[a]André Baião Raposo

# The Lüscher scattering formalism on the $t$-channel cut

Maxwell T. Hansen

###### Abstract

The Lüscher scattering formalism, the standard approach for relating the
discrete finite-volume energy spectrum to two-to-two scattering amplitudes,
fails when analytically continued so far below the infinite-volume two-
particle threshold that one encounters the $t$-channel cut. This is relevant,
especially in baryon-baryon scattering applications, as finite-volume energies
can be observed in this below-threshold regime, and it is not clear how to
make use of them. In this talk, we present a generalization of the scattering
formalism that resolves this issue, allowing one to also constrain scattering
amplitudes on the $t$-channel cut.

##  1 Introduction & motivation

In recent years, there has been considerable progress in the determination of
two-nucleon and other two-baryon scattering amplitudes using numerical lattice
QCD [1, 2, 3, 4, 5, 6, 7, 8, 9]. One of the leading methods in these
calculations is to first extract the finite-volume energy spectrum and
subsequently the scattering amplitudes via the Lüscher formalism and its
extensions [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In such calculations,
each finite-volume energy level constrains or predicts the scattering matrix
for all multi-hadron channels that can physically propagate at that energy.

One limitation in all finite-volume formalisms published to date is the
neglect of volume effects associated with the $t$-channel (also called left-
hand) cuts.111For ref. [21], the issue of the cut may be circumvented by
working in the plane-wave basis, but this is not specifically discussed in
those proceedings. This is most obviously a problem when the lattice
calculation predicts energies that are on top of the cut, as has recently been
seen in ref. [7]. The finite-volume formalism is manifestly not applicable
here, as it leads to predictions of a real-valued K-matrix (equivalently, a
real-valued scattering phase shift) in a region where the latter is known to
be complex.

In this proceedings, we present an extension of the original formalism that
can be applied on the left-hand cut. We begin with a brief review of infinite-
volume scattering theory as well as the standard Lüscher formalism, in
sections 2 and 3, respectively. In section 4, we illustrate how the
$t$-channel cut becomes an issue and, in section 5, we briefly describe our
approach to a solution, the full details of which will be presented in a
publication (to appear). Conclusions and an outlook are given in section 6.

##  2 Two-to-two scattering in infinite-volume

We start by reviewing a few properties of scattering amplitudes in the
infinite-volume context, making no reference yet to the finite-volume
formalism. Considering two-to-two elastic scattering of non-identical mass-
degenerate spin-zero particles with physical mass $M$, we write the total
four-momentum in a given frame as $P=(E,\boldsymbol{P})$ and introduce the
standard Mandelstam invariants $s$ and $t$. Mandelstam $s$ satisfies
$s=P^{2}=E^{2}-\boldsymbol{P}^{2}=(E^{\star})^{2}$, where $E^{\star}$ denotes
the centre-of-mass frame energy. Note we will use $\star$ to denote quantities
boosted to the centre-of-mass frame.

The scattering amplitude, which we write $\mathcal{M}(s,t)$, can be formally
expressed as the sum of all connected and amputated two-to-two Feynman
diagrams, with legs amputated and set on the mass shell (i.e. with external
momenta $p$ having $p^{2}=(p^{0})^{2}-\boldsymbol{p}^{2}=M^{2}$). This all-
orders sum can be organized by introducing the Bethe-Salpeter kernel, defined
as the sum of all connected and amputated two-to-two diagrams that are two-
particle irreducible in the $s$-channel222In other words, the Bethe-Salpeter
kernel is built from diagrams that cannot be separated into two pieces by
cutting through two propagators whose momenta sum to the total four-momentum
$P=(E,\boldsymbol{P})$.. The amplitude is then expressible in terms of the
Bethe-Salpeter kernels and pairs of dressed propagators of the scattering
scalars, as shown in figure 1. Note that all propagators considered are taken
with the standard $i\epsilon$ prescription, and all loop momenta are
integrated over all components.

Figure 1: (a) Diagrammatic representation of the two-to-two scattering
amplitude using Bethe-Salpeter kernels and dressed propagators. (b) Definition
of the Bethe-Salpeter kernel as the sum of all connected and amputated two-to-
two diagrams which are two-particle irreducible in the $s$-channel; we assume
that self-interactions of the scattering particles are
$\mathbb{Z}_{2}$-invariant, such that we consider vertices with even number of
legs only, while dashed lines denote other particles that might couple to the
scattering channels of interest. (c) Definition of the dressed propagator in
terms of bare propagators and self-energy kernels. (d) Definition of the self-
energy as the sum of all one-particle irreducible diagrams, with dashed lines
again denoting other particle types that can couple to the scattering
particles.

It is also instructive to define partial-wave amplitudes according to

$\mathcal{M}(s,t)=\sum_{\ell=0}^{\infty}P_{\ell}(\cos\theta^{\star})\mathcal{M}_{\ell}(s)\,,$
(1)

where $P_{\ell}$ is a Legendre polynomial and $\theta^{\star}$ is the
scattering angle in the centre-of-mass frame, satisfying
$\sin^{2}(\theta^{\star}/2)=-t/(s-4M^{2})$.

Using the all-orders Bethe-Salpeter representation (as discussed below) or
constraints based on the unitarity of the scattering matrix, one can show that
the imaginary part of $\mathcal{M}_{\ell}(s)^{-1}$ is independent of the
details of particle interactions. The real part is then typically
parameterized using the scattering phase shift $\delta_{\ell}(s)$. We may
write

$\displaystyle\text{Im}\,\mathcal{M}_{\ell}(s)^{-1}$
$\displaystyle=-\rho(s)\,\Theta(E^{\star}-2M)\,,$ (2)
$\displaystyle\text{Re}\,\mathcal{M}_{\ell}(s)^{-1}$
$\displaystyle=\rho(s)\cot\delta_{\ell}(s)\equiv\mathcal{K}_{\ell}(s)^{-1}\,,$
(3)

where $\rho(s)\equiv\frac{p^{\star}}{8\pi E^{\star}}$ is the phase-space
factor for non-identical particles, with
$p^{\star}\equiv\frac{1}{2}\sqrt{s-4M^{2}}$ denoting each particle’s centre-
of-mass spatial momentum magnitude, and we have defined the K-matrix
$\mathcal{K}_{\ell}(s)$. This leads to the standard form of the partial-wave
amplitude

$\mathcal{M}_{\ell}(s)=\frac{1}{\mathcal{K}_{\ell}(s)^{-1}-i\rho(s)}=\frac{8\pi
E^{\star}}{p^{\star}\cot\delta_{\ell}-ip^{\star}}\,.$ (4)

One can also reach these results via the Bethe-Salpeter series of figure 1 if
one defines the K-matrix $\mathcal{K}$ by the same series as the amplitude
$\mathcal{M}$, but in which all two-particle loops are evaluated with a
principal-value prescription instead of the $i\epsilon$ prescription. The
definitions of $\mathcal{M}$ and $\mathcal{K}$ can then be related by writing
each $i\epsilon$-loop in terms of its real and imaginary parts. The former
coincides with the principal-value loop integral, and the latter leads to a
Dirac delta function, allowing the loop integral to be exactly evaluated.

We should emphasize that these relations hold only for
$(2M)^{2}<s<(E^{\star}_{\sf inel.})^{2}$ where $E_{\sf inel.}$ is the energy
of the lowest-lying inelastic threshold coupling to the channel of interest.
This fact has received significant attention for energies above $E_{\sf
inel.}$ (in the form of three-particle finite-volume formalisms [22, 23, 24,
25, 26, 27]), but in this work we are concerned with the range $s<(2M)^{2}$.
For these sub-threshold energies, one can analytically continue the amplitude
by taking $-i\rho(s)\to|\rho(s)|$ in order to remain on the physical Riemann
sheet.

Such analytic continuation leads to a real-valued scattering amplitude,
provided that the K-matrix is real. When, however, we have a lighter particle
that couples to the scattering channel of interest, the K-matrix partial waves
become complex-valued due to a sub-threshold branch cut, the so-called
$t$-channel or left-hand cut. Before turning to the consequences of the cut,
we review the standard finite-volume formalism of Lüscher and show that a
real-valued K-matrix is implicitly assumed in the sub-threshold analytic
continuation, and thus that the formalism is not applicable on the left-hand
cut.

##  3 Review of the Lüscher formalism

In this section, we review the derivation of the Lüscher quantization
condition [10], which has been subsequently extended in refs. [11, 12, 13, 14,
15, 16, 17, 18, 19, 20] to include all types of coupled two-particle channels.
We focus here on the case of a single channel with two mass-degenerate but
non-identical spin-zero particles.

Consider a quantum field theory defined in a finite cubic spatial volume of
side-length $L$, with periodic boundary conditions. This system then has a
discrete $L$-dependent energy spectrum, and the energies lying below the
lowest-lying three- or four-particle threshold can be used to extract the
elastic two-to-two scattering amplitude. We follow closely the derivation of
Kim, Sachrajda and Sharpe [13], also used in refs. [17, 28, 29, 30]. We begin
by defining a two-point correlation function

$C_{L}(E,\boldsymbol{P})=\int
dx^{0}\int_{L}d^{3}\boldsymbol{x}\,e^{-iEx^{0}}e^{i\boldsymbol{P}\cdot\boldsymbol{x}}\langle
0|\text{T}\mathcal{A}(x)\mathcal{A}^{\dagger}(0)|0\rangle_{L}\,,$ (5)

where the subscript $L$ in $\int_{L}d^{3}\boldsymbol{x}$ indicates that the
integral runs over the finite volume, $E$ denotes the total energy,
$\boldsymbol{P}$ is the total spatial momentum, and $\mathcal{A}(x)$ and
$\mathcal{A}^{\dagger}(x)$ are annihilation and creation operators carrying
the quantum numbers of the scattering channel of interest.

One can construct a diagrammatic representation for this correlator using the
ingredients already introduced in the previous section, the Bethe-Salpeter
kernel and the dressed propagator pairs. This is known as the skeleton
expansion for the correlator and is shown in figure 2. In finite volume,
spatial loop momenta are discretized as
$\boldsymbol{k}=\frac{2\pi}{L}\boldsymbol{n}$, with
$\boldsymbol{n}\in\mathbb{Z}^{3}$, and we have spatial loop momentum sums
instead of integrals, i.e. we replace
$\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\rightarrow\frac{1}{L^{3}}\sum_{\boldsymbol{k}\in(2\pi/L)\mathbb{Z}^{3}}$
in all loops.

The key observation for deriving the Lüscher formalism is that not all loops
have the same volume dependence: loops with intermediate states that cannot go
on shell in the energy range considered have exponentially suppressed volume
effects $\mathcal{O}(e^{-mL})$, with $m$ being the mass of the lightest
particle coupled to the system, while loops with intermediate states that can
go on shell receive power-like effects $\mathcal{O}(L^{-n})$, for some non-
negative integer $n$. In the elastic regime, i.e. for centre-of-mass frame
energies above the two-particle threshold but below the thresholds of all
other channels, on-shell states come precisely from the loops left explicit in
the skeleton expansion shown in figure 2. Every other loop, implicitly
included in either the Bethe-Salpeter kernels or the dressed propagators, may
be replaced by its infinite-volume counterpart, as the difference, which we
neglect, is exponentially suppressed in $L$. Thus, we effectively replace the
finite-volume Bethe-Salpeter kernels and dressed propagators with their
infinite-volume counterparts.

Figure 2: Skeleton-expansion representation of the finite-volume correlator
$C_{L}(E,\boldsymbol{P})$, in terms of Bethe-Salpeter kernels and dressed
propagators, as defined in figure 1. The end-cap “blobs” stand for functions
in momentum-space originating from the Fourier transforms of the creation and
annihilation operators. As discussed, one can take the kernels and dressed
propagators in the expansion to be the infinite-volume objects, as the
difference between these and their finite-volume counterparts is exponentially
suppressed, and thus only the loops explicitly shown need to be treated as
finite-volume loops.

