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# Risk-Constrained Nonconvex Functional
Resource Allocation has Zero Duality Gap

Dionysis Kalogerias and Spyridon Pougkakiotis
Department of EE – Yale

###### Abstract

We show that risk-constrained dynamic resource allocation problems with
general integrable nonconvex instantaneous service functions exhibit zero
duality gap. We consider risk constraints which involve convex and positively
homogeneous risk measures admitting dual representations with bounded risk
envelopes, strictly generalizing expectations. Beyond expectations, particular
risk measures supported within our setting include the conditional value-at-
risk, the mean-absolute deviation (including the non-monotone case), certain
distributionally robust representations and more generally all real-valued
coherent risk measures on the space ${\cal L}_{1}$. Our proof technique relies
on risk duality in tandem with Uhl’s weak extension of Lyapunov’s convexity
theorem for vector measures taking values in general Banach spaces.

Notation: Bold capital letters (such as ${\bf A}$), or calligraphic letters
(such as ${\cal A}$), or sometimes plain capital letters (such as $A$) will
denote finite-dimensional sets/spaces, such as Euclidean spaces. Double stroke
letters (such as $\mathds{A}$) will denote infinite-dimensional sets/spaces,
such as Banach spaces. Math script letters (such as $\mathscr{A}$) will denote
$\sigma$-algebras. Boldsymbol letters (such as $\boldsymbol{A}$ or
$\boldsymbol{a}$) will denote (random) vectors. The space of $p$-integrable
functions from a measurable space $(\Omega,\mathscr{F})$ equipped with a
finite measure $\mu:\mathscr{F}\rightarrow\mathbb{R}_{+}$ to a Banach space
$\mathds{A}$, with standard notation ${\cal
L}_{p}(\Omega,\mathscr{F},\mu;\mathds{A})$ (Diestel and Uhl, 1977; Shapiro et
al., 2014), is abbreviated as ${\cal L}_{p}(\mu,\mathds{A})$, as it is also
common practice (Diestel and Uhl, 1977). The rest of the notation is standard.

##  1 Introduction and Problem Setting

On some arbitrary base probability space $(\Omega,\mathscr{F},\mu)$, consider
a random element $\boldsymbol{H}:\Omega\rightarrow{\cal
H}\triangleq\mathbb{R}^{N_{\boldsymbol{H}}}$ with induced Borel measure
$\mathrm{P}:\mathscr{B}({\cal H})\rightarrow[0,1]$, modeling some random
phenomenon, which we would like to optimally manage by allocating resources.
In particular, we are interested in the class of nonconvex functional resource
allocation problems formulated as

$\boxed{\begin{array}[]{rl}\infty>\mathsf{P}^{*}=\>\underset{\boldsymbol{x},\boldsymbol{p}(\cdot)}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\end{array},}$ (RCP)

where, with ${\bf C}\triangleq\mathbb{R}^{N}$, $g^{o}:{\bf
C}\rightarrow\mathbb{R}$ and $\boldsymbol{g}:{\bf
C}\rightarrow\mathbb{R}^{N_{\boldsymbol{g}}}$ are concave fixed utility
functions, $\boldsymbol{p}:{\cal H}\rightarrow{\bf
R}\triangleq\mathbb{R}^{N_{\boldsymbol{p}}}$ is the resource allocation
policy, $\boldsymbol{f}:{\bf R}\times{\cal H}\rightarrow{\bf C}$ is a
generally nonconvex instantaneous performance level score, measuring the
quality of a policy $\boldsymbol{p}$ at each realization $\boldsymbol{H}$ in
${\cal H}$ and such that $\boldsymbol{f}(\cdot,\boldsymbol{H})\in{\cal
L}_{1}(\mathrm{P},{\bf C})$ on $\Pi$, and where $\boldsymbol{\rho}:{\cal
L}_{1}(\mathrm{P},{\bf C})\rightarrow{\bf C}$ is a finite-valued vector risk
measure, which we assume that is convex, lower semicontinuous and positively
homogeneous in every dimension (i.e., component-wise), with the standardized
convention that, for each $i\in\mathbb{N}_{N}^{+}$,

$\rho_{i}(\boldsymbol{Z})=\rho_{i}(Z_{i}),\quad\text{for all
}\boldsymbol{Z}\in{\cal L}_{1}(\mathrm{P},{\bf C}).$

As indicated in (RCP), performance risks are further restricted to the finite-
dimensional set ${\cal X}\subseteq{\bf C}$ and resource allocations are
further restricted to the infinite-dimensional set $\Pi$. More specialized
(yet general enough) assumptions on the structure of (RCP) which enable the
development of the results advocated in this paper will be discussed in due
course.

As an example, if $i\in\mathbb{N}_{N}^{+}$ refers to a user of a wireless
system, then the $i$-th entry of $\boldsymbol{\rho}$ may evaluate the risk
associated with the service experienced by the $i$-th user; hereafter,
$\boldsymbol{f}$ will also be called the service function. However, with the
exception of very special scenarios, the service experienced by the $i$-th
user can (and, in general, will) be dependent on the services experienced by
the rest of the users in the system. This is due to each service $f_{i}$ being
a function of the coupling variables $\boldsymbol{p}(\boldsymbol{H})$ and
$\boldsymbol{H}$, which are common to all users in the system. This general
problem structure adheres to a large variety of resource allocation problems
encountered in practice.

Problem (RCP) admits an equivalent and rather useful representation. By the
duality theorem for coherent risk measures (Shapiro et al., 2014, Theorem
6.5), we have that for every risk measure $\rho$ which is convex, lower
semicontinuous and positively homogeneous (i.e., in the class of those
appearing in problem (RCP)), it holds that

$\rho(Z)=\sup_{\zeta\in\mathds{A}}\langle\zeta,Z\rangle\triangleq\sup_{\zeta\in\mathds{A}}\int\zeta(\boldsymbol{h})Z(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h}),\quad\text{for
all }Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R}),$

where the uncertainty set $\mathds{A}$ is the domain of the convex conjugate
of $\rho$ (i.e., its Legendre-Fenchel transform), and it holds that
$\mathds{A}\subseteq{\cal L}_{\infty}(\mathrm{P},\mathbb{R})$ by functional
duality. The set $\mathds{A}$ is also called the risk envelope of $\rho$. In
this paper, we focus on risk envelopes which are subsets of $\\{\zeta\in{\cal
L}_{\infty}(\mathrm{P},\mathbb{R})||\zeta(\cdot)|\leq\gamma,\mathrm{P}\text{-a.e.}\\}$,
where $\gamma>0$ is some arbitrarily large but finite constant, but otherwise
we make no further assumptions. Of course, it follows that, for every
$Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R})$,

$-\rho(-Z)=\inf_{\zeta\in\mathds{A}}\langle\zeta,Z\rangle.$

Under this provisioning, our initial functional program may be equivalently
expressed as

$\boxed{\begin{array}[]{rl}\underset{\boldsymbol{x},\boldsymbol{p}(\cdot)}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq\inf_{\boldsymbol{\zeta}\in\mathds{A}_{\gamma}^{S}}\mathbb{E}\\{\boldsymbol{\zeta}(\boldsymbol{H})\circ\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\end{array},}$ (RCP-E)

where “$\circ$” denotes the Hadamard product, the infimum is understood in a
component-wise manner, and where the service uncertainty set (i.e., risk
envelope) $\mathds{A}_{\gamma}^{S}$ is defined as the Cartesian product
$\mathds{A}_{\gamma}^{S}\triangleq\mathds{A}_{\gamma}^{1}\times\mathds{A}_{\gamma}^{2}\times\ldots\times\mathds{A}_{\gamma}^{N},$
with each $\mathds{A}_{\gamma}^{i}$ satisfying the inclusion

$\mathds{A}_{\gamma}^{i}\subseteq\\{\zeta\in{\cal
L}_{\infty}(\mathrm{P},\mathbb{R})||\zeta(\cdot)|\leq\gamma,\mathrm{P}\text{-a.e.}\\},\quad\forall
i\in\mathbb{N}_{N}^{+}.$

In passing, it may be worth noting that (RCP-E) is equivalent to the semi-
infinite functional program

$\begin{array}[]{rl}\underset{\boldsymbol{x},\boldsymbol{p}(\cdot)}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq\mathbb{E}\\{\boldsymbol{\zeta}(\boldsymbol{H})\circ\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\},\quad\forall\boldsymbol{\zeta}\in\mathds{A}_{\gamma}^{S}\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\end{array},$ (RCP-I)

illustrating the generality of the risk-constrained problem (RCP), as compared
with its risk-neutral counterpart obtained by choosing
$\mathds{A}_{\gamma}^{i}=\\{\zeta(\cdot)=1,\mathrm{P}\text{-a.e.}\\}$ for
$i\in\mathbb{N}_{N}^{+}$, which is of course equivalent with replacing the
vector risk measure $\boldsymbol{\rho}$ by a vector of expectations.

Problem (RCP-E) (which is equivalent to (RCP) under the conditions outlined
above) is very general and several cases of special interest can be formulated
as particular instances. As already discussed, those instances include the
risk-neutral version of (RCP). A nontrivial case of particular value is that
of the Conditional Value-at-Risk at level $\beta\in(0,1]$
($\mathrm{CVaR}^{\beta}$) defined as (Rockafellar and Uryasev, 2000; Shapiro
et al., 2014)

$\mathrm{CVaR}^{\beta}(Z)=\inf_{t\in\mathbb{R}}t+\dfrac{1}{\beta}\mathbb{E}\\{(Z-t)_{+}\\},\quad
Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R}),$

for which (RCP) reduces to

$\boxed{\begin{array}[]{rl}\underset{\boldsymbol{x},\boldsymbol{p}(\cdot)}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq-\mathrm{CVaR}^{\boldsymbol{\beta}}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\end{array},}$ (CVaR)

where
$\boldsymbol{\beta}\triangleq[\beta_{1}\,\beta_{2}\,\ldots\,\beta_{N}]\in(0,1]^{N}$
is a vector containing the $\mathrm{CVaR}$ levels associated with each entry
of the service function $\boldsymbol{f}$. In this case, the equivalent
formulation of (CVaR) in the form of (RCP-E) is valid by choosing
$\gamma=\max_{i\in\mathbb{N}_{N}^{+}}1/\beta_{i}$, and with corresponding risk
envelopes given by (Shapiro et al., 2014, Example 6.19)

$\mathds{A}_{\gamma}^{i}=\\{\zeta\in{\cal
L}_{\infty}(\mathrm{P},\mathbb{R})|\zeta(\cdot)\in[0,1/\beta_{i}],\mathrm{P}\text{-a.e.},\text{
and }\mathbb{E}\\{\zeta\\}=1\\},\quad\forall i\in\mathbb{N}_{N}^{+}.$

Note that in the case of $\mathrm{CVaR}$, the resulting functional program may
also be stated as

$\begin{array}[]{rl}\underset{\boldsymbol{x},\boldsymbol{p}(\cdot)}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq{\displaystyle\sup_{\boldsymbol{t}\in{\bf
C}}\boldsymbol{t}+\dfrac{1}{\boldsymbol{\beta}}\circ\mathbb{E}\\{-(\boldsymbol{t}-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))_{+}\\}}\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\end{array},$

which is equivalent to

$\begin{array}[]{rl}\underset{\boldsymbol{x},\boldsymbol{p}(\cdot),\boldsymbol{t}}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq\boldsymbol{t}+\dfrac{1}{\boldsymbol{\beta}}\circ\mathbb{E}\\{-(\boldsymbol{t}-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))_{+}\\}\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p},\boldsymbol{t})\in{\cal X}\times\Pi\times{\bf
C}\end{array}.$

Another popular special case is that of the mean absolute deviation with
trade-off parameter $\lambda\geq 0$ ($\mathrm{MAD}^{\lambda}$) defined as
(Ogryczak and Ruszczyński, 1999, 2002)

