compared the robust Omega ratio PSP proposed by Kapsos et al. (2014a) to the
robust mean-variance PSP with box and ellipsoidal uncertainty sets and the
robust CVaR PSP proposed by Zhu & Fukushima (2009). Results showed that robust
PSPs are less diversified than their nominal counterparts. However,
improvements were observed in the portfolio performance. Another comparison in
this context has been done by Sehgal & Mehra (2021). They compared PSPs based
on robust Omega ratio, semi-mean absolute deviation ratio, and weighted stable
tail adjusted return ratio (STARR) with their non-robust counterparts. In
these formulations, a budgeted uncertainty set is used for asset returns.
Results showed that the robust formulations outperform the nominal problems
with respect to standard deviation, value at risk (VaR), conditional value at
risk (CVaR), Sharpe ratio, and stable tail adjusted return ratio (STARR).

Sharpe and Omega ratios are based on the absolute volatility of assets.
However, some investors make decisions based on the volatility of an asset
compared to the market and not on the absolute volatility itself. Beta is a
measure of volatility that indicates whether an asset is more or less volatile
compared to the market. Hence, Beta can be used as a decision criteria to
capture the volatility of an asset compared to the market. A asset’s Beta is
calculated by dividing the product of the covariance of the asset returns and
the market returns by the variance of the market returns over a specified
period. However, this measure is subject to uncertainty since all components
of the Beta formula are uncertain parameters. Ghahtarani & Najafi (2013)
proposed a robust multi-objective PSP where the objectives are the portfolio
rate of return and its systematic risk (Beta). A budget uncertainty set to
model the uncertainty of Beta is used. The problem is reformulated as a
tractable goal program. Results show that portfolios selected based on the
robust Beta outperform non-robust Beta portfolios in terms of weight stability
and return volatility.

###  6.5 Robust Scenario-Based Formulation

Unlike Section 5.2 that reviews scenario-based formulations of multi-period
PSPs, this section focuses on the use of discrete scenarios to represent
uncertainty in single-period PSPs. Kouvelis & Yu (1997) proposed a robust
formulation for a discrete scenario-based uncertainty set. It optimizes an
objective function based on the worst possible scenario, which leads to the
worst-case conservative results. Roy (2010) proposed a new definition for
robust scenario-based solutions in which a solution is robust if it exhibits
good performance in most scenarios without ever exhibiting very poor
performance in any scenario. Then, they developed _$bw$ -robustness_, by
taking into consideration minimum acceptable objective value and a target
objective value to achieve, or exceed if possible. Gabrel et al. (2018)
developed a robust scenario-based PSP by using both worst-case scenario and
$bw$-robustness to maximize a portfolio’s return while returns of assets
belong to a discrete scenario-based uncertainty set. Moreover, they introduced
a new robustness criterion called _$pw$ -robustness_, in which instead of
maximizing a proportion of scenarios that their values are greater than or
equal to a threshold, the decision-maker specifies a fixed proportion of
scenarios, and maximizes the value of the soft bound. The $pw$-robustness
formulation is a MIP. To circumvent the computational time issue, Gabrel et
al. (2018) proposed two heuristic methods that can be used to obtain quick
solutions for problems of large sizes.

Some investors might invest based on their preferences of assets, where
ranking information of assets are uncertain. Nguyen & Lo (2012) proposed a
robust ranking mean-variance, which is similar to the classical mean-variance.
However, the ranking of assets is used instead of the return of assets.
Formulations were developed in two cases: the maximum ranking with and without
risk (variance). The ranking of assets belongs to a discrete uncertainty set,
which leads to a MIP solved by a constraint generation method.