Consider one of the two-particle loops shown in the expansion of figure 2. Its
contribution can be written as

$C^{\sf
loop}_{L}(P)=\int\frac{dk^{0}}{2\pi}\frac{1}{L^{3}}\sum_{\boldsymbol{k}}\mathcal{L}(P,k)\,\Delta(k)\,\Delta(P-k)\,\mathcal{R}^{*}(P,k)\,,$
(6)

with $P\equiv(E,\boldsymbol{P})$ and loop momentum
$k\equiv(k^{0},\boldsymbol{k})$. The functions $\mathcal{L}$ and
$\mathcal{R}^{*}$ stand for the objects before and after the given loop,
$\Delta$ is a fully dressed scalar propagator (defined to have unit residue at
each pole). Performing the $k^{0}$-integral and decomposing $\mathcal{L}$ and
$\mathcal{R}$ in spherical harmonics with $k$ individually put on shell, i.e.
setting $k=(\omega({\boldsymbol{k}}),\boldsymbol{k})$ with
$\omega(\boldsymbol{k})\equiv\sqrt{\boldsymbol{k}^{2}+M^{2}}$, we can obtain

$C^{\sf loop}_{L}(P)=\frac{1}{L^{3}}\sum_{\boldsymbol{k}}\mathcal{L}_{\ell
m}(P,|\boldsymbol{k}^{\star}|)\,i\mathcal{S}_{\ell
m;\ell^{\prime}m^{\prime}}(P,\boldsymbol{k};L)\,\mathcal{R}^{*}_{\ell^{\prime}m^{\prime}}(P,|\boldsymbol{k}^{\star}|)+r(P)\,,$
(7)

where we have introduced

$\mathcal{S}_{\ell
m;\ell^{\prime}m^{\prime}}(P,\boldsymbol{k};L)\equiv\frac{4\pi\,Y_{\ell
m}(\hat{\boldsymbol{k}}{}^{\star})\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\boldsymbol{k}}{}^{\star})\,H(\boldsymbol{k}^{\star})}{2\omega(\boldsymbol{k})\,2\omega(\boldsymbol{P}-\boldsymbol{k})\,(E-\omega(\boldsymbol{k})-\omega(\boldsymbol{P}-\boldsymbol{k}))}\left(\frac{|\boldsymbol{k}^{\star}|}{k_{\sf
os}^{\star}}\right)^{\ell+\ell^{\prime}}\,,$ (8)

for later convenience. Note also that sums over the repeated indices $\ell,m$
and $\ell^{\prime},m^{\prime}$ are implied. The term $r(P)$ in eq. (7) is a
sum over a smooth summand, leading to exponentially suppressed finite-volume
corrections, which we may neglect. The summand in the first term and, more
specifically, the quantities $\mathcal{S}_{\ell
m;\ell^{\prime}m^{\prime}}(P,\boldsymbol{k};L)$, contain the pole
corresponding to the two-particle intermediate state going on-shell, as can be
seen explicitly in eq. (8), and thus this term contains all the power-like
volume dependence arising from the loop we are considering.

In eq. (8), we use the notation $k^{\star}_{\sf os}$ for the value of
$|\boldsymbol{k}^{\star}|$ which satisfies the intermediate two-particle state
on-shell condition $E^{\star}=2\omega(\boldsymbol{k}^{\star})$, equivalent to
$E=\omega(\boldsymbol{k})+\omega(\boldsymbol{P}-\boldsymbol{k})$, given by the
simple relation

$k_{\sf
os}^{\star}\equiv\frac{1}{2}\sqrt{s-4M^{2}}=\frac{1}{2}\sqrt{E^{2}-\boldsymbol{P}^{2}-4M^{2}}\,.$
(9)

Note that this relation fixes the magnitude of $\boldsymbol{k}^{\star}$ at the
pole, but not its direction. The barrier factor
$\big{(}|\boldsymbol{k}^{\star}|/k^{\star}_{\sf
os}\big{)}^{\ell+\ell^{\prime}}$ is introduced to ensure that no singularities
arise from the spherical harmonics $Y_{\ell
m}(\hat{\boldsymbol{k}}{}^{\star})$ and
$Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\boldsymbol{k}}{}^{\star})$.

The function $H(\boldsymbol{k}^{\star})$ is a regulator function that takes a
value of 1 for
$4\omega(\boldsymbol{k}^{\star})^{2}=4(|\boldsymbol{k}^{\star}|^{2}+M^{2})<(E^{\star}_{\sf{inel.}})^{2}$
and of 0 for
$4\omega(\boldsymbol{k}^{\star})^{2}=4(|\boldsymbol{k}^{\star}|^{2}+M^{2})>(E^{\star}_{\sf
uv})^{2}$, where again $E_{\sf{inel.}}$ is the lowest lying three- or four-
particle threshold, and $E^{\star}_{\sf{uv}}$ is some chosen high ultraviolet
cut-off. In the region between, $H(\boldsymbol{k}^{\star})$ interpolates
smoothly between the two values. This regulator function is similar to the one
found in the three-body scattering formalism of refs. [22, 23], and
corresponds to a separation of low-energy and high-energy parts of the sum,
such that both parts are regulated and kept finite.333It should be emphasized
that we have renormalization and regularization schemes keeping the overall
result finite, the regulator function here simply ensures we have a separation
of high-energy and low-energy contributions to the sum such that both parts
are finite and that the low-energy part, which contains the relevant singular
behaviour, is tractable when implemented numerically. $E_{\sf uv}$ will simply
be a scheme-dependence of the formalism, but should be kept high, as setting
it too low will lead to enhanced finite-volume effects.

We next reduce eq. (7) by expanding the functions $\mathcal{L}_{\ell
m}(P,|\boldsymbol{k}^{\star}|)$ and
$\mathcal{R}^{*}_{\ell^{\prime}m^{\prime}}(P,|\boldsymbol{k}^{\star}|)$ about
$|\boldsymbol{k}^{\star}|=k^{\star}_{\sf os}$ and subtracting and adding an
integral to reach

$\displaystyle C^{\sf loop}_{L}(P)$ $\displaystyle=\mathcal{L}_{\ell
m}(P,k^{\star}_{\sf os})\,iF_{\ell
m;\ell^{\prime}m^{\prime}}(P;L)\,\mathcal{R}^{*}_{\ell^{\prime}m^{\prime}}(P,k^{\star}_{\sf
os})+r^{\prime}(P)\,,$ $\displaystyle={\mathcal{L}}_{\sf
os}(P)\,iF(P;L)\,{\mathcal{R}}_{\sf os}^{\dagger}(P)+r^{\prime}(P)\,,$ (10)

where we introduce the sum-integral difference

$F_{\ell
m;\ell^{\prime}m^{\prime}}(P;L)=\left[\frac{1}{L^{3}}\sum_{\boldsymbol{k}}-\
{\sf p.v.}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\right]\mathcal{S}_{\ell
m;\ell^{\prime}m^{\prime}}(P,\boldsymbol{k};L)\,.$ (11)

Here, $\sf{p.v.}$ means the integral is evaluated using a principal-value
prescription. The remainder term $r^{\prime}(P)$ differs from $r(P)$, but
still has the property that it contains the sum of a smooth summand together
with integrals. In the second line of eq. (10), we have defined a compact
vector-matrix notation in the angular-momentum index space.

Applying this procedure iteratively to all loops in the skeleton expansion
diagrams, it can be shown that we may write the finite-volume correlator in
the form:

$\displaystyle
C_{L}(P)=\sum_{n=0}^{\infty}A(P)\,iF(P;L)\,\left[i\mathcal{K}(P)\,iF(P;L)\right]^{n}A^{\dagger}(P)+C_{\infty}(P)\,.$
(12)

The quantities in the first term are vectors or matrices in angular momentum
index space: $A(P)$ and $A^{\dagger}(P)$ are smooth vectors loosely
corresponding to the source and sink operators and $\mathcal{K}(P)$ is the
K-matrix introduced in the previous section (note that $\mathcal{K}$ is an
infinite-volume scalar, and thus only depends on $s=P^{2}$, but we keep $P$ as
an argument for compactness of notation). Noting that the above is a geometric
series, we can sum it to obtain

$C_{L}(P)=A(P)\frac{i}{F^{-1}(P;L)+\mathcal{K}(P)}A^{\dagger}(P)+C_{\infty}(P)\,.$
(13)

Given that we neglect exponentially suppressed volume effects, the volume
dependence is entirely contained in the $F(P;L)$ matrix.

Using a spectral representation of the correlator, it is straightforward to
show that it must have poles at the energy levels of the finite-volume
spectrum $E_{n}(\boldsymbol{P};L)$. These poles in $C_{L}(E,\boldsymbol{P})$
can only arise from the $L$-dependent part of the first term, meaning that we
must have

$\det\left[F^{-1}(E_{n}(\boldsymbol{P};L),\boldsymbol{P};L)+\mathcal{K}(E_{n}(\boldsymbol{P};L),\boldsymbol{P})\right]=0\,,$
(14)

at all finite-volume energy levels. This is called the _Lüscher quantization
condition_ , and it can be used to determine $\mathcal{K}$, and hence the
scattering amplitude $\mathcal{M}$, from the knowledge of the finite-volume
spectrum. The matrices involved in the condition (14) are formally infinite-
dimensional, since the set of possible angular momentum indices $\ell m$ is
infinite. For practical use, we must truncate them to the lowest harmonics,
making the approximation that $\mathcal{K}$ vanishes for $\ell>\ell_{\sf
max}$. We should note that this relies on a fast convergence of the partial-
wave expansion of the amplitude, such that keeping the lowest harmonics still
leads to a reasonable reconstruction of the amplitude.

##  4 The $t$-channel problem

The finite-volume spectrum can sometimes include energy levels that drop below
the infinite-volume elastic threshold at $s=(2M)^{2}$. This can occur due to
the appearance of a bound state (such that the $L\to\infty$ limit gives the
bound-state mass) as well as to an attractive scattering state (such that the
energy approaches $2M$ for $L\to\infty$). In many cases, the Lüscher formalism
can be analytically continued from $s>(2M)^{2}$ and the sub-threshold finite-
volume energy then provides an important constraint on the K-matrix below
threshold.

A subtlety arises, however, when the sub-threshold two-to-two scattering
amplitude $\mathcal{M}(s,t)$, and therefore also the K-matrix, has a nearby
$t$-channel cut. This is generically the case in baryon-baryon systems, for
example, where a light meson can be exchanged in the $t$-channel as shown in
figure 3. Taking $m$ and $M$ to be the meson and baryon masses, respectively,
and assuming $m\ll M$, one finds that a pole arises in $\mathcal{M}(s,t)$ at
$t=m^{2}$. Then, for a given fixed choice of the centre-of-mass frame
scattering angle $\theta^{\star}$, this leads to a pole in $s$ at

$s=4M^{2}-\frac{t}{\sin^{2}(\theta^{\star}/2)}\bigg{|}_{t=m^{2}}=4M^{2}-\frac{m^{2}}{\sin^{2}(\theta^{\star}/2)}\,.$
(15)

The analytic structure of the scattering amplitude for such systems is shown
in figure 4(a). From the expression, one sees that the pole position in $s$
varies from $s=4M^{2}-m^{2}$ to $s=-\infty$ as $\theta^{\star}$ is varied from
$0$ to $\pi$. As a result, the angular-momentum projection of the scattering
amplitude leads to a branch cut running over this interval as shown in figure
4(b). Multiple meson exchanges can also occur, leading to additional cuts in
both the fixed-$\theta^{\star}$ and the angular-momentum projected amplitudes.
In the latter case, these run along $s\leq(2M)^{2}-(nm)^{2}$ for $n$ exchanged
mesons.

Figure 3: Meson exchanges contributing to the baryon-baryon scattering
amplitude, expressed diagrammatically as a dressed $t$-channel meson
propagator (with time flowing horizontally). The exchanged meson is a pion in
the case of $NN$ scattering or an $\eta$ meson in the case of $\Lambda\Lambda$
scattering. In the latter case this is not the nearest singularity for
physical quark masses but can become so for heavier-than-physical masses.
Figure 4: (a) Analytic structure of the two-to-two scattering amplitude
$\mathcal{M}(s,t)$ in the complex-$s$ plane, for fixed centre-of-mass
scattering angle $\theta^{\star}$, in the case where a lighter particle of
mass $m$ couples to the scattering particles of mass $M$. We show the
infinite-volume elastic threshold (at $s=(2M)^{2}$) and inelastic threshold
(at $s=(2M+m)^{2}$) and corresponding branch cuts. Below threshold, we see the
$t$-channel exchange pole, corresponding to $t=m^{2}$, and a lower branch cut,
corresponding to two mesons being exchanged in the $t$ channel. (b) Analytic
structure of the amplitude when projected to definite angular momentum. The
so-called $t$-channel or left-hand cut, which runs down from the branch point
at $s=(2M)^{2}-m^{2}$, arises from the $t$-channel pole above.

As stressed above, finite-volume energies can arise in the region of the
branch cuts (as has recently been identified in ref. [7]) and a naive
application of the analytically continued Lüscher formalism fails. In this
work we restrict attention to the region $(2M)^{2}-(2m)^{2}<s<(2M)^{2}-m^{2}$
in which only the single-meson cut arises and derive a modified version of the
scattering formalism that resolves this limitation.

##  5 Proposed solution

The origin of the breakdown in the original formalism can be traced back to
the steps between eq. (7) and (10) in the review of section 3. In the step of
replacing $\mathcal{L}_{\ell m}(P,|\boldsymbol{k}^{\star}|)$ and
$\mathcal{R}^{*}_{\ell^{\prime}m^{\prime}}(P,|\boldsymbol{k}^{\star}|)$ with
the on-shell quantities $\mathcal{L}_{\ell m}(P,k_{\sf os}^{\star})$ and
$\mathcal{R}^{*}_{\ell^{\prime}m^{\prime}}(P,k_{\sf os}^{\star})$, the
derivation assumes that the product of the two-particle pole and a given
difference, e.g. $\mathcal{L}_{\ell
m}(P,|\boldsymbol{k}^{\star}|)-\mathcal{L}_{\ell m}(P,k_{\sf os}^{\star})$, is
a smooth function of $\boldsymbol{k}^{\star}$. This step fails in the sub-
threshold region due to the $t$-channel cut.

