\section{Introduction}

The abundance of most stable nuclei above iron in the universe can be
understood as produced by various types of neutron
capture \cite{mat85,bar06}. However, production of about 40 stable isotopes cannot be explained in this
way, but only through similar proton capture processes
\cite{bur57,arn03,rau13,rei14,pal14}.  The basic ignition fuel is a
large proton flux arising from a stellar explosion.  The sequence of
these reactions are then one proton capture after another until the
proton dripline is reached and further captured protons are
immediately emitted. This dripline nucleus usually must wait to beta-decay to a more stable nucleus which in turn can
capture protons anew.  This is the "rp-process" \cite{sch98,bro02}.  These p-nuclei are
also believed to be produced by other methods: gamma-proton \cite{arn03,woo78} and
neutrino-proton processes \cite{fro06,fro06b}.

The beta-decay waiting time is large for some of these nuclei along
the dripline, which for that reason are denoted waiting points
\cite{oin00}. However, another path is possible to follow for
borromean proton dripline nuclei where two protons, in contrast to
one, are necessary to produce a bound nucleus.  Then two protons can
be captured before beta-decay occurs \cite{gri05,gri01}.  The capture time and the
corresponding mechanism are therefore important for the description of
the outcome of these astrophysical processes \cite{gor95,sch06}.

We focus in this letter on one of these two-proton capture reactions
leading from a prominent waiting point nucleus, $^{68}$Se
\cite{sch07,tu11}, to formation of the borromean proton dripline
nucleus, $^{70}$Kr~($^{68}$Se$+p+p$).  The specific experimental
reaction information is not available, and theoretical estimates are,
at least at the moment, unavoidable \cite{tho04,erl12,pfu12}.

Traditionally, the reactions have been described as sequential
one-proton capture by tunneling through the combined Coulomb plus
centrifugal barrier. The tunneling capture mechanisms have been
discussed as direct, sequential and virtual sequential decay
\cite{jen10,gar05,che07,rod08}. They are all accounted for in the present formulation.
The capture rate depends on temperature through the assumed
Maxwell-Boltzmann energy distribution.  It is then important to know
the energy dependence of the capture cross section for given resonance
energies, and especially in the Gamow window \cite{chr15}.

Clearly the desired detailed description requires a
three-body model which is available and even applied to the present
processes \cite{hov16,gar04}.  However, the crucial proton-core potentials
have so-far been chosen phenomenologically to produce the essentially
unknown, but crucial, single-particle energies.  A new model
involving both core and valence degrees of freedom was recently
constructed to provide mean-field proton potentials derived from the
effective nucleon-nucleon mean-field interaction \cite{hov17,hov17b,hov17c}.  In turn
these potentials produce new and different effective three-body
potentials, which in the present letter is exploited to investigate
the two-proton capture rates.

The techniques are in place all the way from the solution of the
coupled core and valence proton system \cite{hov16,hov17b,hov17c}, over the
self-consistent three-body input and subsequent calculations
\cite{nie01,jen04}, to the capture cross sections and rates
\cite{gar15,die10,die11,die10b}.  We shall first sketch the steps in
the procedure used in the calculations. Then we shall discuss in more
details the numerical results of interest for the astrophysical
network computations, which calculates the abundances of the isotopes in the Universe.

\section{Theoretical description}

The basic formulation and the procedure are described in
\cite{hov16,gar15,gar15b}.  The framework is the three-body technique but based
on the proton-core potential derived through the self-consistently
solved coupled core-plus-valence-protons equations \cite{hov17,hov17b}.  The procedure is
first to select the three-body method, second to formulate how to
calculate the capture rate, and finally to choose the numerical
parameters to be used in the computations.

\subsection{Wave functions}

First, the many-body problem is solved for a mean-field treated core
interacting with two surrounding valence protons \cite{hov16}.  The
details of this recent model are very elaborate, but already applied
on two different neutron dripline nuclei \cite{hov16,hov17}.  It then
suffices to sketch the corresponding details for the present
application.  Briefly, the novel features are to find the mean-field
solution for the core-nucleons in the presence of the external field
from the two valence protons.  In turn, folding the basic
nucleon-nucleon interaction and the core wave function provides the
interaction between each valence proton and the core nucleus.  The
interactions between core and valence nucleons then depend on their
respective wave functions, which are found self-consistently by
iteration.  We emphasize that the same nucleon-nucleon interaction is
used both in the core and for this valence-proton core calculation.
The crucial main ingredient in the three-body solution is this
interaction, which then is provided by the procedure and determined
independent of subsequent applications.

The present application exploits the properties of the derived
three-body solution.  The three-body problem is solved by adiabatic
expansion of the Faddeev equations \cite{gar04} in hyperspherical coordinates. When necessary
the continuum is discretized by a large box confinement \cite{gar15a}.  The
main coordinate is the hyperradius, $\rho$, defined as the mean radial
coordinate in the three-body system \cite{gar15b,hov14}. More
specifically we have
\begin{align}
(2m_n+m_c) \rho^2
= m_n(\bm{r}_{v_1} - \bm{r}_{v_2})^2
+ m_{c} \sum_{i=1}^2 (\bm{r}_{v_i} - \bm{R}_{c})^2
\label{rhodef}
\end{align}
where $m_n$, $m_{c}$, $\bm{r}_{v_1}$, $\bm{r}_{v_2}$ and
$\bm{R}_{c}$ are neutron mass, core mass, valence-proton
coordinates and core center-of-mass coordinate, respectively.  The
three-body wave function is found through this procedure for given
angular momenta as functions of the hyperspherical coordinates for the
required ground state ($\Psi_{J}$).  When necessary the continuum is
discretized by a large box confinement and the discretized continuum
states ($\psi_{j}^{(i)}$) are calculated.

\subsection{Reactions}

The two-proton capture reaction $p+p+c \leftrightarrow A + \gamma$ cross section $\sigma_{ppc}$ and the photodissociation cross section $\sigma^{\lambda}_{\gamma}$ of order ${\lambda}$ are related \cite{gar15b}.
The three-body energy, $E$, and the ground state energy, $E_{gr}$,
determine the photon energy by $E_{\gamma} = E + |E_{gr}|$.  The
dissociation cross section is then given by
\begin{align}
\sigma^{\lambda}_{\gamma}(E_{\gamma})
=& \frac{(2 \pi)^3 (\lambda +1)}{\lambda ((2\lambda +1)!!)^2} \left( \frac{E_{\gamma}}{\hbar c}\right)^{2\lambda-1} \frac{d}{d E}\mathcal{B}({E}\lambda, 0 \rightarrow \lambda), \label{eq siggam}
\end{align}
where the strength function for the ${E}\lambda$ transition,
\begin{align}
 \frac{d}{d E} \mathcal{B}({E}\lambda, 0 \rightarrow \lambda) =
 \sum_i \left| \braket{\psi_{\lambda}^{(i)} | | \hat{\Theta}_{\lambda} | |
 \Psi_{0}} \right|^2  \delta(E-E_i), \label{eq tran}
\end{align}
is given by the reduced matrix elements,
$\braket{\psi_{\lambda}^{(i)} | | \hat{\Theta}_{\lambda} | | \Psi_{0}}$,
where $\hat{\Theta}_{\lambda}$ is the electric multipole operator,
$\psi_{\lambda}^{(i)}$ is the wave function of energy, $E_i$, for all
bound and (discretized) three-body continuum states in the summation.
The capture reaction rate, $R_{ppc}$, is given by Ref.~\cite{die11}
\begin{align}
 R_{ppc}(E) = &  \frac{8 \pi}{(\mu_{cp} \mu_{cp,p})^{3/2}}  \frac{\hbar^3}{c^2}
 \left( \frac{E_{\gamma}}{E} \right)^2 \sigma^{\lambda}_{\gamma}(E_{\gamma}),
\label{eq rate}
\end{align}
where $\mu_{cp}$ and $\mu_{cp,p}$ are reduced masses of proton and
core and proton-plus-core and proton, respectively.  For the astrophysical processes in a gas of temperature, $T$, we have to average the rate in Eq.~(\ref{eq rate}) over the
corresponding Maxwell-Boltzmann distribution, $B(E,T) = \frac{1}{2}
E^2 \exp(-E/T)/T^3$,
\begin{align}
\braket{R_{ppc}(E)} = \frac{1}{2T^3} \int E^2 R_{ppc}(E) \exp(-E/T)
 \, dE, \label{eq ave rate}
\end{align}
where the temperature is in units of energy.

\subsection{Interactions}

The decisive interaction is first of all related to the mean-field
calculation of the core.  We use the Skyrme interaction SLy4 \cite{cha98}  with acceptable global average properties. The
application on one specific nuclear system requires some adjustment to
provide the known borromean character, that is unbound
proton-core $f_{5/2}$ resonance at $0.6$~MeV \cite{san14} and two protons bound to the core.  With a minimum of changes we achieve this by shifting all energies while leaving the
established structure almost completely unaltered.  The simplest
consistent such modification is by scaling all the main Skyrme
strength parameters, $t_i$, by the same factor, $0.9515$.

The density dependence of the Skyrme interaction can be viewed as a
parametrized three-body potential.  To simulate that effect we employ
a short-range Gaussian, $S_0\exp(-\rho^2/b^2)$, which depends on the three-body hyperradial coordinate $\rho$.  We choose $b=6$~fm and leave
$S_0$ to fine-tune each of the $0^+$ and $2^+$ three-body energies. This
is necessary since the keV-scale of binding is crucial for tunneling
through single MeV height barriers.  This level of accuracy is beyond
the present capability of many-body model calculations.  To reproduce
the predicted $0^+$ energy of $-1.34$~MeV \cite{wan12} a three-body
strength $S_0 = -17.5$~MeV is needed. The unknown $2^+$ energy is
varied from almost bound, zero energy, to the top of the barrier by
$S_0$ changing from $-35.05$~MeV to $-26.22$~MeV.

\section{Effective three-body potentials \label{sec:3b}}

The elaborate numerical calculations produce the sets of coupled
``one-body'' effective potentials depending on hyperradius as shown in
Fig.~\ref{fig:potsw} for both $0^{+}$ and $2^{+}$.  The continuations
beyond the $20$~fm in the figure are almost quantitatively Coulomb
plus centrifugal behavior and as such reveal no surprises.  The kinks
and fast bends reflect avoided crossings and related structure
changes.  They are especially abundant at small distances and around
the barriers.  The diagonal non-adiabatic coupling terms as well as
the diagonal structure-less three-body Gaussian potentials are included in the calculations, but not in the figure.  They both leave the structure
essentially unaltered although they may change the energies
of the solutions rather substantially.

\begin{figure}
\centering
\includegraphics[width=1\linewidth]{CompEffPots.pdf}
\vspace*{-6mm}
\caption{The effective, adiabatic potentials for the $0^{+}$ (red,
  solid), and the $2^{+}$ (light-blue, dashed) configuration in
  $^{68}\text{Se}+p+p$ using the SLy4 Skyrme interaction between core
  and valence protons, scaled to reproduce the experimental $f_{5/2}$
  resonance energy of $0.6$~MeV in $^{68}\text{Se}+p$ \cite{san14}. The dotted horizontal line is the $0^+$ energy
  at $-1.34$~MeV from $S_0= -17.5$~MeV.
\label{fig:potsw}}
\end{figure}

The $0^{+}$ ground-state at $-1.34$~MeV, predicted from systematics
\cite{aud12}, is reproduced with the chosen parameters.  The structure
corresponds to the configuration of the pronounced minimum in the
lowest $0^{+}$ potential.  No  $0^{+}$ resonance are produced by the
potentials in Fig.~\ref{fig:potsw}.  The ground state is the final
state in the capture process independent of the specific mechanism.

However, the decisive capture process proceeds within the $2^{+}$
continuum from the large to the short-distance attractive region of
the potentials shown in Fig.~\ref{fig:potsw}.  This lowest minimum is
rather similar to the $0^{+}$ minimum but the non-adiabatic repulsive
terms increase the energy substantially. Unfortunately nothing is
known about a $2^{+}$ resonance which would strongly influence the
capture rate.  Consequently the strength, $S_0$, is used to vary the
position of the $2^{+}$ resonance from almost bound to disappearance
above the barriers.  Both the resonance energy, the height, and the
rather broad Coulomb shape of the barrier strongly influence the
capture process.

The structure of these potentials is substantially simpler than those
obtained in \cite{hov16} where low-lying single-proton states
$p_{3/2}$ and $f_{5/2}$ both appeared.  The present simplification is
an automatic result of the procedure using the nucleon-nucleon
mean-field effective interaction to calculate the proton-core
potential.  This is not an ad hoc assumption, but arises naturally due to
identical interactions for both core and valence particles.  As such
it is a novel deduction embedded in the design of our model.  The
lack of single-particle states of different parity implies that no
$1^-$ three-body resonance states appear in the low-energy region. The transition is then necessarily an $E2$ transition, which contributes to the longer effective lifetime of the system, and could very well be part of the reason this system is a critical waiting point.

\section{Quantitative results}

The all-important core-valence proton potential is derived naturally and unambiguously by our mean-field core treatment, as discussed in the previous sections. As a result the two-proton capture cross section follows directly, only depending on the three-body resonance level. This is discussed in the following section, after which the resulting temperature averaged reaction rates are presented. This is supplemented by a discussion of the reaction mechanism and its implication for the possible reactions.