$\mathrm{MAD}^{\lambda}(Z)=\mathbb{E}\\{Z\\}+\lambda\mathbb{E}\\{|Z-\mathbb{E}\\{Z\\}|\\},\quad
Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R}),$

for which problem (RCP) reduces to

$\boxed{\begin{array}[]{rl}\underset{\boldsymbol{x},\boldsymbol{p}(\cdot)}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq-\mathrm{MAD}^{\boldsymbol{\lambda}}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\end{array},}$ (MAD)

where
$\boldsymbol{\lambda}\triangleq[\lambda_{1}\,\lambda_{2}\,\ldots\,\lambda_{N}]\in\mathbb{R}_{+}^{N}$
is another vector containing the $\mathrm{MAD}$ trade-offs associated with
each entry of the service function $\boldsymbol{f}$. In this case, (MAD) can
be equivalently written in the form of (RCP-E) by choosing
$\gamma=\max_{i\in\mathbb{N}_{N}^{+}}1+2\lambda_{i}$, and risk envelopes
(Shapiro et al., 2014, Example 6.22)

$\mathds{A}_{\gamma}^{i}=\\{\zeta\in{\cal
L}_{\infty}(\mathrm{P},\mathbb{R})|\zeta=1+\zeta^{\prime}-\mathbb{E}\\{\zeta^{\prime}\\},\text{
and }\|\zeta^{\prime}\|_{{\cal L}_{\infty}}\leq\lambda_{i}\\},\quad\forall
i\in\mathbb{N}_{N}^{+}.$

Similar to the case of $\mathrm{CVaR}$, it is easy to see that the
$\mathrm{MAD}$ program takes the form

$\begin{array}[]{rl}\underset{\boldsymbol{x},\boldsymbol{p}(\cdot)}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq\mathbb{E}\\{\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}-\boldsymbol{\lambda}\circ\mathbb{E}\\{|\mathbb{E}\\{\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})|\\}\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\end{array},$

or, equivalently,

$\begin{array}[]{rl}\underset{\boldsymbol{x},\boldsymbol{p}(\cdot),\boldsymbol{t}}{\mathrm{maximize}}&g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\boldsymbol{x}\leq\boldsymbol{t}-\boldsymbol{\lambda}\circ\mathbb{E}\\{|\boldsymbol{t}-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})|\\}\\\
&\boldsymbol{t}=\mathbb{E}\\{\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}\\\
&\boldsymbol{g}(\boldsymbol{x})\geq{\bf 0}\\\
&(\boldsymbol{x},\boldsymbol{p},\boldsymbol{t})\in{\cal X}\times\Pi\times{\bf
C}\end{array}.$

There are numerous other risk measures which are compatible with our
assumptions on the vector risk measure $\boldsymbol{\rho}$. In fact, all
coherent risk measures on the space ${\cal L}_{1}$ (relative to any choice of
the probability measure $\mathrm{P}$) are supported under the adopted
framework. For any such risk measure $\rho$, it is well known that

$\rho(Z)=\sup_{\zeta\in\mathds{A}}\langle\zeta,Z\rangle=\sup_{\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{P}}\in\mathds{A}}\mathbb{E}\bigg{\\{}Z(\boldsymbol{H})\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{P}}(\boldsymbol{H})\bigg{\\}},\quad\text{for
all }Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R}),$

where the risk envelope $\mathds{A}$ takes the special form (Shapiro et al.,
2014, Section 6.3)

$\displaystyle\mathds{A}$ $\displaystyle=\\{\zeta\in{\cal
L}_{\infty}(\mathrm{P},\mathbb{R})|\langle\zeta,Z\rangle\leq\rho(Z)\;\text{for
all }Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R}),\zeta\succeq
0,\mathbb{E}\\{\zeta\\}=1\\}$
$\displaystyle=\bigg{\\{}\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{P}}\in{\cal
L}_{\infty}(\mathrm{P},\mathbb{R})\bigg{|}\bigg{\langle}\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{P}},Z\bigg{\rangle}\leq\rho(Z),\;\text{for
all }Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R})\bigg{\\}}.$

In the above, $\mathrm{d}\mathrm{Q}/\mathrm{d}\mathrm{P}=\zeta$ denotes the
Radon-Nikodym derivative of a probability measure $\mathrm{Q}$ on $({\cal
H},\mathscr{B}({\cal H}))$ relative to $\mathrm{P}$, with the former assumed
absolutely continuous relative to the latter; we use the standard notation
$\mathrm{Q}\ll\mathrm{P}$.

If $\rho$ is also real-valued, it is known that it is continuous in the strong
topology on ${\cal L}_{1}(\mathrm{P},\mathbb{R})$, while its risk envelope
$\mathds{A}$ is a convex bounded and $\mathrm{weakly}^{*}$-closed subset of
the set of densities in ${\cal L}_{\infty}(\mathrm{P},\mathbb{R})$, also
implying that the supremum is attained for some
$\zeta_{Z}^{\star}\in\mathds{A}$ (Shapiro et al., 2014, Proposition 6.6 and
Theorem 6.7). Since, in particular, $\mathds{A}$ is bounded, there must exist
a $\gamma>0$ such that $|\zeta(\cdot)|\leq\gamma,$ $\mathrm{P}\text{-a.e.}$
for every $\zeta\in\mathds{A}$ (note that such a $\gamma$ is necessarily
independent of the choice of $Z$). Therefore, $\mathds{A}$ is consistent with
our setting. It follows that $\rho$ admits a distributionally robust
representation of the form

$\rho(Z)=\sup_{\mathrm{Q}\in\mathfrak{M}}\mathbb{E}_{\mathrm{Q}}\\{Z(\boldsymbol{H})\\},\quad\text{for
all }Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R}),$

where

$\displaystyle\mathfrak{\mathfrak{M}}$
$\displaystyle\triangleq\bigg{\\{}\mathrm{Q}\ll\mathrm{P}\bigg{|}\bigg{\langle}\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{P}},Z\bigg{\rangle}\leq\rho(Z),\;\text{for
all }Z\in{\cal L}_{1}(\mathrm{P},\mathbb{R}),\;\text{with
}\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{P}}(\cdot)\leq\gamma,\mathrm{P}\text{-a.e.}\bigg{\\}}$
$\displaystyle\subseteq\big{\\{}\mathrm{Q}\ll\mathrm{P}\big{|}\mathrm{Q}\leq\gamma\mathrm{P},\text{
on Borel sets}\big{\\}}.$

To further highlight the versatility of (RCP), we would like to emphasize that
one can freely “mix-and-match” any combination of risk measures present in the
constraints (RCP), out of the large variety of those supported under our
assumptions. Such combinations include different choices of risk measures
across the components of the service vector $\boldsymbol{f}$, e.g.,
expectations for some components and $\mathrm{CVaR}$s or $\mathrm{MAD}$s for
others, as well as combinations of different risk measures for each of the
components of $\boldsymbol{f}$; a classical example is mean-$\mathrm{CVaR}$
trade-offs.

##  2 Lagrangian Duality

A promising and celebrated approach for dealing with the explicit constraints
of the risk-aware problem (RCP) is by exploiting Lagrangian duality, which has
been proven essential in analyzing and efficiently solving constrained convex
optimization problems; see, e.g., (Boyd and Vandenberghe, 2004; Ruszczyński,
2006; Bertsekas, 2009). Note, however, that since the services in
$\boldsymbol{f}(\cdot,\boldsymbol{H})$ are nonconcave in general in problem
(RCP) (i.e., with respect to the first argument corresponding to the policy
$\boldsymbol{p}$), most standard results in Lagrangian duality for convex
optimization do not automatically apply. On top of that, one has to
incorporate the structural complexity of the risk measure $\boldsymbol{\rho}$,
which is a nonlinear functional of its argument. As a result, fundamental
properties of expectation (being the most trivial risk measure) which enable
an elegant and straightforward analysis, such as linearity, do not hold for
$\boldsymbol{\rho}$.

The Lagrangian function $\mathsf{L}:{\bf
C}\times\Pi\times\mathbb{R}^{N_{\boldsymbol{g}}}\times{\bf
C}\rightarrow\mathbb{R}$ associated with the risk-constrained problem (RCP) is
defined by scalarizing its constraints as

$\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda})\triangleq
g^{o}(\boldsymbol{x})+\langle\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{g}(\boldsymbol{x})\rangle+\big{\langle}\boldsymbol{\lambda}_{\boldsymbol{\rho}},-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}\big{\rangle},$

where
$\boldsymbol{\lambda}\equiv(\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{\lambda}_{\boldsymbol{\rho}})\in\mathbb{R}^{N_{\boldsymbol{g}}}\times{\bf
C}$ are the multipliers associated with the constraints of (RCP). Then the
dual function $\mathsf{D}:\mathbb{R}^{N_{\boldsymbol{g}}}\times{\bf
C}\rightarrow(-\infty,\infty]$ is defined as

$\mathsf{D}(\boldsymbol{\lambda})\triangleq\sup_{(\boldsymbol{x},\boldsymbol{p})\in{\cal
X}\times\Pi}\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda}).$

If the optimal value of problem (RCP) is $\mathsf{P}^{*}\in\mathbb{R}$, it is
easily understood that that $\mathsf{P}^{*}\leq\mathsf{D}$ on the positive
orthant (i.e., for $\boldsymbol{\lambda}\geq{\bf 0}$), and thus it is most
reasonable to consider the dual problem $\inf_{\boldsymbol{\lambda}\geq{\bf
0}}\mathsf{D}(\boldsymbol{\lambda})$, which is always convex and whose optimal
value

$\mathsf{D}^{*}\triangleq\inf_{\boldsymbol{\lambda}\geq{\bf
0}}\mathsf{D}(\boldsymbol{\lambda})\in(-\infty,\infty]$

serves as the tightest over-estimate of the optimal value of problem (RCP),
$\mathsf{P}^{*}$, when knowing only $\mathsf{D}$. Then, one of the basic
questions in Lagrangian duality is whether we can essentially replace an
original constrained problem with its dual, in the sense that
$\mathsf{P}^{*}\equiv\mathsf{D}^{*}$; in such a case, we say that the problem
has zero duality gap. Referring to problem (RCP), this would imply that it can
be replaced by the minimax problem

$\inf_{\boldsymbol{\lambda}\geq{\bf
0}}\mathsf{D}(\boldsymbol{\lambda})\equiv\inf_{\boldsymbol{\lambda}\geq{\bf
0}}\sup_{(\boldsymbol{x},\boldsymbol{p})\in{\cal
X}\times\Pi}\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda}),$
(1)

whose optimal value is $\mathsf{D}^{*}$. If $\mathsf{D}^{*}$ is attained,
i.e., an optimal multiplier vector exists, then (RCP) is said to exhibit
strong duality. A zero duality gap also implies that problem (RCP) satisfies
the saddle point property (whether a saddle point exists or not), which is
expressed as

$\mathsf{D}^{*}=\inf_{\boldsymbol{\lambda}\geq{\bf
0}}\sup_{(\boldsymbol{x},\boldsymbol{p})\in{\cal
X}\times\Pi}\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda})=\sup_{(\boldsymbol{x},\boldsymbol{p})\in{\cal
X}\times\Pi}\inf_{\boldsymbol{\lambda}\geq{\bf
0}}\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda})=\mathsf{P}^{*}.$

Zero duality gaps are desirable: Because there is a finite number of
constraints, the dual function is finite dimensional even though the original
functional problem (RCP) is infinite dimensional. Additionally, for every
choice of the dual variable $\boldsymbol{\lambda}\geq{\bf 0}$ (and therefore
any optimal multiplier vector), joint maximization of the Lagrangian
$\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda})$ over the pair
$(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi$ is separable. We thus
see that duality transforms a constrained problem into an unconstrained
problem in a principled and favorable way and, provided that the original
constrained stochastic program (in our case (RCP)) exhibits zero duality gap,
presents a general methodological approach to tackle it. While zero duality
gaps are a common and fundamental characteristic of problems in convex
(concave) constrained optimization (under appropriate regularity conditions),
proving zero duality gaps in nonconcave problems such as (RCP) is a much more
delicate and challenging task.

##  3 Main Results

Quite surprisingly, our structural setting concerning our analysis of problem
(RCP) is standard and exactly the same as compared with the relevant
literature on risk-neutral stochastic resource allocation and related problems
(Luo and Zhang, 2008; Ribeiro, 2012; Chamon et al., 2020, 2021).

