∎

11institutetext: Pulak Swain 22institutetext: School of Basic Sciences, Indian
Institute of Technology Bhubaneswar
22email<EMAIL_ADDRESS>33institutetext: Akshay Kumar Ojha 44institutetext:
School of Basic Sciences, Indian Institute of Technology Bhubaneswar
44email<EMAIL_ADDRESS>
# Feasibility Conditions of Robust Portfolio Solutions under Single and
Combined Uncertainties

Pulak Swain  Akshay Kumar Ojha  Pulak Swain *Corresponding author Akshay Kumar
Ojha

(Received: date / Accepted: date)

###### Abstract

In this paper, we derive the feasibility conditions for the robust
counterparts of the uncertain Markowitz model. Our study is based on
ellipsoidal, box, polyhedral uncertainty sets and also the uncertainty sets
obtained from their combinations. We write the uncertain Markowitz problem in
a quadratically constrained convex quadratic programming form and the
uncertain sets are converted into their quadratic forms for the derivation.
The feasibility conditions of robust solutions which we obtained are in the
form of positive semi-definite matrices. The study can be useful for deriving
the robust counterparts of the combined uncertainties, which in general is
difficult to find out.

###### Keywords:

Markowitz portfolio model Box uncertainty Ellipsoidal uncertainty Polyhedral
uncertainty Combined uncertainties Robust optimization Robust feasibility

##  1 Introduction

Markowitz’s theory Markowitz J 1952  for portfolio allocation has been an
evolution in the world of finance. The theory is based on the process of
allocating total wealth among different assets in the best possible way with
the help of the optimization model. Risk and reward are the two major
components in such problems, which are conflicting in nature. An investor
always wants to invest in a manner that he would get maximum reward from the
portfolio by taking the minimum risk. In Markowitz’s model the expected return
and varaince of return are respectively the reward and risks associated with
the portfolio. Later on, some downside risk measures Konno 2002 ; Estrada 2007
such as semivariance, semiabsolute deviation, below target risk, value at risk
and conditional value at risk has been used, mainly when the asset return
distributions are asymmetrical. Moreover, some more models such as capital
asset pricing model, sharpe ratio model were also used to study portfolio
optimization Sharpe 1964 ; Dowd 2000 . However the Markowitz model has the
highest popularity among the researchers due to its simplicity and
transparency while solving the problem. The most important aspect of this
model is the impact on portfolio diversification by the number of assets and
their covariance relationships Markowitz 1959 . With the passage of time, many
studies emerge in the field of portfolio optimization. Meghwani and Thakur
Meghwani 2017  introduced some practical constraints such as, cardinality
constraint, bound constraint, pre-assignment constraint, which can be added in
the problem as per the needs of an investor. In recent years many researchers
make the use of heuristic methods to study portfolio optimization. Cura Cura
2009  applied particle swarm optimization approach to a portfolio optimization
problem and compared the results with those of the genetic algorithm,
simulated annealing and tabu search approaches. Branke et. al. Branke 2009
studied the portfolio optimization with an envelope-based multi-objective
evolutionary algorithm. Meghwani and Thakur Meghwani 2018  proposed a novel
heuristic based repair algorithm for multi-objective portfolio optimization
with practical constraints.

One of the biggest challenges in portfolio optimization is that the portfolio
models are highly sensitive to the uncertainty factor and the model may give a
bad decision if the uncertainty is not handled well. The main reason for
uncertainty in these models is the use of historical data for getting future
decisions. The term ”uncertainty” refers to the ambiguity in constant
coefficients in the optimization problem. This is due to the lack of knowledge
in exact value at the time when the problem is being solved. Previously the
uncertain optimization problems were being dealt with help of dynamic
programming Bellman 1954 , fuzzy optimization Lodwick 2010 , sensitivty
analysis Christopher 2002 . However, in the last two decades robust
optimization approach has been widely used by the researchers due to its
ability to produce completely ”immunized against uncertainty” solutions. In
this approach the problem of worst case realization of uncertain parameters is
solved so that the solution becomes valid for any realization of uncertainty.
One of the primary assumptions in robust optimization is that the uncertain
parameters vary in a prespecified set $\mathscr{U}$. The robust optimization
approach aims to choose the best solution among those “immunized” against data
uncertainty, that is, the solutions should remain feasible for all the
realization of data within $\mathscr{U}$. The ”worst-case-oriented” philosophy
makes it natural to quantify the quality of a robust feasible solution by the
guaranteed value of the original objective. ElGhaoui and Lebret El Ghaoui 1997
first studied the robust solutions for the uncertain least-squares problems,
and Ben-Tal & Nemirovski Ben-Tal 1998  laid the foundation on robust convex
optimization. El-Ghaoui et al. El Ghaoui 1998  studied uncertain semidefinite
problems. They provided sufficient conditions which guarantee that the robust
solution is unique and continuous for the unperturbed problem’s data. Ben-Tal
and Nemirovski Ben-Tal 1999  studied the robust counterparts for several types
of convex optimization problems for ellipsoidal uncertainty. The study showed
that when the uncertainty sets for a linear constraint are ellipsoids, the
robust formulation turns out to be a conic quadratic problem. Now a days
robust optimization has a wide range of applications Gabrel 2014  in various
fields such as inventory & logistics, finance, revenue management, queueing
networks & game theory, machine learning, energy systems etc.

After the introduction of robust optimization, many researchers in the
financial research field also make use of this in their problems. Goldfarb and
Iyengar Goldfarb 2003  used robust optimization to a factor model for
representing return. Tütüncü and Koenig Tutuncu 2004  formulated the robust
counterparts for bound-based uncertainty sets and also they illustrated the
robust efficient frontier. Natarajan et. al. Natarajan 2009  used robust
optimization to generalize different portfolio risk measures and showed that
coherent risk measures can be constructed from the uncertainty sets in robust
optimization models. Fabozzi et al. Fabozzi 2010  applied the robust
optimization approach for mean-variance, value at risk, and conditional value
at risk models. Fliege and Werner Fliege 2014  applied the robust
multiobjective optimization in the portfolio mean-variance problem. Kim et al.
Kim 2018  did an analysis on the performance of robust portfolio for U.S.
equity portfolios and their study highlighted the effectiveness of robust
optimization for controlling uncertainty in efficient investments. Sehgal and
Mehra Sehgal 2019  proposed robust reward-risk ratio portfolio models using
Omega, semi-mean absolute deviation ratio, and weighted stable tail adjusted
return ratio. Their results show that the robust reward-risk models
outperformed their conventional counterpart models. Ismail and Pham Ismail
2019  studied a robust continuous-time Markowitz model with an uncertain
covariance matrix of risky assets. They also obtained a lower bound of Sharpe
ratio of any robust efficient portfolio strategy.

Since the robust optimization framework uses the worst-case parameter
approach, the feasibility of a robust solution is very much important. Because
more often the solutions are infeasible for the worst case realizations of
uncertain parameters. In the existing literature, there has not been any
proper study regarding the feasibility conditions for the robust portfolio
problems and this is the prime objective of this paper. The main contribution
of the paper is as follows. Robust feasibility conditions of Markowitz
portfolio model are derived for several uncertainties such as box,
ellipsoidal, polyhedral, etc. Further, the study is extended for the combined
uncertainties, that is the intersections between any of the two aforementioned
uncertain sets. The uncertain constraints as well as the uncertain sets are
first expressed in quadratic form and then the feasibility conditions are
obtained in the form of positive semidefinite matrices. Since the robust
counterparts of combined uncertainties have not been explored much, the
present study can be used to derive the robust counterparts of uncertain
Markowitz model under these combined uncertainties.

The paper is organized as follows: The introduction part is followed by some
preliminary ideas on the Markowitz portfolio model, robust optimization and
the uncertain sets in Section 2. Section 3 presents the feasibility conditions
for the robust solutions of the Markowitz model under single as well as
combined uncertainty sets. Finally, some concluding remarks have been provided
in Section 4.

##  2 Preliminaries

###  2.1 Markowitz Portfolio Model

Suppose there are $n$ number of assets $S_{1},S_{2},\dots,S_{n}$ with their
respective returns at time $t$ be given by $r_{1t},r_{2t},\dots,r_{nt}$.
Markowitz’s portfolio model is based on two input parameters– expected
portfolio return (acts as the reward) and the variance of portfolio return
(acts as the risk factor). To calculate these, we first need to find the
expected return of each asset $(\mu_{i})$ and the covariance of return between
each pair of assets $(\sigma_{ij})$, which are given by:

$\displaystyle\begin{aligned}
&\mu_{i}=E(r_{i})=\frac{1}{T}\sum\limits_{t=1}^{T}r_{it},\qquad\sigma_{ij}=E[(r_{i}-\mu_{i})(r_{j}-\mu_{j})]=\frac{1}{T}\sum\limits_{t=1}^{T}(r_{it}-\mu_{i})(r_{jt}-\mu_{j})\\\
&\textrm{for }i=1,2,\dots,n\textrm{ and }j=1,2,\dots,n.\end{aligned}$

Let $\bm{\mu}$ be the $(n\times 1)$ vector of all $\mu_{i}$’s and
$\bm{\Sigma}$ be the $(n\times n)$ matrix of all $\sigma_{ij}$’s. The aim is
to form a portfolio which will give our desired return with a minimum risk
associated with it. Let the weight given to asset $S_{i}$ be $x_{i}$. Then the
expected return and variance of return for the portfolio are respectively
given by,

$\displaystyle{\mu_{P}}=\sum_{i}\mu_{i}x_{i}=\bm{\mu}^{\top}\bm{x},\quad{\sigma_{P}}^{2}=\sum_{i,j}{\sigma_{ij}x_{i}x_{j}}=\bm{x}^{\top}\bm{\Sigma}\bm{x}$

where $\bm{x}$ be the $(n\times 1)$ decision vector containing the weights
$x_{i}$’s.
Markowitz’s portfolio model minimizes the variance of portfolio return at a
given level of expected portfolio return (say $\tau$). The mathematical
formulation of the model is given by,

$\displaystyle\begin{aligned}
&\min_{\bm{x}}&&\frac{1}{2}\bm{x}^{\top}\bm{\Sigma}\bm{x}\\\
&\text{s.t.:}&&\bm{\mu}^{\top}\bm{x}\geq\tau,\quad\bm{e}^{\top}\bm{x}=1,\quad\bm{x}\geq\bm{0}\end{aligned}$
(1)

and $\bm{e}$ is the $(n\times 1)$ vector containing $1$’s, that is,
$\bm{e}=(1\ 1\ \dots\ 1)^{\top}$.