To handle this issue, we separate out the problematic $t$-channel exchanges
from the Bethe-Salpeter kernel. We define

$ig^{2}T(\boldsymbol{k}^{\star},\boldsymbol{k}^{\prime\star})\equiv-
ig^{2}\frac{1}{-(\boldsymbol{k}^{\star}-\boldsymbol{k}^{\prime\star})^{2}-m^{2}+i\epsilon}\,,$
(16)

and define a modified kernel by subtracting this from the full Bethe-Salpeter
kernel as shown in figure 5. Here, $g$ denotes the effective baryon-meson-
baryon coupling. We emphasize that $m$ is the physical mass of the meson, and
thus that $-iT$ corresponds to the singular part of the fully-dressed meson
propagator. The difference between the bare and fully dressed propagators is
smooth, and is simply absorbed into the modified kernel.

Examining the modified kernel using, for instance, the cutting rules of time-
ordered perturbation theory, shows that it can be safely evaluated at
$|\boldsymbol{k}^{\star}|=k_{\sf os}^{\star}$ and does not possess a
singularity or cut in the region $(2M)^{2}-(2m)^{2}<s<(2M)^{2}-m^{2}$.
Crucially, we also note that the exchange $ig^{2}T$ is safe if kept partially
off shell, namely if we keep at least one of the magnitudes
$|\boldsymbol{k}^{\star}|$ and $|\boldsymbol{k}^{\prime\star}|$ to be real.

Figure 5: Separation of the standard Bethe-Salpeter kernel into a modified
kernel (square box), which is safe to put on shell, and the problematic
$t$-channel meson exchange, represented by a meson propagator at physical
mass.

Recalling the expression (10) for the contribution of a skeleton expansion
loop, we again emphasize that the finite-volume frame momentum
$\boldsymbol{k}$, and hence its centre-of-mass frame counterpart
$\boldsymbol{k}^{\star}$, are discretized and can be indexed by
$\boldsymbol{n}\in\mathbb{Z}^{3}$. Thus, we can then treat
$\boldsymbol{k}^{\star}$ as an extra index, writing
$\mathcal{L}_{\boldsymbol{k}^{\star}\ell m}\equiv\mathcal{L}_{\ell
m}(P,|\boldsymbol{k}^{\star}|)$, $\mathcal{R}_{\boldsymbol{k}^{\star}\ell
m}\equiv\mathcal{R}_{\ell m}(P,|\boldsymbol{k}^{\star}|)$ and defining

$S_{\boldsymbol{k}^{\star}\ell
m;\boldsymbol{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P;L)\equiv\frac{1}{L^{3}}\delta_{\boldsymbol{k}^{\star}\boldsymbol{k}^{\prime\star}}\mathcal{S}_{\ell
m;\ell^{\prime}m^{\prime}}(P,\boldsymbol{k};L)\,,$ (17)

such that we may rewrite (10) as:

$\displaystyle C^{\sf loop}_{L}(P)$
$\displaystyle=\mathcal{L}_{\boldsymbol{k}^{\star}\ell
m}(P)\,iS_{\boldsymbol{k}^{\star}\ell
m;\boldsymbol{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P)\,\mathcal{R}_{\boldsymbol{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P;L)+r(P)\,,$
(18) $\displaystyle=\mathcal{L}(P)\,iS(P;L)\,\mathcal{R}(P)+r(P)\,.$ (19)

Note that, in the first line, we are now also implicitly summing over the
momentum indices. In the second line, we again employ a compact vector-matrix
notation, but now in the angular momentum plus spatial loop momentum index
space.

Applying this to all loops in the skeleton expansion diagrams and rearranging
by factors of $S(P;L)$, we obtain the finite-volume correlator in the form

$C_{L}(P)=\sum_{n=0}^{\infty}A(P)\,iS(P;L)\left[(i\bar{K}+ig^{2}T(P))\,iS(P;L)\right]^{n}A^{\dagger}(P)+C^{(i)}_{\infty}(P)\,,$
(20)

where all quantities in the first term are vectors or matrices in the angular
momentum plus loop momentum index space. Note that $A(P)$ and $A^{\dagger}(P)$
are different from those in (10). The matrix $ig^{2}T$ is the matrix of
angular momentum projections of the $t$-channel exchange defined in (16), and
$\bar{K}(P)$ is the sum of all possible smooth contributions one can obtain
between $S(P;L)$ matrices. The second term $C^{(i)}_{\infty}(P)$ is a
collection of $L$-independent terms.

From the discussion above, we know it is safe to set
$|\boldsymbol{k}^{\star}|=k^{\star}_{\sf os}$ for $\bar{K}(P)$ and, therefore,
we can expand $\bar{K}(P)$ about the on-shell point. We implement this by
making use of a trivial vector $u$ in the momentum index space, whose elements
are $u_{\boldsymbol{k}^{\star}}=1$, and making the substitution
$\bar{K}(P)=u\bar{\mathcal{K}}(P)u^{\dagger}+\left[\bar{K}(P)-u\bar{\mathcal{K}}(P)u^{\dagger}\right]$.
The matrix $\bar{\mathcal{K}}(P)$ is a matrix in the angular momentum index
space only and corresponds to $\bar{K}(P)$ with the dependence on the
magnitude of spatial momentum (through the momentum index) set to the on-shell
momentum, i.e. with $|\boldsymbol{k}^{\star}|=k^{\star}_{\sf os}$. The
different terms in brackets lead to terms that are sums of smooth summands and
can be shuffled into the remainder term. This yields the following expressions
for the correlator:

$\displaystyle C_{L}(P)$
$\displaystyle=\sum_{n=0}^{\infty}A(P)\,iS(P;L)\left[(iu\bar{\mathcal{K}}(P)u^{\dagger}+ig^{2}T(P))\,iS(P;L)\right]^{n}A^{\dagger}(P)+C^{(ii)}_{\infty}(P)\,,$
(21)
$\displaystyle=A(P)\frac{i}{S^{-1}(P;L)+u\bar{\mathcal{K}}(P)u^{\dagger}+g^{2}T(P)}A^{\dagger}(P)+C^{(ii)}_{\infty}(P)\,.$
(22)

Using the same arguments as in the standard derivation, we can derive a
modified quantization condition:

$\det\left[S^{-1}(E_{n}(\boldsymbol{P},L),\boldsymbol{P})+u\bar{\mathcal{K}}(E_{n}(\boldsymbol{P},L),\boldsymbol{P})u^{\dagger}+g^{2}T(E_{n}(\boldsymbol{P},L),\boldsymbol{P})\right]=0\,,$
(23)

at all finite-volume energy levels $E_{n}(\boldsymbol{P},L)$. Given that
$S^{-1}(E_{n}(\boldsymbol{P},L),\boldsymbol{P})$ and
$T(E_{n}(\boldsymbol{P},L),\boldsymbol{P})$ can be calculated numerically, one
can use the knowledge of the finite-volume spectrum to obtain
$\bar{\mathcal{K}}(E_{n}(\boldsymbol{P},L),\boldsymbol{P})$ as well as the
coupling $g$. This object can then be linked back to the two-to-two scattering
amplitude via integral equations, in a similar vein to the procedure used for
the three-particle scattering formalism refs. [22, 23, 31]. We leave further
discussion to the upcoming paper.

##  6 Summary & Outlook

In this proceedings, we have described our progress in addressing issues
arising in the Lüscher finite-volume scattering formalism [10] and extensions
[11, 12, 13, 14, 15, 16, 17, 18, 19, 20] in the case of sub-threshold finite-
volume energies appearing on the $t$-channel cut. This work is motivated by
recent lattice calculations in baryon-baryon systems that have observed such
energy levels [7].

To present the extension we first reviewed the standard derivation, following
the method of Kim, Sachrajda, and Sharpe [13] for the case on non-identical
spin-zero particles. We then identified the step in the derivation that fails
to correctly account for the sub-threshold cut and provided a modification to
address the issue. Our main result is an adapted quantization condition that
applies above and below elastic threshold, including on the cut associated
with single-meson exchange, though not on lower cuts arising from the exchange
of multiple mesons.

As we have in mind applications to baryon-baryon scattering, the next step,
currently ongoing, is generalizing the derivation to particles with arbitrary
intrinsic spin. Once the theoretical work is concluded, future directions
include numerical tests on mock data (e.g. in the spirit of refs. [19, 32])
and eventually applications to lattice QCD baryon-baryon data.

## Acknowledgements

The authors thank Raúl Briceño, John Bulava, Evgeny Epelbaum, Drew Hanlon,
Arkaitz Rodas, Fernando Romero-López, Maxim Mai, Steve Sharpe, and Hartmut
Wittig for useful discussions including those in the context of the _Bethe
Forum on Multihadron Dynamics in a Box_ that took place at the Bethe Center
for Theoretical Physics in Bonn, Germany. M.T.H. is supported by UKRI Future
Leader Fellowship MR/T019956/1, and both M.T.H and A.B.R. are partly supported
by UK STFC grant ST/P000630/1.

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# Distributionally robust possibilistic optimization problems

Romain Guillaume  Université de Toulouse-IRIT Toulouse, France,
<EMAIL_ADDRESS>Adam Kasperski111Corresponding author Wrocław
University of Science and Technology, Wrocław, Poland
<EMAIL_ADDRESS>Paweł Zieliński  Wrocław University of Science and
Technology, Wrocław, Poland
<EMAIL_ADDRESS>
###### Abstract

In this paper a class of optimization problems with uncertain linear
constraints is discussed. It is assumed that the constraint coefficients are
random vectors whose probability distributions are only partially known.
Possibility theory is used to model the imprecise probabilities. In one of the
interpretations, a possibility distribution (a membership function of a fuzzy
set) in the set of coefficient realizations induces a necessity measure, which
in turn defines a family of probability distributions in this set. The
distributionally robust approach is then used to transform the imprecise
constraints into deterministic counterparts. Namely, the uncertain left-had
side of each constraint is replaced with the expected value with respect to
the worst probability distribution that can occur. It is shown how to
represent the resulting problem by using linear or second order cone
constraints. This leads to problems which are computationally tractable for a
wide class of optimization models, in particular for linear programming.

Keywords: robust optimization; possibility theory; imprecise probabilities;
fuzzy intervals

##  1 Introduction

In this paper we consider the following uncertain optimization problem:

$\begin{array}[]{llll}\min&\boldsymbol{c}^{T}\boldsymbol{x}\\\
&\tilde{\boldsymbol{a}}_{i}^{T}\boldsymbol{x}\leq b_{i}&i\in[m]\\\
&\boldsymbol{x}\in\mathbb{X}.\end{array}$ (1)

In formulation (1), $\boldsymbol{c}\in\mathbb{R}^{n}$ is a vector of objective
function coefficients, $\boldsymbol{b}\in\mathbb{R}^{m}$ is a vector of
constraint right-hand sides, and $\tilde{\boldsymbol{a}}_{i}$ is a vector of
uncertain coefficients of the $i$th constraint. In the following we will use
the notation $[m]=\\{1,\dots,m\\}$. We assume that
$\tilde{\boldsymbol{a}}_{i}$ is a random vector in $\mathbb{R}^{n}$ whose
realization is unknown when a solution to (1) is determined. A particular
realization $\boldsymbol{a}_{i}\in\mathbb{R}^{n}$ of
$\tilde{\boldsymbol{a}}_{i}$ is called a _scenario_. After replacing
$\tilde{\boldsymbol{a}}_{i}$ with scenario $\boldsymbol{a}_{i}$, for each
$i\in[m]$, we get a deterministic counterpart of (1). The true probability
distribution for $\tilde{\boldsymbol{a}}_{i}$ is only partially known. In
particular, the set of possible scenarios
$\mathcal{U}_{i}\subseteq\mathbb{R}^{n}$ is provided, which is the support of
$\tilde{\boldsymbol{a}}_{i}$ or its reasonable approximation. Finally,
$\mathbb{X}$ is a nonempty and bounded subset of $\mathbb{R}^{n}$ which
restricts the domain of decision variables $\boldsymbol{x}$. If $\mathbb{X}$
is a polyhedron, for example,
$\mathbb{X}=\\{\boldsymbol{x}\in\mathbb{R}^{n}:\boldsymbol{L}\leq\boldsymbol{x}\leq\boldsymbol{U}\\}$,
for some $\boldsymbol{L},\boldsymbol{U}\in\mathbb{R}^{n}$, then we get
uncertain linear programming problem. If, additionally,
$\mathbb{X}\subseteq\\{0,1\\}^{n}$, then we get uncertain combinatorial
optimization problem.