\subsection{Cross section}

The incident flux of low-energy protons on the core nucleus may result
in capture.  The corresponding cross section is most easily obtained
from calculation of the inverse reaction, that is photodissociation of
the $0^+$ ground state, $\Psi_{0}$, of $^{70}$Kr. The discretized
continuum states, $\psi_{\lambda}^{(i)}$, are computed and the cross
section is obtained from Eqs.~(\ref{eq tran}) and (\ref{eq siggam}) with
$\lambda =2$.  The two-proton capture cross section of $^{68}$Se,
obtained from Eq.~(\ref{eq siggam}), is shown in Fig.~\ref{fig:cross} as
function of the three-body energy.

\begin{figure}
\centering
\includegraphics[width=1\linewidth]{cross_gam.pdf}
\vspace*{-6mm}
\caption{The electromagnetic $E2$ dissociation cross section,
  $\sigma^{(\lambda=2)}_{\gamma}(E_{\gamma})$, for the proces,
  $^{70}\text{Kr} + \gamma \rightarrow ^{68}\text{Se}+p+p$, as a
  function of photon energy.  The $0^+$ final state energy is
  $-1.34$~MeV and the $2^+$ resonance energies are $E=0.5, \, 1.0, \,
  2.0,$ and $4.0$~MeV, respectively.  The discretized continuum states
  are obtained using box sizes of $\rho_{max} = 150, 200$~fm.
\label{fig:cross}}
\end{figure}

The peaks in the capture cross section occur at experimentally unknown
resonance energies where the tunneling probability is large.  We
therefore vary the energy from $0.5$~MeV to $4.0$~MeV where the widths
of the peaks in the cross section increase with energy as the top of
the barrier is approached.  We emphasize that the crucial quantity is
the resonance energy. This can be tested by varying the number of
adiabatic potentials used in the calculation. This results in
somewhat different resonance energy which however can be
compensated for by use of the three-body potential, which in turn
recover the cross section in Fig.~\ref{fig:cross}.

These features are simply understood as enhanced spatial overlaps
between the $2^+$ continuum states in the resonance region and the
ground state wave function, expressed through Eq.~(\ref{eq tran}).  Beside the resonance contributions we
also find significant, although several orders of magnitude smaller,
background contributions, which incidentally is
independent of the size of the discretization box, as long as it is sufficiently large \cite{gar15a}.

\subsection{Capture rates}

The capture cross sections are the main ingredient in the calculation
of the two-proton absorption rate appropriate for the temperature
dependent astrophysical network computation.  The average rate in
Eq.~(\ref{eq ave rate}) are shown in Fig.~\ref{fig:rate} as function
of temperature.  The Boltzmann smearing factor produces very smooth
curves of the same qualitative behavior.  They are zero at zero
temperature and energy, because the barrier is infinitely thick.  All
rates then increase to a maximum at the Gamow peak where the best
compromise is reached between the decreasing temperature distribution
and the increasing tunneling probability.

\begin{figure}
\centering
\includegraphics[width=1\linewidth]{rate.pdf}
\vspace*{-6mm}
\caption{The reaction rate for the radiative capture process
  $^{68}\text{Se}+p+p \rightarrow ^{70}\text{Kr} + \gamma$, as
  function of temperature for the different $2^+$ resonance energies
  in Fig.~\ref{fig:cross}.  The black dashed curve is the background
  contribution.
\label{fig:rate}}
\end{figure}

The peak contribution moves to higher energy and becomes smoother with
increasing resonance energy. Above temperatures of a few GK the
average rate variation is moderate and the size roughly of order
$\simeq 6 \times 10^{-11}$~cm$^{6}[N_{A} mol]^{-2} s^{-1}$.  A low-lying
resonance energy corresponds to low-lying peak position of larger
height.  We emphasize that the background without resonance
contribution obviously is smaller but only by roughly a factor of two
as soon as the temperature exceeds about $4$~GK ($\sim0.34$~MeV).  In
other words, if temperatures are in the astrophysically interesting
range below about $1$~GK, the size variations are substantial, and
vice versa above a few GK the details from the microscopic origin are
smeared out.

The actual size of the rate may reveal deceivingly little variation at
the relatively high  temperatures. However, the barrier height
and width are all-decisive and both may easily be different for other
systems where the single-particle structure at the Fermi energy is
different and perhaps more complicated as studied in \cite{hov16}.
The relatively large $2^+$ background contribution might suggest
significant corresponding $0^+$ continuum contributions.  However, the
$0^+$ barriers in Fig.~\ref{fig:potsw} are larger and the $0^+
\rightarrow 0^+$ transition as well require processes involving atomic
electrons.  It is again worth emphasizing that a superficially more
complete calculation with for example many coupled potentials would
provide the same rates after adjusting to the same resonance energy.

\begin{figure}
\centering
\includegraphics[width=1\linewidth]{AngDist.pdf}
\vspace*{-6mm}
\caption{The probability of the three-body, $^{68}\text{Se}+p+p$, wave
  function for the lowest allowed potential, integrated over
  directional angles $\left( \sin^2 (\alpha) \cos^2 (\alpha) \int
  |\Phi_n(\alpha, \rho, \Omega_x, \Omega_y)|^2 d\Omega_x
  d\Omega_y\right)$, as a function of hyperradius, $\rho$, and
  hyperangle, $\alpha$, related to the Jacobi coordinate system where
  "x" is between core and proton.  \label{fig:angDist}}
\end{figure}

\subsection{Reaction mechanism}

The rate depends on the capture mechanism.  We are here only concerned
with three-body capture, but a dense environment would enhance
four-body capture processes as discussed in \cite{die10}.  The overall
three-body process is tunneling through a barrier of particles in a
temperature distribution of given density. Once inside the relatively
thick barrier they have essentially only the option of emitting
photons to reach the bound ground-state.  However, the first of this
two-step process can occur through different mechanisms, where the
most obvious possibility is to be captured in different angular
momentum states.  The conservation of angular momentum and parity
quantum numbers are crucial in connection with resonance positions. If
low-lying $1^-$ continuum states are allowed they would be preferred,
and vice versa if prohibited $2^+$ continuum states would be
preferred.  Low-lying resonances enhance the contributions
substantially.  This selection depends strongly on the nucleus under
investigation.

For a given angular momentum of the three-body continuum states, we
still may encounter several qualitatively different ways of absorbing
two protons from the continuum \cite{gar11}. These mechanisms were discussed in
\cite{jen10} for the inverse process of dissociation, that is direct,
sequential and virtual sequential decay.  They are all accounted for
in the present formulation.  In \cite{hov16} we concluded that the
direct process is most probable for very low three-body energy when
two-body subsystems are unbound.  If the energy is larger than stable
two-body substructures such intermediate vehicles enhance the rates
and the mechanism is sequential.

Even when it is energetically forbidden to populate two-body resonance
states it may be advantageous to exploit these structures virtually
while tunneling through an also energetically forbidden barrier.  This
is appropriately named the virtual sequential two-body mechanism.  It
may be appropriate to emphasize that a similar three-body virtual
mechanism is forbidden because the three-body energy is conserved in
contrast to the energy of any two-body subsystem.

The mechanism for the present capture process is revealed in
Fig.~\ref{fig:angDist} where the $2^+$ probability integrated over the directional
angles is shown for the lowest potential as function of hyperradius
and one of the Jacobi angles.  It is a strikingly simple structure for
hyperradii larger than about $15$~fm, which for these coordinates is
equivalent to one proton at that distance from the center of mass of
the combined proton-core system.  Since the Jacobi angle, $\alpha$, is
either close to zero or $\pi/2$, this simply means that one proton is
staying very close to the core for all these hyperrradii.  Eventually
also this proton has to move away from the core since no bound state
exist. But the process is sequential through this substructure which
can be determined to be the proton-core $f_{5/2}$ resonance.

The higher-lying configurations corresponding to the three following
potentials also show precisely the same $f_{5/2}$ structure. This is
explained by combining the compact proton-core $f_{5/2}$ resonance
with one non-interacting (apart from Coulomb and centrifugal) distant
proton in any angular momentum configuration consistent with a $2^+$ structure.
The angular momenta capable of combining with $f_{5/2}$ to produce $2^+$ are $p_{1/2}, \, p_{3/2}, \, f_{5/2}, \, f_{7/2},$ and $h_{9/2}$. This also implies that for temperatures much smaller
than the $f_{5/2}$ resonance energy it would be energetically
advantageous to start the capture process in a configuration
corresponding to direct three-body capture.  The change of structure,
around avoided level crossings, to two-body resonance configurations
would then greatly reduce the barrier and substantially enhance the
capture rate.

\section{Conclusion}

The new model that treats the core and the two valence particles
self-consistently and simultaneously is applied on the waiting point
nucleus ($^{68}$Se) for the astrophysical rp-process. This is done
essentially without any free parameters or phenomenological fitting,
which makes the results much less arbitrary than usual three-body
calculations. Adding two protons, but not one, produces a bound system,
$^{70}$Kr, which is then a borromean nucleus. A moderate overall
scaling of the Skyrme interaction SLy4 reproduces the scarcely known
properties of these dripline nuclei.  Other Skyrme interactions
provide very similar results.

We calculate the radiative two-proton capture rate as function of
temperature for different resonance energies. We investigate the
mechanism and find that sequential capture of one proton after the
other by far is dominating. The first available single-particle
resonance state, $f_{5/2}$, is the vehicle, whereas the other proton
can approach in continuum states of even higher angular momentum. In practice, after tunneling through the barrier into the $2^+$ resonance
state, in practice only $E2$ electric transition to the ground state
is allowed.  Background capture through non-resonance continuum states
also contributes significantly to the capture process.  The sequential
$2^+$ capture mechanism might for other nuclei be replaced by for
example the normally larger $1^-$ capture.

In conclusion, the two-proton capture rates at a waiting point at the
dripline are successfully calculated with a conceptually relatively
simple, but technically advanced, new model.  The same effective
nucleon-nucleon interaction is used for both the nuclear mean-field
and the proton-core calculations.  The temperature dependent rate and
the corresponding capture mechanism are calculated with less ambiguity
than in previous calculations.  A number of applications are now
feasible.

\section*{Acknowledgements}
This work was funded by the Danish Council for Independent Research DFF Natural Science and the DFF Sapere Aude program. This work has been partially supported by the Spanish Ministerio de Economia y Competitividad under Project FIS2014-51971-P.

\section*{References}


\section{Introduction}
\label{sec:intro}

The use of prox-functions in convex programming has become standard in recent decades. Originally, they were introduced in the context of mirror descent methods \cite{nemirovski:1983}.
In order to explain how prox-functions enter into optimization methods, we recall their definition.

\begin{definition}[Prox-function] We say that  $d:\mathbb{R}^n \rightarrow \mathbb{R}\cup\{\infty\}$ is a prox-function on a closed convex set $Q \subset \mathbb{R}^n$ if
\begin{itemize}
    \item[(1)] $d$ is continuous with the domain containing $Q$, i.\,e. $Q \subset \mbox{dom}\, d$.
    \item[(2)] $d$ is strongly convex on $Q$ with respect to a norm $\|\cdot\|$, i.\,e.
    there exists a constant $\beta >0$ such that for all $x, y \in Q$ and $\alpha \in [0,1]$ it holds:
\[
   d(\alpha x +(1-\alpha)y) \leq \alpha d(x) + (1-\alpha) d(y) - \frac{\beta}{2} \alpha (1-\alpha) \|x-y\|^2.
\]
    \item[(3)] The computation of the convex conjugate
    \begin{equation}
        \tag{A}\label{eq:pc}
         d^*(s) = \max_{x \in Q} \, \left\langle s, x \right\rangle - d(x)
         \end{equation}
    is simple, i.\,e. the unique maximizer $x(s)$ can be easily obtained for any $s \in \mathbb{R}^n$.
\end{itemize}
\end{definition}
Auxiliary optimization problem (\ref{eq:pc})
is known to be a key ingredient for minimizing a convex function $f$ on $Q$ by subgradient schemes. Let us take for $s$ subgradients of $f$ or their weighted aggregates. Then, (\ref{eq:pc}) defines a mapping from the dual space with subgradient information $s$ into the primal space with feasible iterates $x(s) \in Q$. This idea leads in particular to primal-dual subgradient methods for convex problems \cite{nesterov:2013}.