######  Assumption 1 (Structural Setting).

The following conditions are in effect: 1) The utilities $g^{o}$ and
$\boldsymbol{g}$ are concave. 2) The service feasible set ${\cal X}$ is
convex. 3) The policy feasible set $\Pi$ is decomposable. 4) The Borel measure
$\mathrm{P}$ is nonatomic111Recall that $\mathrm{P}$ is nonatomic if for any
event $E$ with $\mathrm{P}(E)>0$, an event $E^{\prime}\subseteq E$ exists such
that $\mathrm{P}(E)>\mathrm{P}(E^{\prime})>0$.. 5) Problem (RCP) satisfies
Slater’s condition (i.e., it is strictly feasible).

A useful observation is that the conditions comprising Assumption 1 are fully
compatible, and can be easily satisfied for most practical considerations. For
instance, the space of all (Borel-)measurable functions (policies) from ${\cal
H}$ to ${\bf R}$ as well as all spaces of integrable functions ${\cal
L}_{p}(\mathrm{P},{\bf R}),p\in[1,\infty]$ are decomposable (Rockafellar and
Wets, 2004), and all those examples can be possible choices for the feasible
set $\Pi$. A more specific standard choice of a decomposable feasible set
$\Pi$ which is relevant in applications is the uniform box

$\Pi=\big{\\{}\boldsymbol{p}:{\cal H}\rightarrow{\bf
R}\big{|}\mathrm{ess\,sup}_{\mathrm{P}}\,\|\boldsymbol{p}(\cdot)\|_{\infty}\leq
U\big{\\}},$

where $U>0$ is an appropriate fixed number, or, the more refined rectangular
box

$\Pi=\big{\\{}\boldsymbol{p}:{\cal H}\rightarrow{\bf
R}\big{|}\mathrm{ess\,sup}_{\mathrm{P}}\,p^{i}(\cdot)\leq
U^{i},i\in\mathbb{N}_{N_{\boldsymbol{p}}}^{+}\big{\\}},$

where the $U^{i}$’s are fixed. In a more delicate setting, another also
standard choice for a decomposable $\Pi$ is (see, e.g., (Ribeiro, 2012) and
references therein)

$\Pi=\big{\\{}\boldsymbol{p}:{\cal H}\rightarrow{\bf
R}\big{|}\boldsymbol{p}(\boldsymbol{h})\in{\cal
U}(\boldsymbol{h}),\mathrm{P}\text{-a.e.}\big{\\}},$

where ${\cal U}:{\cal H}\rightrightarrows{\bf R}$ is a closed-valued
multifunction, which is also closed (Shapiro et al., 2014, Section 7.2.3);
this implies in particular that ${\cal U}$ admits at least one measurable
selection (Shapiro et al., 2014, Theorem 7.39). In other words, every feasible
$\boldsymbol{p}\in\Pi$ may be taken to be a well-behaved Borel-measurable
selection of ${\cal U}$. Regarding the rest of the conditions of Assumption 1,
all are standard as mentioned above. We would just like to further point out
that condition (4) holds naturally if the Borel measure $\mathrm{P}$ has a
density with respect with the Lebesgue measure; this is a valid assumption in
numerous practical settings.

We are now in position to state the main result of this work. The detailed
proof is presented next in Section 4, and concerns the rest of this paper.

######  Theorem 1 (Strong Duality of Risk-Constrained Resource Allocation).

Let Assumption 1 be in effect. Then problem (RCP) has zero duality gap, i.e.,
$\mathsf{P}^{*}=\mathsf{D}^{*}$. In fact, (RCP) exhibits strong duality, i.e.,
optimal dual variables exist.

##  4 Proof of Theorem 1

Hereafter, we let Assumption 1 be in effect. We will work with the utility-
constraint set associated with problem (RCP) (or equivalently with problem
(RCP-E))

${\cal
C}\triangleq\left\\{(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\left|\begin{array}[]{l}g^{o}(\boldsymbol{x})\geq\delta_{o}\\\
-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}\geq\boldsymbol{\delta}_{r}\\\
\boldsymbol{g}(\boldsymbol{x})\geq\boldsymbol{\delta}_{d}\end{array},\;\text{for
some }(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\right.\right\\}.$

Following (Chamon et al., 2020, 2021), showing that ${\cal C}$ is convex and
the strict feasibility of problem (RCP) (from Assumption 1) would suffice to
ensure that (RCP) exhibits zero duality gap, as a relatively simple
consequence of the supporting hyperplane theorem; see Section 4.3, or (Chamon
et al., 2020, Theorem 1 and its proof), or (Chamon et al., 2021, Appendix A)
for the details. In fact, this is a special part of a far more general story;
see, e.g., (Flores-Bazán and Mastroeni, 2013). Proving convexity of ${\cal C}$
is nontrivial in the case of (RCP) though, and does not follow from the
analyses presented in the aforementioned articles. This is due to the
nonlinearity of the functionals present in the risk constraints of (RCP), in
sharp contrast to standard problems considered in the literature (e.g., in
(Luo and Zhang, 2008; Ribeiro, 2012; Chamon et al., 2020, 2021)), where the
corresponding constraints evaluate the vector $\boldsymbol{f}$ solely through
linear functionals, i.e., expectations.

The Challenge of Risk: From the discussion above it follows that ultimately we
would like to prove that the set ${\cal C}$ is convex. This would mean that if
$(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\in{\cal C}$ for
$(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi$ and if
$(\delta^{\prime}_{o},\boldsymbol{\delta}^{\prime}_{r},\boldsymbol{\delta}^{\prime}_{d})\in{\cal
C}$ for $(\boldsymbol{x}^{\prime},\boldsymbol{p}^{\prime})\in{\cal
X}\times\Pi$, then, for every $\alpha\in[0,1]$, it should be the case that

$\alpha(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})+(1-\alpha)(\delta^{\prime}_{o},\boldsymbol{\delta}^{\prime}_{r},\boldsymbol{\delta}^{\prime}_{d})\in{\cal
C}.$

In other words, we would have to show that there exists another pair
$(\boldsymbol{x}_{\alpha},\boldsymbol{p}_{\alpha})\in{\cal X}\times\Pi$, such
that

$\displaystyle g^{o}(\boldsymbol{x}_{\alpha})$
$\displaystyle\geq\alpha\delta_{o}+(1-\alpha)\delta^{\prime}_{o},$
$\displaystyle-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}_{\alpha}$
$\displaystyle\geq\alpha\boldsymbol{\delta}_{r}+(1-\alpha)\boldsymbol{\delta}^{\prime}_{r}\quad\text{and}$
$\displaystyle\boldsymbol{g}(\boldsymbol{x}_{\alpha})$
$\displaystyle\geq\alpha\boldsymbol{\delta}_{d}+(1-\alpha)\boldsymbol{\delta}^{\prime}_{d}.$

By choosing
$\boldsymbol{x}_{\alpha}=\alpha\boldsymbol{x}+(1-\alpha)\boldsymbol{x}^{\prime}\in{\cal
X}$ (by assumption, ${\cal X}$ is convex), and appealing to the concavity of
$g^{o}$ and $\boldsymbol{g}$, the proof would be complete if we showed that,
for every $\alpha\in[0,1]$, there is a policy $\boldsymbol{p}_{\alpha}\in\Pi$,
such that

$-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}(\boldsymbol{H}),\boldsymbol{H}))\geq-\alpha\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-(1-\alpha)\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})).$
(2)

Unfortunately, proving that ${\cal C}$ is convex for the risk-constrained
setting seems impossible, and we conjecture that such an assertion is probably
false, at least unless we enforce additional structural conditions (on top of
the fully compatible set of conditions comprising Assumption 1).

Fortunately though, what we can indeed show is convexity of the closure of
${\cal C}$; establishing this key result constitutes the core of the proof of
Theorem 1. Then, convexity of $\mathrm{cl}({\cal C})$ is combined with the
fact is that the point $(\mathsf{P}^{*},{\bf 0},{\bf 0})$ cannot be in the
interior of $\mathrm{cl}({\cal C})$, enabling a separating hyperplane argument
for the pair $(\mathsf{P}^{*},{\bf 0},{\bf 0})$ and $\mathrm{cl}({\cal C})$,
which together with Slater’s condition implies strong duality for problem
(RCP); this step is similar to that in the risk-neutral case, where ${\cal C}$
is a convex set, see, e.g., (Chamon et al., 2020). We believe that
establishing strong duality for (RCP) using only convexity of
$\mathrm{cl}({\cal C})$ is interesting, as it reveals a substantially weaker
chain of self-contained logical arguments sufficient to prove Theorem 1 as
compared with the risk-neutral setting (a special case). For an alternative,
though non-elementary and more involved way of establishing strong duality for
(RCP) –again by exploiting convexity of $\mathrm{cl}({\cal C})$ and Slater’s
condition–, see Appendix B.

In passing, note that (2) is trivially true and thus convexity of ${\cal C}$
is implied in the special case where $\boldsymbol{f}(\cdot,\boldsymbol{h})$ is
concave for every $\boldsymbol{h}\in{\cal H}$ and $\Pi$ is a convex set
(instead of a decomposable one), simply by choosing $\boldsymbol{p}_{\alpha}$
as a convex combination of $\boldsymbol{p}$ and $\boldsymbol{p}^{\prime}$, and
exploiting convexity and positive homogeneity of the vector risk measure
$\boldsymbol{\rho}$; this is a standard case of convex programming.

###  4.1 Preliminaries on Vector Measures in Banach Spaces

Let us first introduce some basic definitions and notation from the study of
vector measures taking values in general, infinite dimensional Banach spaces.
For a comprehensive treatment of the subject, the reader is referred to the
classical monograph (Diestel and Uhl, 1977).

Suppose that $\mathds{X}$ is a possibly infinite-dimensional Banach space. A
vector measure on a measurable space $(\Omega,\mathscr{F})$ is a function
$\boldsymbol{G}:\mathscr{F}\rightarrow\mathds{X}$. A vector measure
$\boldsymbol{G}$ is called countably additive (in the norm topology of
$\mathds{X}$) in the same fashion as a regular real-valued measure. The
variation of a vector measure $\boldsymbol{G}$ is another function on sets
$|\boldsymbol{G}|:\mathscr{F}\rightarrow\mathbb{R}_{+}$ defined as

$|\boldsymbol{G}|(E)\triangleq\sup_{\pi\text{ is a finite partition of
}E}\sum_{A\in\pi}\|\boldsymbol{G}(A)\|_{\mathds{X}}.$

Then $\boldsymbol{G}$ is suggestively said to be of bounded variation whenever
$|\boldsymbol{G}|(\Omega)<\infty$. A vector measure $\boldsymbol{G}$ is called
nonatomic if every event $E\in\mathscr{F}$ such that $\boldsymbol{G}(E)\neq 0$
can be partitioned into events $E^{\prime}$ and $E\backslash E^{\prime}$ such
that $\boldsymbol{G}(E^{\prime})\neq 0$ and $\boldsymbol{G}(E\backslash
E^{\prime})\neq 0$; in other words, every event of non-zero measure can be
split into two events of non-zero measure.

Further, in the following we will be using the concept of a Bochner integral,
which is a now standard extension of the Lebesgue integral for functions
taking values in infinite-dimensional Banach spaces. While we do not provide a
formal description here, the reader is referred to the excellent exposition in
(Diestel and Uhl, 1977, Section II), which is a standard textbook on the
subject.

Lastly, our analysis will be based on the following extension to the
celebrated Convexity Theorem of Lyapunov, due to Uhl (Uhl, 1969, Theorem 1 and
last paragraph before the References section), see also (Diestel and Uhl,
1977, Theorem IX.1.10). This also classical result conveniently generalizes
the convexity theorem to infinite-dimensional Banach spaces, albeit with some
nontrivial provisioning on the topological properties of the range of the
involved vector measure.

######  Theorem 2 ((Uhl, 1969) Weak Lyapunov Theorem for the Strong Topology).

Let $(\Omega,\mathscr{F})$ be a measurable space, and let $\mathds{X}$ be any
Banach space. Let $\boldsymbol{G}:\mathscr{F}\rightarrow\mathds{X}$ be a
countably additive vector measure of bounded variation. If $\boldsymbol{G}$ is
nonatomic and admits a Radon-Nikodym representation, i.e., there exist a
finite measure $\mu:\mathscr{F}\rightarrow\mathbb{R}_{+}$ and a function
$\boldsymbol{f}\in{\cal L}_{1}(\mu,\mathds{X})$ such that

$\boldsymbol{G}(E)=\int_{E}\boldsymbol{f}(\omega)\mathrm{d}\mu(\omega),\quad
E\in\mathscr{F},$

then the norm closure of the range $\boldsymbol{G}(\mathscr{F})$ is convex and
norm-compact.