The method of handling the uncertain constraints and solving (1) depends on
the information available about $\tilde{\boldsymbol{a}}_{i}$. If
$\tilde{\boldsymbol{a}}_{i}$ is a random vector with a known probability
distribution, then the $i$th constraint can be replaced with a stochastic
_chance constraint_ of the form

${\rm P}(\tilde{\boldsymbol{a}}_{i}^{T}\boldsymbol{x}\leq b_{i})\geq
1-\epsilon_{i},$

where $\epsilon_{i}\in[0,1)$ is a given risk level [8, 21]. If the uncertainty
set $\mathcal{U}_{i}$ is the only information provided with
$\tilde{\boldsymbol{a}}_{i}$, then the imprecise constraint
$\tilde{\boldsymbol{a}}_{i}^{T}\boldsymbol{x}\leq b_{i}$ can be replaced with
the following _strict robust constraint_ [2]:

$\max_{\boldsymbol{a}_{i}\in\mathcal{U}_{i}}\boldsymbol{a}_{i}^{T}\boldsymbol{x}\leq
b_{i}.$ (2)

Choosing appropriate uncertainty set $\mathcal{U}_{i}$ is crucial in (2). For
the _discrete uncertainty_ , $\mathcal{U}_{i}$ consists of $K$ explicitly
listed scenarios [25]. These scenarios correspond to some events which
influence the value of $\tilde{\boldsymbol{a}}_{i}$ or can be the result of
sampling the random vector $\tilde{\boldsymbol{a}}_{i}$. For the _interval
uncertainty_ [25], one provides an interval for each component
$\tilde{a}_{ij}$ of
$\tilde{\boldsymbol{a}}_{i}=(\tilde{a}_{i1},\dots,\tilde{a}_{in})$ and
$\mathcal{U}_{i}$ is the Cartesian product of these intervals (a
hyperrectangle in $\mathbb{R}^{n}$). In order to cut off the extreme values of
this hyperrectangle, whose probability of occurrence can be negligible, one
can add a _budget_ to $\mathcal{U}_{i}$. This budget typically limits
deviations of scenarios from some nominal scenario
$\hat{\boldsymbol{a}}_{i}\in\mathcal{U}_{i}$ [5, 29]. In another approach
$\mathcal{U}_{i}$ is specified as an ellipsoid centered at some nominal
scenario $\hat{\boldsymbol{a}}_{i}$ [2]. Using the ellipsoidal uncertainty one
can model correlations between the components of $\tilde{\boldsymbol{a}}_{i}$.
One can also construct $\mathcal{U}_{i}$ from available data [4], which leads
to various _data-deriven robust models_.

Another approach to handle the uncertainty consists in modeling the uncertain
coefficients in (1) by using _fuzzy sets_. There are a lot of concepts of
solving fuzzy optimization problems (see, e.g., [20, 35, 26, 34, 37, 32, 31])
– for a comprehensive description of them we refer the reader to [28]. In one
of the most common class of models, $\tilde{\boldsymbol{a}}_{i}$ is a vector
of fuzzy intervals whose membership functions are interpreted as _possibility
distributions_ for the components of $\tilde{\boldsymbol{a}}_{i}$. A
description of _possibility theory_ that offers a general framework of dealing
with imprecise or incomplete knowledge can be found, for example, in [14]. It
uses two dual _possibility_ and _necessity measures_ to handle the
uncertainty. In the possibilistic setting one can replace the uncertain
constraints in (1) with _fuzzy chance constraints_ of the form [19, 27, 23]:

${\rm N}(\tilde{\boldsymbol{a}}^{T}_{i}\boldsymbol{x}\leq b_{i})\geq
1-\epsilon_{i},$

where ${\rm N}$ is a _necessity measure_ induced by a possibility distribution
for $\tilde{\boldsymbol{a}}_{i}$.

If a partial information about the probability distribution of
$\tilde{\boldsymbol{a}}_{i}$ is available, then the so-called
_distributionally robust approach_ can be used (see, e.g., [16, 39, 10]). In
this approach it is assumed that the true probability distribution ${\rm P}$
for $\tilde{\boldsymbol{a}}_{i}$ belongs to the so-called _ambiguity set_
$\mathcal{P}_{i}$ of probability distributions. For example, one can consider
all probability distributions given some bounds on its mean and covariance
matrix [10]. Alternatively, the ambiguity set can be specified by providing a
family of confidence sets (this approach will be adopted in this paper) [39].
Using the distributionally robust approach, one can consider the following
counterpart of the imprecise constraint
$\tilde{\boldsymbol{a}}_{i}^{T}\boldsymbol{x}\leq b_{i}$:

$\max_{{\rm P}\in\mathcal{P}_{i}}\mathrm{\mathbf{E}}_{\rm
P}[\tilde{\boldsymbol{a}}_{i}^{T}\boldsymbol{x}]\leq b_{i},$ (3)

where the left hand side is the expected value of the random variable
$\tilde{\boldsymbol{a}}_{i}^{T}\boldsymbol{x}$ under a worst probability
distribution ${\rm P}\in\mathcal{P}_{i}$. Observe that (3) reduces to (2) if
$\mathcal{P}_{i}$ contains all probability distributions with the support
$\mathcal{U}_{i}$.

In this paper we propose a new approach in the area of fuzzy optimization.
Will show how the distributionally robust optimization can be naturally used
in the setting of possibility theory. A preliminary version of the paper has
appeared in [17]. We will start by defining a possibility distribution (a
membership function of a fuzzy set) for $\tilde{\boldsymbol{a}}_{i}$. This can
be done by using both available data or experts knowledge. Following the
interpretation of possibility distribution [1, 13], we will define an
ambiguity set of probability distributions $\mathcal{P}_{i}$ for
$\tilde{\boldsymbol{a}}_{i}$. Finally, we will apply constraints of the type
(3) to compute a solution. We will show that the model of uncertainty assumed
leads to a family of linear or second order cone constraints. So, the
resulting problem is tractable for important cases of (1), for example for the
class of linear programming problems.

######  Remark 1.

If one uses (3), then the assumption that $\boldsymbol{c}$ and $b_{i}$,
$i\in[m]$, are precise in (1) causes no loss of generality. Indeed, if
$\tilde{\boldsymbol{c}}$ is imprecise, then we can minimize an auxiliary
variable $t$, subject to the imprecise constraint
$\tilde{\boldsymbol{c}}^{T}\boldsymbol{x}-t\leq 0$ ($t$ reflects the possible
realizations of objective function values). If $\tilde{b}_{i}$ is imprecise,
then the $i$th constraint can be rewritten as
$\tilde{\boldsymbol{a}}_{i}^{T}\boldsymbol{x}-\tilde{b}_{i}x_{n+1}\leq 0$,
where $x_{n+1}$ is an auxiliary variable such that $x_{n+1}=1$. In both cases,
we can replace the imprecise constraint with a one of the form (3).

This paper is organized as follows. In Section 2 we recall basic notions of
possibility theory, in particular its probabilistic interpretation assumed in
this paper. In Section 3 we discuss the discrete uncertainty representation in
which a possibility degree for each scenario is specified. Then the ambiguity
set $\mathcal{P}_{i}$ contains a family of discrete probability distributions
with a finite support $\mathcal{U}_{i}$. We will show that (3) is then
equivalent to a family of linear constraints. In Section 4 we consider the
interval uncertainty representation, in which unknown probability distribution
for $\tilde{\boldsymbol{a}}_{i}$ has a compact continuous support. We propose
a method of defining a continuous possibility distribution for
$\tilde{\boldsymbol{a}}_{i}$, which can be built by using available data or
experts knowledge. We show that (3) is then equivalent to a family of second
order cone constraints.

##  2 Possibility theory and imprecise probabilities

Let $\Omega$ be a set of alternatives. A primitive object of possibility
theory (see, e.g., [1]) is a _possibility distribution_
$\pi:\Omega\rightarrow[0,1]$, which assigns to each element $u\in\Omega$ a
_possibility degree_ $\pi(u)$. We only assume that $\pi$ is _normal_ , i.e.
there is $u\in\Omega$ such that $\pi(u)=1$. Possibility distribution can be
built by using available data or experts opinions (see, e.g., [14]). A
possibility distribution $\pi$ induces the following possibility and necessity
measures in $\Omega$:

$\Pi(A)=\sup_{u\in A}\pi(u),\;A\subseteq\Omega.$ ${\rm
N}(A)=1-\Pi(A^{c})=1-\sup_{u\in A^{c}}\pi(u),\;A\subseteq\Omega,$

where $A^{c}=\Omega\setminus A$ is the complement of $A$. In this paper we
assume that the possibility distribution $\pi$ represents uncertainty in
$\Omega$, i.e. some knowledge about uncertain quantity $\tilde{u}$ taking
values in $\Omega$. We now recall, following [13], a probabilistic
interpretation of the pair $[\Pi,{\rm N}]$ induced by the possibility
distribution $\pi$. Define

$\displaystyle\mathcal{P}(\pi)$ $\displaystyle=\\{{\rm P}:\forall A\text{
measurable }{\rm N}(A)\leq{\rm P}(A)\\}$ $\displaystyle=\\{{\rm P}:\forall
A\text{ measurable }\Pi(A)\geq{\rm P}(A)\\}.$ (4)

In this case $\sup_{{\rm P}\in\mathcal{P}(\pi)}{\rm P}(A)=\Pi(A)$ and
$\inf_{{\rm P}\in\mathcal{P}(\pi)}{\rm P}(A)={\rm N}(A)$ (see [13, 15, 9]).
So, the possibility distribution $\pi$ in $\Omega$ encodes a family of
probability measures in $\Omega$. Any probability measure ${\rm
P}\in\mathcal{P}(\pi)$ is said to be _consistent with_ $\pi$ and for any event
$A\subseteq\Omega$ the inequalities

${\rm N}(A)\leq{\rm P}(A)\leq\Pi(A)$ (5)

hold. A detailed discussion on the expressive power of (5) can also be found
in [38].

Observe that $\mathcal{P}(\pi)$ is nonempty due the assumption that $\pi$ is
normalized. Indeed, the probability distribution such that ${\rm
P}(\\{u\\})=1$ for some $u\in\Omega$ such that $\pi(u)=1$ is in
$\mathcal{P}(\pi)$. To see this consider two cases. If $u\notin A$, then ${\rm
N}(A)=0$ and ${\rm P}(A)\geq{\rm N}(A)=0$. If $u\in A$, then ${\rm
P}(A)=1\geq{\rm N}(A)$.

##  3 Discrete uncertainty model

Consider uncertain constraint $\tilde{\boldsymbol{a}}^{T}\boldsymbol{x}\leq
b$, where $\tilde{\boldsymbol{a}}$ is a random vector in $\mathbb{R}^{n}$ with
an unknown probability distribution (in order to simplify notation we skip the
index $i$ in the constraint). In this section we assume that the support
$\mathcal{U}$ of $\tilde{\boldsymbol{a}}$ is finite, i.e.
$\mathcal{U}=\\{\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{K}\\}$ is the set of
all possible, explicitly listed scenarios (realizations of
$\tilde{\boldsymbol{a}}$) which can occur with a positive probability. We thus
consider the discrete uncertainty model [25, 22]. Let $\mu$ be a membership
function of a fuzzy set in $\mathcal{U}$, $\mu:\mathcal{U}\rightarrow[0,1]$.
We will interpret $\mu$ as a possibility distribution $\pi$ in the scenario
set $\mathcal{U}$, i.e. $\pi=\mu$. In order to simplify presentation, we will
identify each scenario $\boldsymbol{c}_{i}\in\mathcal{U}$ with its index
$i\in[K]$. We can thus assume that $\pi$ is a possibility distribution in the
index set $[K]$, $\pi:[K]\rightarrow[0,1]$. The value of $\pi(i)$ is the
_possibility degree_ for scenario $\boldsymbol{a}_{i}$. Assume also w.l.o.g.
that $\pi(1)\geq\pi(2)\geq\dots\geq\pi(K)$ and thus $\pi(1)=1$.

Given $\pi$, we can now construct an ambiguity set $\mathcal{P}(\pi)$ of
probability distributions for $\tilde{\boldsymbol{a}}$ by using (4). Because
$\mathcal{P}(\pi)$ contains only discrete probability distributions in $[K]$,
we get

$\mathcal{P}(\pi)\subseteq\\{\boldsymbol{p}\in[0,1]^{K}\,:\sum_{i\in[K]}p_{i}=1\\}$

and we can describe $\mathcal{P}(\pi)$ by the following system of linear
constraints:

$\begin{array}[]{llll}\displaystyle\sum_{i\in A}p_{i}\geq 1-\max_{i\notin
A}\pi(i)&\forall\,A\subset[K],|A|\geq 1\\\
\displaystyle\sum_{i\in[K]}p_{i}=1\\\ p_{i}\geq 0&i\in[K]\end{array}$ (6)

In description (6) the first set of constraints model the condition ${\rm
P}(A)\geq 1-\Pi(A^{c})={\rm N}(A)\text{ for all events }A\subset[K]$,
($\Omega=[K]$). Notice that the formulation (6) has exponential number of
constraints. In the following, we will show how to reduce the number of
constraints to $O(K)$.

Let us partition the set of indices $[K]$ into $\ell$ disjoint sets
$I_{1},\dots,I_{\ell}$ such that the elements of $I_{j}$ have the same
possibility degrees and the possibility degrees of the elements in $I_{j}$ are
greater than the possibility degrees of the elements in $I_{k}$ for any $j>k$.
For example, let $\boldsymbol{\pi}=(1,1,0.5,0.5,0.3,0.3,0.3,0.1)$ be a
possibility distribution (a fuzzy set) in $[K]$ for $K=8$ (i.e. a possibility
distribution in the set of eight scenarios). Then $I_{1}=\\{1,2\\}$,
$I_{2}=\\{3,4\\}$, $I_{3}=\\{5,6,7\\}$, $I_{4}=\\{8\\}$. Let $\pi^{j}$ be the
possibility degree of the elements in $I_{j}$. In the example we have
$\pi^{1}=1$, $\pi^{2}=0.5$, $\pi^{3}=0.3$ and $\pi^{4}=0.1$.