###  6.6 Robust Data Envelopment Analysis and Portfolio Selection

One way for evaluating stocks or assets in the financial markets is data
envelopment analysis (DEA), in which the efficiency of stocks or assets is
evaluated based on a set of inputs and outputs (criteria). Based on this
method, units (assets or stocks) are divided into two parts: efficient, and
inefficient. Consequently, DEA calculates the efficacy rate of units. Peykani
et al. (2016) demonstrated that the efficiency of stocks in DEA depends on
inputs and outputs, which are uncertain. Consequently, they proposed robust
DEA with a budget uncertainty set. Results of robust DEA are more robust than
a non-robust DEA formulation with respect to the efficiency of stocks.
However, their formulations can be used only for continuous uncertainty sets.
Peykani et al. (2019) developed a robust DEA for a discrete scenario
formulation with uncertainty, which expands the application of robust DEA to
financial problems in the real-world. However, these robust DEA formulations
provide the efficiency ratio without any detail about the amount of money
invested in each asset while an investor needs to know the proportion of
investment of funds invested in each asset. Peykani et al. (2020) proposed a
two-phase portfolio selection process. At the first stage, the efficiency of
candidate stocks is evaluated by robust DEA. In the second stage, the optimal
portfolio is formed by using robust mean-semi variance-liquidity and robust
mean-absolute deviation-liquidity models. In both phases, budget uncertainty
sets are used for the uncertain parameters. This two-phase formulation
provides two filters (robust DEA, and robust PSP) to find the optimal
portfolio.

Table LABEL:table:50 lists all the reviewed articles (n=142) in a
chronological order and classifies them based on the problem type (PSP),
uncertain parameters (UP), the structure of uncertainty or ambiguity sets used
in the robust formulation (U/A set), the robust optimization method employed
to deal with uncertainty (RO method) and the class of the tractably
reformulated problem (Model).