###  4.2 Core of the Proof: Convexity of $\boldsymbol{\mathrm{cl}({\cal C})}$

We identify with $\mathds{X}$ the Banach space of all real-valued sequences
bounded in the $\sup$ norm, i.e., $\mathds{X}=\ell_{\infty}$. For a feasible
policy $\boldsymbol{p}\in\Pi$, define the vector measure
$\boldsymbol{G}_{\boldsymbol{p}}:\mathscr{B}({\cal H})\rightarrow\mathds{X}$
as

$\boldsymbol{G}_{\boldsymbol{p}}(E)\triangleq\begin{bmatrix}\int_{E}\lambda_{0}(\boldsymbol{h})\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\int_{E}\lambda_{1}(\boldsymbol{h})\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\vdots\\\
\int_{E}\lambda_{n}(\boldsymbol{h})\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\vdots\end{bmatrix}=\begin{bmatrix}\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})\lambda_{0}(\boldsymbol{H})\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}\\\
\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})\lambda_{1}(\boldsymbol{H})\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}\\\
\vdots\\\
\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})\lambda_{n}(\boldsymbol{H})\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}\\\
\vdots\end{bmatrix},\;E\in\mathscr{B}({\cal H}),$

where $\mathds{B}\triangleq\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ is a countable
basis (i.e., a dense subset) on ${\cal L}_{1}(\mathrm{P},\mathbb{R})$. We will
be later using the basis $\mathds{B}$ to approximate elements in
$\mathds{A}_{\gamma}^{i}$, $i\in\mathbb{N}_{N}^{+}$, each of which is a
bounded subset of ${\cal L}_{\infty}(\mathrm{P},\mathbb{R})$, and thus of
${\cal L}_{1}(\mathrm{P},\mathbb{R})$ (since $\mathrm{P}$ is finite);
therefore, without loss of generality we may very well assume that
$|\lambda_{n}|\leq\gamma$ everywhere on ${\cal H}$, and we do so hereafter
(otherwise, just take any countable basis on ${\cal
L}_{1}(\mathrm{P},\mathbb{R})$ and project each of its members to the box
$[-\gamma,\gamma]$; then, for every $\zeta\in\mathds{A}_{\gamma}^{i},$ there
exists a subsequence $\\{\lambda_{n}\\}_{n\in{\cal K}},{\cal
K}\subseteq\mathbb{N}$ converging to $\zeta$ in ${\cal L}_{1}$, and in fact

$0\leq\|\mathrm{proj}_{[-\gamma,\gamma]}(\lambda_{n})-\zeta\|_{{\cal
L}_{1}}=\|\mathrm{proj}_{[-\gamma,\gamma]}(\lambda_{n})-\mathrm{proj}_{[-\gamma,\gamma]}(\zeta)\|_{{\cal
L}_{1}}\leq\|\lambda_{n}-\zeta\|_{{\cal
L}_{1}}\underset{n\rightarrow\infty}{\longrightarrow}0,$

due to nonexpansiveness of the projection map, and so everything works).

For every $E\in\mathscr{B}({\cal H})$, it readily follows that (below the
ordinary absolute value $|\cdot|$ is taken component-wise when presented with
a vector as its input)

$\displaystyle\|\boldsymbol{G}_{\boldsymbol{p}}(E)\|_{\ell_{\infty}}$
$\displaystyle=\sup_{n\in\mathbb{N}}\|\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})\lambda_{n}(\boldsymbol{H})\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}\|_{\infty}$
$\displaystyle\leq\sup_{n\in\mathbb{N}}\|\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})|\lambda_{n}(\boldsymbol{H})||\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})|\\}\|_{\infty}$
$\displaystyle\leq\gamma\|\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})|\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})|\\}\|_{\infty}<\infty,$

verifying that $\boldsymbol{G}_{\boldsymbol{p}}(E)$ is an element of
$\mathds{X}$ for every qualifying $E$. Further, by construction
$\boldsymbol{G}_{\boldsymbol{p}}$ can be represented as a Bochner integral as
(see Appendix A)

$\boldsymbol{G}_{\boldsymbol{p}}(E)=\int_{E}\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h}),\quad
E\in\mathscr{B}({\cal H}),$

where “$\otimes$” denotes the Kronecker product, and where one can verify that
$\boldsymbol{\Lambda}_{\mathds{B}}\triangleq[\lambda_{0}\,\lambda_{1}\,\ldots\,\lambda_{n}\,\ldots]\in\mathds{X}$
(see, e.g., (Diestel and Uhl, 1977, Example II.2.10)); note that
$\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})$
is indeed in $\mathds{X}$ and Bochner integrable as well (follows from
(Diestel and Uhl, 1977, Theorem II.2.2), also see proof of Lemma 1 below).
Then, it follows that $\boldsymbol{G}_{\boldsymbol{p}}$ is both countably
additive and of bounded variation (Diestel and Uhl, 1977, Theorem II.2.4 (iii)
and (iv)). We also use the probabilistic notation

$\boldsymbol{G}_{\boldsymbol{p}}(E)=\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{H})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\},\quad
E\in\mathscr{B}({\cal H}),$

with an understanding that expectation here is in the Bochner sense (i.e.,
expectation of a random element taking values in an infinite-dimensional
Banach space).

Using the construction above, and together with another feasible policy
$\boldsymbol{p}^{\prime}\in\Pi$, we define another vector measure
$\boldsymbol{G}:\mathscr{B}({\cal H})\rightarrow\mathds{X}$ as

$\boldsymbol{G}(E)\triangleq\mathbb{E}\bigg{\\{}\mathds{1}_{E}(\boldsymbol{H})\,\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{H})\otimes\begin{bmatrix}\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\\
\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\end{bmatrix}\bigg{\\}},\quad
E\in\mathscr{B}({\cal H}),$

which will serve as our main construction for the rest of the analysis. The
next key result is concerned with the structure of the range of
$\boldsymbol{G}$, with its proof also verifying that $\boldsymbol{G}$ is
essentially an interleaved concatenation of the vector measures
$\boldsymbol{G}_{\boldsymbol{p}}$ and
$\boldsymbol{G}_{\boldsymbol{p}^{\prime}}$, as initially defined above.

######  Lemma 1.

The norm closure of the range of $\boldsymbol{G}$
$\boldsymbol{G}(\mathscr{B}({\cal
H}))=\\{\boldsymbol{x}\in\mathds{X}|\boldsymbol{x}=\boldsymbol{G}(E),\text{
for some }E\in\mathscr{B}({\cal H})\\}$ is convex and norm-compact.

###### Proof of Lemma 1.

We need to verify the conditions under which Theorem 2 (Uhl) is valid. First,
it is true that

$\displaystyle\mathbb{E}\bigg{\\{}\bigg{\|}\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{H})\otimes\begin{bmatrix}\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\\
\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\end{bmatrix}\bigg{\|}_{\ell_{\infty}}\bigg{\\}}$
$\displaystyle=\mathbb{E}\bigg{\\{}\sup_{n\in\mathbb{N}}\bigg{\|}\lambda_{n}(\boldsymbol{H})\begin{bmatrix}\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\\
\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\end{bmatrix}\bigg{\|}_{\infty}\bigg{\\}}$
$\displaystyle\leq\mathbb{E}\bigg{\\{}\sup_{n\in\mathbb{N}}|\lambda_{n}(\boldsymbol{H})|\bigg{\|}\begin{bmatrix}\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\\
\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\end{bmatrix}\bigg{\|}_{\infty}\bigg{\\}}$
$\displaystyle\leq\gamma\mathbb{E}\bigg{\\{}\bigg{\|}\begin{bmatrix}\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\\
\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\end{bmatrix}\bigg{\|}_{\infty}\bigg{\\}}$
$\displaystyle\leq\gamma\mathbb{E}\bigg{\\{}\bigg{\|}\begin{bmatrix}\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\\
\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\end{bmatrix}\bigg{\|}_{1}\bigg{\\}}<\infty.$

This shows that

$\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{H})\otimes\begin{bmatrix}\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\\
\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\end{bmatrix}\in{\cal
L}_{1}(\mathrm{P},\mathds{X}),$

which implies that $\boldsymbol{G}$ is countably additive (Diestel and Uhl,
1977, Theorem II.2.4 (iii)), has bounded variation (Diestel and Uhl, 1977,
Theorem II.2.4 (iv)) and evidently is Radon-Nikodym representable.

To show that $\boldsymbol{G}$ is nonatomic, consider its primitive
construction (this is justified in the same way as for
$\boldsymbol{G}_{\boldsymbol{p}}$; also see Appendix A) and suppose that
$E\in\mathscr{B}({\cal H})$ is such that

$\boldsymbol{G}(E)\neq{\bf
0}\iff\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})\lambda_{n}(\boldsymbol{H})f_{i}(\widetilde{\boldsymbol{p}}(\boldsymbol{H}),\boldsymbol{H})\\}\neq
0,\;\text{for some }n\in\mathbb{N},i\in\mathbb{N}_{N}^{+}\text{ and for
}\widetilde{\boldsymbol{p}}=\boldsymbol{p}\text{ or }\boldsymbol{p}^{\prime}.$

Note that we necessarily have $\mathrm{P}(E)>0$ (for otherwise
$\boldsymbol{G}(E)={\bf 0}$). Without loss of generality take
$\widetilde{\boldsymbol{p}}=\boldsymbol{p}$ and $n=i=1$. Then, by a lemma of
Blackwell (Blackwell, 1951, Lemma), nonatomicity of $\mathrm{P}$ implies the
existence of a Borel subset $\widetilde{E}\subseteq{\cal H}$ such that
$\mathrm{P}(\widetilde{E})=\mathrm{P}({\cal H})/2=1/2$, for which

$\mathbb{E}\\{\mathds{1}_{E\cap\widetilde{E}}(\boldsymbol{H})\lambda_{1}(\boldsymbol{H})f_{1}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}=\dfrac{1}{2}\mathbb{E}\\{\mathds{1}_{E}(\boldsymbol{H})\lambda_{1}(\boldsymbol{H})f_{1}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}\neq
0.$

Letting $E^{\prime}\triangleq E\bigcap\widetilde{E}$, this necessarily implies
that $\boldsymbol{G}(E^{\prime})\neq{\bf 0}$ as well as
$\boldsymbol{G}(E\backslash E^{\prime})\neq{\bf 0}$; in fact, observe that
necessarily $E^{\prime}\subset E$ and that $\mathrm{P}(E^{\prime})>0$. By
definition, it follows that $\boldsymbol{G}$ has no atoms. Consequently, the
conditions of Theorem 2 are fulfilled. Enough said. ∎

The conclusions of Lemma 1, in particular convexity of norm closure of the
range of $\boldsymbol{G}$, are sufficient to ensure convexity of the (norm)
closure of ${\cal C}$. To see this, let us first consider the range
$\boldsymbol{G}(\mathscr{B}({\cal H}))$ of $\boldsymbol{G}$. Of course
$\boldsymbol{z}=\boldsymbol{G}({\cal H})$ and
$\boldsymbol{z}^{\prime}=\boldsymbol{G}(\emptyset)={\bf 0}$ are both elements
of $\boldsymbol{G}(\mathscr{B}({\cal H}))$. Therefore, Lemma 1 implies that,
for every $\alpha\in[0,1]$, the convex combination
$\alpha\boldsymbol{z}+(1-\alpha)\boldsymbol{z}^{\prime}\equiv\alpha\boldsymbol{z}$
lies in the norm closure of $\boldsymbol{G}(\mathscr{B}({\cal H}))$. In order
words, for each $\alpha\in[0,1]$, there exists a sequence of events
$\\{E_{n}^{\alpha}\in\mathscr{B}({\cal H})\\}_{n\in\mathbb{N}}$ such that

$\lim_{n\rightarrow\infty}\|\alpha\boldsymbol{z}-\boldsymbol{G}(E_{n}^{\alpha})\|_{\ell_{\infty}}=0.$

This in particular implies that (note that limits here are with respect to the
natural norm of $\mathds{X}$)

