102710; and by NRT-HDR: FUTURE
and NSF Award 1922658. The content is solely the responsibility of the authors
and does not necessarily represent the official views of the National
Institutes of Health or National Science Foundation.

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