There are at least two advantages of using prox-funcions in (\ref{eq:pc}):
\begin{itemize}
    \item[(a)]  It is well known that the complexity bounds for optimization methods heavily depend on the size of the feasible set $Q$. This value has been traditionally defined with respect to Euclidean norm. However, the size of $Q$, measured with respect to another norm, can be smaller. Thus, by introducing prox-functions, which are strongly convex with respect to an appropriate norm $\|\cdot\|$, it is possible to take into account a particular geometry of the feasible set $Q$.
    \item[(b)] More interestingly, prox-functions often allow natural interpretations of the iteration steps (\ref{eq:pc}) within the convex optimization framework. This feature is important in order to explain agents' behavioral dynamics as being driven by unintentional optimization.
Let us illustrate this for the well known entropic prox-function
    \[
       d(p)= \sum_{i=1}^{n} p^{(i)} \ln p^{(i)}
    \]
 on the $(n-1)$-dimensional simplex
\[
   \Delta=\left\{p \in \mathbb{R}^n \,\left|\, \sum_{i=1}^{n} p^{(i)}  =1, p^{(i)} \geq 0\right.\right\}.
\]
For the feasible set $Q=\Delta$ the auxiliary optimization problem (A) reads:
\[
\max_{p \in \Delta} \, \sum_{i=1}^{n} s^{(i)} p^{(i)} - \sum_{i=1}^{n} p^{(i)} \ln p^{(i)}.
\]
Its unique solution is given by
\[
    p^{(i)}(s) = \frac{e^{s^{(i)}}}{\displaystyle \sum_{i=1}^{n} e^{s^{(i)}}}, \quad i=1,\ldots,n.
\]
This formula is in accordance with the logit model of discrete choice: $p^{(i)}(s)$ can be viewed as the choice probability of detecting $s^{(i)}$ to be maximal among $s^{(1)}, \ldots, s^{(n)}$.
\end{itemize}

In this paper we introduce prox-functions on the simplex  which allow similar probabilistic interpretation as above. They are derived from the additive random utility models of discrete choice \cite{depalma:1994}. Note that the logit model is just one prominent example within this class. Our main result in Section \ref{sec:prox} states that the convex conjugate of the surplus function, associated with an additive random utility model, is a prox-function on the simplex. For the convex conjugate of the corresponding surplus function we show continuity (Section \ref{ssec:co}), strong convexity (Section \ref{ssec:sc}), and simplicity (Section \ref{ssec:sim}). In particular, we explicitly derive the convexity parameter for the class of
generalized extreme value models \cite{mcfadden:1978}, and specifically of generalized nested logit models \cite{wen:2001}. Section \ref{sec:ea} is devoted to an economic application of discrete choice prox-functions in consumer theory. The discrete choice prox-functions are incorporated into the dual averaging scheme from \cite{nesterov:2013} for consumer's utility maximization.
 This ensures that the update of internal prices for goods' qualities is due to an additive random utility model.
The dual averaging scheme corresponds to a natural consumption cycle which successively leads to an optimal consumption of goods (Section \ref{sec:cyc}). We mention that the proposed consumption cycle generalizes \cite{nesterov:2016} where the update of internal prices is due to the logit model.

{\bf Notation.} Our notation is quite standard. We denote by $\mathbb{R}^n$ the space of $n$-dimensional column vectors $x =
\left(x^{(1)}, \dots , x^{(n)}\right)^T$, by $\mathbb{R}^n_+$ the set of all vectors with nonnegative components.
If the components of $x \in \mathbb{R}^n$ are nonnegative (positive), we write $x \geq 0$ ($x > 0$). For $x \in \mathbb{R}^n$ we write $x^{(-i)} \in \mathbb{R}^{n-1}$ meaning that the $i$-th component of $x$ is missing. Analogously, we write $x^{(-i,j)} \in \mathbb{R}^{n-2}$ meaning that both  $i$-th and $j$-th components of $x$ are missing.
For $x, y \in \mathbb{R}^n$ we introduce the standard scalar product and
(if additionally $y > 0$) the vector division:
\[
\langle x, y \rangle = \sum\limits_{i=1}^n x^{(i)} y^{(i)}, \quad
\frac{x}{y} = \left(\frac{x^{(1)}}{y^{(1)}}, \ldots, \frac{x^{(n)}}{y^{(n)}}\right)^T.
\]
For $x \in \mathbb{R}^n$ we use the following norms:
\[
\|x\|_1 = \sum_{i=1}^{n} \left| x^{(i)}\right|, \quad \|x\|_\infty= \max_{1 \leq i \leq n} \left| x^{(i)}\right|.
\]
Note that they are dual to each other, i.\,e.
\[
  \|x\|_\infty = \sup_{\left\|y \right\|_1 \leq 1} \langle y,x\rangle, \quad
  \|x\|_1 = \sup_{\left\|y \right\|_\infty \leq 1} \langle x,y\rangle.
\]
We denote by $e_j \in \mathbb{R}^n$ the $j$-th coordinate vector of $\mathbb{R}^n$. All components of the vector $e \in \mathbb{R}^n$ are equal to one.
The space of $(n \times n)$-matrices with real-valued entries is denoted by $\mathbb{R}^{n \times n}$. We use the induced matrix norm for $A \in \mathbb{R}^{n\times n}$:
\[
\left\| A\right\|_{\infty,1} = \max_{\|z\|_\infty \leq 1} \left\| A z\right\|_1.
\]
Given a twice differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}$, $\nabla f$ denotes the gradient and $\nabla^2 f$ stands for the Hessian matrix.

\section{Discrete choice prox-functions on the simplex}
\label{sec:prox}

We derive discrete choice prox-functions on the simplex from additive random utility models.

\subsection{Additive random utility models}
\label{sec:aru}

The additive random utility framework has been first introduced in economic context \cite{mcfadden:1978}. It aims to model the discrete choice from a finite number of alternatives $\{1, \ldots, n\}$ by a rational decision-maker prone to some random errors. Accordingly, the $i$-th alternative is endowed with the utility
\[
   u^{(i)} + \epsilon^{(i)},
\]
where $u^{(i)} \in \mathbb{R}$ is its deterministic part and $\epsilon^{(i)}$ is a random error. We denote by
\[
u = \left(u^{(1)}, \ldots, u^{(n)}\right)^T, \quad \epsilon = \left(\epsilon^{(1)}, \ldots, \epsilon^{(n)}\right)^T
\]
the vectors of deterministic utilities and of random utility shocks, respectively.
The following assumption on the stochastic errors is standard, see e.\,g. \cite{depalma:1994}.

\begin{assumption}
\label{ass:rnd}
The random vector $\epsilon$ follows a joint distribution with finite mean that is absolutely continuous with respect to the Lebesgue measure and fully supported on $\mathbb{R}^n$.
\end{assumption}
Since a rational decision-maker chooses alternatives with the maximal utility, the corresponding surplus is given by the expectation
\[
  E(u) = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)} \right).
\]
It is well-known that the surplus function $E$ is convex and differentiable \cite{depalma:1994}. In particular, its partial derivatives can be expressed as choice probabilities:
\begin{equation}
 \label{eq:der1}
  \frac{\partial E(u)}{\partial u^{(i)}} = \mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right), \quad i=1, \ldots, n.
\end{equation}
The latter means that the $i$-th partial derivative of $E$ corresponds to the probability of perceiving the $i$-th alternative as one with the maximal utility among the others. This result is known  as the Williams-Daly-Zachary theorem in the discrete choice literature \cite{mcfadden:1978, mcfadden:1981}. The formula (\ref{eq:der1}) is valid due to the fact that, under Assumption \ref{ass:rnd}, the ties between the alternatives occur with zero-probability, i.\,e.
\[
   \mathbb{P}\left(\epsilon^{(i)} - \epsilon^{(j)} = c \right) =0 \quad
   \mbox{for all } i \not = j \mbox{ and } c \in \mathbb{R}.
\]

\subsection{Convex conjugate of the surplus function}
\label{sec:cc}
We turn our attention to the convex conjugate $E^*:\mathbb{R}^n \rightarrow \mathbb{R}\cup\{\infty\}$ of the surplus function:
\[
   E^*(p) = \sup_{ u \in \mathbb{R}^n} \, \langle p, u \rangle - E(u),
\]
where $p=\left(p^{(1)}, \ldots, p^{(n)}\right)^T \in \mathbb{R}^n$ is the vector of dual variables.

\subsubsection{Continuity}
\label{ssec:co}
We discuss the continuity of the convex conjugate $E^*$ on its domain
\[
  \mbox{dom}\, E^* = \left\{ p \in \mathbb{R}^n \,|\, E^*(p) < \infty \right\}.
\]
For that, we need some elementary properties of the surplus function $E$ listed below.

\begin{lemma}[Elementary properties of $E$]
\label{lem:el}
For the surplus function $E$ it holds:
\begin{itemize}
    \item[(E1)] $E(u+\gamma e) = E(u) + \gamma$ for all $\gamma \in \mathbb{R}, u \in \mathbb{R}^n$.
    \item[(E2)] $E(u) \geq E(v)$ for all $u, v \in \mathbb{R}^n$ with $u \geq v$.
    \item[(E3)] $E(u) \geq \displaystyle \max_{1 \leq i \leq n} u^{(i)} + \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right)$ for all $u \in \mathbb{R}^n$.
\end{itemize}
\end{lemma}
\bf Proof: \rm \par \noindent
\begin{itemize}
    \item[(E1)] The linearity of the expectation provides:
    \[
    E(u+\gamma e) = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} \left(u^{(i)} + \gamma + \epsilon^{(i)}\right) \right) =
    \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} \left( u^{(i)} + \epsilon^{(i)} \right) + \gamma \right)=
    E(u) + \gamma.
    \]
    \item[(E2)] The monotonicity of the expectation provides:
    \[
     E(u) =  \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)} \right) \geq
      \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} v^{(i)} + \epsilon^{(i)} \right) = E(v).
     \]
    \item[(E3)] Due to the finite mean condition from Assumption \ref{ass:rnd}, we have for every $i \in \{1, \ldots, n\}$:
    \[
    E(u) = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)} \right) \geq
    \mathbb{E}_\epsilon \left(u^{(i)} + \epsilon^{(i)} \right)  \geq u^{(i)} + \min_{1 \leq i \leq n}\mathbb{E}_\epsilon \left(\epsilon^{(i)} \right).
    \]
\end{itemize}
\hfill $\Box$ \par \noindent \medskip

\begin{theorem}[Continuity of $E^*$]
\label{th:ct}
   The convex conjugate $E^*$ is continuous on its domain $\mbox{dom}\, E^*$ which coincides with the simplex $\Delta$.
\end{theorem}

\bf Proof: \rm \par \noindent Let us first show that $\mbox{dom}\, E^* \subseteq \Delta$. For $p \in \mathbb{R}^n$ with $\langle p,e\rangle \not =1$ we have:
\[
  E^*(p) \geq \sup_{ \gamma \in \mathbb{R}}\, \langle p,v + \gamma e \rangle - E(v + \gamma e) \overset{\mbox{(E1)}}{=}  \langle p,v \rangle - E (v) + \sup_{ \gamma \in \mathbb{R}} \, \gamma (\langle p,e \rangle - 1) = \infty,
\]
where $v \in \mathbb{R}^n$ is fixed. For $p \in \mathbb{R}^n$ with $p^{(i)} < 0$ for an $i \in \{1, \ldots, n\}$ we have:
\[
 E^*(p) \geq \sup_{ \gamma \leq 0} \, \langle p, \gamma e_i \rangle - E\left(\gamma e_i\right) \overset{\mbox{(E2)}}{\geq}
 \sup_{ \gamma \leq 0} \, \gamma p^{(i)} - E\left(0\right) = \infty,
\]
where $e_i$ denotes the $i$-th coordinate vector.
Secondly, we prove that $\mbox{dom}\, E^* \supseteq \Delta$. For that, it is sufficient to show that $E^*$ is bounded from above on $\Delta$. Due to (E3) from Lemma \ref{lem:el}, it holds:
\begin{equation}
\label{eq:ub}
\begin{array}{rcl}
\displaystyle \sup_{p \in \Delta} \, E^*(p) &=& \displaystyle \sup_{p \in \Delta} \left(\sup_{ u \in \mathbb{R}^n} \, \langle p, u \rangle - E(u) \right) =
  \sup_{ u \in \mathbb{R}^n} \left( \sup_{p \in \Delta} \, \langle p, u \rangle - E(u) \right) \\ \\
&=& \displaystyle
  \sup_{ u \in \mathbb{R}^n} \left(\max_{1 \leq i \leq n} u^{(i)} - E(u)\right) \leq - \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right).
\end{array}
\end{equation}
Further, we discuss the continuity of $E^*$ on the simplex $\Delta$.
Since $E^*$ is convex, it is continuous on the relative interior $\mbox{rint}(\Delta)$ of its domain. The continuity of $E^*$ on the whole domain $\Delta$ can be deduced by an application of the Gale-Klee-Rockafellar theorem.
The Gale-Klee-Rockafellar theorem says that a convex function is upper semi-continuous at every point at
which its domain is polyhedral \cite{gkr:1968}. Note that the domain of $E^*$ -- the $(n-1)$-dimensional simplex $\Delta$ -- is polyhedral. Moreover, the convex conjugate of a function is always lower semi-continuous, so is $E^*$ on $\Delta$. Together, the lower and upper semi-continuity of $E^*$ on $\Delta$ provides the claim. \hfill $\Box$ \par \noindent \medskip

Theorem \ref{th:ct} says that the convex conjugate $E^*$ is finite on the simplex $\Delta$.
The latter can be viewed as the set of probability distributions. Hence, the dual variables $p$ can be interpreted as the probabilities attached to the alternatives $\{1, \ldots, n\}$.

\begin{corollary}[Upper bound for $E^*$]
\label{cor:ub}
The convex conjugate $E^*$ is bounded from above on its domain $\Delta$, namely it holds:
\[
   E^*(p) \leq - \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right) \quad \mbox{for all } p \in \Delta.
\]
\end{corollary}

\bf Proof: \rm \par \noindent The assertion follows from the derivation in (\ref{eq:ub}). \hfill $\Box$ \par \noindent \medskip

\subsubsection{Strong Convexity}
\label{ssec:sc}

We show that the convex conjugate $E^*$ is strongly convex under suitable assumptions, and estimate its convexity parameter.

\begin{definition}[Strong convexity of $E^*$] The convex conjugate $E^*: \Delta \rightarrow \mathbb{R}$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm if for all $p, q \in \Delta$ and $\alpha \in [0,1]$ we have:
\[
  E^*(\alpha p +(1-\alpha)q) \leq \alpha E^*(p) + (1-\alpha) E^*(q) - \frac{\beta}{2} \alpha (1-\alpha) \|p-q\|_1^2.
\]
The positive constant $\beta$ is called the convexity parameter of $E^*$.
\end{definition}