Table 2: Summary of the reviewed articles $\\#$ | Article | PSP type |  UP 111Uncertain Parameters |  U/A 222Uncertainty or Ambiguity set | RO Methods | Model
---|---|---|---|---|---|---
|  |  Mean-Variance |  Minimum Variance |  MAD |  LPM |  Factor-Based |  Utility function |  VaR/CVaR |  Multi-Period |  ALM |  Log-return |  Index tracking |  Ratios |  Others |  Asset return |  Variance-Covariance matrix |  Factor-based parameters |  Distribution function |  Scale parameter (Log-Return) |  Others |  Classical |  Asymmetrical |  Moment-based |  Metric-based |  Discrete |  Others |  RO |  DRO |  Relative robust |  Adaptive robust |  Others |  LP |  NLP |  SOCP |  SDP |  Mixed integer |  Other Convex |  Non-convex
1 |  Ben-Tal et al. (2000) |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
2 |  Costa & Paiva (2002) |  |  |  |  |  |  |  |  |  |  | * |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
3 |  Lauprete et al. (2003) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  | * |  |  |  |  |
4 |  Goldfarb & Iyengar (2003) | * |  |  |  | * |  | * |  |  |  |  | * |  |  |  | * |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
5 |  Ghaoui et al. (2003) |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * | * |  |  |  | * |  |  |  |  |  |  |  |  |  |
6 |  Tütüncü & Koenig (2004) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
7 |  Pinar & Tütüncü (2005) |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  | * |  |  |  | * |  |  |  | *
8 |  Ceria & Stubbs (2006) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  | * |  |  |  |
9 |  Lu (2006) |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |
10 |  Pachamanova (2006) |  |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
11 |  Lutgens et al. (2006) |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  | * |  | * |  |  |  |  |  | * | * |  |  |  |
12 |  Fabozzi et al. (2007) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
13 |  Bienstock (2007) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  | * |  |
14 |  Garlappi et al. (2007) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  | * |  |  |  |  |
15 |  Calafiore (2007) | * |  | * |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  |  |  |  | * |
16 |  Popescu (2007) |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |
17 |  Huang et al. (2007) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  | * |  |  | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |
18 |  Pınar (2007) |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  | * |  |  |  |  | * |  |  |  |  |  |
19 |  Oguzsoy & Güven (2007) | * |  |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  | * |  |
20 |  Van Hest & De Waegenaere (2007) | * |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
21 |  Ma et al. (2008) |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
22 |  Natarajan et al. (2008) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
23 |  Huang et al. (2008) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  | * |  | * |  |  |  | * |  |  |  |  |  |
24 |  Quaranta & Zaffaroni (2008) |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
25 |  Bertsimas & Pachamanova (2008) |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
26 |  Shen & Zhang (2008) |  |  |  |  |  |  |  | * |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  | * |  | * |  |  |  |  |  |  | * |  |  |  |
27 |  Chen & Tan (2009) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
28 |  DeMiguel & Nogales (2009) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  | * |  |  |  |  |
29 |  Schöttle & Werner (2009) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
30 |  Zhu & Fukushima (2009) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  | * |  |  |  | * |  |  | * |  |  |  | * |  | * |  |  |  |
31 |  Natarajan et al. (2009) | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * | * |  |  |  |  |
32 |  Lutgens & Schotman (2010) |  |  |  |  | * |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |
33 |  Natarajan et al. (2010) |  |  |  |  |  | * |  |  |  |  |  |  |  | * | * |  | * |  |  | * |  | * |  |  |  | * | * |  |  |  |  |  | * |  |  |  | *
34 |  Huang et al. (2010) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  | * |  |  | * |  | * |  |  |  |
35 |  Iyengar & Ma (2010) |  |  |  |  | * |  |  |  | * |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
36 |  Iyengar & Ma (2010) |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
37 |  Zymler et al. (2011) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
38 |  Gregory et al. (2011) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
39 |  Gülpınar et al. (2011) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
40 |  Moon & Yao (2011) |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
41 |  Chen et al. (2011a) |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |
42 |  Lu (2011) |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
43 |  Chen et al. (2011b) |  |  |  |  |  |  | * |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
44 |  Hellmich & Kassberger (2011) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  | * |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |
45 |  Guastaroba et al. (2011) |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  | * |  |  |  |
46 |  Kawas & Thiele (2011a) |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
47 |  Kawas & Thiele (2011b) |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
48 |  Zymler et al. (2011) |  |  |  |  |  |  |  |  |  |  |  | * | * | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
49 |  Fonseca & Rustem (2012) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
50 |  Fonseca et al. (2012) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
51 |  Sadjadi et al. (2012) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  | * |  |
52 |  García et al. (2012) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |
53 |  Ling & Xu (2012) |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
54 |  Pflug et al. (2012) | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  | * |  |  |  | * | * |  |  |  |  |
55 |  Lim et al. (2012) |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  | * |  |  |  |  |  |  |  |  |
56 |  Chen & Kwon (2012) |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  | * |  |
57 |  Nguyen & Lo (2012) | * |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  | * |  |
58 |  Hauser et al. (2013) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |
59 |  Kim et al. (2013b) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |
60 |  Kim et al. (2013a) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |
61 |  Gülpinar & Pachamanova (2013) |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
62 |  Glasserman & Xu (2013) |  |  |  |  | * |  |  | * |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |
63 |  Deng et al. (2013) |  |  |  |  |  |  | * |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |
64 |  Ghahtarani & Najafi (2013) |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
65 |  Fliege & Werner (2014) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
66 |  Pınar & Paç (2014) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  | * |  |  | * |  | * |  |  |  | * | * |  |  |  |  |  | * |  |  |  |
67 |  Kim et al. (2014b) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
68 |  Kim et al. (2014c) |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
69 |  Recchia & Scutellà (2014) |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  | * |  |  |  |
70 |  Kakouris & Rustem (2014) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  | * |