######  Proposition 1.

System (6) is equivalent to the following system of constraints:

$\begin{array}[]{llll}\displaystyle\sum_{i\in I_{1}\cup\dots\cup
I_{j}}p_{i}\geq 1-\pi^{j+1}&j\in[\ell-1]\\\
\displaystyle\sum_{i\in[K]}p_{i}=1\\\ p_{i}\geq 0&i\in[K]\end{array}$ (7)

###### Proof.

It is easy to verify that each feasible solution to (6) is also feasible to
(7). Indeed, for $A=I_{1}\cup\dots\cup I_{j}$, $j\in[\ell-1]$, we get
$\max_{i\notin A}\pi(i)=\pi^{j+1}$. So, (7) is a subset of the constraints
(6). Assume that $\boldsymbol{p}$ is feasible to (7). Consider any set
$A\subseteq[K]$, $|A|\geq 1$. If $A=[K]$, then $\boldsymbol{p}$ is feasible to
(6). Assume that $A\subset[K]$ and let $k\in[\ell-1]$ be the first index such
that $I_{k}\not\subseteq A$. If $k=1$, then ${\rm N}(A)=0$ and the constraint
in (6) holds for $A$. Assume that $k>1$. The inequality from (7)

$\sum_{i\in I_{1}\cup\dots\cup I_{k-1}}p_{i}\geq 1-\pi^{k}$

implies

$\sum_{i\in A}p_{i}\geq\sum_{i\in I_{1}\cup\dots\cup I_{k-1}}p_{i}\geq
1-\pi^{k}=1-\max_{i\notin A}\pi(i),$

where the last equality results from the fact that $I_{k}\not\subseteq A$. In
consequence the constraint in (6) holds for $A$. ∎

For the sample possibility distribution
$\boldsymbol{\pi}=(1,1,0.5,0.5,0.3,0.3,0.3,0.1)$, the ambiguity set
$\mathcal{P}(\pi)$ can be described by the following system of constraints:

$\begin{array}[]{llll}p_{1}+p_{2}\geq 0.5\\\ p_{1}+p_{2}+p_{3}+p_{4}\geq
0.7\\\ p_{1}+p_{2}+p_{3}+p_{4}+p_{5}+p_{6}+p_{7}\geq 0.9\\\
p_{1}+p_{2}+p_{3}+p_{4}+p_{5}+p_{6}+p_{7}+p_{8}=1\\\ p_{i}\geq
0&i\in[K]\end{array}$

Using Proposition 1, the value of $\max_{{\rm
P}\in\mathcal{P}(\pi)}\mathrm{\mathbf{E}}_{\rm
P}[\tilde{\boldsymbol{a}}^{T}\boldsymbol{x}]$, for a given $\boldsymbol{x}$,
can be computed by using the following linear program:

$\begin{array}[]{lllll}\max\displaystyle\sum_{i\in[K]}p_{i}\boldsymbol{a}^{T}_{i}\boldsymbol{x}\\\
\displaystyle\sum_{i\in I_{1}\cup\dots\cup I_{j}}p_{i}\geq
1-\pi^{j+1}&j\in[\ell-1]\\\ \displaystyle\sum_{i\in[K]}p_{i}=1\\\ p_{i}\geq
0&i\in[K]\\\ \end{array}$ (8)

######  Theorem 1.

Given $\boldsymbol{x}\in\mathbb{X}$, the constraint

$\max_{{\rm P}\in\mathcal{P}(\pi)}\mathrm{\mathbf{E}}_{\rm
P}[\tilde{\boldsymbol{a}}^{T}\boldsymbol{x}]\leq b$ (9)

is equivalent to the following system of linear constraints:

$\begin{array}[]{lllll}\displaystyle\beta+\sum_{j\in[\ell-1]}\alpha_{j}(\pi^{j+1}-1)\leq
b\\\ \displaystyle\beta-\sum_{j\in[\ell-1]:i\in
I_{j}}\alpha_{j}\geq\boldsymbol{a}_{i}^{T}\boldsymbol{x}&i\in[K]\\\
\alpha_{j}\geq 0&j\in[\ell-1]\\\ \beta\in\mathbb{R}&\end{array}$ (10)

###### Proof.

For a fixed $\boldsymbol{x}\in\mathbb{X}$, (8) is a linear programming
problem, whose dual is:

$\begin{array}[]{lllll}\min&\displaystyle\beta+\sum_{j\in[\ell-1]}\alpha_{j}(\pi^{j+1}-1)\\\
&\displaystyle\beta-\sum_{j\in[\ell-1]:i\in
I_{j}}\alpha_{j}\geq\boldsymbol{a}_{i}^{T}\boldsymbol{x}&i\in[K]\\\
&\alpha_{j}\geq 0&j\in[\ell-1]\\\ &\beta\in\mathbb{R}&\end{array}$ (11)

where $\beta$ and $\alpha_{j}$ are dual variables. Now the strong duality for
linear programming shows that the optimal objective values of (8) and (11) are
the same (see, e.g., [30, Theorem 3.1]). This gives us (10). ∎

Consider two boundary cases of the proposed model. If $\pi(k)=1$ for some
$k\in[K]$, and $\pi(i)=0$ for each $i\neq k$, then $\boldsymbol{a}_{k}$ is the
only scenario which can occur. The constraint (3) reduces then to
$\boldsymbol{a}_{k}^{T}\boldsymbol{x}\leq b$. If $\pi(i)=1$ for all $i\in[K]$,
then $\mathcal{P}(\pi)=\\{\boldsymbol{p}\in[0,1]^{K}:\sum_{i\in[K]}p_{i}=1\\}$
and (3) becomes the strict robust constraint (2) for
$\mathcal{U}=\\{\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{K}\\}$. Theorem 1
yields the following corollary.

######  Corollary 1.

If $\mathbb{X}$ is a polyhedron described by a system of linear constraints
((1) is an uncertain linear programming problem). Then the deterministic
counterpart of (1) with the constraints of the type (9) and in consequence of
the type (10) is a linear programming problem.

Corollary 1 shows that the resulting deterministic counterpart of uncertain
linear programming problem (1), under the discrete uncertainty model, is
polynomially solvable and thus can be solved efficiently by standard off-the-
shelf solvers.

##  4 Interval uncertainty model

Let us again focus on an uncertain constraint
$\tilde{\boldsymbol{a}}^{T}\boldsymbol{x}\leq b$, where
$\tilde{\boldsymbol{a}}=(\tilde{a}_{1},\dots,\tilde{a}_{n})$ is a random
vector in $\mathbb{R}^{n}$ with an unknown probability distribution having now
a compact continuous support $\mathcal{U}\subseteq\mathbb{R}^{n}$. We will
start by describing the support $\mathcal{U}$. Next, we will propose a
continuous possibility distribution $\pi$ in $\mathcal{U}$ which will induce
an ambiguity set of probability distributions $\mathcal{P}(\pi)$ according to
(4). This possibility distribution can be provided by experts or can be
estimated by using available data. Assume that for each component
$\tilde{a}_{j}$ of $\tilde{\boldsymbol{a}}$ a _nominal_ (expected, the most
probable) value $\hat{a}_{j}$ and an interval
$[\hat{a}_{j}-\underline{a}_{j},\hat{a}_{j}+\overline{a}_{j}]$ containing
possible values of $\tilde{a}_{j}$ are known. Let

$\mathcal{B}(0)=[\hat{a}_{1}-\underline{a}_{1},\hat{a}_{1}+\overline{a}_{1}]\times\dots\times[\hat{a}_{n}-\underline{a}_{n},\hat{a}_{n}+\overline{a}_{n}],$

i.e. $\mathcal{B}(0)$ is a hyperrectangle containing possible scenarios
$\boldsymbol{a}$ (realizations of $\tilde{\boldsymbol{a}}$). The components of
$\tilde{\boldsymbol{a}}$ are often correlated. Also, the probability that a
subset of components will take the extreme values can be very small or even 0.
So, $\mathcal{B}(0)$ is typically an overestimation of the support
$\mathcal{U}$ of $\tilde{\boldsymbol{a}}$. In robust optimization some limit
for the deviations of scenarios from the nominal vector $\hat{\boldsymbol{a}}$
is often assumed, i.e. $\delta(\boldsymbol{a})\leq\Gamma$, where
$\delta(\boldsymbol{a})$ is a deviation prescribed. The parameter $\Gamma\geq
0$ is called a _budget_ and it controls the amount of uncertainty in
$\mathcal{U}$. One can define, for example,
$\delta(\boldsymbol{a})=||\boldsymbol{a}-\hat{\boldsymbol{a}}||_{1}$, which
leads to the continuous budgeted uncertainty [29]. In this paper, following
[2, 3] we use

$\delta(\boldsymbol{a})=||\boldsymbol{B}(\boldsymbol{a}-\hat{\boldsymbol{a}})||_{2},$

where $\boldsymbol{B}$ is a given $n\times n$ matrix. Notice that
$\delta(\boldsymbol{a})=(\boldsymbol{a}-\hat{\boldsymbol{a}})^{T}\boldsymbol{\Sigma}(\boldsymbol{a}-\hat{\boldsymbol{a}})$,
where $\boldsymbol{B}=\boldsymbol{\Sigma}^{\frac{1}{2}}$ for some postive
semidefinite matrix $\boldsymbol{\Sigma}$. For example, $\boldsymbol{\Sigma}$
can be an estimation of the covariance matrix for $\tilde{\boldsymbol{a}}$
(see, e.g., [6]). Therefore

$\mathcal{E}(0)=\\{\boldsymbol{a}\in\mathbb{R}^{n}:\delta(\boldsymbol{a})\leq\Gamma\\}$

is an ellipsoid in $\mathbb{R}^{n}$. We postulate that
$\mathcal{B}(0)\cap\mathcal{E}(0)$ contains all possible realizations of
$\tilde{\boldsymbol{a}}$, i.e. ${\rm P}(\mathcal{B}(0)\cap\mathcal{E}(0))=1$.
Hence $\mathcal{B}(0)\cap\mathcal{E}(0)$ is the estimation of the support of
$\tilde{\boldsymbol{a}}$. In the following, we will construct a possibility
distribution for $\tilde{\boldsymbol{a}}$, which can be easily built from
$\hat{a}_{j},\underline{a}_{j},\overline{a}_{j}$, $j\in[n]$, matrix
$\boldsymbol{B}$ and the budget $\Gamma$.

Figure 1: Fuzzy intervals
$\braket{\hat{a},\underline{a},\overline{a}}_{z_{1}-z_{2}}$ and
$\braket{0,0,\Gamma}_{z}$ modeling the possibility distributions for
$\tilde{a}_{j}$ and $\tilde{\delta}$.

Let $\pi_{\tilde{a}_{j}}$ be a possibility distribution for the component
$\tilde{a}_{j}$ of $\tilde{\boldsymbol{a}}$, such that that
$\pi_{\tilde{a}_{j}}(\hat{a}_{j})=1$,
$\pi_{\tilde{a}_{j}}(\hat{a}_{j}-\underline{a}_{j})=\pi_{\tilde{a}_{j}}(\hat{a}_{j}+\overline{a}_{j})=0$,
$\pi_{\tilde{a}_{j}}$ is continuous, strictly increasing in
$[\hat{a}_{j}-\underline{a}_{j},\hat{a}_{j}]$ and continuous strictly
decreasing in $[\hat{a}_{j},\hat{a}_{j}+\overline{a}_{j}]$. One can identify
the possibility distribution $\pi_{\tilde{a}_{j}}$ with membership function
$\mu_{\tilde{a}_{j}}$ of a fuzzy interval
$\braket{\hat{a}_{j},\underline{a}_{j},\overline{a}_{j}}_{z_{1}-z_{2}}$ shown
in Figure 1a (i.e. $\pi_{\tilde{a}_{j}}=\mu_{\tilde{a}_{j}}$). Let

$\tilde{a}_{j}(\lambda)=\\{a_{j}:\pi_{\tilde{a}_{j}}(a_{j})\geq\lambda\\}\;\;\lambda\in(0,1]$

be the $\lambda$-cut of $\tilde{a}_{j}$. It is easy to check that
$\tilde{a}_{j}(\lambda)=[\underline{a}_{j}(\lambda),\overline{a}_{j}(\lambda)]$
is an interval for each fixed $\lambda\in(0,1]$. Set
$[\underline{a}_{j}(0),\overline{a}_{j}(0)]=[\hat{a}_{i}-\underline{a}_{j},\hat{a}_{j}+\overline{a}_{j}]$.
The function $\underline{a}_{j}(\lambda)$, the left profile of
$\pi_{\tilde{a}_{j}}$, is continuous and strictly increasing and the function
$\overline{a}_{j}(\lambda)$, the right profile of $\pi_{\tilde{a}_{j}}$, is
continuous and strictly decreasing for $\lambda\in[0,1]$. For the fuzzy
interval
$\braket{\hat{a}_{j},\underline{a}_{j},\overline{a}_{j}}_{z_{1}-z_{2}}$, the
$\lambda$-cut of $\tilde{a}_{j}$ can be computed by using the following
formula:

$\tilde{a}_{j}(\lambda)=[\hat{a}_{j}-\underline{a}_{j}(1-\lambda^{z_{1}}),\hat{a}_{j}+\overline{a}_{j}(1-\lambda^{z_{2}})]\;\lambda\in[0,1].$

We will assume that $\underline{a}_{j},\overline{a}_{j},z_{1},z_{2}>0$. Then
$\tilde{a}_{j}(\lambda)$ are nondegenerate intervals (they have nonempty
interiors) for each $\lambda\in[0,1)$. The numbers $z_{1}$ and $z_{2}$ control
the amount of uncertainty for $\tilde{a}_{j}$ below and over $\hat{a}_{j}$,
respectively. In particular, when $z_{1},z_{2}\rightarrow 0$, then
$\tilde{a}_{j}$ becomes $\hat{a}_{j}$. On the other hand, if
$z_{1},z_{2}\rightarrow\infty$, then $\tilde{a}_{j}$ becomes the closed
interval $[\hat{a}_{j}-\underline{a}_{j},\hat{a}_{j}+\overline{a}_{j}]$ (see
Figure 1).