The strong convexity of $E^*$ is closely related to the strong smoothness of $E$.

\begin{definition}[Strong smoothness of $E$] The surplus function $E: \mathbb{R}^n \rightarrow \mathbb{R}$ is $L$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$ if for all $u, v \in \mathbb{R}^n$ we have:
\[
  E(u+v) \leq E(u) + \langle \nabla E(u), v \rangle + \frac{L}{2} \|v\|_\infty^2.
\]
The positive constant $L$ is called the smoothness parameter of $E$.
\end{definition}

The following duality result between the strong convexity of $E^*$ and the strong smoothness of $E$ can be easily deduced from \cite[Theorem 6]{kak:2009} shown there in the general setting.

\begin{lemma}[Strong convex/smooth duality]
\label{lem:sd}
The convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm if and only if
the surplus function $E$ is $\frac{1}{\beta}$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$.
\end{lemma}

\bf Proof: \rm \par \noindent We apply \cite[Theorem 6]{kak:2009} which says that a closed and convex function is $\beta$-strongly convex with respect to a norm if and only if its convex conjugate is $\frac{1}{\beta}$-strongly smooth with respect to the dual norm. For that, we note that $E^*$ is proper and lower semi-continuous, hence, closed. Moreover, by the Fenchel-Moreau theorem we have
\[
   E^{**} = E,
\]
since $E$ is, in particular, a proper, lower semi-continuous, and convex function. Finally, the dual of the $\|\cdot\|_1$ norm is the maximum norm $\|\cdot\|_\infty$.
\hfill $\Box$ \par \noindent \medskip

In view of Lemma \ref{lem:sd}, we may focus on the strong smoothness of $E$. For the characterization of the latter property, we use the fact that the surplus function $E$ is twice differentiable. Let us compute the second order partial derivatives of $E$. Recall that its $i$-th partial derivative can be written as the choice probability
\[
 \begin{array}{rcl}
        \displaystyle \frac{\partial E(u)}{\partial u^{(i)}} &=& \displaystyle \mathbb{P} \left(\epsilon^{(-i)} - \epsilon^{(i)} \leq u^{(i)} -u^{(-i)} \right) \\ \\ &=& \displaystyle \int_{-\infty}^{u^{(i)} - u^{(-i)}} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i)} +x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i)}, \end{array}
\]
where $f_\epsilon$ is the probability density function of the random utility shocks $\epsilon$. Here, the inner integral is the probability density function of the $(n-1)$-dimensional vector of random differences $\epsilon^{(-i)}-\epsilon^{(i)}$ \cite{depalma:1994}. By differentiating this formula with respect to $u^{(j)}$ for $j\not =i$, we obtain mixed partial derivatives of $E$:
\[
   \frac{\partial^2 E(u)}{\partial u^{(i)} \partial u^{(j)}} = -
   \int_{-\infty}^{u^{(i)} - u^{(-i,j)}} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, u^{(i)}-u^{(j)}+x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)}.
\]
This integral can be interpreted as the probability density that $\epsilon^{(j)} - \epsilon^{(i)}=u^{(i)}-u^{(j)}$, and $\epsilon_{-i,j} - \epsilon^{(i)} \leq u^{(i)}-u^{(-i,j)}$, i.\,e. both alternatives $i$ and $j$ yield the maximal utility.
Analogously, we obtain the second order partial derivative of $E$ with respect to $u^{(i)}$:
\[
   \frac{\partial^2 E(u)}{\partial u^{{(i)}2}} = \sum_{j \not =i}
\int_{-\infty}^{u^{(i)} - u^{(-i,j)}} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, u^{(i)}-u^{(j)}+x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)}.
\]

\begin{lemma}[$C^2$-characterization of strong smoothness of $E$]
\label{lem:c2} The surplus function $E$ is $L$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$ if for all $u \in \mathbb{R}^n$ it holds:
\[
  \left\| \nabla^2 E(u)\right\|_{\infty,1} \leq L.
\]

\bf Proof: \rm \par \noindent For any $u, v \in \mathbb{R}^n$ we have:
\[
\nabla E(u) - \nabla E(v) = \int_0^1 \diff \nabla E(v + \tau (u-v)) =
\int_0^1 \nabla^2 E(v + \tau (u-v)) \cdot (u-v) \diff{\tau}.
\]
Hence, the gradient of $E$ is Lipschitz continuous:
 \[
 \begin{array}{rcl}
        \left\|\nabla E(u) - \nabla E(v)\right\|_1 &\leq& \displaystyle
   \int_0^1 \left\| \nabla^2 E(v + \tau (u-v))\cdot(u-v) \right\|_1 \diff{\tau}  \\ \\
   &\leq& \displaystyle
   \int_0^1 \left\| \nabla^2 E(v + \tau (u-v))\right\|_{\infty,1} \cdot \left\|u-v\right\|_\infty \diff{\tau} \leq   L \cdot \left\|u-v\right\|_\infty.
\end{array}
    \]
Further, we have:
\[
  E(u+v) - E(u) = \int_0^1 \diff E(u+\tau v) =
  \int_0^1 \langle \nabla E(u+\tau v), v\rangle \diff \tau.
\]
Due to the Lipschitz continuity of $\nabla E$, we obtain:
\[
\begin{array}{rcl}
  E(u+v) - E(u) - \langle \nabla E(u), v\rangle &=& \displaystyle \int_0^1 \langle \nabla E(u+\tau v) - \nabla E(u), v\rangle \diff \tau \\ \\
   &\leq& \displaystyle \int_0^1 \left\| \nabla E(u+\tau v) - \nabla E(u) \right\|_1 \cdot \left\|v\right\|_\infty \diff \tau \\ \\
   &\leq& \displaystyle
   \int_0^1 L \cdot \left\| u+\tau v - u \right\|_\infty \cdot \left\|v\right\|_\infty \diff \tau = \frac{L}{2} \left\|v\right\|^2_\infty.
  \end{array}
\]
\hfill $\Box$ \par \noindent \medskip
\end{lemma}

Now, let us consider the set $\mathcal{A}$ of symmetric matrices $A=\left(a_{ij}\right) \in \mathbb{R}^{n\times n}$ satisfying:
\begin{itemize}
    \item[(A1)] $\displaystyle a_{ii} \geq 0$ for all $i = 1, \ldots, n$, and $a_{ij} \leq 0$ for all $i \not = j$,
    \item[(A2)] $\displaystyle a_{ii} + \sum_{j\not = i}^{} a_{ij} = 0$ for all $i = 1, \ldots, n$.
\end{itemize}
Note that the set $\mathcal{A}$ is closed under matrix addition and multiplication by nonnengative scalars, i\,e. for any $A,B \in \mathcal{A}$ and $\lambda \geq 0$ it holds:
\[
   A+B, \lambda A \in \mathcal{A}.
\]
Moreover, the Hessian matrix $\nabla^2 E(u)$ of the surplus function is an element of $\mathcal{A}$.

\begin{lemma}[Representation of $\|\cdot\|_{\infty,1}$]
\label{lem:nr}
For $A \in \mathcal{A}$ it holds:
\begin{equation}
    \|A\|_{\infty,1} = 4 \max \, \left\{ \left.\sum_{i,j \in K} a_{ij} \,\right|\, K \subset \{1, \ldots, n\}, |K| \leq \left\lfloor \frac{n}{2} \right\rfloor  \right\}.
\end{equation}

\bf Proof: \rm \par \noindent The maximum in the definition of the matrix norm
\[
  \|A\|_{\infty,1} = \max_{\|z\|_\infty \leq 1} \left\| A z\right\|_1
\]
is attained at some vertex of the feasible set $\left\{z \in \mathbb{R}^n \,\left|\, \|z\|_\infty \leq 1 \right. \right\}$. These are the vectors
\[
   z_K = e_K - e_{K^c},
\]
where $e_K \in \mathbb{R}^n$ is the indicator vector of the subset $K\subset \{1, \ldots, n\}$, i.\,e. $e_K^{(i)}=1$ if $i \in K$, and $e_K^{(i)}=0$ if $i \not \in K$.
We may restrict the choice of $K$ by the condition $|K| \leq \left\lfloor \frac{n}{2} \right\rfloor$, since
\[
  \left\| A z_{K^c}\right\|_1 =
  \left\| A \left(-z_{K}\right)\right\|_1 =
\left\| A z_{K}\right\|_1.
\]
For such a fixed subset $K$ we compute
\[
\begin{array}{rcl}
 \displaystyle \left\| A z_{K}\right\|_1 &=& \displaystyle \sum_{i \in K} \left| a_{ii} + \sum_{j \in K\backslash \{i\}} a_{ij} - \sum_{j \in K^c} a_{ij} \right|+  \sum_{i \in K^c} \left| \sum_{j \in K} a_{ij} - a_{ii} - \sum_{j \in K^c\backslash \{i\}} a_{ij} \right| \\ \\
   &\overset{\mbox{(A2)}}{=}& \displaystyle 2 \sum_{i \in K} \left| a_{ii} + \sum_{j \in K\backslash \{i\}} a_{ij} \right|+ 2 \sum_{i \in K^c} \left| \sum_{j \in K} a_{ij} \right| \\ \\
   &\overset{\mbox{(A1)}}{=}& \displaystyle 2 \sum_{i \in K} \left(a_{ii} + \sum_{j \in K\backslash \{i\}} a_{ij} \right)- 2 \sum_{i \in K^c} \sum_{j \in K} a_{ij} \\ \\
   &=& \displaystyle
   2 \sum_{j \in K} \left(a_{jj} + \sum_{i \in K\backslash \{j\}} a_{ij} - \sum_{i \in K^c}  a_{ij} \right) \\ \\
   &\overset{\mbox{(A2)}}{=}& \displaystyle
   2 \sum_{j \in K} \left(2 a_{jj} + 2 \sum_{i \in K\backslash \{j\}} a_{ij} \right) =

   4 \sum_{i,j \in K} a_{ij}.
  \end{array}
\]
\hfill $\Box$ \par \noindent \medskip
\end{lemma}

We use Lemma \ref{lem:nr} to estimate the $\|\cdot\|_{\infty,1}$ norm on a particular subset of $A$, which will appear in what follows.

\begin{corollary}
\label{cor:pp}
For $p\in \Delta$ we define the matrix $R={\rm diag \,}{p} - p\cdot p^T$. It holds for the latter:
\[
R \in \mathcal{A}, \quad \|R\|_{\infty,1} \leq 1.
\]
\bf Proof: \rm \par \noindent The symmetric matrix $R=(r_{ij})$ fulfills (A1), since in view of $p \in \Delta$ it holds for all $i = 1, \ldots, n$, and $i \not = j$:
\[
   r_{ii}=p^{(i)}\left(1-p^{(i)}\right) \geq 0, \quad r_{ij}=-p^{(i)}p^{(j)} \leq 0.
\]
It also fulfills (A2) due to the following derivation:
\[
   R \cdot e = {\rm diag \,}{p}\cdot e - p\cdot p^T \cdot e = p - p= 0.
\]
Let us fix a subset of indices $K \subset \left\{1, \ldots,n \right\}$.
We estimate the following expression from Lemma \ref{lem:nr} uniformly for all $p \in \Delta$:
\[
    \sum_{i,j \in K} r_{ij} = \sum_{i \in K} p^{(i)} \left( 1- \sum_{j \in K} p^{(j)} \right).
\]
For that, let us solve the maximization problem
\[
    \max \, \sum_{i \in K} p^{(i)} \left( 1- \sum_{i \in K} p^{(i)} \right) \quad \mbox{s. t.} \quad \sum_{i \in K} p^{(i)} \leq 1, \quad p^{(i)} \geq 0 \mbox{ for all } i \in K.
\]
Without loss of generality, we may assume that for its solution holds:
\[
  \sum_{i \in K} p^{(i)} < 1, \quad p^{(i)} > 0 \mbox{ for all } i \in K.
\]
Otherwise, the optimal value vanishes or we pass over to a smaller subset of indices. The first order optimality condition reads:
\[
  1- \sum_{i \in K} p^{(i)} - \sum_{i \in K} p^{(i)} =0.
\]
Hence, we get $\displaystyle \sum_{i \in K} p^{(i)} = \frac{1}{2}$,
and the optimal value is
\[
  \sum_{i \in K} p^{(i)} \left( 1- \sum_{i \in K} p^{(i)} \right) =
  \sum_{i \in K} p^{(i)} \left( 1- \frac{1}{2} \right) = \frac{1}{4}.
\]
The application of Lemma \ref{lem:nr} provides the assertion. \hfill $\Box$ \par \noindent \medskip
\end{corollary}

\begin{lemma}[Estimation of $\|\cdot\|_{\infty,1}$]
\label{lem:tr}
For $A \in \mathcal{A}$ it holds:
\[
  \|A\|_{\infty,1} \leq 2 \tr{A},
\]
where $\tr{A}$ denotes the trace of the matrix $A$.

