Data-Driven Distributionally Robust Risk-Averse Two-Stage Stochastic Linear Programming over Wasserstein Ball

Abstract

In this paper, we consider a data-driven distributionally robust two-stage stochastic linear optimization problem over 1-Wasserstein ball centered at a discrete empirical distribution. Differently from the traditional two-stage stochastic programming which involves the expected recourse function as the preference criterion and hence is risk-neutral, we take the conditional value-at-risk (CVaR) as the risk measure in order to model its effects on decision making problems. We mainly explore tractable reformulations for the proposed robust two-stage stochastic programming with mean-CVaR criterion by analyzing the first case where uncertainties are only in the objective function and then the second case where uncertainties are only in the constraints. We demonstrate that the first model can be exactly reformulated as a deterministic convex programming. Furthermore, it is shown that under several different support sets, the resulting convex optimization problems can be converted into computationally tractable conic programmings. Besides, the second model is generally NP-hard since checking constraint feasibility can be reduced to a norm maximization problem over a polytope. However, even with the case of uncertainty in constraints, tractable conic reformulations can be established when the extreme points of the polytope are known. Finally, we present numerical results to discuss how to control the risk for the best decisions and illustrate the computational effectiveness and superiority of the proposed models.

Appendix: Proofs

We first draw the following auxiliary result which will be used throughout the proofs of Corollaries $3.1-3.3$.

Lemma A.1

Suppose that ${\Xi }\subseteq \mathbb {R}^{m}$ is a convex set, then for any $\varvec{y}_{1},\varvec{y}_{2}\in \mathbb {R}^{m}$ and finite $\eta \ge 0$, it holds that

$$\begin{aligned} \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{y}_{1}-\eta \left\| \varvec{\xi }-\varvec{y}_{2}\right\| _{p}\right) =\inf _{\left\| \varvec{z}\right\| _{q}\le \eta }\left( \varvec{z}^{\top }\varvec{y}_{2}+ \sup _{\varvec{\xi }\in {\Xi }}\left( (\varvec{y}_{1}-\varvec{z})^{\top } \varvec{\xi }\right) \right) , \end{aligned}$$

where $\left\| \cdot \right\| _{q}$ is the dual norm of $\left\| \cdot \right\| _{p}$, i.e., $\frac{1}{p}+\frac{1}{q}=1$.

Proof

Note that

$$\begin{aligned}&\quad ~\sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{y}_{1}-\eta \left\| \varvec{\xi }-\varvec{y}_{2}\right\| _{p}\right) \\&\quad =\sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{y}_{1}- \sup _{\left\| \varvec{z}\right\| _{q}\le \eta }{\varvec{z}}^{\top }(\varvec{\xi }-\varvec{y}_{2})\right) \\&\quad =\sup _{\varvec{\xi }\in {\Xi }}\inf _{\left\| \varvec{z}\right\| _{q}\le \eta }\left( \varvec{\xi }^{\top }\varvec{y}_{1}-\varvec{z}^{\top } (\varvec{\xi }-\varvec{y}_{2})\right) \\&\quad =\inf _{\left\| \varvec{z}\right\| _{q}\le \eta }\left( \varvec{z}^{\top }\varvec{y}_{2}+ \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{y}_{1}- \varvec{z}^{\top }\varvec{\xi }\right) \right) , \end{aligned}$$

where the first equality is due to the definition of the dual norm and the third equality follows from the general minimax theorem [34, Corollary 3.3], which can be applied because the set $\left\{ \varvec{z}\in \mathbb {R}^{m}:{\left\| \varvec{z}\right\| _{q}\le \eta }\right\}$ is compact convex for any finite $\eta \ge 0$. This completes the proof. $\square$