The deviation $\delta(\boldsymbol{a})$ depends on an unknown realization of
$\tilde{\boldsymbol{a}}$, so we can treat the deviation as an uncertain
quantity $\tilde{\delta}$. We can now build the following possibility
distribution for $\tilde{\delta}$: $\pi_{\tilde{\delta}}(0)=1$,
$\pi_{\tilde{\delta}}$ is continuous and strictly decreasing in $[0,\Gamma]$
and $\pi_{\tilde{\delta}}(\delta)=0$ if $\delta\notin[0,\Gamma]$. Let

$\tilde{\delta}(\lambda)=\\{\delta\in\mathbb{R}^{n}:\pi_{\tilde{\delta}}(\delta)\geq\lambda\\}=[0,\overline{\delta}(\lambda)]\;\;\lambda\in(0,1]$

be the $\lambda$-cut of $\tilde{\delta}$. Fix $\overline{\delta}(0)=\Gamma$.
We can identify $\pi_{\tilde{\delta}}$ with membership function
$\mu_{\tilde{\delta}}$ of a fuzzy interval $\tilde{\delta}$ given, for
example, in the form of $\braket{0,0,\Gamma}_{z}$ (see Figure 1b). The
$\lambda$-cut of $\tilde{\delta}$ is then
$\tilde{\delta}(\lambda)=[0,\Gamma(1-\lambda^{z})]$ for $\lambda\in[0,1]$.

We now build the following joint possibility distribution
$\pi:\mathbb{R}^{n}\rightarrow[0,1]$, for $\tilde{\boldsymbol{a}}$:

$\pi(\boldsymbol{a})=\min\\{\pi_{\tilde{a}_{1}}(a_{1}),\dots,\pi_{\tilde{a}_{n}}(a_{n}),\pi_{\tilde{\delta}}(\delta(\boldsymbol{a}))\\}.$
(12)

The value of $\pi(\boldsymbol{a})$ is the possibility degree for scenario
$\boldsymbol{a}\in\mathbb{R}^{n}$. The first part of the possibility
distribution, i.e.
$\min\\{\pi_{\tilde{a}_{1}}(a_{1}),\pi_{\tilde{a}_{2}}(a_{2}),\dots,\pi_{\tilde{a}_{n}}(a_{n})\\}$,
is built from the possibility distributions of non-interacting components
[11]. The second part $\pi_{\tilde{\delta}}(\delta(\boldsymbol{a}))$ is added
to model interactions between the components of $\tilde{\boldsymbol{a}}$.
Define the $\lambda$-cut of $\tilde{\boldsymbol{a}}$

$\mathcal{C}(\lambda)=\\{\boldsymbol{a}\in\mathbb{R}^{n}:\pi(\boldsymbol{a})\geq\lambda\\},\;\;\lambda\in(0,1].$

It is easily seen that

$\displaystyle\mathcal{C}(\lambda)=\\{\boldsymbol{a}\in\mathbb{R}^{n}\,:\,$
$\displaystyle
a_{j}\in[\underline{a}_{j}(\lambda),\overline{a}_{j}(\lambda)],\;j\in[n],$
$\displaystyle||\boldsymbol{B}(\boldsymbol{a}-\hat{\boldsymbol{a}})||_{2}\leq\overline{\delta}(\lambda)\\}\;\lambda\in(0,1].$

Set $\mathcal{C}(0)=\mathcal{B}(0)\cap\mathcal{E}(0)$. Observe that
$\mathcal{C}(1)=\\{\hat{\boldsymbol{a}}\\}$. Furthermore
$\mathcal{C}(\lambda)$, $\lambda\in[0,1]$, is a family of nested sets, i.e.
$\mathcal{C}(\lambda_{1})\subset\mathcal{C}(\lambda_{2})$ if
$\lambda_{1}>\lambda_{2}$. By the continuity and monotonicity of
$\underline{a}_{j}(\lambda)$, $\overline{a}_{j}(\lambda)$ and
$\overline{\delta}(\lambda)$ we get

${\rm N}(\mathcal{C}(\lambda))=1-\lambda,\;\;\lambda\in[0,1].$ (13)

Notice also that $\mathcal{C}(\lambda)$ is a closed convex set for each
$\lambda\in[0,1]$ and has nonempty interior for all $\lambda\in[0,1)$.

######  Proposition 2.

The following equality

$\mathcal{P}(\pi)=\\{{\rm P}\in\mathcal{PM}(\mathbb{R}^{n}):{\rm
P}(\mathcal{C}(\lambda))\geq 1-\lambda,\lambda\in[0,1]\\}$

holds, where $\mathcal{PM}(\mathbb{R}^{n})$ is the set of all probability
measures on $\mathbb{R}^{n}$.

###### Proof.

Let $\mathcal{P}^{\prime}(\pi)=\\{{\rm P}\in\mathcal{PM}(\mathbb{R}^{n}):{\rm
P}(\mathcal{C}(\lambda))\geq 1-\lambda,\lambda\in[0,1]\\}$. We need to show
that $\mathcal{P}^{\prime}(\pi)=\mathcal{P}(\pi)$. Observe first that
$\mathcal{P}(\pi)\subseteq\mathcal{P}^{\prime}(\pi)$. To see this choose any
probability distribution ${\rm P}\in\mathcal{P}(\pi)$. Since
$\mathcal{C}(\lambda)\subseteq\mathbb{R}^{n}$ is an event for each
$\lambda\in[0,1]$, we get ${\rm P}(\mathcal{C}(\lambda))\geq{\rm
N}(\mathcal{C}(\lambda))=1-\lambda$ for each $\lambda\in[0,1]$ (see equation
(13)). In consequence ${\rm P}\in\mathcal{P}^{\prime}(\pi)$. We now show that
$\mathcal{P}^{\prime}(\pi)\subseteq\mathcal{P}(\pi)$. Let ${\rm
P}\in\mathcal{P}^{\prime}(\pi)$ and consider event $A\subseteq\mathbb{R}^{n}$.
If $\mathcal{C}(1)=\\{\hat{\boldsymbol{a}}\\}\not\subseteq A$, then ${\rm
N}(A)=0$ and ${\rm P}(A)\geq{\rm N}(A)=0$. Suppose $\mathcal{C}(1)\subseteq A$
and let

$\lambda^{*}=\inf\\{\lambda\in[0,1]:\mathcal{C}(\lambda)\subseteq A\\}.$

Then ${\rm P}(A)\geq{\rm P}(\mathcal{C}(\lambda^{*}))\geq 1-\lambda^{*}={\rm
N}(\mathcal{C}(\lambda^{*})).$ Choose $\lambda^{\prime}:=\lambda^{*}-\epsilon$
for arbitrarily small $\epsilon>0$. As
$\mathcal{C}(\lambda^{\prime})\not\subseteq A$, there is scenario
$\boldsymbol{a}$ such that $\boldsymbol{a}\notin A$ and
$\pi(\boldsymbol{a})\geq\lambda^{\prime}$. But $\boldsymbol{a}\in A$ for each
scenario $\boldsymbol{a}$ such that $\pi(\boldsymbol{a})\geq\lambda^{*}$.
Consequently $\sup_{\boldsymbol{a}\notin
A}\pi(\boldsymbol{a})\in[\lambda^{\prime},\lambda^{*}]$. Because
$\epsilon\rightarrow 0$, ${\rm N}(A)=1-\lambda^{*}$. Hence ${\rm P}(A)\geq{\rm
N}(A)$ and ${\rm P}\in\mathcal{P}(\pi)$. ∎

In order to construct a tractable reformulation of (3) for $\mathcal{P}(\pi)$
we need to use a discretization of $\mathcal{P}(\pi)$. Choose integer
$\ell\geq 0$ and set $\Lambda=\\{0,1,\dots,\ell\\}$. Define
$\lambda_{i}=i/\ell$, $i\in\Lambda$. We consider the following ambiguity set:

$\mathcal{P}^{\ell}(\pi)=\\{{\rm P}\in\mathcal{PM}(\mathbb{R}^{n}):{\rm
P}(\mathcal{C}(\lambda_{i}))\geq 1-\lambda_{i}:i\in\Lambda\\}.$ (14)

Set $\mathcal{P}^{\ell}(\pi)$ is a discrete approximation of
$\mathcal{P}(\pi)$. It is easy to verify that
$\mathcal{P}(\pi)\subseteq\mathcal{P}^{\ell}(\pi)$ for any $\ell\geq 0$. By
fixing sufficiently large constant $\ell$, we obtain arbitrarily close
approximation of $\mathcal{P}(\pi)$.

Figure 2: The ambiguity set $\mathcal{P}^{2}(\pi)$, ${\rm
P}(\mathcal{C}(0))\geq 1$, ${\rm P}(\mathcal{C}(0.5))\geq 0.5$, ${\rm
P}(\mathcal{C}(1))\geq 0$

######  Theorem 2.

The constraint

$\max_{{\rm P}\in\mathcal{P}^{\ell}(\pi)}\mathrm{\mathbf{E}}_{\rm
P}[\tilde{\boldsymbol{a}}^{T}\boldsymbol{x}]\leq b$ (15)

is equivalent to the following system of second-order cone constraints:

$\begin{array}[]{llll}\displaystyle
w+\sum_{i\in\Lambda}(\lambda_{i}-1)v_{i}\leq b\\\
\displaystyle\gamma_{i}\overline{\delta}(\lambda_{i})+\sum_{j\in[n]}\alpha_{ij}(\overline{a}_{j}(\lambda_{i})-\hat{a}_{j})+\sum_{j\in[n]}\beta_{ij}(\hat{a}_{j}-\underline{a}_{j}(\lambda_{i}))+\\\
\begin{array}[]{lll}+\hat{\boldsymbol{a}}^{T}\boldsymbol{x}\leq\displaystyle
w-\sum_{j\leq i}v_{j}&i\in\Lambda\\\
\alpha_{ij}-\beta_{ij}+\boldsymbol{B}^{T}_{j}\boldsymbol{u}_{i}=x_{j}&i\in\Lambda,j\in[n]\\\
||\boldsymbol{u}_{i}||_{2}\leq\gamma_{i}&i\in\Lambda\\\
\alpha_{ij},\beta_{ij}\geq 0&i\in\Lambda,j\in[n]\\\ v_{i},\gamma_{i}\geq
0&i\in\Lambda\\\
w\in\mathbb{R},\boldsymbol{u}_{i}\in\mathbb{R}^{n}&i\in\Lambda\end{array}\end{array}$
(16)

where $\Lambda=\\{0,1,\dots,\ell\\}$ and $\boldsymbol{B}_{j}$ is the $j$th
column of $\boldsymbol{B}$.