###  2.2 Uncertainty in Optimization Problems and the Robust Optimization
Approach

An uncertain optimization problem is defined as the optimization problem in
which the constant coefficients are not exactly known; the only thing we know
is that they are close to some nominal values. The general form of an
uncertain optimization problem is given by,

$\displaystyle\begin{aligned} &\min_{\bm{x}}&&f(\bm{x},\bm{u})\\\
&\textrm{s.t.:}&&c(\bm{x},\bm{u})\leq
0,\quad\bm{u}\in\mathscr{U}\end{aligned}$ (2)

where $\bm{x}$ is the vector of decision variables, $\bm{u}$ is the vector of
uncertain parameters and $\mathscr{U}$ is the uncertain set.
The uncertain set can be affinely parameterized by a perturbation vector in a
given perturbation set $\mathcal{Z}$ as follows:

$\displaystyle\begin{aligned}
\mathscr{U}=\left\\{\bm{u}:\bm{u}=\bm{u}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{u}^{(j)},\
\bm{\zeta}=(\zeta_{1},\zeta_{2},\dots,\zeta_{n})\
\in\mathcal{Z}\subset\mathbb{R}^{n}\right\\}\end{aligned}$ (3)

where $\bm{u}^{(0)}$ is the nominal vector of the uncertain parameters,
$\bm{u}^{(j)}$ are the basic shifts, and $\bm{\zeta}$ is the perturbation
vector.
For an example, suppose we have three uncertain parameters $u_{1}$, $u_{2}$
and $u_{3}$ in our problem. Let the nominal values are respectively
$u_{1}^{(0)}=3$, $u_{2}^{(0)}=4$ and $u_{3}^{(0)}=5$. And let the values of
$u_{1}$, $u_{2}$ and $u_{3}$ can perturb upto $0.02$, $0.04$ and $0.03$
respectively. So we can write the uncertainty set as,

$\displaystyle\begin{aligned} \mathscr{U}=\left\\{\begin{bmatrix}u_{1}\\\
u_{2}\\\ u_{3}\end{bmatrix}:\begin{bmatrix}3\\\ 4\\\
5\end{bmatrix}+\zeta_{1}\begin{bmatrix}0.02\\\ 0\\\
0\end{bmatrix}+\zeta_{2}\begin{bmatrix}0\\\ 0.04\\\
0\end{bmatrix}+\zeta_{3}\begin{bmatrix}0\\\ 0\\\ 0.03\end{bmatrix},\
\zeta_{1},\zeta_{2},\zeta_{3}\in[-1,1]\right\\}\end{aligned}$

Some important concepts and definitions of robust optimization relevant to the
present study are enlisted as follows:

######  Definition 1 (Robust Feasible Solution)

Ben-Tal 2009  A vector $\bm{x}$ is said to be the robust feasible solution of
the uncertain problem (2) if it satisfies the uncertain constraints for all
realizations of the uncertain set $\mathscr{U}$, that is, if $\bm{x}$
satisfies

$\displaystyle\begin{aligned} c(\bm{x},\bm{u})\leq
0,\quad\forall\bm{u}\in\mathscr{U}\end{aligned}$

######  Definition 2 (Robust Value)

Ben-Tal 2009  For a given candidate solution $\bm{x}$, the robust value
$\widehat{f}(\bm{x})$ of the objective in problem (2) is the largest value of
$f(\bm{x},\bm{u})$ over all realizations of the data from the uncertain set,
that is

$\displaystyle\begin{aligned}
\widehat{f}(\bm{x})=\max_{\bm{u}\in\mathscr{U}}f(\bm{x},\bm{u})\end{aligned}$

It is to be noted that if the problem (2) was a maximization problem, then the
robust value would have been
$\widehat{f}(\bm{x})=\min_{\bm{u}\in\mathscr{U}}f(\bm{x},\bm{u})$.

######  Definition 3 (Robust Counterpart)

Ben-Tal 2009  The robust counterpart of the uncertain problem (2) is the
optimization problem

$\displaystyle\begin{aligned}
&\min_{\bm{x}}&&\left\\{\max_{\bm{u}\in\mathscr{U}}f(\bm{x},\bm{u})\right\\}\\\
&\text{s.t.:}&&c(\bm{x},\bm{u})\leq\bm{0},\quad\forall\bm{u}\in\mathscr{U}\end{aligned}$
(4)

of minimizing the robust value of the objective over all robust feasible
solutions to the uncertain problem.

In other words, the robust counterpart is the optimization problem with the
worst case realization of uncertain parameters.

######  Definition 4 (Robust Optimal Solution)

Ben-Tal 2009  The optimal solution of the robust counterpart problem
(LABEL:eq4.2b) or is said to be the robust optimal solution of the uncertain
problem (2).

However, sometimes the robust counterpart problems are very difficult to
solve. So the tractability of robust counterparts should be taken into
consideration when solving uncertain problems.

###  2.3 Several Uncertainties

Robust optimization assumes the uncertain sets in the form of bounded convex
sets. Some of the most used uncertainties are given as,
(i) Box Uncertainty:

$\displaystyle\mathscr{U}^{(B)}=\left\\{\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq\delta_{B}\right\\}=\left\\{{\zeta_{j}}:\sup_{j}|{\zeta_{j}}|\leq\delta_{B}\right\\}$

(ii) Ellipsoidal Uncertainty:

$\displaystyle\mathscr{U}^{(E)}=\left\\{\bm{\zeta}:||\bm{\zeta}||_{2}\leq\delta_{E}\right\\}=\left\\{{\zeta_{j}}:\sum_{j}({\zeta_{j}})^{2}\leq(\delta_{E})^{2}\right\\}$

(iii) Polyhedral Uncertainty:

$\displaystyle\mathscr{U}^{(P)}=\left\\{\bm{\zeta}:||\bm{\zeta}||_{1}\leq\delta_{P}\right\\}=\left\\{{\zeta_{j}}:\sum_{j}|{\zeta_{j}}|\leq\delta_{P}\right\\}$

where $\delta_{B}$, $\delta_{E}$ and $\delta_{P}$ are respectively the radii
of box, eliipsoidal and polyhedral uncertain sets.
Li and Floudas Li 2012  used the combinations of the above uncertainties.
Since the intersection of two bounded convex sets is also bounded and convex,
these combinations can be used as the uncertain sets. The combined sets are
enlisted as follows.
(i) ”Box $\cap$ Ellipsoidal” Uncertainty Set:

$\displaystyle\mathscr{U}^{(BE)}=\left\\{\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq\delta_{B},\
||\bm{\zeta}||_{2}\leq\delta_{E}\right\\}=\left\\{{\zeta_{j}}:\sup_{j}|{\zeta_{j}}|\leq\delta_{B},\
\sum_{j}({\zeta_{j}})^{2}\leq(\delta_{E})^{2}\right\\}$

(ii) ”Box $\cap$ Polyhedral” Uncertainty Set:

$\displaystyle\mathscr{U}^{(BP)}=\left\\{\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq\delta_{B},\
||\bm{\zeta}||_{1}\leq\delta_{P}\right\\}=\left\\{{\zeta_{j}}:\sup_{j}|{\zeta_{j}}|\leq\delta_{B},\
\sum_{j}|{\zeta_{j}}|\leq\delta_{P}\right\\}$

(iii) ”Ellipsoidal $\cap$ Polyhedral” Uncertainty Set:

$\displaystyle\mathscr{U}^{(EP)}=\left\\{\bm{\zeta}:||\bm{\zeta}||_{2}\leq\delta_{E},\
||\bm{\zeta}||_{1}\leq\delta_{P}\right\\}=\left\\{{\zeta_{j}}:\sum_{j}({\zeta_{j}})^{2}\leq(\delta_{E})^{2},\
\sum_{j}|{\zeta_{j}}|\leq\delta_{P}\right\\}$

##  3 Feasibility Conditions of Robust Markowitz Model with Single and
Combined Uncertainties

The study of Ben-Tal and Nemirovski Ben-Tal 1999  on convex optimization
problems includes an uncertain linear program with quadratic constraints given
by,

$\displaystyle\begin{aligned} &\min_{x}&&\bm{c}^{\top}\bm{x}\\\
&\textrm{s.t.:}&&-\bm{x}^{\top}[\bm{A}^{i}]^{\top}\bm{A}^{i}\bm{x}+2[\bm{b}^{i}]^{\top}\bm{x}+\gamma^{i}>0,\quad\bm{(}\bm{A}^{i},\bm{b}^{i},\gamma^{i})\in\mathscr{U}_{i}\\\
&&&i=1,2,\dots,m\end{aligned}$ (5)

where $\mathscr{U}_{i}$ is the $i^{th}$ component of a bounded ellipsoid given
by,

$\displaystyle\begin{aligned}
&\mathscr{U}_{i}=\left\\{(\bm{A}^{i},\bm{b}^{i},\gamma^{i})=(\bm{A}^{i(0)},\bm{b}^{i(0)},\gamma^{i(0)})+\sum_{j=1}^{k}\zeta_{j}(\bm{A}^{i(j)},\bm{b}^{i(j)},\gamma^{i(j)}):\zeta_{1}^{2}+\zeta_{2}^{2}+\dots+\zeta_{k}^{2}\leq
1\right\\}\end{aligned}$ (6)

The robust counterpart of the problem obtained from the study is equivalent to
an explicit semidefinite programming problem (SDP). The result is given as
follows.