\bf Proof: \rm \par \noindent For any $z\in \mathbb{R}^n$ it holds:
\[
  \left\| A z\right\|_1 = \sum_{i=1}^{n} \left| \sum_{j=1}^{n} a_{ij} z^{(j)} \right|
\leq \sum_{i=1}^{n} \sum_{j=1}^{n} \left|a_{ij}\right| \cdot \left|z^{(j)}\right|
\leq \sum_{i=1}^{n} \sum_{j=1}^{n} \left|a_{ij}\right| \cdot \left\|z\right\|_\infty.
\]
Additionally, we have:
\[
 \sum_{i=1}^{n} \sum_{j=1}^{n} \left|a_{ij}\right|
 \overset{\mbox{(A1)}}{=} \sum_{i=1}^{n} \left( a_{ii} - \sum_{j\not = i}^{} a_{ij} \right) \overset{\mbox{(A2)}}{=} \sum_{i=1}^{n} \left( a_{ii} + a_{ii} \right) = 2 \tr{A}.
\]
Overall, we obtain the inequality:
\[
  \|A\|_{\infty,1} = \max_{\|z\|_\infty \leq 1} \left\| A z\right\|_1 \leq
  \max_{\|z\|_\infty \leq 1} 2 \tr{A} \left\|z\right\|_\infty = 2 \tr{A}.
\]
\hfill $\Box$ \par \noindent \medskip
\end{lemma}

Now, we are ready to state the general result on the strong convexity of $E^*$. It is given in terms of the differences $\epsilon^{(j)} - \epsilon^{(i)}$, $i \not = j$, of random utility shocks. We recall that the density function of $\epsilon^{(j)} - \epsilon^{(i)}$ can be written as
\[
  g_{i,j} \left(z\right)= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, z +x^{(i)},x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)}.
\]
Any point $\bar z_{i,j} \in \mathbb{R}$ which maximizes the density function $g_{i,j}$ is called a mode of the random variable $\epsilon^{(j)} - \epsilon^{(i)}$.

\begin{theorem}[Strong convexity of $E^*$]
\label{th:sc}
   Let the differences $\epsilon^{(j)} - \epsilon^{(i)}$  of random utility shocks have modes $\bar z_{i,j} \in \mathbb{R}$, $i \not = j$.
   Then, the corresponding convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm, where the convexity parameter is given by
\[
  \beta = \frac{1}{\displaystyle 2\sum_{i=1}^{n} \sum_{j\not =i} g_{i,j} \left(\bar z_{i,j}\right)}.
\]
\bf Proof: \rm \par \noindent
We estimate the second order derivative of the surplus function $E$ with respect to $u^{(i)}$:
\[
\begin{array}{rcl}
 \displaystyle \frac{\partial^2 E(u)}{\partial u^{{(i)}2}} &=& \displaystyle \sum_{j \not =i} \int_{-\infty}^{u^{(i)} - u^{(-i,j)}} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, u^{(i)}-u^{(j)}+x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)}\\ \\
   &\leq& \displaystyle \sum_{j \not =i} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, u^{(i)}-u^{(j)}+x^{(i)}, x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)} \\ \\
   &=& \displaystyle \sum_{j \not =i}  g_{i,j} \left(u^{(i)}-u^{(j)}\right) \leq
   \sum_{j \not =i}  g_{i,j} \left(\bar z_{i,j}\right).
  \end{array}
\]
Hence, we obtain for the trace of $\nabla^2 E$ the following inequality:
\[
  \tr{\nabla^2 E(u)} = \sum_{i=1}^{n} \frac{\partial^2 E(u)}{\partial u^{{(i)}2}} \leq
  \sum_{i=1}^{n} \sum_{j\not =i} g_{i,j} \left(\bar z_{i,j}\right).
\]
In view of Lemma \ref{lem:tr}, whose application is justified by the fact that the matrix $\nabla^2 E(u)$ fulfills (A1)--(A2), it holds:
\[
  \left\|\nabla^2 E(u)\right\|_{\infty,1} \leq 2 \tr{\nabla^2 E(u)} \leq 2 \sum_{i=1}^{n} \sum_{j\not =i} g_{i,j} \left(\bar z_{i,j}\right) = \frac{1}{\beta}.
\]
The latter provides, due to Lemma \ref{lem:c2}, that the surplus function $E$ is $\frac{1}{\beta}$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$. Finally, we apply Lemma \ref{lem:sd} to conclude that the convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm.\hfill $\Box$ \par \noindent \medskip
\end{theorem}

\begin{remark}[Existence of modes] \textup{We note that the condition on the existence of modes in Theorem \ref{th:sc} cannot be weakened. This can be seen already in case of two alternatives, i.\,e. $n=2$. Then, the second order derivative of the surplus function $E$ is
\[
\nabla^2 E(u) = \left( \begin{array}{rr}
    g_{1,2}\left(u^{(1)}-u^{(2)}\right) & -g_{1,2}\left(u^{(1)}-u^{(2)}\right) \\
    -g_{2,1}\left(u^{(2)}-u^{(1)}\right) & g_{2,1}\left(u^{(2)}-u^{(1)}\right)
\end{array}\right).
\]
After a moment of reflection we realize that
\[
  g_{1,2}\left(u^{(1)}-u^{(2)}\right) = g_{2,1}\left(u^{(2)}-u^{(1)}\right).
\]
Due to Lemma \ref{lem:nr}, it holds:
\[
  \left\|\nabla^2 E(u)\right\|_{\infty,1} = 4 g_{1,2}\left(u^{(1)}-u^{(2)}\right).
\]
From now on we assume that $g_{1,2}$ is continuous.
Hence, the reverse implication in Lemma \ref{lem:c2} becomes valid, and $E$ is strongly smooth -- or, equivalently, $E^*$ is strongly convex -- if and only if the density function $g_{1,2}$ is bounded on $\mathbb{R}$. The latter property can be characterized by the existence of a mode $\bar z_{1,2}$ of $\epsilon^{(2)}-\epsilon^{(1)}$. Additionally, the convexity parameter of $E^*$ from Theorem \ref{th:sc} can be expressed as
\[
  \beta = \frac{1}{\displaystyle 2\sum_{i=1}^{n} \sum_{j\not =i} g_{i,j} \left(\bar z_{i,j}\right)} = \frac{1}{\displaystyle 4  g_{1,2}\left(\bar z_{1,2}\right)} = \frac{1}{\left\|\nabla^2 E\left(\bar u\right)\right\|_{\infty,1}},
\]
where the utilities $\bar u = \left(\bar u^{(1)},\bar u^{(2)} \right)^T$ are chosen to satisfy $\bar u^{(1)}-\bar u^{(2)}=\bar z_{1,2}$. This formula provides that the convexity parameter of $E^*$ cannot be larger than $\beta$.}\hfill $\Box$ \par \noindent \medskip
\end{remark}

The estimation of the convexity parameter of $E^*$ in Theorem \ref{th:sc} is rather pessimistic when the number of alternatives increases. The dependence of $\beta$ on $n$ becomes explicit in case of independent and identically distributed random utility shocks.

\begin{corollary}[Strong convexity of $E^*$ for IID utility shocks]
\label{cor:iid}
   Let the random utility shocks $\epsilon^{(i)}$, $i=1,\ldots,n$, be independent and identically distributed with the common probability density function $f$ having a mode $\bar z \in \mathbb{R}$.
   Then, the corresponding convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm, where the convexity parameter is given by
\[
  \beta = \frac{1}{\displaystyle 2n(n-1) f\left(\bar z\right)}.
\]
\bf Proof: \rm \par \noindent
We estimate the probability distribution function of $\epsilon^{(j)}-\epsilon^{(i)}$:
\[
\begin{array}{rcl}
 \displaystyle g_{i,j} \left(z\right) &=&
  \displaystyle \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_\epsilon \left(y^{(-i,j)} +x^{(i)}, z +x^{(i)},x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)} \\ \\
  &=&
  \displaystyle
  \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \prod_{k \not = i,j} f\left(y_k +x^{(i)}\right)  f\left(z+x^{(i)}\right) f\left(x^{(i)}\right) \diff x^{(i)} \diff y^{(-i,j)} \\ \\
  &=&
  \displaystyle
  \int_{-\infty}^{\infty} f\left(z+x^{(i)}\right) f\left(x^{(i)}\right) \prod_{k \not = i,j}\int_{-\infty}^{\infty}  f\left(y_k +x^{(i)}\right) \diff y_{k}   \diff x^{(i)}.
  \end{array}
\]
For all $k\not =i,j$, and $x^{(i)} \in \mathbb{R}$ we have:
\[
  \int_{-\infty}^{\infty}  f\left(y_k +x^{(i)}\right) \diff y_{k} = 1.
\]
Moreover, $f\left(z+x^{(i)}\right) \leq f\left(\bar z \right)$ for all $x^{(i)} \in \mathbb{R}$. Altogether, we get
\[
 \displaystyle g_{i,j} \left(z\right) \leq f\left(\bar z \right).
\]
The application of Theorem \ref{th:sc} yields the assertion.
\hfill $\Box$ \par \noindent \medskip
\end{corollary}

In what follows we concentrate on some special distributions of random utility shocks widely used in the discrete choice literature. For them we obtain better estimations of the convexity parameter of $E^*$, in particular, not dependent on the number of alternatives. First, we consider the class of generalized extreme value models \cite{mcfadden:1978}. The vector $\epsilon= \left(\epsilon^{(1)}, \ldots, \epsilon^{(n)}\right)^T$ of random utility shocks defines a generalized extreme value model (GEV) if it follows the joint distribution given by the probability density function
\[
  f_\epsilon\left(y^{(1)}, \ldots, y^{(n)}\right) = \frac{\partial^n \exp\left(-G\left(e^{-y^{(1)}},\ldots, e^{-y^{(n)}}\right)\right)}{\partial y^{(1)} \cdots \partial y^{(n)}},
\]
where the generating function $G:\mathbb{R}^n_+ \rightarrow \mathbb{R}_+$ has the following properties:
\begin{itemize}
\item[(G1)] $G$ is homogeneous of degree $\nicefrac{1}{\mu} > 0$.
\item[(G1)] $G\left(x^{(1)}, \ldots, x^{(i)}, \ldots, x^{(n)}\right) \rightarrow \infty$ as $x^{(i)} \rightarrow \infty$, $i=1, \ldots,n$.
\item[(G3)]
For the partial derivatives of $G$ with respect to $k$ distinct variables it holds:
\[
\frac{\partial^k G\left(x^{(1)},\ldots, x^{(n)}\right)}{\partial x^{\left(i_1\right)} \cdots \partial x^{\left(i_k\right)}} \geq 0 \mbox{ if } k \mbox{ is odd}, \quad
\frac{\partial^k G\left(x^{(1)},\ldots, x^{(n)}\right)}{\partial x^{\left(i_1\right)} \cdots \partial x^{\left(i_k\right)}} \leq 0 \mbox{ if } k \mbox{ is even}.
\]
\end{itemize}
It is well known from \cite{mcfadden:1978} that the surplus function for GEV is
\[
   E(u) = \mu \ln G\left(e^{u}\right) + \mu \gamma,
\]
where $\gamma$ is Euler's constant and we set $e^u=\left(e^{u^{(1)}}, \ldots, e^{u^{(n)}}\right)^T$ for the sake of brevity.
The choice probability of the $i$-th alternative is given by the $i$-th partial derivative of the GEV surplus function $E$:
\[
  \mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right)=\frac{\partial E(u)}{\partial u^{(i)}} = \mu \frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}}\cdot \frac{e^{u^{(i)}}}{G\left(e^{u}\right)}.
\]

We state a sufficient condition for the strong convexity of $E^*$ in terms of the generating function $G$.