71 |  Han et al. (2017) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  | * |
72 |  Zhu et al. (2014) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  | * | * |  |  |  |  |  | * | * |  |  |  | * |  | * |  |  |  |
73 |  Liu & Chen (2014) |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |
74 |  Flor & Larsen (2014) |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |
75 |  Pae & Sabbaghi (2014) |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
76 |  Gülpinar et al. (2014) |  |  |  |  |  |  |  |  |  | * |  |  | * |  |  |  |  |  | * | * | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
77 |  Ling et al. (2014) |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  | * |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |
78 |  Kapsos et al. (2014a) |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  | * |  |  |  | * |  |  | * |  |  |  | * |  | * |  |  |  |
79 |  Maillet et al. (2015) |  | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  |  | * |  |  |  |
80 |  Kim et al. (2015) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |
81 |  Doan et al. (2015) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  | * |  |  |  | * |  |  |  |  |  |
82 |  Rezaie et al. (2015) |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
83 |  Marzban et al. (2015) |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
84 |  Liu et al. (2015) |  |  |  |  |  | * |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
85 |  Desmettre et al. (2015) |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  |
86 |  Pınar (2016) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
87 |  Li et al. (2016) |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
88 |  Rujeerapaiboon et al. (2016) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |
89 |  Fernandes et al. (2016) |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  |
90 |  Yu (2016) |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  | * |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  |
91 |  Gülpınar et al. (2016) |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
92 |  Peykani et al. (2016) |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
93 |  Xidonas et al. (2017a) |  | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  | * |  |
94 |  Xidonas et al. (2017b) |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  |  |  |  | *
95 |  Belhajjam et al. (2017) | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * | * |  |  |  |
96 |  Yu et al. (2017) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  | * |  |  | * |  |  |  |  |  |
97 |  Cong & Oosterlee (2017) | * |  |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
98 |  Ling et al. (2017) |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |
99 |  citeplatanakis2017asset |  |  |  |  | * |  |  |  | * |  |  |  |  | * |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
100 |  Bo & Capponi (2017) |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |
101 |  Kawas & Thiele (2017) |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  | * |  |  |  |  |  | * | * |  |  |  | * |  |  |  |  |  |
102 |  Gülpınar & Çanakoḡlu (2017) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  |  | * | * | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
103 |  Sharma et al. (2017) |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  | * |  |  |  | * |  |  | * |  |  |  | * |  | * |  |  |  |
104 |  Ding et al. (2018) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  |  |  |  | * |
105 |  Simões et al. (2018) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |
106 |  Kapsos et al. (2018) |  | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  | * |  | * |  |  |  |  |  | * |  |  |  |  |
107 |  Chen & Zhou (2018) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  | * |  |  | * |  |  |  |
108 |  Ghahtarani & Najafi (2018) |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
109 |  Hasuike & Mehlawat (2018) |  |  |  |  | * |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |
110 |  Ghahtarani et al. (2018) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  | * |  |  |  |  |  |
111 |  Paç & Pınar (2018) | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  | * |  |  |  | * | * |  |  |  |  |
112 |  Liu & Chen (2018) |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |
113 |  Gabrel et al. (2018) |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |
114 |  Lu et al. (2019) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
115 |  Chen & Wei (2019) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  | * |  |  | * |  |  |  |
116 |  Bai et al. (2019) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  | * |  |  | * |  |  |  |
117 |  Dai & Wang (2019) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |
118 |  Plachel (2019) |  | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |
119 |  Liu et al. (2019) |  |  |  |  |  |  | * | * |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  | * |  |  |  |  |  |  |  | *
120 |  Kang et al. (2019) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |
121 |  Goel et al. (2019) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  | * |  |  |  | * |  |  |  |  |  |
122 |  Kara et al. (2019) |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  |
123 |  Ling et al. (2019) |  |  |  | * | * |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
124 |  Yu et al. (2019) |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  | * |  |  |  | * |  |  | * |  |  |  | * |  |  |  |  |  |
125 |  Peykani et al. (2019) |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  |  | * |  | * |  |  |  |  | * |  |  |  |  |  |
126 |  Khodamoradi et al. (2020) | * |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  |  |  | * |  |
127 |  Lee et al. (2020) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
128 |  Peykani et al. (2020) | * |  | * |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
129 |  Georgantas et al. (2021) | * |  |  |  |  |  | * |  |  |  |  | * |  | * |  |  | * |  |  | * |  |  |  |  |  | * | * |  |  |  | * |  |  |  |  |  |
130 |  Kouaissah (2021) |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
131 |  Chakrabarti (2021) |  | * |  |  |  |  |  |  |  |  |  | * |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  | * |  |  | * | * |  |  |  |  |
132 |  Benati & Conde (2021) |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  |  |  |  | *
133 |  Ashrafi & Thiele (2021) |  |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
134 |  Jiang & Wang (2021) | * |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |
135 |  Yin et al. (2021) | * |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * | * |  |  |  |
136 |  Baviera & Bianchi (2021) | * |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  | * |  |  |  |  | * |  |  |  |  |
137 |  Huang et al. (2021) |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  | * |  |  |  |  |  |
138 |  Sehgal & Mehra (2021) |  |  | * |  |  |  | * |  |  |  |  | * |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  |  |
139 |  Caçador et al. (2021) | * | * |  |  |  |  |  |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  | * |  |  |  |  |  |  |  |  | *
140 |  Gajek & Krajewska (2021) |  |  |  |  |  |  | * |  | * |  |  |  |  |  |  |  | * |  |  |  |  |  |  | * |  |  | * |  |  |  |  | * |  |  |  |  |
141 |  Zhao et al. (2021) |  |  |  |  |  |  | * |  |  |  |  |  |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  | * |  |  |  |  |
142 |  Swain & Ojha (2021) | * | * |  |  |  |  |  |  |  |  |  |  |  | * | * |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |  | * |  |  |  |  |