###### Proof.

The proof is adapted from [39]. The left hand side of (15) can be expressed as
the _problem of moments_ (see, e.g., [36, 39]):

$\begin{array}[]{lll}\max&\displaystyle\int_{\mathcal{C}(\lambda_{0})}\boldsymbol{a}^{T}\boldsymbol{x}\;{\rm
d}\mu(\boldsymbol{a})\\\
&\displaystyle\int_{\mathcal{C}(\lambda_{0})}\boldsymbol{1}_{[\boldsymbol{a}\in\mathcal{C}(\lambda_{i})]}\;{\rm
d}\mu(\boldsymbol{a})\geq 1-\lambda_{i}&i\in\Lambda\\\
&\displaystyle\int_{\mathcal{C}(\lambda_{0})}{\rm d}\mu(\boldsymbol{a})=1\\\
&\mu\in\mathcal{M}_{+}(\mathbb{R}^{n}),\end{array}$ (17)

where $\mathcal{M}_{+}(\mathbb{R}^{n})$ is the set of all nonnegative measures
on $\mathbb{R}^{n}$. Notice that the second (equality) constraint implies
$\mu$ is a probability measure supported on $\mathcal{C}(\lambda_{0})$. The
dual of the problem of moments takes the following form (see, e.g., [24]):

$\begin{array}[]{llll}\min&\displaystyle
w+\sum_{i\in\Lambda}(\lambda_{i}-1)v_{i}\\\ &\displaystyle
w-\sum_{i\in\Lambda}\boldsymbol{1}_{[\boldsymbol{a}\in\mathcal{C}(\lambda_{i})]}v_{i}\geq\boldsymbol{a}^{T}\boldsymbol{x}&\forall\boldsymbol{a}\in\mathcal{C}(\lambda_{0})\\\
&v_{i}\geq 0&i\in\Lambda\\\ &w\in\mathbb{R}&\end{array}$ (18)

Strong duality implies that the optimal objective values of (17) and (18) are
the same (see, e.g., [24, Theorem 1, Corollary 1]). Define

$\overline{\mathcal{C}}(\lambda_{i})=\mathcal{C}(\lambda_{i})\setminus\mathcal{C}(\lambda_{i+1}),\;i\in[\ell-1]$

and $\overline{\mathcal{C}}(\lambda_{\ell})=\mathcal{C}(\lambda_{\ell})$.
Hence $\overline{C}(\lambda_{i})$, $i\in\Lambda$, form a partition of the
support $\mathcal{C}(\lambda_{0})$ into $\ell+1$ disjoint sets. Model (18) can
be then rewritten as follows:

$\begin{array}[]{llll}\min&\displaystyle
w+\sum_{i\in\Lambda}(\lambda_{i}-1)v_{i}\\\ &\displaystyle
w-\sum_{i\in\Lambda}\boldsymbol{1}_{[\boldsymbol{a}\in\mathcal{C}(\lambda_{i})]}v_{i}\geq\boldsymbol{a}^{T}\boldsymbol{x}&\forall\boldsymbol{a}\in\overline{\mathcal{C}}(\lambda_{i}),i\in\Lambda\\\
&v_{i}\geq 0&i\in\Lambda\\\ &w\in\mathbb{R}&\end{array}$ (19)

which is equivalent to

$\begin{array}[]{llll}\min&\displaystyle
w+\sum_{i\in\Lambda}(\lambda_{i}-1)v_{i}\\\ &\displaystyle w-\sum_{j\leq
i}v_{j}\geq\boldsymbol{a}^{T}\boldsymbol{x}&\forall\boldsymbol{a}\in\overline{\mathcal{C}}(\lambda_{i}),i\in\Lambda\\\
&v_{i}\geq 0&i\in\Lambda\\\ &w\in\mathbb{R}&\end{array}$ (20)

The $i$th constraint in (20) can be reformulated as

$\max_{\boldsymbol{a}\in\overline{\mathcal{C}}(\lambda_{i})}\boldsymbol{a}^{T}\boldsymbol{x}\leq
w-\sum_{j\leq i}v_{j}.$ (21)

By the linearity of $\boldsymbol{a}^{T}\boldsymbol{x}$ in $\boldsymbol{a}$,
the left hand side is maximized at the boundary of
$\overline{\mathcal{C}}(\lambda_{i})$, which coincides with the boundary of
$\mathcal{C}(\lambda_{i})$. Hence (21) is equivalent to

$\max_{\boldsymbol{a}\in\mathcal{C}(\lambda_{i})}\boldsymbol{a}^{T}\boldsymbol{x}\leq
w-\sum_{j\leq i}v_{j}.$

The left hand side of this inequality, by definition of
$\mathcal{C}(\lambda_{i})$, yields the following problem:

$\begin{array}[]{lll}\max&\boldsymbol{a}^{T}\boldsymbol{x}\\\
&a_{j}\leq\overline{a}_{j}(\lambda_{i})&j\in[n]\\\
&-a_{j}\leq-\underline{a}_{j}(\lambda_{i})&j\in[n]\\\
&\displaystyle||\boldsymbol{B}(\boldsymbol{a}-\hat{\boldsymbol{a}})||_{2}\leq\overline{\delta}(\lambda_{i})&\\\
&\boldsymbol{a}\in\mathbb{R}^{n}&\end{array}$

Let us substitute $\boldsymbol{y}=\boldsymbol{a}-\hat{\boldsymbol{a}}$, which
yields:

$\begin{array}[]{lll}\max&\boldsymbol{y}^{T}\boldsymbol{x}+\hat{\boldsymbol{a}}^{T}\boldsymbol{x}\\\
&y_{j}\leq-\hat{a}_{j}+\overline{a}_{j}(\lambda_{i})&j\in[n]\\\
&-y_{j}\leq-\underline{a}_{j}(\lambda_{i})+\hat{a}_{j}&j\in[n]\\\
&\displaystyle||\boldsymbol{B}\boldsymbol{y}||_{2}\leq\overline{\delta}(\lambda_{i})&\\\
&\boldsymbol{y}\in\mathbb{R}^{n}&\end{array}$

Introducing new variables $\boldsymbol{z}$ leads to the following model:

$\begin{array}[]{lll}\max&\boldsymbol{y}^{T}\boldsymbol{x}+\hat{\boldsymbol{a}}^{T}\boldsymbol{x}\\\
&y_{j}\leq-\hat{a}_{j}+\overline{a}_{j}(\lambda_{i})&j\in[n]\\\
&-y_{j}\leq-\underline{a}_{j}(\lambda_{i})+\hat{a}_{j}&j\in[n]\\\
&\boldsymbol{B}\boldsymbol{y}-\boldsymbol{z}=\boldsymbol{0}&\\\
&\displaystyle||\boldsymbol{z}||_{2}\leq\overline{\delta}(\lambda_{i})&\\\
&\boldsymbol{y},\boldsymbol{z}\in\mathbb{R}^{n}&\end{array}$ (22)

The dual to (22) is (see the Appendix):

$\begin{array}[]{lll}\min&\displaystyle\gamma\overline{\delta}(\lambda_{i})+\sum_{j=1}^{n}\alpha_{j}(\overline{a}_{j}(\lambda_{i})-\hat{a}_{j})+\sum_{j=1}^{n}\beta_{j}(\hat{a}_{j}-\underline{a}_{j}(\lambda_{i}))+\\\
&+\hat{\boldsymbol{a}}^{T}\boldsymbol{x}\\\
&\begin{array}[]{ll}\alpha_{j}-\beta_{j}+\boldsymbol{B}^{T}_{j}\boldsymbol{u}=x_{j}&j\in[n]\\\
||\boldsymbol{u}||_{2}\leq\gamma\\\ \alpha_{j},\beta_{j},\gamma\geq
0&j\in[n]\\\ \boldsymbol{u}\in\mathbb{R}^{n}&\end{array}\end{array}$ (23)

where $\boldsymbol{B}_{j}$ is the $j$th column of $\boldsymbol{B}$. The strong
duality (see Appendix) implies that (22) and (23) have the same optimal
objective function values. Model (23) together with (20) yield (16). ∎

Theorem 2 leads to the following corollary:

######  Corollary 2.

If $\mathbb{X}$ is a polyhedron described by a system of linear constraints
(i.e. (1) is an uncertain linear programming problem), then the deterministic
counterpart of (1) with the constraints of the type (15) is a second-order
cone program.

Corollary 2 shows that the resulting deterministic counterpart of uncertain
linear programming problem (1), under the interval uncertainty model, is
polynomially solvable (see, e.g., [7]) and thus can solve efficiently by some
standard off-the-shelf solvers like IBM ILOG CPLEX [18].

Consider a sample of optimization problem (1) in which the value of the
objective function $\tilde{a}_{1}x_{1}+\tilde{a}_{2}x_{2}$ is uncertain and
two constraints: $x_{1}\geq 2.74$ and $x_{2}\geq 3.3$ are precisely known. The
uncertainty of the coefficients $\tilde{a}_{1}$ and $\tilde{a}_{2}$ is modeled
by fuzzy intervals $\tilde{a}_{1}=\braket{3,2.5,2.5}_{1-0.32}$ and
$\tilde{a}_{2}=\braket{2,1,1}_{1-1}$, respectively, regarded as possibility
distributions for their values. A possibility distribution for uncertain
deviation $\tilde{\delta}$ is prescribed by fuzzy interval
$\tilde{\delta}=\braket{0,0,6}_{1}$, $\Gamma=6$, and
$\boldsymbol{B}=\left[\begin{array}[]{ll}2&2.5\\\ 1&-3\end{array}\right]$. By
(12), we obtain a joint possibility distribution $\pi$ for
$\tilde{\boldsymbol{a}}=(\tilde{a}_{1},\tilde{a}_{2})$. Fix $\ell=2$, which
leads to the ambiguity set $\mathcal{P}^{2}(\pi)$ for
$\tilde{\boldsymbol{a}}=(\tilde{a}_{1},\tilde{a}_{2})$ shown in Figure 2,
built according to (14). In particular, the set $\mathcal{C}(0)$, being the
intersection of $[0.5,5.5]\times[1,3]$ and the ellipse
$\\{\boldsymbol{a}:||\boldsymbol{B}(\boldsymbol{a}-\hat{\boldsymbol{a}})||_{2}\leq
6\\}$ contains all possible scenarios. Applying now the distributionally
robust approach in the possibilistic setting gives:

$\begin{array}[]{lll}\min&\displaystyle\max_{{\rm
P}\in\mathcal{P}^{\ell}(\pi)}\mathrm{\mathbf{E}}_{\rm
P}[\tilde{a}_{1}x_{1}+\tilde{a}_{2}x_{2}]\\\ &x_{1}\geq 2.74\\\ &x_{2}\geq
3.3\end{array}$ (24)

Remark 1 shows how to convert the above problem with the uncertain objective
function to the one with the precise objective function and the uncertain
constraint. Theorem 2 (see Corollary 2) now leads to a second-order cone
program that is equivalent to (24). An optimal solution of the program is
$x_{1}=2.74$ and $x_{2}=3.3$. Let us focus on evaluating the objective
function

$\max_{{\rm P}\in\mathcal{P}^{2}(\pi)}\mathrm{\mathbf{E}}_{\rm
P}[2.74\tilde{a}_{1}+3.3\tilde{a}_{2}].$

The worst probability distribution ${\rm P}$ in $\mathcal{P}^{2}(\pi)$ assigns
probability 0.5 to scenario $(5.15,2.68)\in\mathcal{C}(0)$ and probability 0.5
to scenario $(3.5,2.5)$ in $\mathcal{C}(0.5)$. These two points can be
obtained by maximizing the linear function $2.74a_{1}+3.3a_{2}$ over the
convex sets $\mathcal{C}(0)$ and $\mathcal{C}(0.5)$, respectively (see Figure
2). We thus get $\max_{{\rm P}\in\mathcal{P}^{2}(\pi)}\mathrm{\mathbf{E}}_{\rm
P}[2.74\tilde{a}_{1}+3.3\tilde{a}_{2}]=0.5\cdot(2.74\cdot 5.15+3.3\cdot
2.68)+0.5\cdot(2.74\cdot 3.5+3.3\cdot 2.5)=20.39$.

##  5 Risk aversion modeling

In this section we will show how the ambiguity set $\mathcal{P}^{\ell}(\pi)$
can be relaxed to take individual decision maker risk aversion into account.
Observe that the lower bounds on the probabilities for the sets
$\mathcal{C}(\lambda_{i})$ in $\mathcal{P}_{\pi}^{\ell}$ change uniformly. In
consequence, a worst probability distribution uniformly assigns probabilities
to points on the boundaries of
$\mathcal{C}(\lambda_{0}),\dots,\mathcal{C}(\lambda_{\ell-1})$. An example is
shown in Figure 2, where (for $\ell=2$) probability 0.5 is assigned to points
on the boundaries of $\mathcal{C}(0)$ and $\mathcal{C}(0.5)$. It has been
observed that risk averse decision makers can perceive worst coefficient
realizations as more probable (see, e.g., [33, 12]). We now propose an method
of taking this risk-aversion into account. Let

$g_{\rho}(z)=\frac{1}{1-\rho}(1-\rho^{z})$

for $\rho\in(0,1)$ and $z\in[0,1]$. The function $g_{\rho}(z)$ is continuous,
concave, increasing in $[0,1]$ and such that $g_{\rho}(0)=0$ and
$g_{\rho}(1)=1$. If $\rho\rightarrow 0$, then $g_{\rho}(z)\rightarrow 1$ for
each $z\in(0,1]$. On the other hand, if $\rho\rightarrow 1$, then
$g_{\rho}(z)\rightarrow z$ for each $z\in[0,1]$, so $g_{\rho}(z)$ tends to a
linear function (see Fig. 3).

Figure 3: Function $g_{\rho}(z)=\frac{1}{1-\rho}(1-\rho^{z})$.

Fix $\rho\in(0,1)$ and consider the following ambiguity set:

$\mathcal{P}^{\ell,\rho}(\pi)=\left\\{{\rm
P}\in\mathcal{PM}(\mathbb{R}^{n}):{\rm P}(\mathcal{C}(\lambda_{i}))\geq
1-g_{\rho}(\lambda_{i}),\;i\in\Lambda\right\\}.$

Since $1-\lambda_{i}\geq 1-g_{\rho}(\lambda_{i})$, we still get a valid bound
${\rm P}(\mathcal{C}(\lambda_{i}))\geq{\rm
N}(\mathcal{C}(\lambda_{i}))=1-\lambda_{i}\geq 1-g_{\alpha}(\lambda_{i})$.
Notice that the lower bound can be now smaller (i.e. larger probabilities can
be assigned to worse scenarios), which reflects the probability distortion.