\begin{theorem}[Strong convexity of $E^*$ for GEV]
\label{th:gev}
   Let a generating function $G$ for GEV satisfy the following inequality for all $x=\left(x^{(1)}, \ldots, x^{(n)}\right)^T\in \mathbb{R}^n_+$:
\[
   \sum_{i=1}^{n} \frac{\partial^2 G(x)}{\partial x^{{(i)}2}} \cdot x^{{(i)}2} \leq M \cdot G(x)
\]
with some constant $M \in \mathbb{R}$. Then, the corresponding convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm, where the convexity parameter is given by
\[
  \beta = \frac{1}{2(\mu M +1)-\nicefrac{1}{\mu}}.
\]
\bf Proof: \rm \par \noindent
For the second order partial derivative of $E$ with respect to $u^{(i)}$ it holds:
\[
\begin{array}{rcl}
  \displaystyle \frac{\partial^2 E(u)}{\partial u^{{(i)}2}} &=& \displaystyle \mu \left( \frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}}\cdot \frac{e^{u^{(i)}}}{G\left(e^{u}\right)}  \left(1-\frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}} \cdot\frac{e^{u^{(i)}}}{G\left(e^{u}\right)} \right)
   +
    \frac{\partial G^2\left(e^{u}\right)}{\partial x^{{(i)}2}}\cdot\frac{\left(e^{u^{(i)}}\right)^2}{G\left(e^{u}\right)} \right) \\ \\
    &=& \displaystyle  \frac{1}{\mu}\frac{\partial E(u)}{\partial u^{(i)}} \left( 1 - \frac{\partial E(u)}{\partial u^{(i)}}\right) + \left(1-\frac{1}{\mu}\right)\frac{\partial E(u)}{\partial u^{(i)}}
   + \mu
    \frac{\partial G^2\left(e^{u}\right)}{\partial x^{{(i)}2}}\cdot\frac{\left(e^{u^{(i)}}\right)^2}{G\left(e^{u}\right)}.
    \end{array}
\]
Analogously, we obtain the mixed partial derivative of $E$ for $j\not = i$:
\[
\begin{array}{rcl}
  \displaystyle
   \frac{\partial^2 E(u)}{\partial u^{(i)} \partial u^{(j)}} &=& \displaystyle
   \mu \left( -\frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}} \frac{e^{u^{(i)}}}{G\left(e^{u}\right)} \cdot  \frac{\partial G\left(e^{u}\right)}{\partial x^{(j)}} \frac{e^{u^{(j)}}}{G\left(e^{u}\right)}
   +
    \frac{\partial G^2\left(e^{u}\right)}{\partial x^{(i)} \partial x^{(j)}}\cdot \frac{e^{u^{(i)}} e^{u^{(j)}}}{G\left(e^{u}\right)} \right) \\ \\
     &=&\displaystyle -\frac{1}{\mu}\frac{\partial E(u)}{\partial u^{(i)}} \cdot  \frac{\partial E(u)}{\partial u^{(j)}}
   + \mu
    \frac{\partial G^2\left(e^{u}\right)}{\partial x^{(i)} \partial x^{(j)}}\cdot\frac{e^{u^{(i)}} e^{u^{(j)}}}{G\left(e^{u}\right)}.
\end{array}
\]
Equivalently, we have in matrix form:
\[
  \nabla^2 E(u) = \frac{1}{\mu} R(u) + S(u),
\]
where
\[
 \begin{array}{rcl}
 R(u) &= & \displaystyle {\rm diag \,}{\nabla E(u)} - \nabla E(u) \cdot \nabla^T E(u), \\ \\
 S(u) &=& \displaystyle \left(1-\frac{1}{\mu}\right) {\rm diag \,}{\nabla E(u)} + \mu \frac{{\rm diag \,}{e^u} \cdot \nabla^2  G\left(e^u\right)\cdot {\rm diag \,}{e^u}}{G\left(e^u\right)}.
\end{array}
\]
Since $\nabla E(u) \in \Delta$, we may apply Corollary \ref{cor:pp} to derive that
\[
 R(u) \in \mathcal{A}, \quad \|R(u)\|_{\infty,1} \leq 1.
\]
In view of $\nabla^2E(u), R(u) \in \mathcal{A}$, the matrix $S(u)$ fulfills (A2):
\[
   S(u) \cdot e = \underbrace{\nabla^2 E(u)\cdot e}_{=0} - \frac{1}{\mu} \underbrace{R(u)\cdot e}_{=0} = 0.
\]
The off-diagonal entries of $S(u)$ are nonpositive due to (G3), hence, $S(u)$ fulfills also (A1). We apply Lemma \ref{lem:tr} to estimate the $\|\cdot\|_{\infty,1}$ norm of $S(u)\in \mathcal{A}$:
\[
\begin{array}{rcl}
    \displaystyle
  \|S(u)\|_{\infty,1} &\leq& 2 \tr{S(u)} \\ \\ &=& \displaystyle 2 \left(\left(1-\frac{1}{\mu}\right) \sum_{i=1}^{n} \frac{\partial E(u)}{\partial u^{(i)}} + \mu \sum_{i=1}^{n}  \frac{\partial G^2\left(e^{u}\right)}{\partial x^{{(i)}2}}\cdot\frac{\left(e^{u^{(i)}}\right)^2}{G\left(e^{u}\right)}\right) \\ \\
  &\leq& \displaystyle 2 \left(\left(1-\frac{1}{\mu}\right) + \mu M \right).
\end{array}
\]
Overall, we have:
\[
   \|\nabla^2 E(u)\|_{\infty,1} \leq \frac{1}{\mu} \|R(u)\|_{\infty,1} + \|S(u)\|_{\infty,1} \leq \frac{1}{\mu} + 2 \left(\left(1-\frac{1}{\mu}\right) + \mu M \right)=\frac{1}{\beta}.
\]
\end{theorem}
The latter provides, due to Lemma \ref{lem:c2}, that the surplus function $E$ is $\frac{1}{\beta}$-strongly smooth with respect to the maximum norm $\|\cdot\|_\infty$. Finally, we apply Lemma \ref{lem:sd} to conclude that the convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm.\hfill $\Box$ \par \noindent \medskip

Now, we consider generalized nested logit models (GNL) introduced in \cite{wen:2001}. GNL is a particular class of GEV models defined by the generating function
\[
   G(x)= \sum_{\ell \in L} \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}}.
\]
Here, $L$ is a generic set of nests. The parameters $\sigma_{i\ell} \geq 0$ denote the shares of the $i$-th alternative with which it is attached to the $\ell$-th nest. For any fixed $i \in \{1, \ldots,n\}$ they sum up to one:
\[
  \sum_{\ell \in L} \sigma_{i\ell}  = 1,
\]
and $\sigma_{i\ell}=0$ means that the $\ell$-th nest does not contain the $i$-th alternative. Hence, the set of alternatives within the $\ell$-th nest is
\[
  N_\ell = \left\{i \,|\, \sigma_{i\ell} >0\right\}.
\]
The nest parameters $\mu_\ell > 0$ describe the variance of the random errors while choosing alternatives within the $\ell$-th nest. Analogously, $\mu >0$ describes the variance of the random errors while choosing among the nests. For the function $G$ to fulfill (G1)-(G3) we require:
\[
   \mu_\ell \leq \mu \quad \mbox{for all } \ell \in L.
\]
The underlying choice process can be viewed to comprise two stages:
\begin{itemize}
    \item[(1)] the probability of choosing the $\ell$-th nest is
\[
     q_\ell = \frac{e^{\nicefrac{v_\ell}{\mu}}}{\displaystyle
     \sum_{\ell \in L}e^{\nicefrac{v_\ell}{\mu}}},
\]
where
\[
v_\ell = \mu_\ell \ln \left( \sum_{i =1}^{n} \left(\sigma_{i\ell} \cdot e^{u^{(i)}}\right)^{\nicefrac{1}{\mu_\ell}} \right)
\]
stands for the utility attached to the $\ell$-th nest;
 \item[(2)] the probability of choosing the $i$-th alternative within the $\ell$-th nest is
\[
     p_{i\ell} = \frac{\left(\sigma_{i\ell} \cdot e^{u^{(i)}}\right)^{\nicefrac{1}{\mu_\ell}}}{\displaystyle
     \sum_{i=1}^{n} \left(\sigma_{i\ell} \cdot e^{u^{(i)}}\right)^{\nicefrac{1}{\mu_\ell}}}.
\]
\end{itemize}
Overall, the choice probability of the $i$-th alternative according to GNL amounts to
\[
  \mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right)= \mu \frac{\partial G\left(e^{u}\right)}{\partial x^{(i)}}\cdot\frac{e^{u^{(i)}}}{G\left(e^{u}\right)}= \sum_{\ell \in L} q_\ell \cdot p_{i\ell}.
\]

\begin{corollary}[Strong convexity of $E^*$ for GNL]
\label{cor:gnl}
   For GNL the corresponding convex conjugate $E^*$ is $\beta$-strongly convex with respect to the $\|\cdot\|_1$ norm, where the convexity parameter is given by
\[
  \beta = \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-\nicefrac{1}{\mu}}.
\]
\bf Proof: \rm \par \noindent Let us estimate the constant $M$ from Theorem \ref{th:gev}. We have:
\[
  \frac{\partial G\left(x\right)}{\partial x^{(i)}}= \frac{1}{\mu}\sum_{\ell \in L} \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-1}
   \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-1} \cdot \sigma_{i\ell},
\]
and, further:
\[
 \begin{array}{rcl}
 \displaystyle \frac{\partial^2 G(x)}{\partial x^{{(i)}2}} &=& \displaystyle \frac{1}{\mu}\sum_{\ell \in L}
 \frac{1}{\mu_\ell}\left(\frac{\mu_\ell}{\mu}-1\right)
 \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-2}
   \left(\left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-1} \cdot\sigma_{i\ell}\right)^2 \\ \\
  &+& \displaystyle \frac{1}{\mu}\sum_{\ell \in L} \left(\frac{1}{\mu_\ell}-1 \right) \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-1}
  \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-2}  \cdot\sigma_{i\ell}^2.
 \end{array}
\]
Due to $\mu_\ell \leq \mu$, $\ell \in L$, we get:
\[
  \frac{\partial^2 G(x)}{\partial x^{{(i)}2}} \leq \frac{1}{\mu}\sum_{\ell \in L} \left(\frac{1}{\mu_\ell}-1 \right) \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-1}
   \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-2} \cdot\sigma_{i\ell}^2.
\]
We multiply these terms by $x^{{(i)}2}$ and sum up over $i=1, \ldots, n$:
\[
\begin{array}{rcl}
 \displaystyle
   \sum_{i=1}^{n} \frac{\partial^2 G(x)}{\partial x^{{(i)}2}} \cdot x^{{(i)}2} &\leq& \displaystyle
   \frac{1}{\mu}\sum_{\ell \in L} \left(\frac{1}{\mu_\ell}-1 \right) \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}-1} \\ \\ &&\displaystyle \cdot
   \sum_{i=1}^{n}\left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}-2} \cdot  \sigma_{i\ell}^2 \cdot x^{{(i)}2} \\ \\
  &=& \displaystyle \frac{1}{\mu}\sum_{\ell \in L} \left(\frac{1}{\mu_\ell}-1 \right) \left( \sum_{i =1}^{n} \left(\sigma_{i\ell}\cdot x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\nicefrac{\mu_\ell}{\mu}}\\ \\
   &\leq& \displaystyle\frac{1}{\mu} \max_{\ell \in L} \left( \frac{1}{\mu_\ell} -1\right) \cdot G(x)= \frac{1}{\mu}  \left( \frac{1}{\displaystyle \min_{\ell \in L}\mu_\ell} -1\right) \cdot G(x).
 \end{array}
\]
Hence,  we may apply Theorem \ref{th:gev} with
\[
  M = \frac{1}{\mu}  \left( \frac{1}{\displaystyle \min_{\ell \in L}\mu_\ell} -1\right).
\]
\hfill $\Box$ \par \noindent \medskip
\end{corollary}

\begin{example}[Multinomial logit] \textup{Let in GNL there be just one nest, i.\,e. $L = \{1\}$, and $\mu_1 = \mu$. Then, the generating function
\[
   G(x)= \sum_{i =1}^{n} \left(x^{(i)}\right)^{\nicefrac{1}{\mu}}
\]
leads to the multinomial logit (ML). The corresponding surplus function is
\[
   E(u) = \mu \ln \sum_{i =1}^{n} e^{\nicefrac{u^{(i)}}{\mu}} + \mu \gamma,
\]
and the choice probabilities are
\[
   \mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right)= \frac{e^{\nicefrac{u^{(i)}}{\mu}}}{\displaystyle
     \sum_{i=1}^{n}e^{\nicefrac{u^{(i)}}{\mu}}}, \quad i=1, \ldots, n.
\]
Note that ML can be deduced from the IID random utility shocks $\epsilon^{(i)}$, $i=1, \ldots,n$, each of them following the Gumbel distribution with zero mode and variance $\nicefrac{\mu \pi}{\sqrt{6}}$ \cite{depalma:1994}. For ML the convex conjugate of the surplus function can be explicitly given:
\[
   E^*(p) = \mu \sum_{i=1}^{n} p^{(i)} \ln p^{(i)} - \mu \gamma = \mu H(p) - \mu \gamma,
\]
where $H$ is the (negative) entropy. It is well known that $H$ is $1$-strongly convex with respect to the $\|\cdot\|_1$ norm due to the Pinsker inequality. Hence, $E^*$ is $\mu$-strongly convex with respect to the $\|\cdot\|_1$ norm. The same result also follows from Corollary \ref{cor:gnl} with the convexity parameter
\[
  \beta = \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-\nicefrac{1}{\mu}}= \frac{1}{\nicefrac{2}{\mu}-\nicefrac{1}{\mu}}= \mu.
\]
} \hfill $\Box$ \par \noindent \medskip
\end{example}

\begin{example}[Nested logit]
\textup{Let in GNL for every alternative $i \in \{1,\ldots,n\}$ there be a unique nest $\ell_i \in L$ with $\sigma_{i \ell_i}=1$, and $\mu=1$. Then,
the nests $N_\ell=\left\{i \,|\, \ell_i=\ell\right\}$ are mutually exclusive, and
the generating function
\[
   G(x)=\sum_{\ell \in L} \left( \sum_{i \in N_\ell} x^{{(i)}\nicefrac{1}{\mu_\ell}} \right)^{\mu_\ell}
\]
leads to the nested logit (NL). The corresponding surplus function is
\[
   E(u) = \mu \ln \sum_{\ell \in L} \left( \sum_{i \in N_\ell} e^{\nicefrac{u^{(i)}}{\mu_\ell}} \right)^{\mu_\ell} + \mu \gamma,
\]
and the choice probabilities for $i \in N_\ell$, $\ell \in L$ are
\[
   \mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right)=
   \frac{e^{\mu_\ell \ln \sum_{i \in N_\ell} e^{\nicefrac{u^{(i)}}{\mu_\ell}}}}{\displaystyle
     \sum_{\ell \in L} e^{\mu_\ell \ln \sum_{i \in N_\ell} e^{\nicefrac{u^{(i)}}{\mu_\ell}}}}\cdot   \frac{e^{\nicefrac{u^{(i)}}{\mu_\ell}}}{\displaystyle
     \sum_{i\in N_\ell} e^{\nicefrac{u^{(i)}}{\mu_\ell}}}.
\]
For NL the convex conjugate of the surplus function is explicitly given in \cite{fosgerau:2017}:
\[
   E^*(p) = \sum_{\ell \in L} \mu_\ell \sum_{i\in N_\ell} p^{(i)} \ln p^{(i)}
   + \sum_{\ell \in L} \left(1-\mu_\ell\right) \left(\sum_{i\in N_\ell} p^{(i)} \right) \ln \left(\sum_{i\in N_\ell} p^{(i)} \right)
   - \mu\gamma.
\]
By examining the first part of this formula, it can be shown that $E^*$ is  $\displaystyle \left(\min_{\ell \in L} \mu_\ell\right)$-strongly convex with respect to the $\|\cdot\|_1$ norm. Corollary \ref{cor:gnl} provides the convexity parameter, which is of the same order:
\[
  \beta = \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-\nicefrac{1}{\mu}}= \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-1} > \frac{1}{2} \min_{\ell \in L} \mu_\ell.
\]
} \hfill $\Box$ \par \noindent \medskip
\end{example}