###### Theorem 3.1

Ben-Tal 1999  The robust counterpart of the uncertain problem (LABEL:eq4.3)
associated with the uncertain set
$\mathscr{U}=\mathscr{U}_{1}\times\mathscr{U}_{2}\times\dots\times\mathscr{U}_{m}$,
$\mathscr{U}_{i}$’s being given by (LABEL:eq4.3aa), is equivalent to the SDP
problem

$\displaystyle\begin{aligned} &\min_{x}&&\bm{c}^{\top}\bm{x}\\\
&\textrm{w.r.t.}&&\bm{x}\in\mathbb{R}^{n},\
\lambda^{1},\dots,\lambda^{n}\in\mathbb{R}\\\
&\textrm{s.t.:}&&\small\left[\begin{array}[]{c|c c c
c|c}\gamma^{i(0)}+2\bm{x}^{\top}\bm{b}^{i(0)}-\lambda^{i}&\dfrac{\gamma^{i(1)}}{2}+\bm{x}^{\top}\bm{b}^{i(1)}&\dfrac{\gamma^{i(1)}}{2}+\bm{x}^{\top}\bm{b}^{i(1)}&\dots&\dfrac{\gamma^{i(1)}}{2}+\bm{x}^{\top}\bm{b}^{i(1)}&[\bm{A}^{i(1)}\bm{x}]^{\top}\\\
\hline\cr\dfrac{\gamma^{i(1)}}{2}+\bm{x}^{\top}\bm{b}^{i(1)}&\lambda^{i}&&&&[\bm{A}^{i(1)}\bm{x}]^{\top}\\\
\dfrac{\gamma^{i(2)}}{2}+\bm{x}^{\top}\bm{b}^{i(2)}&&\lambda^{i}&&&[\bm{A}^{i(2)}\bm{x}]^{\top}\\\
\vdots&&&\ddots&&\vdots\\\
\dfrac{\gamma^{i(k)}}{2}+\bm{x}^{\top}\bm{b}^{i(k)}&&&&\lambda^{i}&[\bm{A}^{i(k)}\bm{x}]^{\top}\\\
\hline\cr\bm{A}^{i(0)}\bm{x}&\bm{A}^{i(1)}\bm{x}&\bm{A}^{i(2)}\bm{x}&\dots&\bm{A}^{i(k)}\bm{x}&I_{l_{i}}\end{array}\right]\succeq
0\\\ &&&\text{for }i=1,2,\dots,m\text{ and $I_{l_{i}}$ is the identity matrix
of order $l_{i}$}.\end{aligned}$ (7)

During the derivation of the robust counterpart, the ellipsoidal uncertain
sets $\mathscr{U}_{i}$’s, and the uncertain constraints are derived in
quadratic form. Consequently the robust counterpart (LABEL:eq4.3a) is obtained
with the help of the well known S-lemma.

######  Lemma 1 (S-Lemma)

Ben-Tal 2009  Let $\bm{A}$, $\bm{B}$ be symmetric matrices of the same size
such that, there exists $\bm{\bar{z}}$ satisfying
$\bm{\bar{z}}^{\top}\bm{A}\bm{\bar{z}}\geq 0$. Then the implication

$\displaystyle\begin{aligned} \bm{z}^{\top}\bm{A}\bm{z}\geq
0\implies\bm{z}^{\top}\bm{B}\bm{z}\geq 0\end{aligned}$

holds true if and only if

$\displaystyle\begin{aligned} \exists\ \lambda\geq
0\quad:\quad\bm{B}\succeq\lambda\bm{A}\end{aligned}$

This approach can be used for deriving the robust feasibility conditions of
uncertain quadratic constraint(s) for ellipsoidal uncertainty. In fact, the
feasibility conditions for the other uncertainties can be obtained, if they
are expressed in quadratic forms. We aim to derive the feasibility conditions
of uncertain Markowitz portfolio model under several single and combined
uncertainties given in Section 2.3.

###  3.1 Construction of the Uncertain Markowitz Portfolio Problem

The uncertainty in the covariance of asset returns does not affect the optimal
solution much as compared to the uncertainty in expected asset returns does
Pulak 2021 . So we assume that the uncertainty is only due to the assets’
expected returns. Then the the uncertain Markowitz portfolio model is given
by,

$\displaystyle\begin{aligned}
&\min_{\bm{x}}&&\frac{1}{2}\bm{x}^{\top}\bm{\Sigma}\bm{x}\\\
&\textrm{s.t.:}&&\bm{\mu}^{\top}\bm{x}\geq\tau,\quad\bm{\mu}\in\mathscr{U}_{\bm{\mu}}\\\
&&&\bm{e}^{\top}\bm{x}=1,\quad\bm{x}\geq\bm{0}\end{aligned}$ (8)

where the uncertain set corresponding to $\mathscr{U}_{\mu}$ is given by,

$\displaystyle\begin{aligned}
\mathscr{U}_{\mu}=\left\\{\bm{\mu}:\bm{\mu}=\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)},\
\bm{\zeta}=(\zeta_{1},\zeta_{2},\dots,\zeta_{n})^{\top}\in\mathcal{Z}\right\\}\end{aligned}$

The problem (3) becomes a QCQP problem by converting the uncertain constraint
into a quadratic constraint, which is given by,

$\displaystyle\begin{aligned}
&\min_{\bm{x}}&&\frac{1}{2}\bm{x}^{\top}\bm{\Sigma}\bm{x}\\\
&\textrm{s.t.:}&&\bm{x}^{\top}\bm{\mu}\bm{\mu}^{\top}\bm{x}-\tau^{2}\geq
0,\quad\bm{\mu}\in\mathscr{U}_{\bm{\mu}}\\\
&&&\bm{e}^{\top}\bm{x}=1,\quad\bm{x}\geq\bm{0}.\end{aligned}$ (9)

Next we derive the robust feasibility conditions for single as well as
combined uncertainties.

###  3.2 Feasibility Conditions for Single Uncertainties

Since the robust counterpart result discussed earlier is for ellipsoidal
uncertainty, we begin with the ellipsoidal uncertainty case.

####  3.2.1 Ellipsoidal Uncertainty

When the uncertain set in the problem (LABEL:eq4.7) is a unit ellipsoid, it is
represented by,

$\displaystyle\begin{aligned}
\mathscr{U}_{\mu}^{(E)}=\left\\{\bm{\mu}:\bm{\mu}=\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)},\
||\bm{\zeta}||_{2}\leq 1\right\\}\end{aligned}$ (10)

Then any solution $\bm{x}$ is robust feasible for the uncertain Markowitz
potfolio model with ellipsoidal uncertainty, if and only if we have,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{2}\leq 1\end{aligned}$ (11)

Now to make the solution $\bm{x}$ feasible over an ellipsoid of radius
$\delta_{E}$, a transformation $\zeta_{j}\to\dfrac{\zeta_{j}}{\delta_{E}}$ is
applied in (LABEL:eq4.8), which then becomes

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{E}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{E}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}{\delta_{E}}^{2}\geq
0,\qquad\forall\bm{\zeta}:||\bm{\zeta}||_{2}\leq\delta_{E}\end{aligned}$ (12)

Equivalently the above condition is written as,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{E}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{E}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{E}^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{2}^{2}\leq\delta_{E}^{2}\end{aligned}$
(13)

The value of ${||\bm{\zeta}||_{2}}^{2}$ is
$|{\zeta_{1}}|^{2}+|{\zeta_{2}}|^{2}+\dots+|{\zeta_{n}}|^{2}$ which can be
written in a quadratic form as, $\bm{\zeta}^{\top}\bm{\zeta}$. So the
condition (LABEL:eq4.10) is equivalent to,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{E}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{E}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{E}^{2}\geq
0,\quad\forall\bm{\zeta}:\delta_{E}^{2}-\bm{\zeta}^{\top}\bm{\zeta}\geq
0\end{aligned}$ (14)

We claim that, the left hand side of the left expression in (LABEL:eq4.11) is
a homogeneous quadratic form of the vector
$(\delta_{E},\zeta_{1},\zeta_{2},\dots,\zeta_{n})$. We prove this as follows.
To begin with, the vectors used in (LABEL:eq4.10) need to be expressed
componentwise for more clarity.
Now we have,