As $\mathcal{P}_{\pi}^{\ell}\subseteq\mathcal{P}^{\ell,\rho}_{\pi}$ for each
$\rho\in(0,1)$ and
$\mathcal{P}^{\ell,\rho}_{\pi}\subseteq\mathcal{P}^{\ell,\rho^{\prime}}_{\pi}$
if $\rho\geq\rho^{\prime}$, decreasing $\rho$ leads to more conservative
constraint (3). If $\rho\rightarrow 1$, then
$\mathcal{P}^{\ell,\rho}_{\pi}=\mathcal{P}_{\pi}^{\ell}$. On the other hand,
if $\rho\rightarrow 0$, then $\mathcal{P}^{\ell,\rho}_{\pi}$ is the set of all
probability measures with support $\mathcal{C}(0)$, and the constraint (3)
becomes

$\max_{\boldsymbol{a}\in\mathcal{C}(0)}\boldsymbol{a}^{T}\boldsymbol{x}\leq
b,$

which is equivalent to the strict robust constraint (2). Theorem 2 remains
valid for the ambiguity set $\mathcal{P}^{\ell,\rho}_{\pi}$. It is enough to
replace the constant $\lambda_{i}$ with the constant $g_{\rho}(\lambda_{i})$
in the first inequality in (16).

##  6 Application to portfolio selection

In this section we will apply the model constructed in Section 4 (see also
Section 5) to a portfolio selection problem. We are given a set of $n$ assets
whose returns form a random vector
$\tilde{\boldsymbol{a}}=(\tilde{a}_{1},\dots,\tilde{a}_{n})$. A portfolio is a
vector $\boldsymbol{x}\in[0,1]^{n}$, where $x_{i}$ is the share of the $i$th
asset. Then $\tilde{\boldsymbol{a}}^{T}\boldsymbol{x}$ is the uncertain return
of portfolio $\boldsymbol{x}$ (accordingly, the quantity
$-\tilde{\boldsymbol{a}}^{T}\boldsymbol{x}$ is the uncertain loss for
$\boldsymbol{x}$). The probability distribution for $\tilde{\boldsymbol{a}}$
is unknown. However, we have a sample of past realizations
$\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{K}$ of $\tilde{\boldsymbol{a}}$.
Using this sample we can estimate the mean return $\hat{\boldsymbol{a}}$ and
the covariance matrix $\boldsymbol{\Sigma}$ for $\tilde{\boldsymbol{a}}$. The
sample mean and covariance matrix for 7 assets, computed for a sample of 30
subsequent observations in Polish stock market, are

$\hat{\boldsymbol{a}}=[0.057,-0.378,0.324,-0.799,-0.873,-0.271,-0.323]$

${\footnotesize\boldsymbol{\Sigma}=\left[\begin{array}[]{rrrrrrrr}7.469&0.149&0.099&0.076&2.225&0.044&1.649\\\
&0.967&0.865&-0.578&-1.558&0.053&-0.143\\\
&&3.714&-0.454&-1.265&1.188&0.320\\\ &&&2.188&-0.529&-0.152&0.525\\\
&&&&18.168&-1.561&4.558\\\ &&&&&12.745&1.391\\\
&&&&&&5.371\end{array}\right]}$

We fix $\boldsymbol{B}=\boldsymbol{\Sigma}^{\frac{1}{2}}$, $\ell=100$,
$\underline{a}_{j}=\overline{a}_{j}=6\cdot\sigma_{j}$, where
$\sigma_{j}=\sqrt{\Sigma_{jj}}$ for each $j\in[n]$. Using Chebyshev’s
inequality one can show that
$\tilde{a}_{j}\in[\hat{a}_{j}-\underline{a}_{j},\hat{a}_{j}+\overline{a}_{j}]$
with high probability. We now use the fuzzy intervals
$\braket{\hat{a}_{j},\underline{a}_{j},\overline{a}_{j}}_{1-1}$ to model the
possibility distributions for the asset returns. We use linear membership
functions. One can, however, further refine the information for the return
values using different shapes, i.e. changing the parameters $z_{1}$ and
$z_{2}$. We use the fuzzy interval $\braket{0,\Gamma}_{1}$ for
$\Gamma\in[0,50]$ to model the possibility distribution for the deviation
$\tilde{\delta}$. We consider the following problem:

$\begin{array}[]{lllll}\displaystyle\min&\displaystyle\max_{{\rm
P}\in\mathcal{P}^{\ell}(\pi)}\mathrm{\mathbf{E}}_{\rm
P}[-\tilde{\boldsymbol{a}}^{T}\boldsymbol{x}]\\\
&\boldsymbol{1}^{T}\boldsymbol{x}=1\\\
&\boldsymbol{x}\geq\boldsymbol{0}\end{array}$ (25)

Figure 4: Optimal objective values for various $\Gamma$.

Figure 4 shows the optimal objective value of (25) for $\Gamma=0,1,\dots,50$.
In the first boundary case, $\Gamma=0$, there is no uncertainty and in the
optimal portfolio all is allocated to the asset with the largest expected
return (smallest expected loss). In the second boundary case, when $\Gamma\geq
48$, there is no limit on deviation from $\hat{\boldsymbol{a}}$ and all is
allocated to the asset with the smallest $\hat{a}_{j}+\overline{a}_{j}$. For
the intermediate values of $\Gamma$, we get a family of various portfolios in
which a diversification is profitable. One such portfolio is shown in Figure
4.

Figure 5 shows the effect of taking the individual risk aversion into account.
In this figure the optimal objective value of (25) for $\rho\in(0,1)$ is shown
(we fix $\Gamma=20$). For smaller values of $\rho$ the objective value of (25)
is larger. One can see that the optimal portfolio is adjusted (see Figure 5)
to take larger sets of admissible probability distributions into account.

Figure 5: Optimal objective values for various $\rho$ and $\Gamma=20$.

##  7 Conclusions

In this paper we have proposed a method of unifying the fuzzy (possibilistic)
optimization with the distributionally robust approach. Using the connections
between the possibility and probability theories one can build a set of
probability distributions for uncertain constraint coefficient, based on a
possibilistic information. The uncertain constraint, the left-hand side, can
be then replaced with the expected value with respect to the worst probability
distribution that can occur. In practical applications the true probability
distributions for the uncertain data are often unknown. However, in most cases
some information about the possible scenarios (parameter realizations) is
available. In this case the possibility theory is an attractive choice,
because it is flexible and allows to take both available data and experts’
opinions into account. In this paper we have proposed some methods of defining
possibility distributions for scenarios, which is easy to apply in practice.
Furthermore, the resulting deterministic counterpart of the problem can be
solved efficiently for a large class of problems (for example for linear
programming).

The key element of our model is the definition of a joint possibility
distribution in scenario set. This step is quite flexible, as there are only
minor restrictions on possibility distribution function which must be imposed.
The support of this possibility distribution should contain all possible
scenarios and a budget should control the magnitude of deviations of scenarios
from some nominal (expected) one. We believe that such possibility
distribution can be a good tool to handle the uncertainty in optimization
problems.

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## Appendix

###### The proof of (23).

Let us rewrite (22) as follows:

$\begin{array}[]{lll}\min&-\boldsymbol{y}^{T}\boldsymbol{x}-\hat{\boldsymbol{a}}^{T}\boldsymbol{x}\\\
&y_{j}\leq-\hat{a}_{j}+\overline{a}_{j}(\lambda_{i})&j\in[n]\\\
&-y_{j}\leq-\underline{a}_{j}(\lambda_{i})+\hat{a}_{j}&j\in[n]\\\
&\boldsymbol{B}\boldsymbol{y}-\boldsymbol{z}=\boldsymbol{0}&\\\
&\displaystyle||\boldsymbol{z}||_{2}\leq\overline{\delta}(\lambda_{i})&\\\
&\boldsymbol{y},\boldsymbol{z}\in\mathbb{R}^{n}&\end{array}$ (26)

and write the Lagrangian function

$\begin{array}[]{lllll}L(\boldsymbol{y},\boldsymbol{z},\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{u},\gamma)=&-\boldsymbol{y}^{T}\boldsymbol{x}-\hat{\boldsymbol{a}}^{T}\boldsymbol{x}+\\\
&\displaystyle\sum_{j\in[n]}\alpha_{j}(y_{j}+\hat{a}_{j}-\overline{a}_{j}(\lambda_{i}))+\\\
&\displaystyle\sum_{j\in[n]}\beta_{j}(-y_{j}+\underline{a}_{j}(\lambda_{i})-\hat{a}_{j})+\\\
&(\boldsymbol{B}\boldsymbol{y}-\boldsymbol{z})^{T}\boldsymbol{u}+\gamma(||\boldsymbol{z}||_{2}-\overline{\delta}(\lambda_{i})),\end{array}$

where $\alpha_{j},\beta_{j},\gamma\geq 0$ and
$\boldsymbol{u}\in\mathbb{R}^{n}$ are the Lagrangian multipliers. After
rearranging the terms we get

$\begin{array}[]{lll}L(\boldsymbol{y},\boldsymbol{z},\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{u},\gamma)=&-\hat{\boldsymbol{a}}^{T}\boldsymbol{x}+\\\
&\displaystyle\sum_{j\in[n]}(-x_{j}+\alpha_{j}-\beta_{j}+\boldsymbol{B}^{T}_{j}\boldsymbol{u})y_{j}+\\\
&(\gamma||\boldsymbol{z}||_{2}-\boldsymbol{z}^{T}\boldsymbol{u})-\\\
&\displaystyle\sum_{j\in[n]}\alpha_{j}(\hat{a}_{j}-\overline{a}_{j}(\lambda_{i}))+\\\
&\displaystyle\sum_{j\in[n]}\beta_{j}(\underline{a}_{j}(\lambda_{i})-\hat{a}_{j})-\gamma\overline{\delta}(\lambda_{i}).\end{array}$

Problem (26) is convex and the set of feasible solutions is bounded and has
nonempty interior. Hence it has an optimal solution with the finite objective
function value $v^{*}$. The problem also satisfies the strong duality (see,
e.g., [7]), i.e.

$\max_{\boldsymbol{\alpha}\geq\boldsymbol{0},\boldsymbol{\beta}\geq\boldsymbol{0},\boldsymbol{u},\gamma\geq
0}\min_{\boldsymbol{y},\boldsymbol{z}}L(\boldsymbol{y},\boldsymbol{z},\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{u},\gamma)=v^{*}.$
(27)

Since $y_{j}$, $j\in[n]$, are unrestricted, the equality:

$-x_{j}+\alpha_{j}-\beta_{j}+\boldsymbol{B}^{T}_{j}\boldsymbol{u}=0,\;j\in[n]$
(28)

must hold. We now show that
$\min_{\boldsymbol{z}}(\gamma||\boldsymbol{z}||_{2}-\boldsymbol{z}^{T}\boldsymbol{u})=0$
if $||\boldsymbol{u}||_{2}\leq\gamma$ and $-\infty$, otherwise. Assume, that
$||\boldsymbol{u}||_{2}\leq\gamma$. By Cauchy-Shwartz inequality,
$\boldsymbol{z}^{T}\boldsymbol{u}\leq||\boldsymbol{z}||_{2}||\boldsymbol{u}||_{2}\leq\gamma||\boldsymbol{z}||_{2}$.
Hence $\gamma||\boldsymbol{z}||_{2}-\boldsymbol{z}^{T}\boldsymbol{u}\geq 0$
and
$\min_{\boldsymbol{z}}(\gamma||\boldsymbol{z}||_{2}-\boldsymbol{z}^{T}\boldsymbol{u})=0$.
Assume that $||\boldsymbol{u}||_{2}>\gamma$ and take
$\boldsymbol{z}=s\boldsymbol{u}$ for some $s>0$. Then
$\gamma||\boldsymbol{z}||_{2}-\boldsymbol{z}^{T}\boldsymbol{u}=\gamma
s||\boldsymbol{u}||_{2}-s\boldsymbol{u}^{T}\boldsymbol{u}=s||\boldsymbol{u}||_{2}(\gamma-||\boldsymbol{u}||_{2})$,
which can be made arbitrarily small by increasing $s$. Hence, it must be

$||\boldsymbol{u}||_{2}\leq\gamma.$ (29)

Now, using (28) and (29), the problem (21) can be rewritten as

$\begin{array}[]{lllll}\max&\displaystyle-\hat{\boldsymbol{a}}^{T}\boldsymbol{x}+\sum_{j=1}^{n}\alpha_{j}(\hat{a}_{j}-\overline{a}_{j}(\lambda_{i}))+\sum_{j=1}^{n}\beta_{j}(\underline{a}_{j}(\lambda_{i})-\hat{a}_{j})-\\\
&\gamma\overline{\delta}(\lambda_{i})\\\
&\begin{array}[]{lll}-x_{j}+\alpha_{j}-\beta_{j}+\boldsymbol{B}^{T}_{j}\boldsymbol{u}=0&j\in[n]\\\
||\boldsymbol{u}||_{2}\leq\gamma&\\\ \alpha_{j},\beta_{j},\gamma\geq
0&j\in[n]\\\ \boldsymbol{u}\in\mathbb{R}^{n}&\end{array}\end{array}$

which is equivalent to (23). ∎