\begin{example}[GNL's from literature] \textup{For other specifications of GNL -- except of ML and NL -- the explicit form of the convex conjugate $E^*$ is not known yet. In this case Corollary \ref{cor:gnl} can be applied in order to estimate the convexity parameter of $E^*$.
\begin{itemize}
    \item[(i)] {\bf Ordered GEV} \cite{small:1987} is a GNL model with
\[
  L=\{ 1, \ldots, n+m\}, \quad \mu=1,
\]
\[
 \sigma_{i\ell}>0 \mbox{ for all } \ell \in \{i, \ldots,i+m\}, \quad
 \sigma_{i\ell}=0 \mbox{ for all } \ell \in L \backslash \{i, \ldots,i+m\}.
\]
There are $n+m$ overlapping nests $N_\ell=\left\{i \,|\, \ell-m \leq i \leq \ell\right\}$, and every alternative lies exactly in $m+1$ of them, namely $i \in N_\ell$ for $\ell=i, \ldots, i+m$. Then,
the generating function is
\[
   G(x)=\sum_{\ell=1}^{n+m} \left( \sum_{i \in N_\ell} \left(\sigma_{i\ell} x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\mu_\ell}.
\]
    \item[(ii)] {\bf Paired combinatorial logit} \cite{koppelman:2000} is a GNL model with
\[
  L=\{ (i,j)\in \{1,\ldots, n\} \,|\,  i \not = j\}, \quad \mu=1,
\]
\[
\sigma_{i\ell}=
\left\{ \begin{array}{cl}
     \displaystyle\frac{1}{2(n-1)} & \mbox{if } \ell = (i,j), (j,i) \mbox{ with } j \not =i,  \\
     0 & \mbox{else.}
\end{array}
\right.
\]
There are $n^2-n$ nests corresponding to the pairs of alternatives, and every alternative lies in $2(n-1)$ of them. Then,
the generating function is
\[
   G(x)=\sum_{\ell=(i,j), i \not =j} \left( \left(\sigma_{i\ell} x^{(i)}\right)^{\nicefrac{1}{\mu_\ell}} + \left(\sigma_{j\ell} x^{(j)}\right)^{\nicefrac{1}{\mu_\ell}} \right)^{\mu_\ell}.
\]
 \item[(iii)] {\bf Principles of differentiation GEV} \cite{bresnahan:1997} is a GNL model with
\[
  L = \mathop{\dot{\bigcup}}_{d \in D} L_d, \quad \mu=1, \quad
  \mu_\ell = \mu_d \mbox{ for all } \ell \in L_d,
\]
\[
 \sigma_{i\ell}=
\left\{ \begin{array}{cl}
     \displaystyle \sigma_d & \mbox{if } i \in N_{\ell d} \mbox{ and } \ell \in L_d, \\
     0 & \mbox{else,}
\end{array}
\right.
\]
where
\[
  \{1,\ldots,n\}=\mathop{\dot{\bigcup}}_{\ell \in L_d} N_{\ell d}.
\]
The set $D$ represents the dimensions of alternatives. Along the $d$-th dimension alternatives can be clustered into the disjoint nests $N_{\ell d}$, $\ell \in L_d$. The shares $\sigma_{i\ell}$ of all alternatives within these nests depend only on the dimension $d$. The parameters $\mu_\ell$ also coincide for all $\ell \in L_d$.
Then, the generating function is
\[
   G(x)=\sum_{d \in D} \sigma_d \sum_{\ell \in L_d} \left( \sum_{i\in N_{\ell d}} \left(x^{(i)}\right)^{\nicefrac{1}{\mu_d}} \right)^{\mu_d}.
\]
\end{itemize}
Note that the convexity parameter of $E^*$ for GNL specifications (i)-(iii) depends only on the smallest of the nest parameters $\mu_\ell$, $\ell \in L$:
\[
  \beta = \frac{1}{\frac{2}{\displaystyle \min_{\ell \in L} \mu_\ell}-1}.
\]
\hfill $\Box$ \par \noindent \medskip
}
\end{example}

Let us relate our results on the strong convexity of $E^*$ to those existing in the literature.

\begin{remark}[Cross moment model]
In CMM \cite{mishra:2012} the random vector $\epsilon \sim (0,\Sigma)$
follows a joint distribution with zero mean and a given covariance matrix $\Sigma$. Additionally, it maximizes the surplus function:
\[
Z(u) = \max_{\epsilon \sim (0,\Sigma)} \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)} \right).
\]
Assuming that the covariance matrix $\Sigma$ is positive definite, the following dual representation of $Z$ has been derived in \cite{li:2019}:
\[
      Z(u) =  \max_{ p \in \Delta} \, \langle p, u \rangle + \mbox{tr} \left( \left(\Sigma^{\nicefrac{1}{2}}\left( \mbox{diag}(p) - p p^T \right) \Sigma^{\nicefrac{1}{2}}\right)^{\nicefrac{1}{2}} \right).
\]
Moreover, the solution $p\in \Delta$ of the latter optimization problem provides the choice probabilities corresponding to the random error $\epsilon(u)\sim (0,\Sigma)$ that maximizes the surplus function above, i.\,e.
\[
   p^{(i)} = \mathbb{P} \left( u^{(i)} + \epsilon^{(i)}(u) = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}(u)\right), \quad i=1, \ldots, n.
\]
Hence, the convex conjugate of $Z$ is
\[
   Z^*(p) = - \mbox{tr} \left( \left(\Sigma^{\nicefrac{1}{2}}\left( \mbox{diag}(p) - p p^T \right) \Sigma^{\nicefrac{1}{2}}\right)^{\nicefrac{1}{2}} \right).
\]
In \cite{li:2019}, $Z^*$ is shown to be strongly convex on the simplex w.r.t. Euclidean norm (however, its convexity parameter is not given explicitly). We point out that $Z^*$ can be therefore used as a discrete choice prox-function on the simplex as well. \hfill $\Box$ \par \noindent \medskip
\end{remark}

\subsubsection{Simplicity}
\label{ssec:sim}

In view of the discussion in Section \ref{ssec:sc}, we assume that the convex conjugate $E^*$ is strongly convex.

\begin{theorem}[Simplicity]
\label{th:sim}
   The unique maximizer of the optimization problem
\begin{equation}
    \label{eq:sim}
    E(u) =  \sup_{ p \in \Delta} \, \langle p, u \rangle - E^*(p)
\end{equation}
is given by the choice probabilities
\[
    p^{(i)} = \mathbb{P} \left( u^{(i)} + \epsilon^{(i)} = \max_{1 \leq i \leq n} u^{(i)} + \epsilon^{(i)}\right), \quad i=1, \ldots, n.
\]
\bf Proof: \rm \par \noindent Let a vector $u \in \mathbb{R}^n$ of deterministic utilities be given. The vector $p$ of corresponding probabilities lies in the relative interior of the simplex $\Delta$, i.\,e. $p \in \mbox{rint}(\Delta)$. This is due to the fact that the distribution of the random vector $\epsilon$ is fully supported on $\mathbb{R}^n$, see \cite{norets:2013}. Hence, we have:
\[
  p = \nabla E(u),
\]
or, equivalently, by the convex duality \cite{rockafellar:1970}:
\[
  u \in \partial E^*(p).
\]
The latter is the first order optimality condition for  the optimization problem (\ref{eq:sim}). Additionally, note that
the dual representation of the surplus function is valid due to the Fenchel-Moreau theorem.
\hfill $\Box$ \par \noindent \medskip
\end{theorem}

\begin{corollary}[Lower bound for $E^*$]
\label{cor:lb}
The unique minimizer $p_0$ of the convex conjugate $E^*$ consists of the choice probabilities with respect to the zero-utility, i.\,e.
\[
    p^{(i)}_0 = \mathbb{P} \left( \epsilon^{(i)} = \max_{1 \leq i \leq n} \epsilon^{(i)}\right), \quad i=1, \ldots, n.
\]
Moreover, it holds:
\[
    E^*\left(p\right) \geq
    E^*\left(p_0\right) = -  \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n}\epsilon^{(i)} \right) \quad \mbox{for all } p \in \Delta.
\]
\bf Proof: \rm \par \noindent
Setting $u=0$ in Theorem \ref{th:sim} yields the assertion.
\hfill $\Box$ \par \noindent \medskip
\end{corollary}

\section{Economic application}
\label{sec:ea}

We apply discrete choice prox-functions for the natural adjustment of consumer's demand.

\subsection{Lancaster's approach to consumer theory}
\label{sec:lan}

We briefly describe the Lancaster's approach to consumer theory as presented in \cite{lancaster:1966}. For that, let $x \in \mathbb{R}^m_+$ denote the consumer's demand vector of $m$ goods. Every such demand vector $x$ generates the vector $z \in \mathbb{R}^n$ of $n$ qualities (sometimes called characteristics):
\[
     z = Q x,
\]
where $Q=\left(q_{ij}\right) \in \mathbb{R}^{n\times m}$ is a fixed quality matrix. Its entries $q_{ij}$ denote the amounts of the $i$-th quality while consuming one unit of the $j$-th good. Further, the consumer assigns utility $u(z)$ to the goods' qualities $z$, and tries to maximize it by adjusting the demand $x$. Hereby, the budget constraint need to be satisfied, i.\,e.
\[
   \langle \pi, x \rangle \leq w,
\]
where $\pi \in \mathbb{R}^m_{+}$ is a fixed vector of positive goods' prices, i.\,e. $\pi >0$, and $w >0$ is a fixed available budget. Overall, the Lancaster's approach to consumer theory consists in solving the following maximization problem:
\[
    \max_{\scriptsize \begin{array}{c} x \geq 0 \end{array}} \, u(z) \quad \mbox{s.t.} \quad z = Q x, \quad \langle \pi, x \rangle \leq w.
\]
In what follows we focus on the Leontieff utility function by setting
\[
   u(z)=\min \left\{ \frac{z^{(1)}}{\sigma^{(1)}}, \ldots, \frac{z^{(n)}}{\sigma^{(n)}}\right\},
\]
where $\sigma=\left(\sigma^{(1)}, \ldots, \sigma^{(n)}\right)^T\in \mathbb{R}^n_{+}$ is a fixed vector of positive quality standards, i.\,e. $\sigma >0$.
Additionally,
we assume that there exists a feasible demand vector which delivers positive Leontieff utility. Otherwise, the Lancaster's consumption problem is trivially solved by driving the demand to zero. On the contrary, under the proposed assumption the whole budget $w$ will be spent at any optimal demand. Hence, the budget constraint can be taken tight without loss of generality. The primal Lancaster's consumption problem becomes
\[
   \mbox{P:}\quad \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w   \end{array}} U(x),
\]
with the concave objective function
\[
U(x)= u(Qx)=\min_{1\leq i \leq n} \frac{(Qx)^{(i)}}{\sigma^{(i)}}.
\]
The optimization problem (P) consists in adjusting demand $x$ by spending the budget $w$ in order to maximize the worst ratio of consumed qualities $(Qx)^{(i)}$ in relation to their desired standards $\sigma^{(i)}$ taking over $i=1, \ldots, n$.

Let us derive a dual optimization problem for (P). For that, we introduce dual variables
\[
\lambda=\left(\lambda^{(1)}, \ldots, \lambda^{(n)}\right)^T \in \mathbb{R}^n,
\]
which can be interpreted as internal prices of qualities. Due to the duality of linear programming, we have:
\[
\begin{array}{rcl}
   \displaystyle
   \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w   \end{array}} U(x) &=& \displaystyle
   \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w   \end{array}}  \min_{1\leq i \leq n} \frac{(Qx)^{(i)}}{\sigma^{(i)}} \\ \\ &=&  \displaystyle
   \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w   \end{array}}  \min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1   \end{array}} \langle Qx, \lambda \rangle \\ \\
   &=& \displaystyle
    \min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1   \end{array}}   \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w   \end{array}} \left\langle x, Q^T\lambda \right\rangle \\ \\ &=& \displaystyle
    \min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1   \end{array}}  w \max_{1\leq j \leq m} \frac{\left(Q^T \lambda\right)^{(j)}}{\pi^{(j)}} = \displaystyle
    \min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1   \end{array}} \Phi(\lambda).
\end{array}
\]
The dual Lancaster's consumption problem becomes
\[
   \mbox{D:}\quad \min_{\scriptsize \begin{array}{c} \lambda \geq 0\\\langle \sigma, \lambda \rangle = 1   \end{array}}  \Phi(\lambda),
\]
with the convex objective function
\[
\Phi(\lambda)=w \max_{1\leq j \leq m} \frac{\left(Q^T \lambda\right)^{(j)}}{\pi^{(j)}}.
\]
The optimization problem (D) consists in adjusting internal prices of qualities $\lambda$ in order to minimize the best ratio of quality estimates  $\left(Q^T \lambda \right)^{(j)}$ of goods in relation to their market prices $\pi^{(j)}$ taking over $j=1, \ldots, m$.
Note that usually the number $n$ of goods  is considerably bigger than the number $m$ of their qualities. From a consumer it will be thus more plausible to expect that (D) is successively solved rather than (P).

