Abstract
In this paper, we study riskaverse models for multicriteria optimization problems under uncertainty. We use a weighted sumbased scalarization and take a robust approach by considering a set of scalarization vectors to address the ambiguity and inconsistency in the relative weights of each criterion. We model the risk aversion of the decision makers via the concept of multivariate conditional valueatrisk (CVaR). First, we introduce a model that optimizes the worstcase multivariate CVaR and show that its optimal solution lies on a particular type of stochastic efficient frontier. To solve this model, we develop a finitely convergent delayed cut generation algorithm for finite probability spaces. We also show that the proposed model can be reformulated as a compact linear program under certain assumptions. In addition, for the cut generation problem, which is in general a mixedinteger program, we give a stronger formulation than the existing ones for the equiprobable case. Next, we observe that similar polyhedral enhancements are also useful for a related class of multivariate CVaRconstrained optimization problems that has attracted attention recently. In our computational study, we use a budget allocation application to benchmark our proposed maximin type riskaverse model against its riskneutral counterpart and a related multivariate CVaRconstrained model. Finally, we illustrate the effectiveness of the proposed solution methods for both classes of models.
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Acknowledgements
We thank the two referees and the associate editor for their valuable comments that improved the presentation. Simge Küçükyavuz and Xiao Liu are supported, in part, by National Science Foundation Grants 1732364 and 1733001. Nilay Noyan acknowledges the support from Bilim Akademisi—The Science Academy, Turkey, under the BAGEP Program.
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Appendices
Appendix A: Stochastic dominance
In this section, we review the wellknown stochastic dominance relations, which are essential for the stochastic Pareto optimality definitions presented in Sect. 2.
The stochastic dominance relations are fundamental concepts in comparing random variables (Mann and Whitney 1947; Lehmann 1955) and have been widely used in economics and finance (see, e.g., Levy 1992). Different from the approaches based on risk measures, in a stochastic dominance based approach, the random variables are compared by a pointwise comparison of some performance functions (constructed from their distribution functions when the order is greater than zero). We note that the lower order dominance relations (\(k=0,1,\) and 2) are the most common ones (referred to as ZSD, FSD, and SSD, respectively). We provide the formal definitions below and refer the reader to Müller and Stoyan (2002) and Shaked and Shanthikumar (1994) for further details.

We say that a random variable X dominates another random variable Y in the zeroth order if \(X\ge Y\) everywhere, i.e., \(X(\omega )\ge Y(\omega )\) for all \(\omega \in \varOmega \).

An integrable random variable X dominates another integrable Y in the first order (or X is stochastically larger than Y) if \(F_1(X,\eta ):=\mathbbm {P}(X \le \eta ) \le F_1(Y,\eta ):=\mathbbm {P}(Y \le \eta )\) for all \(\eta \in \mathbbm {R}\).

For \(k\ge 2\) we say that a kintegrable random variable X (i.e., \(\in \mathcal {L}^k\)) dominates another kintegrable random variable Y in the kth order if \(F_k(X,\eta ) \le F_k(Y,\eta )\) for all \(\eta \in \mathbbm {R}\), where \(F_k(X,\eta )=\int _{\infty }^\eta F_{k1}(X,t)~\mathrm {d}t\) for all \(\eta \in \mathbbm {R}\).

For \(k=0\), if \(X(\omega )> Y(\omega )\) for all \(\omega \in \varOmega \), we will refer to the relation as “strong ZSD” and denote it by \(X\succ _{(0)}Y\). For \(k \ge 1\), if all the inequalities \(F_k(X,\eta ) \le F_k(Y,\eta )\) are strict, then we refer to the relation as “strong kSD” and denote it by \(X\succ _{(k)}Y\). We remark that the notion of “strong kSD” is not analogous to the notion of strict kSD, which requires that at least one of the inequalities defining the dominance relation is strict.
Appendix B: A class of facets of \(\textit{conv}(\mathcal S)\)
Recall that
Before we study the facets of \(\textit{conv}(\mathcal S)\), we first need to establish its dimension.
Proposition B.1
\(\textit{Conv}(\mathcal S)\) is a polyhedron with dimension \(n + d + m  1\).