$\displaystyle\begin{aligned}
&\bm{\mu}^{(0)}\delta_{E}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}&&=\begin{bmatrix}\mu_{1}^{(0)}\delta_{E}\\\
\mu_{2}^{(0)}\delta_{E}\\\ \vdots\\\
\mu_{n}^{(0)}\delta_{E}\end{bmatrix}+\sum_{j=1}^{n}\zeta_{j}\begin{bmatrix}0\\\
\vdots\\\ \mu_{j}^{(j)}\\\ \vdots\\\ 0\end{bmatrix}\\\
&&&=\begin{bmatrix}\mu_{1}^{(0)}\delta_{E}+\zeta_{1}\mu_{1}^{(1)}\\\
\mu_{2}^{(0)}\delta_{E}+\zeta_{2}\mu_{1}^{(2)}\\\ \vdots\\\
\mu_{n}^{(0)}\delta_{E}+\zeta_{n}\mu_{n}^{(n)}\end{bmatrix}\end{aligned}$ (15)

Simplifying the left expression of (LABEL:eq4.11) in component form, we get,

$\displaystyle\begin{aligned}
\biggl{[}\left(\mu_{1}^{(0)}\delta_{E}x_{1}+\mu_{2}^{(0)}\delta_{E}x_{2}+\dots+\mu_{n}^{(0)}\delta_{E}x_{n}\right)+\left(\zeta_{1}\mu_{1}^{(1)}x_{1}+\zeta_{2}\mu_{2}^{(2)}x_{2}+\dots+\zeta_{n}\mu_{n}^{(n)}x_{n}\right)\biggr{]}^{2}-\tau^{2}\delta_{E}^{2}\geq
0\end{aligned}$ (16)

Equivalently it can be written as,

$\displaystyle\begin{aligned}
&\left(\mu_{1}^{(0)}\delta_{E}x_{1}+\dots+\mu_{n}^{(0)}\delta_{E}x_{n}\right)^{2}-\tau^{2}\delta_{E}^{2}+\zeta_{1}^{2}\left(\mu_{1}^{(1)}x_{1}\right)^{2}+\dots+\zeta_{n}^{2}\left(\mu_{n}^{(n)}x_{n}\right)^{2}+2\sum_{i\neq
j}\zeta_{i}\zeta_{j}\left(\mu_{i}^{(i)}x_{i}\right)\left(\mu_{j}^{(j)}x_{j}\right)\\\
&\qquad\qquad+2\left[\zeta_{1}\mu_{1}^{(1)}x_{1}\left(\mu_{1}^{(0)}\delta_{E}x_{1}+\dots+\mu_{n}^{(0)}\delta_{E}x_{n}\right)+\dots+\zeta_{n}\mu_{n}^{(n)}x_{n}\left(\mu_{1}^{(0)}\delta_{E}x_{1}+\dots+\mu_{n}^{(0)}\delta_{E}x_{n}\right)\right]\geq
0\end{aligned}$ (17)

Now writing this in vector form, we get,

$\displaystyle\begin{aligned}
&\delta_{E}^{2}\left[\left({\bm{\mu}^{(0)}}^{\top}\bm{x}\right)^{2}-\tau^{2}\right]+\zeta_{1}^{2}\left({\bm{\mu}^{(1)}}^{\top}\bm{x}\right)^{2}+\dots+\zeta_{n}^{2}\left({\bm{\mu}^{(n)}}^{\top}\bm{x}\right)^{2}+2\sum_{i\neq
j}\zeta_{i}\zeta_{j}\left({\bm{\mu}^{(i)}}^{\top}\bm{x}\right)\left({\bm{\mu}^{(j)}}^{\top}\bm{x}\right)\\\
&\qquad\qquad+2\left[\delta_{E}\zeta_{1}\left({\bm{\mu}^{(1)}}^{\top}\bm{x}\right)\left({\bm{\mu}^{(0)}}^{\top}\bm{x}\right)+\dots+\delta_{E}\zeta_{n}\left({\bm{\mu}^{(n)}}^{\top}\bm{x}\right)\left({\bm{\mu}^{(0)}}^{\top}\bm{x}\right)\right]\geq
0\end{aligned}$ (18)

The above expression can be written as,

$\displaystyle\begin{aligned} \scriptsize{\begin{bmatrix}\delta_{E}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\
\zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\end{bmatrix}\begin{bmatrix}\delta_{E}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\geq 0\end{aligned}$
(19)

Thus the left expression of (LABEL:eq4.11) is given by,

$\displaystyle\begin{aligned} &(\bm{z}_{E})^{\top}\bm{A}(\bm{z}_{E})\geq
0,\end{aligned}$ (20)

where

$\displaystyle\begin{aligned} \bm{z}_{E}=\begin{bmatrix}\delta_{E}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix},\text{ and
}\bm{A}=\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\end{bmatrix}\end{aligned}$
(21)

are respectively an $(n+1)$ dimensional vector and a symmetric matrix of order
$(n+1)\times(n+1)$.
Moreover, the left hand side of (LABEL:eq4.11d) is a quadratic form of
$(\delta_{E},\zeta_{1},\zeta_{2},\dots,\zeta_{n})$, which proves our claim.
Let us denote this quadratic form as,
$Q_{\bm{x}}^{(E)}(\delta_{E},\bm{\zeta})$.
Now writing the left hand side of the right expression of (LABEL:eq4.11) in
quadratic form, we have,

$\displaystyle\begin{aligned}
&\delta_{E}^{2}-\bm{\zeta}^{\top}\bm{\zeta}&&=\delta_{E}^{2}-(\zeta_{1}^{2}+\zeta_{2}^{2}+\dots+\zeta_{n}^{2})\\\
&&&={\begin{bmatrix}\delta_{E}\\\ \zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\
\zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}1&0&0&\dots&0\\\
0&-1&0&\dots&0\\\ 0&0&-1&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\dots&-1\end{bmatrix}\begin{bmatrix}\delta_{E}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\end{aligned}$ (22)

Let us denote this quadratic form as, $P^{(E)}(\delta_{E},\bm{\zeta})$ and the
right expression in (LABEL:eq4.11) can be written as,

$\displaystyle\begin{aligned} &&&{\begin{bmatrix}\delta_{E}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\
\zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}1&0&0&\dots&0\\\
0&-1&0&\dots&0\\\ 0&0&-1&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\dots&-1\end{bmatrix}\begin{bmatrix}\delta_{E}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\geq 0\end{aligned}$ (23)

This means the right expression of (LABEL:eq4.11) is given by,

$\displaystyle\begin{aligned} &(\bm{z}_{E})^{\top}\bm{B}^{(E)}(\bm{z}_{E})\geq
0,\end{aligned}$ (24)

where

$\displaystyle\begin{aligned} \bm{B}^{(E)}=\begin{bmatrix}1&0&0&\dots&0\\\
0&-1&0&\dots&0\\\ 0&0&-1&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\dots&-1\end{bmatrix}\end{aligned}$ (25)

is a symmetric matrix of order $(n+1)\times(n+1)$.
Now using the expressions (LABEL:eq4.11f) and (LABEL:eq4.11j), we can write
(LABEL:eq4.11) as,

$\displaystyle\begin{aligned} &(\bm{z}_{E})^{\top}\bm{A}(\bm{z}_{E})\geq
0,\quad\forall(\bm{z}_{E}):(\bm{z}_{E})^{\top}\bm{B}^{(E)}(\bm{z}_{E})\geq
0\end{aligned}$ (26)

Then using Lemma 1 we have, the matrix $\bm{A}-\lambda\bm{B}^{(E)}$ is
positive semidefinite. Thus the following matrix inequality holds.

$\displaystyle\begin{aligned}
\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq\bm{0}\end{aligned}$
(27)

Thus the feasibility condition of the robust counterpart problem (LABEL:eq4.5)
for ellipsoidal uncertainty can be expressed in the form of a positive
semidefinite matrix. The result is given as the following theorem.

###### Theorem 3.2

Consider the uncertain Markowitz portfolio problem (LABEL:eq4.5) under
ellipsoidal uncertainty. The solution vector $\bm{x}$ to the problem is
feasible for the robust constraint in (LABEL:eq4.5) if and only if there
exists a $\lambda\geq 0$ such that the matrix

$\displaystyle\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}$

is positive semidefinite.