$\lim_{n\rightarrow\infty}\|\alpha\boldsymbol{G}_{\boldsymbol{p}}({\cal
H})-\boldsymbol{G}_{\boldsymbol{p}}(E_{n}^{\alpha})\|_{\ell_{\infty}}=0,$

and by a symmetric argument,

$\lim_{n\rightarrow\infty}\|(1-\alpha)\boldsymbol{G}_{\boldsymbol{p}^{\prime}}({\cal
H})-\boldsymbol{G}_{\boldsymbol{p}^{\prime}}((E_{n}^{\alpha})^{c})\|_{\ell_{\infty}}=0.$

Now, we may define the sequence of policies

$\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H})=\mathds{1}_{E_{n}^{\alpha}}(\boldsymbol{H})\boldsymbol{p}(\boldsymbol{H})+\mathds{1}_{{\cal
H}\backslash
E_{n}^{\alpha}}(\boldsymbol{H})\boldsymbol{p}^{\prime}(\boldsymbol{H})=\begin{cases}\boldsymbol{p}(\boldsymbol{H}),&\text{if
}\boldsymbol{H}\in E_{n}^{\alpha}\\\
\boldsymbol{p}^{\prime}(\boldsymbol{H}),&\text{if }\boldsymbol{H}\in{\cal
H}\backslash E_{n}^{\alpha}\end{cases},\quad n\in\mathbb{N}.$

Of course, it holds that $\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H})\in\Pi$
for all $n\in\mathbb{N}$ because $\Pi$ is decomposable. Then, it follows that

$\displaystyle\|\boldsymbol{G}_{\boldsymbol{p}_{\alpha}^{n}}({\cal
H})-\alpha\boldsymbol{G}_{\boldsymbol{p}}({\cal
H})-(1-\alpha)\boldsymbol{G}_{\boldsymbol{p}^{\prime}}({\cal
H})\|_{\ell_{\infty}}$
$\displaystyle=\|\boldsymbol{G}_{\boldsymbol{p}_{\alpha}^{n}}(E_{n}^{\alpha})+\boldsymbol{G}_{\boldsymbol{p}_{\alpha}^{n}}((E_{n}^{\alpha})^{c})-\alpha\boldsymbol{G}_{\boldsymbol{p}}({\cal
H})-(1-\alpha)\boldsymbol{G}_{\boldsymbol{p}^{\prime}}({\cal
H})\|_{\ell_{\infty}}$
$\displaystyle=\|\boldsymbol{G}_{\boldsymbol{p}}(E_{n}^{\alpha})+\boldsymbol{G}_{\boldsymbol{p}^{\prime}}((E_{n}^{\alpha})^{c})-\alpha\boldsymbol{G}_{\boldsymbol{p}}({\cal
H})-(1-\alpha)\boldsymbol{G}_{\boldsymbol{p}^{\prime}}({\cal
H})\|_{\ell_{\infty}}$
$\displaystyle\leq\|\boldsymbol{G}_{\boldsymbol{p}}(E_{n}^{\alpha})-\alpha\boldsymbol{G}_{\boldsymbol{p}}({\cal
H})\|_{\ell_{\infty}}+\|\boldsymbol{G}_{\boldsymbol{p}^{\prime}}((E_{n}^{\alpha})^{c})-(1-\alpha)\boldsymbol{G}_{\boldsymbol{p}^{\prime}}({\cal
H})\|_{\ell_{\infty}},$

which implies that

$\lim_{n\rightarrow\infty}\|\boldsymbol{G}_{\boldsymbol{p}_{\alpha}^{n}}({\cal
H})-\alpha\boldsymbol{G}_{\boldsymbol{p}}({\cal
H})-(1-\alpha)\boldsymbol{G}_{\boldsymbol{p}^{\prime}}({\cal
H})\|_{\ell_{\infty}}=0.$

Equivalently, we have shown that for every $\varepsilon>0$, there exists a
positive number $N(\varepsilon)>0$, such that for every $n>N(\varepsilon)$,

$\displaystyle\|\mathbb{E}\\{\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{H})\otimes\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H}),\boldsymbol{H})\\}$
$\displaystyle-\alpha\mathbb{E}\\{\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{H})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}-(1-\alpha)\mathbb{E}\\{\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{H})\otimes\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\\}\|_{\ell_{\infty}}\leq\varepsilon.$

Evidently, $N(\varepsilon)$ is uniform over the individual elements of the
countable basis $\mathds{B}$. We can rewrite the preceding expression as

$\displaystyle|\mathbb{E}\\{\lambda_{m}(\boldsymbol{H})f_{i}(\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H}),\boldsymbol{H})\\}$
$\displaystyle-\alpha\mathbb{E}\\{\lambda_{m}(\boldsymbol{H})f_{i}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}-(1-\alpha)\mathbb{E}\\{\lambda_{m}(\boldsymbol{H})f_{i}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\\}|\leq\varepsilon,$

for every pair $(m,i)\in\mathbb{N}\times\mathbb{N}_{N}^{+}$. Now, for each
choice of $\zeta\in\mathds{A}_{\gamma}^{i}$, $i\in\mathbb{N}_{N}^{+}$, we can
extract a subsequence $\\{\lambda_{m}\\}_{m\in{\cal K}},{\cal
K}\subseteq\mathbb{N}$ converging to $\zeta$ in ${\cal L}_{1}$. A consequence
of this is the existence of a further sub-subsequence
$\\{\lambda_{m}\\}_{m\in{\cal K}^{\prime}}$, ${\cal K}^{\prime}\subseteq{\cal
K}$ such that

$\lambda_{m}\underset{{\cal K}^{\prime}\ni
m\rightarrow\infty}{\longrightarrow}\zeta,\quad\mathrm{P}\text{-a.e.}$

Then, by dominated convergence, we have for each $i$-th element of the service
vector $\boldsymbol{f}$,

$\displaystyle\mathbb{E}\\{\lambda_{m}(\boldsymbol{H})f_{i}(\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H}),\boldsymbol{H})\\}$
$\displaystyle\underset{{\cal K}^{\prime}\ni
m\rightarrow\infty}{\longrightarrow}\mathbb{E}\\{\zeta(\boldsymbol{H})f_{i}(\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H}),\boldsymbol{H})\\},$
$\displaystyle\mathbb{E}\\{\lambda_{m}(\boldsymbol{H})f_{i}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}$
$\displaystyle\underset{{\cal K}^{\prime}\ni
m\rightarrow\infty}{\longrightarrow}\mathbb{E}\\{\zeta(\boldsymbol{H})f_{i}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}\quad\text{and}$
$\displaystyle\mathbb{E}\\{\lambda_{m}(\boldsymbol{H})f_{i}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\\}$
$\displaystyle\underset{{\cal K}^{\prime}\ni
m\rightarrow\infty}{\longrightarrow}\mathbb{E}\\{\zeta(\boldsymbol{H})f_{i}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\\}.$

Therefore, we have that, for every $\varepsilon>0$ there exists a positive
number $N(\varepsilon)>0$, such that for every $n>N(\varepsilon)$ and for
every $\zeta\in\mathds{A}_{\gamma}^{i}$,222Note that the basic fact that
enables interchanging the order of limits relative to $n$ and $m$ above is
that convergence over $n$ is uniform over $m$, i.e., the index of the elements
in the dense set $\mathds{B}$. Then, we reiterate the same procedure for every
$\zeta$ in each of the risk envelopes
$\mathds{A}_{\gamma}^{i},i\in\mathbb{N}_{N}^{+}$, by extracting a different
subsequence out of $\mathds{B}$ each time.

$\displaystyle|\mathbb{E}\\{\zeta(\boldsymbol{H})f_{i}(\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H}),\boldsymbol{H})\\}$
$\displaystyle-\alpha\mathbb{E}\\{\zeta(\boldsymbol{H})f_{i}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})\\}-(1-\alpha)\mathbb{E}\\{\zeta(\boldsymbol{H})f_{i}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})\\}|\leq\varepsilon,\quad
i\in\mathbb{N}_{N}^{+}.$

The last expression implies in particular that

$\displaystyle\bigg{|}\inf_{\zeta\in\mathds{A}_{\gamma}^{i}}\mathbb{E}\\{\zeta(\boldsymbol{H})f_{i}(\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H}),\boldsymbol{H})\\}$
$\displaystyle-\inf_{\zeta\in\mathds{A}_{\gamma}^{i}}\mathbb{E}\\{\zeta(\boldsymbol{H})[\alpha
f_{i}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})+(1-\alpha)f_{i}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})]\\}\bigg{|}\leq\varepsilon,\quad
i\in\mathbb{N}_{N}^{+},$

which is the same as

$\bigg{\|}\inf_{\boldsymbol{\zeta}\in\mathds{A}_{\gamma}^{S}}\mathbb{E}\\{\boldsymbol{\zeta}(\boldsymbol{H})\circ\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H}),\boldsymbol{H})\\}-\inf_{\boldsymbol{\zeta}\in\mathds{A}_{\gamma}^{S}}\mathbb{E}\\{\boldsymbol{\zeta}(\boldsymbol{H})\circ[\alpha\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})+(1-\alpha)\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H})]\\}\bigg{\|}_{\infty}\leq\varepsilon.$

By risk duality, we obtain that, for every $\varepsilon>0$, there is
$N(\varepsilon)>0$ such that for every $n>N(\varepsilon)$, it is true that

$\|-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{n}(\boldsymbol{H}),\boldsymbol{H}))+\boldsymbol{\rho}(-\alpha\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})-(1-\alpha)\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H}))\|_{\infty}\leq\varepsilon.$

This implies that, for every choice of $\boldsymbol{p}\in\Pi$,
$\boldsymbol{p}^{\prime}\in\Pi$, for every $\alpha\in[0,1]$ and for every
$\varepsilon>0$, there exists at least one policy
$\boldsymbol{p}_{\alpha}^{\varepsilon}\in\Pi$ (in fact, a whole family of such
policies) such that

$-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{\varepsilon}(\boldsymbol{H}),\boldsymbol{H}))+\boldsymbol{\rho}(-\alpha\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})-(1-\alpha)\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H}))\geq-\varepsilon{\bf
1}.$

This fact together with convexity and positive homogeneity of
$\boldsymbol{\rho}$ further implies that

$\displaystyle-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{\varepsilon}(\boldsymbol{H}),\boldsymbol{H}))$
$\displaystyle\geq-\boldsymbol{\rho}(-\alpha\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H})-(1-\alpha)\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H}))-\varepsilon{\bf
1}$
$\displaystyle\geq-\boldsymbol{\rho}(-\alpha\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{\rho}(-(1-\alpha)\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H}))-\varepsilon{\bf
1}$
$\displaystyle=-\alpha\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-(1-\alpha)\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H}))-\varepsilon{\bf
1}.$

Let us now see how the preceding fact implies that $\mathrm{cl}({\cal C})$ is
a convex set. Let
$(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\in{\cal C}$ for
$(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi$ and let
$(\delta^{\prime}_{o},\boldsymbol{\delta}^{\prime}_{r},\boldsymbol{\delta}^{\prime}_{d})\in{\cal
C}$ for $(\boldsymbol{x}^{\prime},\boldsymbol{p}^{\prime})\in{\cal
X}\times\Pi$. Also choose a sequence
$\\{\varepsilon_{n}>0\\}_{n\in\mathbb{N}}$ decreasing to zero, say
$\varepsilon_{n}\triangleq 1/(n+1),n\in\mathbb{N}$. By our discussion above,
we have actually shown that, for every $\alpha\in[0,1]$, it holds that, for
every $n$,

$\alpha(\delta_{o},\boldsymbol{\delta}_{r}-\varepsilon_{n}{\bf
1},\boldsymbol{\delta}_{d})+(1-\alpha)(\delta^{\prime}_{o},\boldsymbol{\delta}^{\prime}_{r}-\varepsilon_{n}{\bf
1},\boldsymbol{\delta}^{\prime}_{d})\in{\cal C}.$

To verify this claim, observe that, for every $n\in\mathbb{N}$, there exists a
policy $\boldsymbol{p}_{\alpha}^{\varepsilon_{n}}\in\Pi$ such that for the
pair
$(\boldsymbol{x}_{\alpha}=\alpha\boldsymbol{x}+(1-\alpha)\boldsymbol{x}^{\prime},\boldsymbol{p}_{\alpha}^{\varepsilon_{n}})\in{\cal
X}\times\Pi$ (by assumption, ${\cal X}$ is convex) it is true that