Moreover, figures provide some statistics about the reviewed papers.

Figure 4: Distribution of articles based on PSP type Figure 5: Distribution of
articles based on optimization problem types Figure 6: Distribution of
articles based on uncertain parameters Figure 7: Distribution of articles
based on uncertainty sets Figure 8: Distribution of articles based on RO
methods

Figures 4, 5 show that $\%26$ of published articles used mean-variance and
$\%23$ of published articles proposed robust VaR/CVaR formulations. Moreover,
majority of robust PSPs leads to SOCP, and LP. The third robust counterpart
type is NLP with $\%19$. Figures 6, 7 show that $\%46$ of published articles
consider asset return as uncertain parameters. Another important
classification of articles is based on type of uncertainty set. This figure
also shows the distribution of uncertainty sets in published articles. It
demonstrates that about $\%55$ of articles use classical uncertainty sets
include box, ellipsoidal, budgeted, and polyhedral. Figure 8 illustrates the
distribution of RO methods in published articles. This figure shows that
mostly classical RO and DRO are used in articles.

##  7 Conclusions and Future Research Directions

Portfolio selection has been a fertile area for applying modern RO techniques
as evident by the large number of robust PSP articles published in the last
two decades. The inherent uncertainty about future asset returns, the
abundance of public data available and the risk-averse nature of most
investors make RO an appealing approach in this area. As shown in this review
paper, a wide range of robust PSP variants was studied, from a “plain vanilla”
single-period, mean-variance PSP with a simple box uncertainty set (e.g.,
Tütüncü & Koenig (2004)) to formulations that consider advanced risk measures
(e.g., Ghahtarani et al. (2018), Huang et al. (2010)), adaptive uncertainty
sets (e.g., Yu (2016)), real-life investment strategies (e.g., Pflug et al.
(2012), Paç & Pınar (2018)) and dynamic portfolio balancing (e.g., Ling et al.
(2019), Cong & Oosterlee (2017)). This variety of modeling assumptions and
approaches and the overlaps among them make it difficult to develop a unifying
framework for robust PSPs, yet we adopted a multi-dimensional classification
scheme that depends on the risk measure to be optimized, the type of uncertain
parameters, the approach used to capture uncertainty and the the planning
horizon (i.e., single- vs. multi-period).