####  3.2.2 Box Uncertainty

When the uncertain set in the problem (LABEL:eq4.7) is a unit box, it is
represented by,

$\displaystyle\begin{aligned}
\mathscr{U}_{\mu}^{(B)}=\left\\{\bm{\mu}:\bm{\mu}=\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)},\
||\bm{\zeta}||_{\infty}\leq 1\right\\}\end{aligned}$ (28)

Then any solution $\bm{x}$ is robust feasible for the uncertain Markowitz
potfolio model with box uncertainty, if and only if we have,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq 1\end{aligned}$ (29)

Now to make the solution $\bm{x}$ feasible over a box of radius $\delta_{B}$,
a transformation $\zeta_{j}\to\dfrac{\zeta_{j}}{\delta_{B}}$ is applied in
(LABEL:eq4.23), which then becomes

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{B}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{B}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}{\delta_{B}}^{2}\geq
0,\qquad\forall\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq\delta_{B}\\\
\end{aligned}$ (30)

Equivalently the above condition is written as,

$\displaystyle\begin{aligned}
&&&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{B}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{B}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{B}^{2}\geq
0,\quad\forall\bm{\zeta}:{||\bm{\zeta}||_{\infty}}^{2}\leq\delta_{B}^{2}\\\
&\iff&&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{B}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{B}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{B}^{2}\geq
0,\quad\forall\bm{\zeta}:\delta_{B}^{2}-{||\bm{\zeta}||_{\infty}}^{2}\geq
0\end{aligned}$ (31)

Analogous to the ellipsoidal uncertainty case, the left hand side of the left
expression of (LABEL:eq4.25) can be written in quadratic form
$Q_{\bm{x}}^{(B)}(\delta_{B},\bm{\zeta})$, which is given by,

$\displaystyle\begin{aligned} &(\bm{z}_{B})^{\top}\bm{A}(\bm{z}_{B})\geq
0,\end{aligned}$ (32)

where

$\displaystyle\begin{aligned} \bm{z}_{B}=\begin{bmatrix}\delta_{B}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix},\text{ and
}\bm{A}=\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\end{bmatrix}\end{aligned}$
(33)

are respectively an $(n+1)$ dimensional vector and a symmetric matrix of order
$(n+1)\times(n+1)$.
Next the left hand side of the right expression of (LABEL:eq4.25) needs to be
written in quadratic form. The value of ${||\bm{\zeta}||_{\infty}}^{2}$ equals
to $\sup_{j}\zeta_{j}^{2}$. Let us denote it as ${\zeta_{M}}^{2}$. Now writing
the left hand side of the right expression of (LABEL:eq4.25) in quadratic
form, we have,

$\displaystyle\begin{aligned}
&\delta_{B}^{2}-{||\bm{\zeta}||_{\infty}}^{2}&&=\delta_{B}^{2}-\zeta_{M}^{2}\\\
&&&={\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\ \vdots\\\ \zeta_{M}\\\
\vdots\\\ \zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}1&\ 0&\ \dots&\ 0&\
\dots&\ 0\\\ 0&\ 0&\ \dots&\ 0&\ \dots&\ 0\\\ \vdots&\ \vdots&\ \ddots&\
\vdots&\ \ddots&\ \vdots\\\ 0&\ 0&\ \dots&\ -1&\ \dots&\ 0\\\ \vdots&\
\vdots&\ \ddots&\ \vdots&\ \ddots&\ \vdots\\\ 0&\ 0&\dots&\ 0&\dots&\
0\end{bmatrix}\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\ \vdots\\\ \zeta_{M}\\\
\vdots\\\ \zeta_{n}\end{bmatrix}\end{aligned}$ (34)

Let us denote this quadratic form as, $P^{(B)}(\delta_{B},\bm{\zeta})$. Now
the right expression of (LABEL:eq4.25) can be written as,

$\displaystyle\begin{aligned} &(\bm{z}_{B})^{\top}\bm{B}^{(B)}(\bm{z}_{B})\geq
0,\end{aligned}$ (35)

where

$\displaystyle\begin{aligned} \bm{B}^{(B)}=\begin{bmatrix}1&\ 0&\ \dots&\ 0&\
\dots&\ 0\\\ 0&\ 0&\ \dots&\ 0&\ \dots&\ 0\\\ \vdots&\ \vdots&\ \ddots&\
\vdots&\ \ddots&\ \vdots\\\ 0&\ 0&\ \dots&\ -1&\ \dots&\ 0\\\ \vdots&\
\vdots&\ \ddots&\ \vdots&\ \ddots&\ \vdots\\\ 0&\ 0&\dots&\ 0&\dots&\
0\end{bmatrix}\end{aligned}$ (36)

is a symmetric matrix of order $(n+1)\times(n+1)$. More specifically,
$\bm{B^{(B)}}$ is a diagonal matrix with its first and $M^{th}$ diagonal
elements are $1$ and $-1$ respectively, and the other diagonal elements are
zeros.
Now using the expressions (LABEL:eq4.25aa) and (LABEL:eq4.25b), we can write
(LABEL:eq4.25) as,

$\displaystyle\begin{aligned} &(\bm{z}_{B})^{\top}\bm{A}\bm{(}z_{B})\geq
0,\quad\forall(\bm{z}_{B}):(\bm{z}_{B})^{\top}\bm{B}^{(B)}(\bm{z}_{B})\geq
0\end{aligned}$ (37)

Then using Lemma 1 we have, the matrix $\bm{A}-\lambda\bm{B}^{(B)}$ is
positive semidefinite. Thus the following matrix inequality holds.

$\displaystyle\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&\
({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(M)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&\
({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&\ \dots&\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(M)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(M)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\ \vdots&\
\vdots&\ \ddots&\ \vdots&\ \ddots&\ \vdots\\\
({\bm{\mu}^{(M)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&\
({\bm{\mu}^{(M)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(M)}}^{\top}\bm{x})^{2}+\lambda&\ \dots&\
({\bm{\mu}^{(M)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\ \vdots&\
\vdots&\ \ddots&\ \vdots&\ \ddots&\ \vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(M)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\end{bmatrix}\succeq\bm{0}$ (38)

This can be written as the following result.

###### Theorem 3.3

Consider the uncertain portfolio problem (LABEL:eq4.5) under box uncertainty.
The solution vector $\bm{x}$ to the problem is feasible for the robust
constraint in (LABEL:eq4.5) if and only if there exists a $\lambda\geq 0$ such
that the matrix

$\displaystyle\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&\
({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(M)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&\
({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&\ \dots&\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(M)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(M)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\ \vdots&\
\vdots&\ \ddots&\ \vdots&\ \ddots&\ \vdots\\\
({\bm{\mu}^{(M)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&\
({\bm{\mu}^{(M)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(M)}}^{\top}\bm{x})^{2}+\lambda&\ \dots&\
({\bm{\mu}^{(M)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\ \vdots&\
\vdots&\ \ddots&\ \vdots&\ \ddots&\ \vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(M)}}^{\top}\bm{x})&\ \dots&\
({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\\\ \end{bmatrix}$

is positive semidefinite, where $M$ is the index of $\sup_{j}\zeta_{j}^{2}$.

####  3.2.3 Polyhedral Uncertainty

When the uncertain set in the problem (LABEL:eq4.7) is a unit polyhedron, it
is represented by,

$\displaystyle\begin{aligned}
\mathscr{U}_{\mu}^{(P)}=\left\\{\bm{\mu}:\bm{\mu}=\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)},\
||\bm{\zeta}||_{1}\leq 1\right\\}\end{aligned}$ (39)

Then any solution $\bm{x}$ is robust feasible for the uncertain Markowitz
potfolio model with polyhedral uncertainty, if and only if we have,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{1}\leq 1\end{aligned}$ (40)

Now to make the solution $\bm{x}$ feasible over a polyhedron of radius
$\delta_{P}$, a transformation $\zeta_{j}\to\dfrac{\zeta_{j}}{\delta_{P}}$ is
applied in (LABEL:eq4.23), which then becomes

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{P}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{P}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}{\delta_{P}}^{2}\geq
0,\qquad\forall\bm{\zeta}:||\bm{\zeta}||_{1}\leq\delta_{P}\\\ \end{aligned}$
(41)

Equivalently the above condition is written as,

$\displaystyle\begin{aligned}
&&&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{P}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{P}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{P}^{2}\geq
0,\quad\forall\bm{\zeta}:{||\bm{\zeta}||_{1}}^{2}\leq\delta_{P}^{2}\\\
&\iff&&\bm{x}^{\top}\left[\bm{\mu}^{(0)}\delta_{P}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}\delta_{P}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{P}^{2}\geq
0,\quad\forall\bm{\zeta}:\delta_{P}^{2}-{||\bm{\zeta}||_{1}}^{2}\geq
0\end{aligned}$ (42)

The left hand side of the left expression of (LABEL:eq4.16) can be written in
quadratic form $Q_{\bm{x}}^{(P)}(\delta_{P},\bm{\zeta})$, which is given by,

$\displaystyle\begin{aligned} &(\bm{z}_{P})^{\top}\bm{A}(\bm{z}_{P})\geq
0,\end{aligned}$ (43)

where

$\displaystyle\begin{aligned} \bm{z}_{P}=\begin{bmatrix}\delta_{P}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix},\text{ and
}\bm{A}=\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\end{bmatrix}\end{aligned}$
(44)

are respectively an $(n+1)$ dimensional vector and a symmetric matrix of order
$(n+1)\times(n+1)$.
The term ${||\bm{\zeta}||_{1}}^{2}$ in the right expression of (LABEL:eq4.16)
can be written as a quadratic form as $\bm{\zeta}^{\top}S\bm{\zeta}$, where
$S$ is an $(n\times n)$ symmetric matrix with $1$ as each entry and is given
by,

$\displaystyle\begin{aligned} S=\begin{bmatrix}1&&1&&1&&\dots&&1\\\
1&&1&&1&&\dots&&1\\\ \vdots&&\vdots&&\vdots&&\ddots&&\vdots\\\
1&&1&&1&&\dots&&1\\\ \end{bmatrix}\end{aligned}$ (45)

and hence the left hand side of the right expression of (LABEL:eq4.16) can be
written in quadratic form as,

$\displaystyle\begin{aligned}
&\delta_{P}^{2}-{||\bm{\zeta}||_{1}}^{2}&&={\begin{bmatrix}\delta_{P}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\
\zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}1&0&0&\dots&0\\\
0&-1&-1&\dots&-1\\\ 0&-1&-1&\dots&-1\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&-1&-1&\dots&-1\end{bmatrix}\begin{bmatrix}\delta_{P}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\end{aligned}$ (46)

Let us denote this quadratic form as, $P^{(P)}(\delta_{P},\bm{\zeta})$. Now
the right expression of (LABEL:eq4.16) can be written as,