\subsection{Dual Averaging Scheme}
\label{sec:da}
We rewrite the dual optimization problem (D) by introducing new variables
\[
    p^{(i)} = \sigma^{(i)} \lambda^{(i)}, \quad i =1, \ldots, n.
\]
In terms of the variables $p=\left(p^{(1)},\ldots,p^{(n)} \right)^T$ we equivalently obtain the following auxiliary optimization problem on the simplex:
\[
   \mbox{A:}\quad \min_{\scriptsize \begin{array}{c} p \in \Delta   \end{array}}  \Psi\left(p\right),
\]
where the objective function is
\[
\Psi(p)= \Phi\left(\frac{p}{\sigma}\right).
\]
For solving (A) we apply the dual averaging scheme from \cite{nesterov:2013}. For that, we use the family of prox-functions on the simplex discussed in Section \ref{sec:prox}:
\[
   d(p)=E^*(p) - E^*\left(p_0\right).
\]
Note that $d$ is continuous on the simplex $\Delta$ (Theorem \ref{th:ct}), strongly convex with convexity parameter $\beta>0$ (Theorems \ref{th:sc} and \ref{th:gev}, Corollary \ref{cor:gnl}). The computation of its convex conjugate
$
   d^*(u) = E(u) - E\left(0\right)
$
is simple (Theorem \ref{th:sim}). Additionally, Corollary \ref{cor:lb} provides us with the prox-center of $\Delta$:
\[
   p_0 = \mbox{arg} \min_{p \in \Delta} \, d(p).
\]
In view of $d\left(p_0\right)=0$, the dual averaging scheme can be initialized by $p_0$.

\[
   \begin{tabular}{|c|}
      \hline \\
      \begin{tabular}{c}
         {\bf Dual Averaging Scheme for (A)}
      \end{tabular}\\ \hline \\
      \begin{tabular}{ll}
            {\bf 1.} Compute $\nabla \Psi\left(p_k\right)$. \\ \\
            {\bf 2.} Set $\displaystyle s_{k+1}=\frac{1}{k+1}\sum_{\ell=0}^{k} \nabla \Psi\left(p_\ell\right)$. \\ \\
            {\bf 3.} Update
            $\displaystyle p_{k+1} = \mbox{arg} \min_{p \in \Delta} \left\{ \left\langle s_{k+1}, p \right\rangle + \frac{d(p)}{\sqrt{k+1}}  \right\}$.\\
      \end{tabular}
      \\ \\ \hline
   \end{tabular}
\]
The convergence properties of the dual averaging scheme follow from \cite[Theorem 1]{nesterov:2013}. For $k\geq 0$ it holds:
\begin{equation}
    \label{eq:ineq}
  \delta_k\leq \frac{D}{\sqrt{k+1}}+\frac{M^2}{2 \beta} \cdot \frac{1}{k+1}\left(1+\sum_{\ell=1}^{k} \frac{1}{\sqrt{\ell}}\right),
\end{equation}
where
\[
  \delta_k = \max_{p \in \Delta} \, \frac{1}{k+1} \sum_{\ell=0}^{k} \left\langle \nabla \Psi\left(p_\ell\right), p_\ell - p \right\rangle, \quad
  M = \max_{p \in \Delta} \left\|\nabla \Psi\left(p\right) \right\|_\infty, \quad D\geq \max_{p \in \Delta} \, d(p).
\]

\subsection{Consumption cycle}
\label{sec:cyc}
Let us state the dual averaging scheme from Section \ref{sec:da} in terms of the primal and dual Lancaster consumption problem (P) and (D), respectively.

\begin{itemize}
    \item[\bf 1.] We compute
    \[
       \nabla \Psi\left(p_k\right) = \nabla \Phi\left(\frac{p_k}{\sigma}\right) = \frac{\nabla \Phi \left(\lambda_k\right)}{\sigma},
    \]
    where $\lambda_k$ denotes the internal prices of qualities at the $k$-th iteration. It holds:
    \[
      \nabla \Phi \left(\lambda_k\right) =
      \nabla \left( w \max_{1\leq j \leq m} \frac{\left(Q^T \lambda_k\right)^{(j)}}{\pi^{(j)}} \right) = w \frac{Q y_k}{\pi},
    \]
    where the sharing vector $y_k \in \Delta$ fulfills
    \[
    y^{(j)}_k = 0 \quad \mbox{for } j \not \in J\left(\lambda_k\right)
    \]
    with the active index set
    \[
       J\left(\lambda_k\right) = \left\{ j \in \{1, \ldots, m\} \, \left| \,  \frac{\left(Q^T \lambda_k\right)^{(j)}}{\pi^{(j)}}=\max_{1\leq j \leq m} \frac{\left(Q^T \lambda_k\right)^{(j)}}{\pi^{(j)}} \right.\right\}.
    \]
    In other words, $J\left(\lambda_k\right)$ contains goods with the best quality/price ratio estimated by means of internal prices $\lambda_k$.
    We set the demand at the $k$-th iteration as
    \[
       x_k = w\frac{y_k}{\pi}.
    \]
    Note that $x_k$ is feasible for (P), since
    \[
       \left\langle \pi,x_k\right\rangle = \left\langle \pi,w\frac{y_k}{\pi}\right\rangle = w \left\langle e,y_k\right\rangle =w.
    \]
    Moreover, the demand is concentrated on the goods with the best quality/price ratio, i.\,e. $x^{(j)}_k \not = 0$ if and only if $j \in J\left(\lambda_k\right)$. Overall, we obtain:
    \[
       \nabla \Psi\left(p_k\right) = \frac{Q x_k}{\sigma},
    \]
    which are the ratios of the consumed qualities $Q x_k$ in relation to their standards $\sigma$.
    \item[\bf 2.] We set
    \[
      s_{k+1}=\frac{1}{k+1}\sum_{\ell=0}^{k} \nabla \Psi\left(p_\ell\right) = \frac{1}{k+1}\sum_{\ell=0}^{k} \frac{Q x_\ell}{\sigma}
      = \frac{Q \bar x_k}{\sigma}
    \]
    with the average demand
    \[
       \bar x_k = \frac{1}{k+1}\sum_{\ell=0}^{k} x_\ell.
    \]
    Again, $s_{k+1}$ relates the average consumption $Q \bar x_k$ to the standards $\sigma$.
    \item[\bf 3.] We update
    \[
    p_{k+1} = \mbox{arg} \min_{p \in \Delta} \left\{ \left\langle s_{k+1}, p \right\rangle + \frac{d(p)}{\sqrt{k+1}} \right\}.
    \]
    Due to Theorem \ref{th:sim}, we equivalently obtain for $i=1, \ldots, n$:
    \[
      p_{k+1}^{(i)} =  \mathbb{P} \left(  s_{k+1}^{(i)} - \frac{\epsilon^{(i)}}{\sqrt{k+1}} = \min_{1 \leq i \leq n}   s_{k+1}^{(i)} - \frac{\epsilon^{(i)}}{\sqrt{k+1}}\right).
    \]
    For the internal prices we have:
    \[
      \lambda_{k+1}^{(i)} =  \frac{1}{\sigma^{(i)}} \mathbb{P} \left( s_{k+1}^{(i)} - \frac{\epsilon^{(i)}}{\sqrt{k+1}} = \min_{1 \leq i \leq n}  s_{k+1}^{(i)} - \frac{\epsilon^{(i)}}{\sqrt{k+1}}\right).
    \]
    Thus, the internal price $\lambda_{k+1}^{(i)}$ of the $i$-th quality is proportional to the probability of detecting its average consumption $\left(Q \bar x_k\right)^{(i)}$ as the lowest one in comparison to the standard $\sigma^{(i)}$. Moreover, this detecting process is in accordance with additive random utility models allowing behavioral interpretations. E.\,g., the additive random errors $\frac{\epsilon^{(i)}}{\sqrt{k+1}}$ are diminishing with respect to the iteration number $k$, which accounts for the learning effect.
    We further mention that $\lambda_{k+1}$ is feasible for (D), since
    \[
       \left\langle \sigma,\lambda_{k+1}\right\rangle = \left\langle e,p_{k+1}\right\rangle =1.
    \]
    So are the average internal prices
\[
 \bar \lambda_k = \frac{1}{k+1}\sum_{\ell=0}^{k} \lambda_\ell.
\]
\end{itemize}

Let us examine the convergence properties of the proposed consumption cycle.

\begin{theorem}[Consumption cycle]
The duality gap between (P) and (D) evaluated at the average demand and the average internal prices is closing at the optimal rate $\mbox{O}\left(\frac{1}{\sqrt{k+1}}\right)$. Namely, it holds for $k \geq 0$:
\[
0 \leq \Phi\left( \bar \lambda_k\right) - U\left(\bar x_k\right) \leq
\left(D+\frac{M^2}{\beta}\right) \frac{1}{ \sqrt{k+1}},
\]
where
\[
   M = w \max_{\scriptsize \begin{array}{c} 1 \leq i \leq n\\ 1 \leq j \leq m   \end{array}} \frac{\left|q_{i,j}\right|}{\sigma^{(i)} \cdot \pi^{(j)}}, \quad
   D = \mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n}\epsilon^{(i)} \right) - \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right).
\]
\end{theorem}
\bf Proof: \rm \par \noindent
We compute the gap bound
\[
 \begin{array}{rcl}
 \displaystyle \delta_k &=& \displaystyle \max_{p \in \Delta} \, \frac{1}{k+1} \sum_{\ell=0}^{k} \left\langle \nabla \Psi\left(p_\ell\right), p_\ell - p \right\rangle =
  \frac{1}{k+1} \sum_{\ell=0}^{k} w \left\langle \frac{Q y_\ell}{\pi}, \lambda_\ell\right\rangle - \min_{p \in \Delta} \left\langle s_{k+1}, p \right\rangle \\ \\
  &=& \displaystyle  \frac{1}{k+1} \sum_{\ell=0}^{k} w \sum _{j \in J\left(\lambda_\ell \right)} y_\ell^{(j)} \cdot \frac{\left(Q^T \lambda_\ell\right)^{(j)}}{\pi^{(j)}} - \min_{1 \leq i \leq n} s_{k+1}^{(i)} \\ \\ &=&\displaystyle \frac{1}{k+1} \sum_{\ell=0}^{k} w \max_{1 \leq j \leq m} \frac{\left(Q^T \lambda_\ell\right)^{(j)}}{\pi^{(j)}}- \min_{1 \leq i \leq n} \frac{\left(Q \bar x_k\right)^{(i)}}{\sigma^{(i)}} = \frac{1}{k+1} \sum_{\ell=0}^{k} \Phi\left( \lambda_\ell\right) - U\left(\bar x_k\right).
\end{array}
 \]
The uniform bound for subgradients of $\Psi$ is given by
\[
M=\max_{p \in \Delta} \left\|\nabla \Psi\left(p\right) \right\|_\infty = \max_{\scriptsize \begin{array}{c} x \geq 0\\\langle \pi, x \rangle = w   \end{array}} \left\| \frac{Q x}{\sigma} \right\|_\infty =  w \max_{\scriptsize \begin{array}{c} 1 \leq i \leq n\\ 1 \leq j \leq m   \end{array}} \frac{\left|q_{i,j}\right|}{\sigma^{(i)} \cdot \pi^{(j)}}.
\]
By applying Corollaries
\ref{cor:ub} and \ref{cor:lb}, we estimate the constant $D$:
\[
  \max_{p \in \Delta} \, d(p) = \max_{p \in \Delta} \,
  E^*(p) - E^*\left(p_0\right) \leq  - \min_{1 \leq i \leq n} \mathbb{E}_\epsilon \left(\epsilon^{(i)} \right) +\mathbb{E}_\epsilon \left(\max_{1 \leq i \leq n}\epsilon^{(i)} \right)= D.
\]
It is straightforward to see that
\[
  \frac{1}{k+1}\left(1+\sum_{\ell=1}^{k} \frac{1}{\sqrt{\ell}}\right) \leq \frac{2}{\sqrt{k+1}}.
\]
Finally, by using (\ref{eq:ineq}) we get:
\[
\begin{array}{rcl}
  \displaystyle \Phi\left( \bar \lambda_k\right) - U\left(\bar x_k\right) &\leq& \displaystyle \frac{1}{k+1} \sum_{\ell=0}^{k} \Phi\left( \lambda_\ell\right) - U\left(\bar x_k\right) = \delta_k \\ \\ &\leq& \displaystyle
\frac{D}{\sqrt{k+1}}+\frac{M^2}{2 \beta} \cdot \frac{1}{k+1}\left(1+\sum_{\ell=1}^{k} \frac{1}{\sqrt{\ell}}\right)=\left(D+\frac{M^2}{\beta}\right) \frac{1}{ \sqrt{k+1}}.
\end{array}
\]
\hfill $\Box$ \par \noindent \medskip