Proof
Note that \(\textit{conv}(S)\) is a polyhedron, because \(\varvec{\beta }\in \{0,1\}^n\). Next, we show that the dimension of \(\textit{conv}(\mathcal S) \) is \(n + d + m  1\). Clearly, in the original constraints defining \(\mathcal S\), there are two linearly independent equalities: \(\sum _{j \in [d]} c_j = 1\), \(\sum _{i \in [n]} \beta _i = k.\) In addition, there are nd equalities: \(\gamma _{ij} = c_j \beta _i\), for all \(i \in [n]\) and \(j \in [d]\). Hence, \(\textit{dim} (\textit{conv}(\mathcal S)) \le n + m + d  1.\)
Consider the following set of points:
where \(\mathbf {u}_v\), for all \(v \in [n]\) are any affinely independent vectors with k elements equal to 1 and the remaining elements equal to 0. These vectors exist because the dimension of the following system:
is \(n1\). Clearly, this set of points is feasible and affinely independent. In addition, the cardinality of this set is \(n + m + d \). Hence, \(dim (conv( \mathcal S) ) \ge n + m + d  1\), which completes the proof.
Proposition B.2
For any \(i \in [n]\), \(s \in [m]\), and \(t \in [m]{\setminus } \{s\}\), inequality (25) is facetdefining for \(conv(\mathcal S)\) if and only if \(s \in [m], t\in [m]{\setminus } \{s\}\) are such that \(y_{sj}< y_{tj}\) and \(y_{si} > y_{ti}\) for some \(i,j\in [d]\).
Proof
To show the necessity, we first note that if there exists a pair \(s \in [m], t\in [m]{\setminus } \{s\}\) such that \(y_{sj}\ge y_{tj}\) or \(y_{sj}\le y_{tj}\) for all \(j\in [d]\), in other words, when the realizations under a scenario are dominated by the realizations under another scenario, then the corresponding inequality (25) is dominated. To see this, suppose that \(y_{sj}\le y_{tj}\) for all \(j\in [d]\) for some pair \(\forall ~s \in [m], \forall ~t\in [m]{\setminus } \{s\}\). Then the corresponding inequality (25) is dominated by the original inequality \(\mathbf {c}^\top \mathbf {y}_s\ge \eta w_s\), because the coefficients of \(\gamma _{ij}\) are \(y_{tj}y_{sj}\ge 0\), and \(\gamma _{ij},w_t\ge 0\). Now consider the case that \(y_{sj}\ge y_{tj}\) for all \(j\in [d]\) for some pair \(\forall ~s \in [m], \forall ~t\in [m]{\setminus } \{s\}\). Then the corresponding inequality (25) is dominated by the original inequality \(\mathbf {c}^\top \mathbf {y}_t\ge \eta w_t\). To see this, observe that we can rewrite inequality (25) for this choice of s and t as, \(\mathbf {c}^\top \mathbf {y}_t + \sum _{j\in [d]}(y_{sj}y_{tj})(c_j\gamma _{ij}) \ge \eta w_tw_s\). It is now easy to see that the inequality is dominated, because \(y_{sj}y_{tj}\ge 0\), \(c_j\ge \gamma _{ij}\) and \(w_s\ge 0\).
To show sufficiency, we need to show that for any given \(i \in [n]\), \(s \in [m]\), and \(t \in [m]{\setminus } \{s\}\), there are \(n + m + d  1\) affinely independent points that satisfy (25) at equality. From the necessity condition, we only need to consider the cases for which there exists an index \(j_1 \in [d]\), such that \(y_{sj_1} < y_{tj_1} \), and there exists an index \(j_2 \in [d]\), such that \(y_{sj_2} > y_{tj_2} \). In order to simplify the notation, and without loss of generality, throughout the rest of the proof, we let \(j_1 = 1\), and \(j_2 = 2\), or equivalently, \(y_{s1} < y_{t1}\), and \(y_{s2} > y_{t2}\).