$\displaystyle\begin{aligned} &(\bm{z}_{P})^{\top}\bm{B}^{(P)}\bm{(}z_{P})\geq
0,\end{aligned}$ (47)

where

$\displaystyle\begin{aligned} \bm{B}^{(P)}=\begin{bmatrix}1&0&0&\dots&0\\\
0&-1&-1&\dots&-1\\\ 0&-1&-1&\dots&-1\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&-1&-1&\dots&-1\end{bmatrix}\end{aligned}$ (48)

is a symmetric matrix of order $(n+1)\times(n+1)$.
Moreover, the left expression of (LABEL:eq4.16) is given by,

$\displaystyle\begin{aligned} &(\bm{z}_{P})^{\top}\bm{A}(\bm{z}_{P})\geq
0,\end{aligned}$ (49)

where $\bm{A}$ is the $(n+1)\times(n+1)$ symmetric matrix given in (21).
Now using the expressions (LABEL:eq4.18) and (LABEL:eq4.20), we can write
(LABEL:eq4.16) as,

$\displaystyle\begin{aligned} &(\bm{z}_{P})^{\top}\bm{A}(\bm{z}_{P})\geq
0,\quad\forall\bm{z}_{P}:(\bm{z}_{P})^{\top}\bm{B}^{(P)}(\bm{z}_{P})\geq
0\end{aligned}$ (50)

Then applying Lemma 1, we have the matrix $\bm{A}-\lambda\bm{B}^{(P)}$ is
positive semidefinite and hence the following matrix inequality holds.

$\displaystyle\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq\bm{0}$

The above result is given as the following theorem.

###### Theorem 3.4

Consider the uncertain portfolio problem (LABEL:eq4.5) under polyhedral
uncertainty. The solution vector $\bm{x}$ to the problem is feasible for the
robust constraint in (LABEL:eq4.5) if and only if there exists a $\lambda\geq
0$ such that the matrix

$\displaystyle\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}$

is positive semidefinite.

###  3.3 Feasibility Conditions for the Combined Uncertainties

Being the intersection of two convex sets, each combined uncertain set is
itself a convex set and can be represented as a quadratic form. Now we derive
the feasibility conditions for the combined uncertainties.

####  3.3.1 ”Box $\cap$ Ellipsoidal” Uncertainty

When the uncertain set in the problem (LABEL:eq4.7) is the intersection of a
unit box and a unit ellipsoid, it is represented by,

$\displaystyle\begin{aligned}
\mathscr{U}_{\mu}=\left\\{\bm{\mu}:\bm{\mu}=\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)},\
||\bm{\zeta}||_{\infty}\leq 1,\ ||\bm{\zeta}||_{2}\leq
1\right\\}\end{aligned}$ (51)

Then any solution $\bm{x}$ is robust feasible for the uncertain Markowitz
potfolio model with ”box $\cap$ ellipsoidal” uncertainty, if and only if we
have,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq 1,\
||\bm{\zeta}||_{2}\leq 1\end{aligned}$ (52)

Before proceeding further, we need to find the intersection between the box
and ellipsoidal sets of radii $\delta_{B}$ and $\delta_{E}$ respectively.
Now the box uncertain set of radius $\delta_{B}$ is given by,

$\displaystyle\left\\{\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq\delta_{B}\right\\}\
\implies\left\\{\zeta_{j}:|\zeta_{j}|\leq\delta_{B},\quad\forall j\right\\}$

Further, the ellipsoidal uncertain set of radius $\delta_{E}$ is given by,

$\displaystyle\left\\{\bm{\zeta}:||\bm{\zeta}||_{2}\leq\delta_{E}\right\\}\
\implies\left\\{\zeta_{j}:\sum_{j=1}^{n}(\zeta_{j})^{2}\leq(\delta_{E})^{2}\right\\}$

So the intersection of the two sets is written as,

$\displaystyle\begin{aligned}
&&&\left\\{\zeta_{j}:\sum_{j=1}^{n}(\zeta_{j})^{2}\leq\min\left\\{n\delta_{B}^{2},\delta_{E}^{2}\right\\}\right\\}\\\
&\text{ or,
}&&\left\\{\bm{\zeta}:||\bm{\zeta}||_{2}^{2}\leq\min\left\\{n\delta_{B}^{2},\delta_{E}^{2}\right\\}\right\\}\end{aligned}$

Now there arises two cases.
Case I: When
$\min\left\\{n\delta_{B}^{2},\delta_{E}^{2}\right\\}=n\delta_{B}^{2}$
For this case the intersection set satisfies
$||\bm{\zeta}||_{2}\leq\sqrt{n}\delta_{B}$, and it can be achieved by making a
transformation $\zeta_{j}\to\dfrac{\zeta_{j}}{\sqrt{n}\delta_{B}}$ in the
constraint (LABEL:eq4.38), which then becomes,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}(\sqrt{n}\delta_{B})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(\sqrt{n}\delta_{B})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}n\delta_{B}^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{2}^{2}\leq
n\delta_{B}^{2}\end{aligned}$ (53)

Equivalently the above condition is written as,

$\displaystyle\begin{aligned}
\bm{x}^{\top}\left[\bm{\mu}^{(0)}(\sqrt{n}\delta_{B})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(\sqrt{n}\delta_{B})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}n\delta_{B}^{2}\geq
0,\quad\forall\bm{\zeta}:n\delta_{B}^{2}-\bm{\zeta}^{\top}\bm{\zeta}\geq
0\end{aligned}$ (54)

Similar derivations as used in the single uncertainty cases can convert the
left hand side of the left expression of (LABEL:eq4.25) into quadratic form
$Q_{\bm{x}}^{(BE)}(\delta_{B},\bm{\zeta})$, which is given by,

$\displaystyle\begin{aligned}
&(\bm{z}_{B})^{\top}\bm{A}^{\prime}(\bm{z}_{B})\geq 0,\end{aligned}$ (55)

where

$\displaystyle\begin{aligned} &\bm{z}_{B}=\begin{bmatrix}\delta_{B}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\text{ is an $(n+1)$
dimensional vector, and }\\\
&\bm{A}^{\prime}=\begin{bmatrix}(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-n\tau^{2}&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\end{bmatrix}\end{aligned}$
(56)

is a symmetric matrix of order $(n+1)\times(n+1)$.
Now writing the left hand side of the right expression of (54) in quadratic
form, we have,

$\displaystyle\begin{aligned}
&n\delta_{B}^{2}-\bm{\zeta}^{\top}\bm{\zeta}&&=n\delta_{B}^{2}-(\zeta_{1}^{2}+\zeta_{2}^{2}+\dots+\zeta_{n}^{2})\\\
&&&={\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\
\zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}n&0&0&\dots&0\\\
0&-1&0&\dots&0\\\ 0&0&-1&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\dots&-1\end{bmatrix}\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\end{aligned}$ (57)

Let us denote this quadratic form as, $P_{1}^{(BE)}(\delta_{B},\bm{\zeta})$
and the right expression in (54) can be written as,

$\displaystyle\begin{aligned} &&&{\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\
\zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}n&0&0&\dots&0\\\
0&-1&0&\dots&0\\\ 0&0&-1&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\dots&-1\end{bmatrix}\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\geq 0\end{aligned}$ (58)

That means,

$\displaystyle\begin{aligned}
&(\bm{z}_{B})^{\top}\bm{B}_{1}^{(BE)}(\bm{z}_{B})\geq 0,\end{aligned}$ (59)

where

$\displaystyle\begin{aligned}
\bm{B}_{1}^{(BE)}=\begin{bmatrix}n&0&0&\dots&0\\\ 0&-1&0&\dots&0\\\
0&0&-1&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\dots&-1\end{bmatrix}\end{aligned}$ (60)

is a symmetric matrix of order $(n+1)\times(n+1)$.
Now using the expressions (LABEL:eq4.40a) and (LABEL:eq4.41a), we can write
(54) as,

$\displaystyle\begin{aligned}
&(\bm{z}_{B})^{\top}\bm{A}^{\prime}(\bm{z}_{B})\geq
0,\quad\forall(\bm{z}_{B}):(\bm{z}_{B})^{\top}\bm{B}_{1}^{(BE)}(\bm{z}_{B})\geq
0\end{aligned}$ (61)

Then by Lemma 1 we have, the matrix $\bm{A}^{\prime}-\lambda\bm{B}_{1}^{(BE)}$
is positive semidefinite. Thus the following matrix inequality holds.
Therefore the matrix

$\displaystyle\begin{aligned}
\scriptsize\begin{bmatrix}(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-n\tau^{2}-n\lambda&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq\bm{0}\end{aligned}$
(62)

Case II: When
$\min\left\\{n\delta_{B}^{2},\delta_{E}^{2}\right\\}=\delta_{E}^{2}$
For this case the intersection set satisfies
$||\bm{\zeta}||_{2}\leq\delta_{E}$, and it can be achieved by making a
transformation $\zeta_{j}\to\dfrac{\zeta_{j}}{\delta_{E}}$ in the constraint
(LABEL:eq4.38), which then becomes,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}(\delta_{E})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(\delta_{E})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{E}^{2}\geq
0,\qquad\forall\bm{\zeta}:||\bm{\zeta}||_{2}^{2}\leq\delta_{E}^{2}\end{aligned}$
(63)

This is equivalent to the ellipsoidal uncertainty case given in Theorem 3.2.
So proceeding in a similar manner, the following matrix inequality holds.