Proof of Corollary 3.1

Note that the support set ${\Xi }=\{\varvec{\xi }\in \mathbb {R}^{m}:({\varvec{\xi }-\varvec{\xi }_{0}})^{\top }\varvec{W}^{-1}({\varvec{\xi }-\varvec{\xi }_{0}})\le 1\}$ with $\varvec{W}\in \mathbb {S}_{++}^{m}$ is convex. Then by Lemma A.1, we can reformulate (12b) as

$$\begin{aligned} s_{i}&\ge \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{Q} \varvec{\hat{y}}^{i}-\eta \left\| \varvec{\xi }-\varvec{\zeta }^{i}\right\| _{p}\right) +({\varvec{q}^{0}})^{\top }\varvec{\hat{y}}^{i}\nonumber \\&=\inf _{\left\| \varvec{\hat{z}}_{i}\right\| _{q}\le \eta }\left( \varvec{\hat{z}}_{i}^{\top }\varvec{\zeta }^{i}+ \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{Q}\varvec{\hat{y}}^{i}- \varvec{\hat{z}}_{i}^{\top }\varvec{\xi }\right) \right) +(\varvec{q}^{0})^{\top } \varvec{\hat{y}}^{i}. \end{aligned}$$

(37)

We now discuss the reformulation of the inner subproblem $\sup _{\varvec{\xi }\in {\Xi }}$ in (37). For this problem, there exists $\varvec{\xi }=\varvec{\xi }_{0}$ such that $({\varvec{\xi }-\varvec{\xi }_{0}})^{\top }\varvec{W}^{-1}({\varvec{\xi } -\varvec{\xi }_{0}})<1$, which implies Slater’s condition is satisfied, and hence, the strong duality holds. Then we have

$$\begin{aligned} \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{Q}\varvec{\hat{y}}^{i}- \varvec{\hat{z}}_{i}^{\top }\varvec{\xi }\right) =\inf _{\tilde{\nu }_{i}\ge 0}\sup _{\varvec{\xi }}\left( \varvec{\xi }^{\top } \varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i}^{\top }\varvec{\xi }- \hat{\nu }_{i}\left( ({\varvec{\xi }-\varvec{\xi }_{0}})^{\top } \varvec{W}^{-1}({\varvec{\xi }-\varvec{\xi }_{0}})-1\right) \right) , \end{aligned}$$

which can be rewritten as

$$\begin{aligned} \begin{aligned} \min _{\hat{\nu }_{i}\ge 0,\hat{u}_{i}}\quad&\hat{u}_{i}\\ \text{ s.t. }\quad&\varvec{\xi }^{\top }\varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i}^{\top } \varvec{\xi }-\hat{\nu }_{i}\left( ({\varvec{\xi }-\varvec{\xi }_{0}})^{\top } \varvec{W}^{-1}({\varvec{\xi }-\varvec{\xi }_{0}})-1\right) \le \hat{u}_{i}~~\forall \varvec{\xi }\in \mathbb {R}^{m}. \end{aligned} \end{aligned}$$

(38)

Note that the semi-infinite constraint in (38) is equivalent to linear matrix inequality constraint as below:

$$\begin{aligned} \begin{aligned}&\left( \varvec{\xi }^{\top },1\right) \begin{bmatrix} \hat{\nu }_{i}\varvec{W}^{-1}&{}-\frac{1}{2}(2\hat{\nu }_{i}\varvec{W}^{-1} \varvec{\xi }_{0}+\varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i})\\ &{}&{}\\ -\frac{1}{2}(2\hat{\nu }_{i}\varvec{W}^{-1}\varvec{\xi }_{0}+\varvec{Q}\varvec{\hat{y}}^{i}- \varvec{\hat{z}}_{i})^{\top }&{}\hat{u}_{i}+\hat{\nu }_{i}{\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1} \varvec{\xi }_{0}-\hat{\nu }_{i} \end{bmatrix}\\&\quad \left( \varvec{\xi }^{\top },1\right) ^{\top }\ge 0~~\forall \varvec{\xi }\in \mathbb {R}^{m}\\&\iff \begin{bmatrix} \hat{\nu }_{i}\varvec{W}^{-1}&{}-\frac{1}{2}(2\hat{\nu }_{i}\varvec{W}^{-1} \varvec{\xi }_{0}+\varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i})\\ &{}&{}\\ -\frac{1}{2}(2\hat{\nu }_{i}\varvec{W}^{-1}\varvec{\xi }_{0}+\varvec{Q} \varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i})^{\top }&{}\hat{u}_{i}+\hat{\nu }_{i} {\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1}\varvec{\xi }_{0}-\hat{\nu }_{i} \end{bmatrix}\succeq 0,\nonumber \end{aligned} \end{aligned}$$