$\displaystyle-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{\varepsilon_{n}}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}_{\alpha}$
$\displaystyle=-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{\varepsilon_{n}}(\boldsymbol{H}),\boldsymbol{H}))-\alpha\boldsymbol{x}-(1-\alpha)\boldsymbol{x}^{\prime}$
$\displaystyle\geq-\alpha\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-(1-\alpha)\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H}))-\alpha\boldsymbol{x}-(1-\alpha)\boldsymbol{x}^{\prime}-\varepsilon_{n}{\bf
1}$
$\displaystyle=-\alpha\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-\alpha\boldsymbol{x}-(1-\alpha)\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}^{\prime}(\boldsymbol{H}),\boldsymbol{H}))-(1-\alpha)\boldsymbol{x}^{\prime}-\varepsilon_{n}{\bf
1}$
$\displaystyle\geq\alpha\boldsymbol{\delta}_{r}+(1-\alpha)\boldsymbol{\delta}^{\prime}_{r}-\varepsilon_{n}{\bf
1},$

or, equivalently,

$-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{\varepsilon_{n}}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}_{\alpha}\geq\alpha(\boldsymbol{\delta}_{r}-\varepsilon_{n}{\bf
1})+(1-\alpha)(\boldsymbol{\delta}^{\prime}_{r}-\varepsilon_{n}{\bf 1}).$

This means that the pair
$(\boldsymbol{x}_{\alpha},\boldsymbol{p}_{\alpha}^{\varepsilon_{n}})\in{\cal
X}\times\Pi$ is such that

$\displaystyle g^{o}(\boldsymbol{x}_{\alpha})$
$\displaystyle\geq\alpha\delta_{o}+(1-\alpha)\delta^{\prime}_{o},$
$\displaystyle-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}_{\alpha}^{\varepsilon_{n}}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}_{\alpha}$
$\displaystyle\geq\alpha(\boldsymbol{\delta}_{r}-\varepsilon_{n}{\bf
1})+(1-\alpha)(\boldsymbol{\delta}^{\prime}_{r}-\varepsilon_{n}{\bf
1})\quad\text{and}$ $\displaystyle\boldsymbol{g}(\boldsymbol{x}_{\alpha})$
$\displaystyle\geq\alpha\boldsymbol{\delta}_{d}+(1-\alpha)\boldsymbol{\delta}^{\prime}_{d},$

verifying our claim. But, evidently,

$\lim_{n\rightarrow\infty}\alpha(\delta_{o},\boldsymbol{\delta}_{r}-\varepsilon_{n}{\bf
1},\boldsymbol{\delta}_{d})+(1-\alpha)(\delta^{\prime}_{o},\boldsymbol{\delta}^{\prime}_{r}-\varepsilon_{n}{\bf
1},\boldsymbol{\delta}^{\prime}_{d})=\alpha(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})+(1-\alpha)(\delta^{\prime}_{o},\boldsymbol{\delta}^{\prime}_{r},\boldsymbol{\delta}^{\prime}_{d}),$

which of course implies that

$\alpha(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})+(1-\alpha)(\delta^{\prime}_{o},\boldsymbol{\delta}^{\prime}_{r},\boldsymbol{\delta}^{\prime}_{d})\in\mathrm{cl}({\cal
C}),$

as a limit point of elements in ${\cal C}$. Finally, convexity of
$\mathrm{cl}({\cal C})$ follows in light of the next elementary result, whose
proof we also present for completeness.

######  Proposition 1.

Suppose that a set ${\cal A}\subseteq\mathbb{R}^{N}$ has the property that all
convex combinations of any two points in ${\cal A}$ belong to its closure,
i.e.,

$\boldsymbol{x}\in{\cal A}\quad\text{and}\quad\boldsymbol{y}\in{\cal
A}\implies\forall\alpha\in[0,1],\quad\alpha\boldsymbol{x}+(1-\alpha)\boldsymbol{y}\in\mathrm{cl}({\cal
A}).$

Then $\mathrm{cl}({\cal A})$ is (closed) convex.

###### Proof of Proposition 1.

Let $\boldsymbol{x}\in\mathrm{cl}({\cal A})$ and
$\boldsymbol{y}\in\mathrm{cl}({\cal A})$. Then, we can find sequences
$\\{\boldsymbol{x}_{n}\in{\cal A}\\}_{n\in\mathbb{N}}$ and
$\\{\boldsymbol{y}_{n}\in{\cal A}\\}_{n\in\mathbb{N}}$ such that

$\boldsymbol{x}=\lim_{n\rightarrow\infty}\boldsymbol{x}_{n}\quad\text{and}\quad\boldsymbol{y}=\lim_{n\rightarrow\infty}\boldsymbol{y}_{n},$

where the limits are interpreted in the standard Euclidean sense. So,

$\displaystyle\alpha\boldsymbol{x}+(1-\alpha)\boldsymbol{y}$
$\displaystyle=\alpha\lim_{n\rightarrow\infty}\boldsymbol{x}_{n}+(1-\alpha)\lim_{n\rightarrow\infty}\boldsymbol{y}_{n}$
$\displaystyle=\lim_{n\rightarrow\infty}\alpha\boldsymbol{x}_{n}+(1-\alpha)\boldsymbol{y}_{n}.$

However, by assumption for the set ${\cal A}$ it holds that, for every
$n\in\mathbb{N}$,

$\alpha\boldsymbol{x}_{n}+(1-\alpha)\boldsymbol{y}_{n}\in\mathrm{cl}({\cal
A}),$

and the sequence converges; therefore it must converge in $\mathrm{cl}({\cal
A})$, and the limit is $\alpha\boldsymbol{x}+(1-\alpha)\boldsymbol{y}$.
Therefore, $\mathrm{cl}({\cal A})$ is closed convex. ∎

By a trivial application of Proposition 1, we obtain that $\mathrm{cl}({\cal
C})$ is a (closed) convex set, and the proof is now complete. $\blacksquare$

###  4.3 Convexity of $\boldsymbol{\mathrm{cl}({\cal C})}$ Implies Strong
Duality

Let us now finish the proof of Theorem 1 by exploiting the convexity of the
closure of the utility-constraint set

${\cal
C}=\left\\{(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\left|\begin{array}[]{l}g^{o}(\boldsymbol{x})\geq\delta_{o}\\\
-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}\geq\boldsymbol{\delta}_{r}\\\
\boldsymbol{g}(\boldsymbol{x})\geq\boldsymbol{\delta}_{d}\end{array},\;\text{for
some }(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\right.\right\\},$

the expression of which we repeat here for convenience, together with
condition (5) of Assumption 1, namely that problem (RCP) satisfies Slater’s
condition. Let us recall the Lagrangian associated with problem (RCP), i.e.,

$\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda})=g^{o}(\boldsymbol{x})+\langle\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{g}(\boldsymbol{x})\rangle+\big{\langle}\boldsymbol{\lambda}_{\boldsymbol{\rho}},-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}\big{\rangle},$

where $(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda})\in{\bf
C}\times\Pi\times\mathbb{R}^{N_{\boldsymbol{g}}}\times{\bf C}$, for which we
already know that

$\inf_{\boldsymbol{\lambda}\geq{\bf
0}}\sup_{(\boldsymbol{x},\boldsymbol{p})\in{\cal
X}\times\Pi}\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda})=\mathsf{D}^{*}\geq\mathsf{P}^{*}.$

Strong duality of (RCP) will be immediate if we can show that
$\mathsf{D}^{*}\leq\mathsf{P}^{*}$.

Our discussion substantially extends (Chamon et al., 2020, Theorem 1 and its
proof) and (Chamon et al., 2021, Appendix A), and is eventually based on a
standard application of the supporting hyperplane theorem (see e.g.,
(Bertsekas, 2009, Proposition 1.5.1)), which we outline below for
completeness. Note that the same technique would be applicable in case our
initial problem (RCP) was originally convex. The reason is that the only fact
needed at this point is the “top-level” convexity of the $\mathrm{cl}({\cal
C})$, which would come for free if (RCP) was itself a convex program. Before
we assemble everything altogether, we need an additional technical result.

######  Lemma 2 (Point on the Shell).

$(\mathsf{P}^{*},{\bf 0},{\bf 0})$ is not in the interior of
$\mathrm{cl}({\cal C})$.

###### Proof of Lemma 2.

First, we observe that the point $(\mathsf{P}^{*},{\bf 0},{\bf 0})$ cannot be
in the interior of ${\cal C}$, for otherwise there would exist an
$\varepsilon>0$ such that $(\mathsf{P}^{*}+\varepsilon,{\bf 0},{\bf
0})\in{\cal C}$, contradicting $\mathsf{P}^{*}$ being the optimal value of the
initial constrained problem. In fact, if
$(\delta_{o}^{+},\boldsymbol{\delta}_{r}^{+},\boldsymbol{\delta}_{d}^{+})>{\bf
0}$, every perturbation of the form

$(\mathsf{P}^{*},{\bf 0},{\bf
0})+(\delta_{o}^{+},\boldsymbol{\delta}_{r}^{+},\boldsymbol{\delta}_{d}^{+})=(\mathsf{P}^{*}+\delta_{o}^{+},\boldsymbol{\delta}_{r}^{+},\boldsymbol{\delta}_{r}^{+})$

cannot be in ${\cal C}$ either; if it was, this would imply the existence of a
pair $(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi$ attaining a
strictly larger objective than $\mathsf{P}^{*}$, while also keeping all the
constraints inactive. Therefore, it follows that the open orthant (an open
convex set)

${\cal
D}=\\{(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})|(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})=(\mathsf{P}^{*}+\delta_{o}^{+},\boldsymbol{\delta}_{r}^{+},\boldsymbol{\delta}_{r}^{+}),\quad(\delta_{o}^{+},\boldsymbol{\delta}_{r}^{+},\boldsymbol{\delta}_{r}^{+})>{\bf
0}\\}$

is disjoint from ${\cal C}$.

Let us show that, in fact, ${\cal D}$ and $\mathrm{cl}({\cal C})$ are also
disjoint. Note that ${\cal D}$ has a nonempty interior in the standard
Euclidean topology of ${\cal C}$. Therefore, for every element of ${\cal D}$
there exists an open ball of the same dimension as that of ${\cal C}$ entirely
contained in ${\cal D}$ and having that element of ${\cal D}$ as its center.
Next, if an element of ${\cal D}$, say $\boldsymbol{d}$, was in
$\mathrm{cl}({\cal C})$, there should exist a sequence contained entirely in
${\cal C}$ converging to $\boldsymbol{d}$. However, since $\boldsymbol{d}$ is
contained in an open ball, say ${\cal B}_{\boldsymbol{d}}$, entirely contained
in ${\cal D}$ and of the same dimension as that of ${\cal C}$, every sequence
in ${\cal C}$ converging to $\boldsymbol{d}$ must break into that open ball.
This implies that the aforementioned sequence in ${\cal C}$ converging to
$\boldsymbol{d}$ must have elements also in ${\cal B}_{\boldsymbol{d}}$. This
is absurd, since ${\cal C}\bigcap{\cal B}_{\boldsymbol{d}}=\emptyset$ (due to
the fact that ${\cal C}\bigcap{\cal D}=\emptyset$).

Now suppose that $(\mathsf{P}^{*},{\bf 0},{\bf 0})$ is in the interior of
$\mathrm{cl}({\cal C})$. Then, there must exist another open ball, say ${\cal
B}_{*}$, of the same dimension as that of ${\cal C}$, completely contained in
$\mathrm{cl}({\cal C})$ and centered at $(\mathsf{P}^{*},{\bf 0},{\bf 0})$.
This means that there exists a positive perturbation vector
$(\delta_{o}^{+},\boldsymbol{\delta}_{r}^{+},\boldsymbol{\delta}_{r}^{+})>{\bf
0}$ such that the point
$(\mathsf{P}^{*}+\delta_{o}^{+},\boldsymbol{\delta}_{r}^{+},\boldsymbol{\delta}_{r}^{+})$
is in ${\cal B}_{*}$. This is also absurd, because
$(\mathsf{P}^{*}+\delta_{o}^{+},\boldsymbol{\delta}_{r}^{+},\boldsymbol{\delta}_{r}^{+})\in{\cal
D}$ and ${\cal D}\bigcap\mathrm{cl}({\cal C})=\emptyset$, as shown above. ∎

The complete argument based on proper application supporting hyperplane
theorem now follows.