$\displaystyle\begin{aligned}
\scriptsize\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq\bm{0}\end{aligned}$
(64)

The results of the above two cases can be given as follows.

###### Theorem 3.5

Consider the uncertain portfolio problem (LABEL:eq4.5) under the uncertainty
set obtained from the combination of box and ellipsoidal uncertainty sets. Let
the radii of the box and ellipsoidal sets are respectively $\delta_{B}$ and
$\delta_{E}$. Then the solution vector $\bm{x}$ to this problem is feasible
for the robust constraint in (LABEL:eq4.5) if and only if there exists a
$\lambda\geq 0$ such that the following matrix inequalities hold:

$\displaystyle\begin{aligned}
&\left\\{\begin{array}[]{ll}\begin{bmatrix}(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-n\tau^{2}-n\lambda&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})(\sqrt{n}{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\\\
\end{bmatrix}\succeq 0,&\mbox{ if
}\min\left\\{n\delta_{B}^{2},\delta_{E}^{2}\right\\}=n\delta_{B}^{2}\\\ \\\
\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq
0,&\mbox{ if
}\min\left\\{n\delta_{B}^{2},\delta_{E}^{2}\right\\}=\delta_{E}^{2}\end{array}\right.\end{aligned}$

####  3.3.2 ”Box $\cap$ Polyhedral” Uncertainty

When the uncertain set in the problem (LABEL:eq4.7) is the intersection of a
unit box and a unit polyhedron, it is represented by,

$\displaystyle\begin{aligned}
\mathscr{U}_{\mu}=\left\\{\bm{\mu}:\bm{\mu}=\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)},\
||\bm{\zeta}||_{\infty}\leq 1,\ ||\bm{\zeta}||_{1}\leq
1\right\\}\end{aligned}$ (65)

Then any solution $\bm{x}$ is robust feasible for the uncertain Markowitz
potfolio model with ”box $\cap$ polyhedral” uncertainty, if and only if we
have,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\geq
0,\qquad\forall\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq 1,\
||\bm{\zeta}||_{1}\leq 1\end{aligned}$ (66)

First we need to find the intersection between box and polyhedral uncertain
sets of radii $\delta_{B}$ and $\delta_{P}$ respectively.
Now the box uncertain set of radius $\delta_{B}$ is given by,

$\displaystyle\left\\{\bm{\zeta}:||\bm{\zeta}||_{\infty}\leq\delta_{B}\right\\}\
\implies\left\\{\zeta_{j}:|\zeta_{j}|\leq\delta_{B},\quad\forall j\right\\}$

Further, the polyhedral uncertain set of radius $\delta_{P}$ is given by,

$\displaystyle\left\\{\bm{\zeta}:||\bm{\zeta}||_{1}\leq\delta_{P}\right\\}\
\implies\left\\{\zeta_{j}:\sum_{j=1}^{n}|\zeta_{j}|\leq\delta_{P}\right\\}$

So we can write the combined uncertainty as intersection between the two sets.
This can be written as,

$\displaystyle\left\\{\zeta_{j}:\sum_{j=1}^{n}|\zeta_{j}|\leq\min\left\\{n\delta_{B},\delta_{P}\right\\}\right\\}\
\implies\left\\{\bm{\zeta}:||\bm{\zeta}||_{1}\leq\min\left\\{n\delta_{B},\delta_{P}\right\\}\right\\}$

Now there arises two cases.
Case I: When $\min\left\\{n\delta_{B},\delta_{P}\right\\}=n\delta_{B}$
For this case the intersection set satisfies $||\bm{\zeta}||_{1}\leq
n\delta_{B}$, and it can be achieved by making a transformation
$\zeta_{j}\to\dfrac{\zeta_{j}}{n\delta_{B}}$ in the constraint (LABEL:eq4.31),
which then becomes,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}(n\delta_{B})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(n\delta_{B})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}n^{2}\delta_{B}^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{1}\leq n\delta_{B}\end{aligned}$ (67)

Equivalently the above condition is written as,

$\displaystyle\begin{aligned}
\bm{x}^{\top}\left[\bm{\mu}^{(0)}(n\delta_{B})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(n\delta_{B})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}n^{2}\delta_{B}^{2}\geq
0,\quad\forall\bm{\zeta}:n^{2}\delta_{B}^{2}-\bm{\zeta}^{\top}S\bm{\zeta}\geq
0\end{aligned}$ (68)

where $S$ is an $(n\times n)$ matrix with $1$ as each entry given by (45).
The left hand side of the left expression of (68) into quadratic form
$Q_{\bm{x}}^{(BP)}(\delta_{B},\bm{\zeta})$, which is given by,

$\displaystyle\begin{aligned}
&(\bm{z}_{B})^{\top}\bm{A}^{\prime\prime}(\bm{z}_{B})\geq 0,\end{aligned}$
(69)

where

$\displaystyle\begin{aligned} &\bm{z}_{B}=\begin{bmatrix}\delta_{B}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\text{ is an $(n+1)$
dimensional vector, and }\\\
&\bm{A}^{\prime\prime}=\begin{bmatrix}(n{\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-n^{2}\tau^{2}&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\end{bmatrix}\end{aligned}$
(70)

is a symmetric matrix of order $(n+1)\times(n+1)$.
Now writing the left hand side of the right expression of (68) in quadratic
form, we have,

$\displaystyle\begin{aligned}
&n^{2}\delta_{B}^{2}-\bm{\zeta}^{\top}S\bm{\zeta}&&={\begin{bmatrix}\delta_{B}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\
\zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}n^{2}&0&0&\dots&0\\\
0&-1&-1&\dots&-1\\\ 0&-1&-1&\dots&-1\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&-1&-1&\dots&-1\end{bmatrix}\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\end{aligned}$ (71)

Let us denote this quadratic form as, $P_{1}^{(BP)}(\delta_{B},\bm{\zeta})$
and the right expression in (68) can be written as,

$\displaystyle\begin{aligned} &&&{\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\
\zeta_{n}\end{bmatrix}}^{\top}\begin{bmatrix}n^{2}&0&0&\dots&0\\\
0&-1&-1&\dots&-1\\\ 0&-1&-1&\dots&-1\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&-1&-1&\dots&-1\end{bmatrix}\begin{bmatrix}\delta_{B}\\\ \zeta_{1}\\\
\zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix}\geq 0\end{aligned}$ (72)

That means,

$\displaystyle\begin{aligned}
&(\bm{z}_{B})^{\top}\bm{B}_{1}^{(BP)}(\bm{z}_{B})\geq 0,\end{aligned}$ (73)

where

$\displaystyle\begin{aligned}
\bm{B}_{1}^{(BP)}=\begin{bmatrix}n^{2}&0&0&\dots&0\\\ 0&-1&-1&\dots&-1\\\
0&-1&-1&\dots&-1\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&-1&-1&\dots&-1\end{bmatrix}\end{aligned}$ (74)

is a symmetric matrix of order $(n+1)\times(n+1)$.
Now using the expressions (LABEL:eq4.33a) and (LABEL:eq4.33e), we can write
(68) as,

$\displaystyle\begin{aligned}
&(\bm{z}_{B})^{\top}\bm{A}^{\prime\prime}(\bm{z}_{B})\geq
0,\quad\forall(\bm{z}_{B}):(\bm{z}_{B})^{\top}\bm{B}_{1}^{(BP)}(\bm{z}_{B})\geq
0\end{aligned}$ (75)

Then by Lemma 1 we have, the matrix
$\bm{A}^{\prime\prime}-\lambda\bm{B}_{1}^{(BP)}$ is positive semidefinite.
Thus the following matrix inequality holds.

$\displaystyle\begin{aligned}
\scriptsize\begin{bmatrix}(n{\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-n^{2}\tau^{2}-n^{2}\lambda&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq\bm{0}\end{aligned}$
(76)

Case II: When $\min\left\\{n\delta_{B},\delta_{P}\right\\}=\delta_{P}$
For this case the intersection set satisfies
$||\bm{\zeta}||_{1}\leq\delta_{P}$, and it can be achieved by making a
transformation $\zeta_{j}\to\dfrac{\zeta_{j}}{\delta_{P}}$ in the constraint
(LABEL:eq4.31), which then becomes,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}(\delta_{P})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(\delta_{P})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{P}^{2}\geq
0,\qquad\forall\bm{\zeta}:||\bm{\zeta}||_{1}\leq\delta_{P}\end{aligned}$ (77)

This is equivalent to the polyhedral uncertainty case given in Theorem 3.4. So
proceeding in a similar manner, the following matrix inequality holds.

$\displaystyle\begin{aligned}
\scriptsize\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq\bm{0}\end{aligned}$
(78)

The results of the above two cases can be given as follows.