First, we construct a set of points:
where if \(u_{vi} = 0\), then \({\tilde{\mathbf {e}}}_v = \mathbf {e}_1\) and \(\rho ^1_v = y_{s1}\), else if \(u_{vi} = 1\), then \({\tilde{\mathbf {e}}}_v = \mathbf {e}_2\) and \(\rho ^1_v = y_{t2}\). In addition, \(\xi ^1_{vs} = \xi ^1_{vt}= 0\), and \(\xi ^1_{vl} = \max \{ {\tilde{M}}_s, {\tilde{M}}_t \}\) for all \(v \in [n] \) and \(l \in [m] {\setminus } \{s, t\}\). Clearly, the set of points defined in (39) are affinely independent feasible points, and satisfy (25) at equality. Next, we construct a set of points:
where \({\tilde{\mathbf {u}}}_j\) is any feasible point of (38) with \({\tilde{u}}_{ji} = 0\) if \(y_{sj} \le y_{tj} \), and \({\tilde{u}}_{ji} = 1\) otherwise (i.e., if \(y_{sj} \ge y_{tj} \)), for all \(j \in [d] {\setminus } \{1, 2\}\). In addition, \(\rho ^2_j = \min \{ y_{sj}, y_{tj}\}\), for all \(j \in [d] {\setminus } \{1, 2\}\). Furthermore, \(\xi ^2_{js} = \xi ^2_{jt}= 0\), and \(\xi ^2_{jl} = \max \{ {\tilde{M}}_s, {\tilde{M}}_t \}\) for all \(j \in [d] {\setminus } \{1, 2\} \) and \(l \in [m] {\setminus } \{s, t\}\). It is easy to see that the set of points defined in (40) are feasible, affinely independent from (39), and satisfy (25) at equality.
Furthermore, we construct the following set of points:
where \({\bar{\mathbf {u}}}_1\) is any feasible point of (38) with \({\bar{u}}_{1i} = 0\), and \({\bar{\mathbf {u}}}_2\) is any feasible point of (38) with \({\bar{u}}_{2i} = 1\). In addition, \(\xi ^3_{ss} = y_{t1}  y_{s1}\), \(\xi ^3_{st} = 0\), and \(\xi ^3_{sl} = \max \{ {\tilde{M}}_s, {\tilde{M}}_t \}\) for all \(l \in [m] {\setminus } \{s,t\}\). Similarly, \(\xi ^3_{ts} = 0\), \(\xi ^3_{tt} = y_{s2}  y_{t2}\), and \(\xi ^3_{tl} = \max \{ {\tilde{M}}_s, {\tilde{M}}_t \}\) for all \(l \in [m] {\setminus } \{s,t\}\). Clearly, the set of points defined by (41) are affinely independent feasible points which satisfy (25) at equality.
Finally, we construct the single point:
where \(\mathbf {c}^* = (c^*_1, c^*_2, 0, \ldots , 0)\), and the parameters \((c^*_1, c^*_2, \eta ^*)\) are uniquely defined by the following linear system:
or equivalently, \(c^*_1 = \frac{y_{s2}  y_{t2}}{y_{s2}  y_{t2} + y_{t1}  y_{s1} }\), \(c^*_2 = 1  c^*_1\), and \(0<c^*_1 ,c^*_2<1 \). In addition, \(\xi ^4_s = \xi ^4_t = 0\), and \(\xi ^4_l =\max \{ {\tilde{M}}_s, {\tilde{M}}_t \}\), for all \(l \in [m] {\setminus }\{s,t\}\).
Clearly, PT \(^4\) is affinely independent from the points defined by (39), since the following matrix:
has full rank (due to \( y_{t2} < y_{s2})\). In addition, it is easy to check that (42) is affinely independent from the points defined by (40) and (41). Furthermore, it is also a feasible point which satisfies (25) at equality.
From (39)–(42), we obtain \(n + m +d  1\) affinely independent feasible points which satisfy (25) at equality. Hence, inequalities (25) are facet defining. \(\square \)
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Liu, X., Küçükyavuz, S. & Noyan, N. Robust multicriteria riskaverse stochastic programming models. Ann Oper Res 259, 259–294 (2017). https://doi.org/10.1007/s104790172526z
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DOI: https://doi.org/10.1007/s104790172526z
Keywords
 Stochastic programming
 Risk aversion
 Robust optimization
 Multicriteria optimization
 Stochastic Pareto optimality
 Conditional valueatrisk
 Cut generation
 Mixedinteger programming
 McCormick envelopes
 RLT technique