######  Theorem 3 (Supporting Hyperplane Theorem).

Let ${\cal A}\subseteq\mathbb{R}^{n}$ be a nonempty convex set. If
$\boldsymbol{\delta}^{*}\in\mathbb{R}^{n}$ is not in the interior of ${\cal
A}$, then there exists a hyperplane passing through $\boldsymbol{\delta}^{*}$
such that ${\cal A}$ is in one of its closed halfspaces. In other words, there
exists a vector $\text{$\boldsymbol{\lambda}$}\neq{\bf 0}$ such that, for
every $\boldsymbol{\delta}\in{\cal A}$, it holds that
$\langle\text{$\boldsymbol{\lambda}$},\boldsymbol{\delta}^{*}\rangle\geq\langle\text{$\boldsymbol{\lambda}$},\boldsymbol{\delta}\rangle$.

######  Remark 1.

Note that the inequality in the supporting hyperplane theorem can be
equivalently reversed by flipping the sign of the support vector
$\boldsymbol{\lambda}$.

We start by observing that since (RCP) satisfies Slater’s condition, it
follows that ${\cal C}$ is nonempty, and the same follows for
$\mathrm{cl}({\cal C})$. So $\mathrm{cl}({\cal C})$ is a nonempty (closed)
convex set. By Lemma 2, we also know that the point $(\mathsf{P}^{*},{\bf
0},{\bf 0})$ is not in the interior of $\mathrm{cl}({\cal C})$. We can then
apply the supporting hyperplane theorem on the pair $\mathrm{cl}({\cal C})$
and $(\mathsf{P}^{*},{\bf 0},{\bf 0})$, implying existence of a vector of
multipliers
$(\lambda_{o},\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{\lambda}_{\boldsymbol{\rho}})\neq{\bf
0}$ such that, for every
$(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\in\mathrm{cl}({\cal
C})$, it is true that

$\lambda_{o}\delta_{o}+\langle\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{\delta}_{r}\rangle+\langle\boldsymbol{\lambda}_{\boldsymbol{\rho}},\boldsymbol{\delta}_{d}\rangle\leq\lambda_{o}\mathsf{P}^{*}.$

It readily follows that, in fact,
$(\lambda_{o},\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{\lambda}_{\boldsymbol{\rho}})\geq{\bf
0}$. Indeed, if any component of
$(\lambda_{o},\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{\lambda}_{\boldsymbol{\rho}})$
was negative, then we could choose
$(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\in{\cal
C}\subseteq\mathrm{cl}({\cal C})$ such that the corresponding inner product
becomes arbitrarily large (note that ${\cal C}$ is unbounded below),
eventually violating the inequality above, regardless of the sign of
$\mathsf{P}^{*}$.

The second fact we may show is that $\lambda_{o}\neq 0$, which implies that
$\lambda_{o}>0$. Again, if $\lambda_{o}=0$, then we would have that

$\langle\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{\delta}_{r}\rangle+\langle\boldsymbol{\lambda}_{\boldsymbol{\rho}},\boldsymbol{\delta}_{d}\rangle\leq
0.$

But
$(\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{\lambda}_{\boldsymbol{\rho}})\neq{\bf
0}$ (i.e., there is at least one nonzero component) and
$(\boldsymbol{\lambda}_{\boldsymbol{g}},\boldsymbol{\lambda}_{\boldsymbol{\rho}})\geq{\bf
0}$ (shown above), and problem (RCP) satisfies Slater’s condition, so the
preceding inequality is also absurd.

As a result, we may divide by $\lambda_{o}>0$, showing that there is
$(\boldsymbol{\lambda}_{\boldsymbol{g}}^{\star}\triangleq\boldsymbol{\lambda}_{\boldsymbol{g}}/\lambda_{o},\boldsymbol{\lambda}_{\boldsymbol{\rho}}^{\star}\triangleq\boldsymbol{\lambda}_{\boldsymbol{\rho}}/\lambda_{o})\geq{\bf
0}$, such that, for every
$(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\in\mathrm{cl}({\cal
C})$, it holds that

$\delta_{o}+\langle\boldsymbol{\lambda}_{\boldsymbol{g}}^{\star},\boldsymbol{\delta}_{r}\rangle+\langle\boldsymbol{\lambda}_{\boldsymbol{\rho}}^{\star},\boldsymbol{\delta}_{d}\rangle\leq\mathsf{P}^{*}.$

By construction of the set $\mathrm{cl}({\cal C})$ (as the closure of ${\cal
C}$), we obtain in particular that

$\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda}^{\star})=g^{o}(\boldsymbol{x})+\langle\boldsymbol{\lambda}_{\boldsymbol{g}}^{\star},\boldsymbol{g}(\boldsymbol{x})\rangle+\big{\langle}\boldsymbol{\lambda}_{\boldsymbol{\rho}}^{\star},-\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))-\boldsymbol{x}\big{\rangle}\leq\mathsf{P}^{*},$

for every pair $(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi$, since
all such pairs correspond to points included in ${\cal
C}\subseteq\mathrm{cl}({\cal C})$. This further implies that we are allowed to
maximize both sides of the inequality over all possible
$(\boldsymbol{x},\boldsymbol{p})$, yielding

$\displaystyle-\infty<\mathsf{D}^{*}$
$\displaystyle=\inf_{\boldsymbol{\lambda}\geq{\bf
0}}\sup_{(\boldsymbol{x},\boldsymbol{p})\in{\cal
X}\times\Pi}\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda})$
$\displaystyle\leq\sup_{(\boldsymbol{x},\boldsymbol{p})\in{\cal
X}\times\Pi}\mathsf{L}(\boldsymbol{x},\boldsymbol{p},\boldsymbol{\lambda}^{\star})\leq\mathsf{P}^{*},$

and we are done. $\blacksquare$

##  Appendix A Bochner Integral Representation of
$\boldsymbol{\boldsymbol{G}_{\boldsymbol{p}}}$

To verify the seemingly obvious equivalence

$\boldsymbol{G}_{\boldsymbol{p}}(E)=\int_{E}\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h}),\quad
E\in\mathscr{B}({\cal H}),$

we first note that, since
$\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})$
is Bochner integrable –and specifically in ${\cal
L}_{1}(\mathrm{P},\mathds{X}),\mathds{X}=\ell_{\infty}$–, by definition of the
general Bochner integral there is a sequence of simple $\mathds{X}$-valued
measurable functions $\\{\boldsymbol{g}^{m}:{\cal
H}\rightarrow\mathds{X}\\}_{m\in\mathbb{N}}$ such that (Diestel and Uhl, 1977,
Definition II.2.1)

$\lim_{m\rightarrow\infty}\int\|\boldsymbol{g}^{m}(\boldsymbol{h})-\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\|_{\ell_{\infty}}\mathrm{d}\mathrm{P}(\boldsymbol{h})=0.$

Then, it is true that, for every
$(n^{o},i^{o})\in\mathbb{N}\times\mathbb{N}_{N}^{+}$,

$\displaystyle\int\|\boldsymbol{g}^{m}(\boldsymbol{h})-\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\|_{\ell_{\infty}}\mathrm{d}\mathrm{P}(\boldsymbol{h})$
$\displaystyle=\int\sup_{n\in\mathbb{N}}\max_{i\in\mathbb{N}_{N}^{+}}|g_{n,i}^{m}(\boldsymbol{h})-\lambda_{n}(\boldsymbol{h})f_{i}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})|\mathrm{d}\mathrm{P}(\boldsymbol{h})$
$\displaystyle\geq\int|g_{n^{o},i^{o}}^{m}(\boldsymbol{h})-\lambda_{n^{o}}(\boldsymbol{h})f_{i^{o}}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})|\mathrm{d}\mathrm{P}(\boldsymbol{h})\geq
0,$

which reveals that each component sequence
$\\{g_{n^{o},i^{o}}^{m}\\}_{m\in\mathbb{N}}$ is such that

$\lim_{m\rightarrow\infty}\int|g_{n^{o},i^{o}}^{m}(\boldsymbol{h})-\lambda_{n^{o}}(\boldsymbol{h})f_{i^{o}}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})|\mathrm{d}\mathrm{P}(\boldsymbol{h})=0.$

This of course further implies that, for every
$(n,i)\in\mathbb{N}\times\mathbb{N}_{N}^{+}$,

$\lim_{m\rightarrow\infty}\int_{E}g_{n,i}^{m}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})=\int_{E}\lambda_{n}(\boldsymbol{h})f_{i}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h}),\quad
E\in\mathscr{B}({\cal H}).$

Using properties of $\mathds{X}$ as a vector space (i.e., addition and scalar
multiplication), it follows that for every simple function
$\boldsymbol{g}:{\cal H}\rightarrow\mathds{X}$, we may write, for some fixed
vectors $\\{\boldsymbol{z}^{k}\in\mathds{X}\\}_{k\in\mathbb{N}_{K}^{+}}$ and
events $\\{{\cal G}_{k}\in\mathscr{B}({\cal H})\\}_{k\in\mathbb{N}_{K}^{+}}$
(and with the obvious definition of the Bochner integral for simple
functions),

$\displaystyle\int_{E}\boldsymbol{g}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})$
$\displaystyle=\int_{E}\sum_{k=1}^{K}\boldsymbol{z}^{k}\mathds{1}_{{\cal
G}_{k}}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})$
$\displaystyle\equiv\sum_{k=1}^{K}\boldsymbol{z}^{k}\mathrm{P}(E\,{\textstyle\bigcap}\,{\cal
G}_{k})=\begin{bmatrix}\sum_{k=1}^{K}z_{0,1}^{k}\mathrm{P}(E\bigcap{\cal
G}_{k})\\\ \vdots\\\ \sum_{k=1}^{K}z_{0,N}^{k}\mathrm{P}(E\bigcap{\cal
G}_{k})\\\ \sum_{k=1}^{K}z_{1,1}^{k}\mathrm{P}(E\bigcap{\cal G}_{k})\\\
\vdots\end{bmatrix}=\begin{bmatrix}\int_{E}g_{0,1}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\vdots\\\
\int_{E}g_{0,N}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\int_{E}g_{1,1}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\vdots\end{bmatrix}.$

In other words, the desired property that we would like to show for the
Bochner integral holds for all simple functions. Using this basic fact, again
by definition (Diestel and Uhl, 1977, Definition II.2.1), it holds that, for
each $E\in\mathscr{B}({\cal H})$,

$\displaystyle\int_{E}\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})$
$\displaystyle=\lim_{m\rightarrow\infty}\int_{E}\boldsymbol{g}^{m}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})=\lim_{m\rightarrow\infty}\begin{bmatrix}\int_{E}g_{0,1}^{m}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\int_{E}g_{0,2}^{m}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\vdots\\\
\int_{E}g_{0,N}^{m}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\int_{E}g_{1,1}^{m}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\\\
\vdots\end{bmatrix},$

where the limit is with respect to the natural norm of $\mathds{X}$.
Therefore, for every $(n,i)\in\mathbb{N}\times\mathbb{N}_{N}^{+}$, we have
that

$\displaystyle 0$
$\displaystyle=\lim_{m\rightarrow\infty}\bigg{\|}\int_{E}\boldsymbol{g}^{m}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})-\int_{E}\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\bigg{\|}_{\ell_{\infty}}$
$\displaystyle\geq\lim_{m\rightarrow\infty}\bigg{|}\int_{E}g_{n,i}^{m}(\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})-\bigg{[}\int_{E}\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\bigg{]}_{n,i}\bigg{|}$
$\displaystyle=\bigg{|}\int_{E}\lambda_{n}(\boldsymbol{h})f_{i}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})-\bigg{[}\int_{E}\boldsymbol{\Lambda}_{\mathds{B}}(\boldsymbol{h})\otimes\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{h}),\boldsymbol{h})\mathrm{d}\mathrm{P}(\boldsymbol{h})\bigg{]}_{n,i}\bigg{|}\geq
0.$

Enough said. $\blacksquare$

##  Appendix B Strong Duality of (RCP) via Convexity of
$\boldsymbol{\mathrm{cl}({\cal C})}$:
Alternative Proof

It is possible to establish strong duality of problem (RCP) by exploiting a
characterization of strong duality in general infinite dimensional cone-
constrained nonconvex programming by Flores-Bazán and Mastroeni (Flores-Bazán
and Mastroeni, 2013, Theorem 3.2). While the overall argument is indeed
elegant, we believe that it is not as transparent and significantly less
elementary in comparison with our discussion in Section 4.3. Nevertheless, we
find the development valuable.

