Robust multicriteria risk-averse stochastic programming models

Abstract

In this paper, we study risk-averse models for multicriteria optimization problems under uncertainty. We use a weighted sum-based 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 value-at-risk (CVaR). First, we introduce a model that optimizes the worst-case 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 mixed-integer 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 CVaR-constrained optimization problems that has attracted attention recently. In our computational study, we use a budget allocation application to benchmark our proposed maximin type risk-averse model against its risk-neutral counterpart and a related multivariate CVaR-constrained model. Finally, we illustrate the effectiveness of the proposed solution methods for both classes of models.

Appendices

Appendix A: Stochastic dominance

In this section, we review the well-known 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 point-wise 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 k-integrable random variable X (i.e., $\in \mathcal {L}^k$) dominates another k-integrable random variable Y in the kth order if $F_k(X,\eta ) \le F_k(Y,\eta )$ for all $\eta \in \mathbbm {R}$, where $F_k(X,\eta )=\int _{-\infty }^\eta F_{k-1}(X,t)~\mathrm {d}t$ 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

$$\begin{aligned} \mathcal S :=&\bigg \{(\varvec{\gamma }, \mathbf {c}, \varvec{\beta }, \eta , \mathbf {w}) \in \mathbbm {R}_+^{n \times d}\times \mathbbm {R}_+^d \times \{0,1\}^n \times \mathbbm {R}\times \mathbbm {R}^m_+ \; |&\;\varvec{\gamma }= \varvec{\beta }\mathbf {c}^\top , ~\sum _{j \in [d]} c_j = 1,~\\&\sum _{i \in [n]} \beta _i = k,\mathbf {c}^\top \mathbf {y}_l \ge \eta - w_l,~\forall ~l \in [m]\bigg \}. \end{aligned}$$

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:

$$\begin{aligned}&\left( \mathbf {u}_v\mathbf {e}_1^\top , \mathbf {e}_1, \mathbf {u}_v, 0, 0\right)&\forall ~v \in [n], \\&\left( \mathbf {u}_1\mathbf {e}_j^\top , \mathbf {e}_j, \mathbf {u}_1, 0, 0\right)&\forall ~j \in [d] {\setminus } \{1\}, \\&\left( \mathbf {u}_1\mathbf {e}_1^\top , \mathbf {e}_1, \mathbf {u}_1, 0, e_l\right)&\forall ~l \in [m], \\&\left( \mathbf {u}_1\mathbf {e}_1^\top , \mathbf {e}_1, \mathbf {u}_1, -1, 0\right) , \end{aligned}$$

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:

$$\begin{aligned}&\varvec{\beta }\in \{0,1\}^n, \quad \sum _{i \in [n] } \beta _i = k, \end{aligned}$$

(38)

is $n-1$. 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 facet-defining 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, $\mathbf {c}^\top \mathbf {y}_t + \sum _{j\in [d]}(y_{sj}-y_{tj})(c_j-\gamma _{ij}) \ge \eta -w_t-w_s$. 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:

$$\begin{aligned}&\mathbf PT ^1_v= \left( \mathbf {u}_v \tilde{\mathbf {e}}_v^\top , {\tilde{\mathbf {e}}}_v, \mathbf {u}_v, \rho ^1_v ,\varvec{\xi }^1_v\right) ,&\forall ~v \in [n], \end{aligned}$$

(39)

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:

$$\begin{aligned}&\mathbf PT ^2_j= \left( \tilde{\mathbf {u}}_j\mathbf {e}_j^\top , \mathbf {e}_j, {\tilde{\mathbf {u}}}_j, \rho ^2_j, \varvec{\xi }^2_j \right) ,&\forall ~j \in [d] {\setminus } \{1,2\}, \end{aligned}$$

(40)

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:

$$\begin{aligned}&\mathbf PT ^3_s = \left( {\bar{\mathbf {u}}}_1\mathbf {e}_1^\top , \mathbf {e}_1, {\bar{\mathbf {u}}}_1, y_{t1}, \varvec{\xi }^3_s\right) \end{aligned}$$

(41a)

$$\begin{aligned}&\mathbf PT ^3_t= \left( {\bar{\mathbf {u}}}_2\mathbf {e}_2^\top , \mathbf {e}_2, {\bar{\mathbf {u}}}_2, y_{s2}, \varvec{\xi }^3_t\right) \end{aligned}$$

(41b)

$$\begin{aligned}&\mathbf PT ^3_l= \mathbf PT ^3_s + \left( 0, 0, 0, 0, \mathbf {e}_l\right) ,&\forall ~l \in [m] {\setminus } \{s, t\}, \end{aligned}$$

(41c)

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:

$$\begin{aligned} \mathbf PT ^4= \left( \mathbf {u}_1 {\mathbf {c}^*}^\top , \mathbf {c}^*, \mathbf {u}_1, \eta ^*, \varvec{\xi }^4\right) , \end{aligned}$$

(42)

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:

$$\begin{aligned}&c^*_1 + c^*_2 = 1 \\&y_{s1}c^*_1 + y_{s2}c^*_2 = \eta ^* \\&y_{t1}c^*_1 + y_{t2}c^*_2 = \eta ^*, \end{aligned}$$

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:

$$\begin{aligned} \begin{bmatrix} 1&0&y_{s1} \\ 0&1&y_{t2} \\ c^*_1&c^*_2&\eta ^*=y_{s1}c^*_1 + y_{s2}c^*_2 \end{bmatrix}, \end{aligned}$$

(43)

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. $\square$