###### Theorem 3.6

Consider the uncertain portfolio problem (LABEL:eq4.5) under the uncertainty
set obtained from the combination of box and polyhedral uncertainty sets. Let
the radii of the box and polyhedral sets are respectively $\delta_{B}$ and
$\delta_{P}$. Then the solution vector $\bm{x}$ to this problem is feasible
for the robust constraint in (LABEL:eq4.5) if and only if there exists a
$\lambda\geq 0$ such that the following matrix inequalities hold:

$\displaystyle\begin{aligned}
&\left\\{\begin{array}[]{ll}\begin{bmatrix}(n{\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-n^{2}\tau^{2}-n^{2}\lambda&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&(n{\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})(n{\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq
0,&\mbox{ if }\min\left\\{n\delta_{B},\delta_{P}\right\\}=n\delta_{B}\\\ \\\
\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})+\lambda\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})+\lambda&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})+\lambda&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\\\
\end{bmatrix}\succeq 0,&\mbox{ if
}\min\left\\{n\delta_{B},\delta_{P}\right\\}=\delta_{P}\end{array}\right.\end{aligned}$

####  3.3.3 ”Ellipsoidal $\cap$ Polyhedral” Uncertainty

When the uncertain set in the problem (LABEL:eq4.7) is the intersection of a
unit box and a unit polyhedron, it is represented by,

$\displaystyle\begin{aligned}
\mathscr{U}_{\mu}=\left\\{\bm{\mu}:\bm{\mu}=\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)},\
||\bm{\zeta}||_{1}\leq 1,\ ||\bm{\zeta}||_{2}\leq 1\right\\}\end{aligned}$
(79)

Then any solution $\bm{x}$ is robust feasible for the uncertain Markowitz
potfolio model with ”ellipsoidal $\cap$ polyhedral” uncertainty, if and only
if we have,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}-\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\geq
0,\qquad\forall\bm{\zeta}:||\bm{\zeta}||_{1}\leq 1,\ ||\bm{\zeta}||_{2}\leq
1\end{aligned}$ (80)

First we need to find the intersection between ellipsoidal and polyhedral
uncertain sets of radii $\delta_{E}$ and $\delta_{P}$ respectively.
Now the ellipsoidal uncertain set of radius $\delta_{E}$ is given by,

$\displaystyle\left\\{\bm{\zeta}:||\bm{\zeta}||_{2}\leq\delta_{E}\right\\}\implies\left\\{\zeta_{j}:\sum_{j=1}^{n}(\zeta_{j})^{2}\leq\delta_{E}^{2}\right\\}$

Further, the polyhedral uncertain set of radius $\delta_{P}$ is given by,

$\displaystyle\left\\{\bm{\zeta}:||\bm{\zeta}||_{1}\leq\delta_{P}\right\\}\implies\left\\{\zeta_{j}:\left(\sum_{j=1}^{n}|\zeta_{j}|\right)^{2}\leq\delta_{P}^{2}\right\\}$

So we can write the combined uncertainty as intersection between the two sets.
This can be written as,

$\displaystyle\begin{aligned}
&&&\left\\{\zeta_{j}:\sum_{j=1}^{n}(\zeta_{j})^{2}\leq\min\\{\delta_{E}^{2},\delta_{P}^{2}-2\sum_{i\neq
j}{|\zeta_{i}||\zeta_{j}|}\\}\right\\}\\\
&\implies&&\left\\{\zeta_{j}:\sum_{j=1}^{n}(\zeta_{j})^{2}\leq\min\\{\delta_{E}^{2},\delta_{P}^{2}\\}\right\\},\end{aligned}$

since each $|\zeta_{j}|\geq 0$, which implies $\delta_{P}^{2}-2\sum_{i\neq
j}{|\zeta_{i}||\zeta_{j}|}\leq\delta_{P}^{2}$.
So the combined uncertain set is given by,

$\displaystyle\begin{aligned}
&&&\left\\{\bm{\zeta}:||\bm{\zeta}||_{2}\leq\min\\{\delta_{E},\delta_{P}\\}\right\\}\end{aligned}$

Now there arises two cases.
Case I: When $\min\left\\{\delta_{E},\delta_{P}\right\\}=\delta_{E}$
For this case the intersection set satisfies
$||\bm{\zeta}||_{2}\leq\delta_{E}$, and it can be achieved by making a
transformation $\zeta_{j}\to\dfrac{\zeta_{j}}{\delta_{E}}$ in the constraint
(LABEL:eq4.44), which then becomes,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}(\delta_{E})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(\delta_{E})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{E}^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{2}\leq\delta_{E}\end{aligned}$ (81)

Equivalently the above constraint is written as,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}(\delta_{E})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(\delta_{E})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{E}^{2}\geq
0,\ \forall\bm{\zeta}:\delta_{E}^{2}-\bm{\zeta}^{\top}\bm{\zeta}\geq
0\end{aligned}$ (82)

This is equivalent to the ellipsoidal uncertainty case given in Theorem 3.2.
So proceeding in a similar manner, the following matrix inequality holds.

$\displaystyle\begin{aligned}
\scriptsize\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq\bm{0}\end{aligned}$
(83)

Case II: When $\min\left\\{\delta_{E},\delta_{P}\right\\}=\delta_{P}$
For this case the intersection set satisfies
$||\bm{\zeta}||_{2}\leq\delta_{P}$, and it can be achieved by making a
transformation $\zeta_{j}\to\dfrac{\zeta_{j}}{\delta_{P}}$ in the constraint
(LABEL:eq4.44), which then becomes,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}(\delta_{P})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(\delta_{P})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{P}^{2}\geq
0,\quad\forall\bm{\zeta}:||\bm{\zeta}||_{2}\leq\delta_{P}\\\ \end{aligned}$
(84)

Equivalently the above constraint is written as,

$\displaystyle\begin{aligned}
&\bm{x}^{\top}\left[\bm{\mu}^{(0)}(\delta_{P})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]\left[\bm{\mu}^{(0)}(\delta_{P})+\sum_{j=1}^{n}\zeta_{j}\bm{\mu}^{(j)}\right]^{\top}\bm{x}-\tau^{2}\delta_{P}^{2}\geq
0,\ \forall\bm{\zeta}:\delta_{P}^{2}-\bm{\zeta}^{\top}\bm{\zeta}\geq
0\end{aligned}$ (85)

This is equivalent to the case of ellipsoidal uncertain set with radius
$\delta_{P}$. So the similar derivations as Theorem 3.2 reduces the constraint
(LABEL:eq4.48) to,

$\displaystyle\begin{aligned} &(\bm{z}_{P})^{\top}\bm{A}\bm{(}z_{P})\geq
0,\quad\forall(\bm{z}_{P}):(\bm{z}_{P})^{\top}\bm{B}_{2}^{(EP)}(\bm{z}_{P})\geq
0\end{aligned}$ (86)

where $z_{P}$, $A$, and $B_{2}^{(EP)}$ are respectively given by,

$\displaystyle\begin{aligned} &\bm{z}_{P}=\begin{bmatrix}\delta_{P}\\\
\zeta_{1}\\\ \zeta_{2}\\\ \vdots\\\ \zeta_{n}\end{bmatrix},\
\bm{A}=\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}\end{bmatrix}\\\
&\bm{B}^{(EP)}=\begin{bmatrix}1&0&0&\dots&0\\\ 0&-1&0&\dots&0\\\
0&0&-1&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\dots&-1\end{bmatrix}\end{aligned}$ (87)

Then using Lemma 1 we have, the matrix $\bm{A}-\lambda\bm{B}^{(EP)}$ is
positive semidefinite. Thus the following matrix inequality holds.

$\displaystyle\begin{aligned}
\scriptsize\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq\bm{0}\end{aligned}$
(88)

Therefore the feasibility condition over ”ellipsoidal $\cap$ polyhedral”
uncertainty remains the same for both the cases and is given by the following
result.

###### Theorem 3.7

Consider the uncertain portfolio problem (LABEL:eq4.5) under the uncertainty
set obtained from the combination of ellipsoidal and polyhedral uncertainty
sets. Let the radii of the ellipsoidal and polyhedral sets are respectively
$\delta_{E}$ and $\delta_{P}$. Then the solution vector $\bm{x}$ to this
problem is feasible for the robust constraint in (LABEL:eq4.5) if and only if
there exists a $\lambda\geq 0$ such that the following matrix inequality hold:

$\displaystyle\scriptsize\begin{bmatrix}({\bm{\mu}^{(0)}}^{\top}\bm{x})^{2}-\tau^{2}-\lambda&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(0)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(1)}}^{\top}\bm{x})^{2}+\lambda&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(1)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(2)}}^{\top}\bm{x})^{2}+\lambda&\dots&({\bm{\mu}^{(2)}}^{\top}\bm{x})({\bm{\mu}^{(n)}}^{\top}\bm{x})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(0)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(1)}}^{\top}\bm{x})&({\bm{\mu}^{(n)}}^{\top}\bm{x})({\bm{\mu}^{(2)}}^{\top}\bm{x})&\dots&({\bm{\mu}^{(n)}}^{\top}\bm{x})^{2}+\lambda\end{bmatrix}\succeq
0.$

##  4 Conclusion

In this paper, we have obtained the feasibility conditions for the robust
counterparts of the uncertain Markowitz model in the form of nonlinear
semidefinite programming. We represented our uncertainty sets in the form of
quadratic functions and that helped us to write our uncertain constraint in
quadratic forms of the perturbation vector $\bm{\zeta}$. We can also express
the robust counterpart problem in form of a quadratic semidefinite programming
problem by replacing the uncertain constraint with the obtained feasibility
condition. In this way, we will also be able to obtain the robust counterparts
for the combined uncertainty sets, which in general is difficult to find out.
With this approach, we can make use of the solution algorithms, convergence
analysis for non-linear semidefinite programming given in Correa 2004 ; Freund
2007 .

In addition, the study of combined uncertainty sets is more important because
the robust solutions under these sets are less conservative as compared to the
individual uncertainty sets. There is a scope of further research in obtaining
the robust counterparts of some other portfolio models in the semidefinite
programming form by using this approach.

## Declaration on Conflict of Interest

The authors hereby declare that they have no conflicts of interest.

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