In what follows, $\mathds{S}$ is a Hausdorff topological vector space,
$\mathds{Y}$ is a real Hausdorff locally convex topological vector space with
topological dual $\mathds{\mathds{Y}}^{*}$ and bilinear pairing
$\langle\cdot,\cdot\rangle_{\mathds{Y}^{(*)}}$,
$\mathds{P}\subseteq\mathds{Y}$ is a nonempty closed convex cone with possibly
empty (topological) interior, and $\mathds{C}$ is a nonempty subset of
$\mathds{S}$. Under this setting and given operators
$F:\mathds{\mathds{C}}\rightarrow\mathbb{R}$ and
$\boldsymbol{G}:\mathds{\mathds{C}}\rightarrow\mathds{Y}$, let us consider the
infinite dimensional cone-constrained program

$\begin{array}[]{rl}-\infty<\mathsf{F}^{*}\triangleq\>\underset{\boldsymbol{\tau}}{\mathrm{minimize}}&F(\boldsymbol{\tau})\\\
\mathrm{subject\,to}&\boldsymbol{G}(\boldsymbol{\tau})\in-\mathds{P}\\\
&\boldsymbol{\tau}\in\mathds{C}\end{array},$ (CCP)

whose feasible set is hereafter assumed to be nonempty; thus
$\mathsf{F}^{*}\in\mathbb{R}$. In direct analogy to the finite dimensional
setting, the Lagrangian dual associated to (CCP) is

$\sup_{\boldsymbol{\lambda}\in\mathds{P}^{*}}\inf_{\boldsymbol{\tau}\in\mathds{C}}F(\boldsymbol{\tau})+\langle\boldsymbol{\lambda},\boldsymbol{G}(\boldsymbol{\tau})\rangle_{\mathds{Y}^{(*)}},$

where $\mathds{P}^{*}$ is the positive polar cone of $\mathds{P}$. We also
define the image space set (Giannessi, 2005)

${\cal E}_{*}\triangleq\begin{bmatrix}F(\mathds{C})\\\
\boldsymbol{G}(\mathds{C})\end{bmatrix}-\begin{bmatrix}\mathsf{F}^{*}\\\ {\bf
0}\end{bmatrix}+\begin{bmatrix}\mathbb{R}_{+}\\\ \mathds{P}\end{bmatrix},$

where set addition is in the Minkowski sense. Under this general setting, the
following powerful result holds, providing necessary and sufficient conditions
for strong duality in infinite dimensions.

######  Theorem 4 ((Flores-Bazán and Mastroeni, 2013) Characterization of
Strong Duality).

Strong duality holds for (CCP) if and only if

$\mathrm{cl}[\mathrm{cone}(\mathrm{conv}({\cal
E}_{*}))]\bigcap\begin{bmatrix}-\mathbb{R}_{++}\\\ \\{{\bf
0}\\}\end{bmatrix}=\emptyset.$

It readily follows that the risk-constrained problem (RCP) is an instance of
(CCP) and can be expressed as

$\begin{array}[]{rl}-\mathsf{P}^{*}=\>\underset{\boldsymbol{x},\boldsymbol{p}(\cdot)}{\mathrm{minimize}}&-g^{o}(\boldsymbol{x})\\\
\mathrm{subject\,to}&\begin{bmatrix}\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))+\boldsymbol{x}\\\
-\boldsymbol{g}(\boldsymbol{x})\end{bmatrix}\in-\mathbb{R}_{+}^{N+N_{\boldsymbol{g}}}\\\
&(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\end{array},$ (RCP)

with the identifications $\mathds{S}=\mathbb{R}^{N}\times{\cal
L}_{1}(\mathrm{P},{\bf R})$ –assuming for simplicity that $\Pi\subseteq{\cal
L}_{1}(\mathrm{P},{\bf R})$–, $\mathds{Y}=\mathbb{R}^{N+N_{\boldsymbol{g}}}$,
$\mathds{P}=\mathbb{R}_{+}^{N+N_{\boldsymbol{g}}}$, $\mathds{C}={\cal
X}\times\Pi$, and of course $\mathsf{F}^{*}=-\mathsf{P}^{*}$. In this case,
the set ${\cal E}_{*}$ takes the particular form

$\displaystyle{\cal E}_{*}$
$\displaystyle=\left\\{(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\left|\begin{array}[]{ll}\delta_{o}&=-g^{o}(\boldsymbol{x})-(-\mathsf{P}^{*})+z_{o}\\\
\boldsymbol{\delta}_{r}&=\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))+\boldsymbol{x}+\boldsymbol{z}_{r}\\\
\boldsymbol{\delta}_{d}&=-\boldsymbol{g}(\boldsymbol{x})+\boldsymbol{z}_{d}\end{array},\;\text{for
some}\begin{array}[]{l}(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\\\
(z_{o},\boldsymbol{z}_{r},\boldsymbol{z}_{d})\in\mathbb{R}_{+}^{1+N+N_{\boldsymbol{g}}}\end{array}\right.\right\\}$
$\displaystyle=\left\\{(\delta_{o},\boldsymbol{\delta}_{r},\boldsymbol{\delta}_{d})\left|\begin{array}[]{l}-g^{o}(\boldsymbol{x})+\mathsf{P}^{*}\leq\delta_{o}\\\
\boldsymbol{\rho}(-\boldsymbol{f}(\boldsymbol{p}(\boldsymbol{H}),\boldsymbol{H}))+\boldsymbol{x}\leq\boldsymbol{\delta}_{r}\\\
-\boldsymbol{g}(\boldsymbol{x})\leq\boldsymbol{\delta}_{d}\end{array},\;\text{for
some }(\boldsymbol{x},\boldsymbol{p})\in{\cal X}\times\Pi\right.\right\\},$

or, equivalently,

${\cal E}_{*}=-\bigg{[}{\cal C}-\begin{bmatrix}\mathsf{P}^{*}\\\ {\bf
0}\end{bmatrix}\bigg{]}\triangleq-{\cal C}_{*}.$

It follows that ${\cal E}_{*}$ and our utility-constraint set ${\cal C}$ are
essentially equivalent. From Theorem 4 we can readily see that strong duality
of (RCP) will follow if we can show that

$\mathrm{cl}[\mathrm{cone}(\mathrm{conv}({\cal
C}_{*}))]\bigcap\begin{bmatrix}\mathbb{R}_{++}\\\ \\{{\bf
0}\\}\end{bmatrix}=\emptyset.$

Next we show that this is indeed the case as a consequence of the convexity of
$\mathrm{cl}({\cal C})$, and provided that Slater’s constraint qualification
holds. First, note that convexity of $\mathrm{cl}({\cal C})$ is equivalent to
convexity of $\mathrm{cl}({\cal C}_{*})$ (the latter set is just a translation
of the former set), and since

$\displaystyle\mathrm{conv}({\cal
C}_{*})\subseteq\mathrm{conv}(\mathrm{cl}({\cal C}_{*}))=\mathrm{cl}({\cal
C}_{*})$ $\displaystyle\subseteq\mathrm{cone}(\mathrm{cl}({\cal C}_{*}))$
$\displaystyle\implies\mathrm{cone}(\mathrm{conv}({\cal C}_{*}))$
$\displaystyle\subseteq\mathrm{cone}(\mathrm{cl}({\cal C}_{*})),$

it will suffice to show that

$\mathrm{cl}[\mathrm{cone}(\mathrm{cl}({\cal
C}_{*}))]\bigcap\begin{bmatrix}\mathbb{R}_{++}\\\ \\{{\bf
0}\\}\end{bmatrix}=\emptyset,$

because $\mathrm{cl}[\mathrm{cone}(\mathrm{cl}({\cal
C}_{*}))]\supseteq\mathrm{cl}[\mathrm{cone}(\mathrm{conv}({\cal C}_{*}))]$. In
other words, if suffices to show that

$\forall\varepsilon>0,\quad(\varepsilon,{\bf 0},{\bf
0})\notin\mathrm{cl}[\mathrm{cone}(\mathrm{cl}({\cal C}_{*}))].$

Under Slater’s condition, there exist a number $\eta^{S}>0$ and vectors
$\boldsymbol{\delta}_{r}^{S}>{\bf 0}$ and $\boldsymbol{\delta}_{d}^{S}>{\bf
0}$ such that

$(-\eta^{S},\boldsymbol{\delta}_{r}^{S},\boldsymbol{\delta}_{d}^{S})\in{\cal
C}_{*}\subseteq\mathrm{cl}({\cal
C}_{*})\subseteq\mathrm{cl}[\mathrm{cone}(\mathrm{cl}({\cal C}_{*}))].$

Now, suppose that there is an $\varepsilon>0$ such that $(\varepsilon,{\bf
0},{\bf 0})\in\mathrm{cl}[\mathrm{cone}(\mathrm{cl}({\cal C}_{*}))]$. Because
of convexity of the latter set, it holds that, for every $\alpha\in[0,1]$,

$\boldsymbol{z}_{\alpha}=(\alpha\varepsilon-(1-\alpha)\eta^{S},(1-\alpha)\boldsymbol{\delta}_{r}^{S},(1-\alpha)\boldsymbol{\delta}_{d}^{S})\in\mathrm{cl}[\mathrm{cone}(\mathrm{cl}({\cal
C}_{*}))],$

and we can choose $\alpha$ such that

$1>\alpha>\dfrac{\eta^{S}}{\varepsilon+\eta^{S}}\iff\alpha\varepsilon-(1-\alpha)\eta^{S}>0\quad\implies\quad\boldsymbol{z}_{\alpha}>{\bf
0}.$

Since $\boldsymbol{z}_{\alpha}$ is in the closure of
$\mathrm{cone}(\mathrm{cl}({\cal
C}_{*}))=\\{\boldsymbol{x}|\boldsymbol{x}=\beta\boldsymbol{y},\boldsymbol{y}\in\mathrm{cl}({\cal
C}_{*}),\beta\geq 0\\}$ (note that $\mathrm{cl}({\cal C}_{*})$ is convex),
there exists a sequence of points
$\\{\boldsymbol{z}_{\alpha}^{n}\\}_{n\in\mathbb{N}}$ entirely contained in
$\mathrm{cone}(\mathrm{cl}({\cal C}_{*}))$ which converges to
$\boldsymbol{z}_{\alpha}$; that is, each member of such a sequence must be of
the form

$\boldsymbol{z}_{\alpha}^{n}=\beta^{n}\boldsymbol{y}^{n},\quad\text{for some
}\text{$\boldsymbol{y}$}^{n}\in\mathrm{cl}({\cal C}_{*})\text{ and
}\beta^{n}\geq 0,\quad n\in\mathbb{N}.$

But because $\boldsymbol{z}_{\alpha}$ is strictly positive, such a sequence
must also be eventually strictly positive, which implies the existence of an
index $n_{o}$ such that

${\bf
0}<\boldsymbol{z}_{\alpha}^{n_{o}}\iff\boldsymbol{z}_{\alpha}^{n_{o}}=\beta^{n_{o}}\boldsymbol{y}^{n_{o}},\quad\text{for
some }{\bf 0}<\text{$\boldsymbol{y}$}^{n_{o}}\in\mathrm{cl}({\cal
C}_{*})\text{ and }\beta^{n_{o}}>0.$

This is impossible, because $\mathrm{cl}({\cal C}_{*})$ cannot contain
strictly positive points; if this could happen, then there would exist another
sequence entirely contained in ${\cal C}_{*}$ and converging to that strictly
positive point, implying the existence of at least one strictly positive point
in ${\cal C}_{*}$, which is absurd –this is essentially the same limiting
argument as that used right above. Consequently, the condition of Theorem 4 is
verified, confirming that problem (RCP) exhibits strong duality.
$\blacksquare$

As a final remark, the reader might notice the similarity of our arguments
above with those in the proof of Lemma 2; of course, this is not at all
coincidental.

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