Despite the surge of interest about robust PSPs in the research community,
this area has received little attention from practitioners. A possible reason
for such a rift between theory and practice is that research in this area was
often driven by advancements in operations research methods rather than being
in response to the real needs of the financial industry. Moreover, the value
of using robust approaches might not be readily apparent to practitioners who
are accustomed to classical PSP models. Therefore, experimental studies, like
those presented in Kim et al. (2013b), Kim et al. (2014b), Kim et al. (2013a),
Kim et al. (2018b), Kim et al. (2015), Schöttle & Werner (2009), are crucial
for bridging this gap. The fact that tractable reformulations of most robust
counterparts are more complex, both conceptually and computationally, than
their corresponding deterministic formulations might make robust
reformulations less attractive for practitioners (e.g., Kouvelis & Yu (1997),
Hauser et al. (2013), Simões et al. (2018), Lim et al. (2012), Huang et al.
(2010)).

Nevertheless, the perception of robust optimization as an overly conservative
portfolio selection approach is probably the major obstacle to its wide
adoption by investment professionals. The reader can easily notice that this
issue has received a lot of attention in the robust PSP research. Approaches
proposed in the literature to attenuate the conservatism of robust
formulations include: using controllable uncertainty sets (e.g., ellipsoid
(Fabozzi et al., 2007) or budget (Liu et al., 2015)), data-driven approaches
(e.g., Doan et al. (2015), Bienstock (2007)), alternative risk measures (e.g.,
relative log-return (Lim et al., 2012) or (Huang et al., 2010)),
distributionally robust optimization (e.g., Ling et al. (2014)) and regime-
dependent robust models (e.g., Liu & Chen (2014), Yu (2016)). While these
approached can be effective in controlling conservatism and providing well-
balanced solutions, they often lead to models that are challenging to handle.

Given that asset returns do not generally behave like independent random
variables, but are instead dependent on common factors and have significant
temporal correlations, trying to capture the variability of returns directly
often leads to large uncertainty sets and hence conservative solutions.
Instead, robust factor models deals with the uncertainty in the independent
factors themselves, thus lead less conservative formulations. However, as Lu
(2006) noted, selecting the suitable factors for the model and adjusting their
weights are still worthy of further investigation. Another promising direction
is to use _dynamic uncertainty sets_ , that incorporate time-series models to
capture auto-correlations in asset returns. Dynamic sets have been shown to
result in less pessimistic solutions compared to static ones in other
applications (Lorca & Sun, 2014, 2016).

An important advantage of financial markets is the abundance of historical
data that can be used to build uncertainty and ambiguity sets for uncertain
parameters. Although data-driven robust formulations for a few variants of the
PSP have been proposed in the literature (e.g., Bienstock (2007), Kawas &
Thiele (2017), Rujeerapaiboon et al. (2016), Doan et al. (2015), Lotfi &
Zenios (2018), Liu et al. (2019), Kang et al. (2019)), this is still a
promising area for future research given the recently-proposed techniques for
constructing and sizing uncertainty sets to achieve desirable properties (see
e.g., Bertsimas & Brown (2009), Bertsimas et al. (2018)). In a related matter,
and as noted by Kang et al. (2019) in the context of robust CVaR optimization,
it is still unclear which ambiguity sets should be used for DRO PSPs and how
they should be sized to provide the best out-of-sample performance. With the
plethora of ambiguity set structures proposed in recent year, investigating
new variants of the distributionally robust PSPs is a plausible research
direction.

Another promising research direction is the application of “soft” robust
optimization methods to financial problems. A drawback of classical robust
optimization is that it tries to capture most possible realizations of the
parameters within the uncertainty set, which usually results in large sets and
conservative solutions. Alternatively, one can construct smaller uncertainty
sets that include only a subset of these possible realization and allow robust
constraints to be violated, yet with penalties. Examples of these approaches
include Globalized Robust Optimization (Ben-Tal et al., 2017), Robustness
Optimization (Long et al., 2019) and Almost Robust Optimization (Baron et al.,
2019). Soft robust optimization methods are still scarcely applied in the PSP
literature (see Recchia & Scutellà (2014)), but have the potential for
providing a trade-off between robustness and the quality of solutions.

## References

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