which in turn allows us to reformulate problem (38) as

$$\begin{aligned} \begin{aligned} \min _{\hat{\nu }_{i}\ge 0,\hat{u}_{i}}\quad&\hat{u}_{i}\\ \text{ s.t. }\quad&\begin{bmatrix} \hat{\nu }_{i}\varvec{W}^{-1}&{}-\frac{1}{2}(2\hat{\nu }_{i} \varvec{W}^{-1}\varvec{\xi }_{0}+\varvec{Q}\varvec{\hat{y}}^{i} -\varvec{\hat{z}}_{i})\\ &{}&{}\\ -\frac{1}{2}(2\hat{\nu }_{i}\varvec{W}^{-1}\varvec{\xi }_{0}+ \varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i})^{\top }&{}\hat{u}_{i} +\hat{\nu }_{i} {\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1}\varvec{\xi }_{0}-\hat{\nu }_{i} \end{bmatrix}\succeq 0. \end{aligned} \end{aligned}$$

(39)

By replacing the inner subproblem $\sup _{\varvec{\xi }\in {\Xi }}$ in (37) by (39), we obtain

$$\begin{aligned} \begin{aligned} s_{i}&\ge \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{Q} \varvec{\hat{y}}^{i}-\eta \left\| \varvec{\xi }-\varvec{\zeta }^{i}\right\| _{p}\right) +({\varvec{q}^{0}})^{\top }\varvec{\hat{y}}^{i}\\&=\inf _{\varvec{\hat{z}}_{i},\hat{\nu }_{i},\hat{u}_{i}} \left( \varvec{\hat{z}}_{i}^{\top } \varvec{\zeta }^{i}+\hat{u}_{i}\right) +(\varvec{q}^{0})^{\top }\varvec{\hat{y}}^{i}\\&\qquad \text{ s.t. } \begin{bmatrix} \hat{\nu }_{i}\varvec{W}^{-1}&{}-\frac{1}{2}(2\hat{\nu }_{i} \varvec{W}^{-1}\varvec{\xi }_{0}+\varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i})\\ &{}&{}\\ -\frac{1}{2}(2\hat{\nu }_{i}\varvec{W}^{-1}\varvec{\xi }_{0}+ \varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i})^{\top }&{}\hat{u}_{i}+ \hat{\nu }_{i}{\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1}\varvec{\xi }_{0}-\hat{\nu }_{i}\\ \end{bmatrix}\succeq 0\\ \nonumber&\qquad \quad ~\;\left\| \varvec{\hat{z}}_{i}\right\| _{q}\le \eta \\&\qquad \quad ~\;\hat{\nu }_{i}\ge 0. \end{aligned} \end{aligned}$$

Similarly, for the reformulation of (12c), we present the following parallel conclusion.

$$\begin{aligned} \begin{aligned} s_{i}+\frac{\lambda \beta }{\alpha }&\ge \sup _{\varvec{\xi }\in {\Xi }} \left( (1+\frac{\lambda }{\alpha })\varvec{\xi }^{\top }\varvec{Q}\varvec{\tilde{y}}^{i}-\eta \left\| \varvec{\xi }-\varvec{\zeta }^{i}\right\| _{p}\right) +(1+\frac{\lambda }{\alpha })({\varvec{q}^{0}})^{\top }\varvec{\tilde{y}}^{i}\\&=\inf _{\varvec{\tilde{z}}_{i},\tilde{\nu }_{i},\tilde{u}_{i}} \left( \varvec{\tilde{z}}_{i}^{\top }\varvec{\zeta }^{i}+\tilde{u}_{i}\right) + (1+\frac{\lambda }{\alpha })({\varvec{q}^{0}})^{\top }\varvec{\tilde{y}}^{i}\\&\qquad \text{ s.t. } \begin{bmatrix} \tilde{\nu }_{i}\varvec{W}^{-1}&{}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!-\frac{1}{2} (2\tilde{\nu }_{i}\varvec{W}^{-1}\varvec{\xi }_{0}+(1+\frac{\lambda }{\alpha }) \varvec{Q}\varvec{\tilde{y}}^{i}-\varvec{\tilde{z}}_{i})\\ &{}&{}\\ -\frac{1}{2}(2\tilde{\nu }_{i}\varvec{W}^{-1}\varvec{\xi }_{0}+ (1+\frac{\lambda }{\alpha })\varvec{Q}\varvec{\tilde{y}}^{i}-\varvec{\tilde{z}}_{i})^{\top }&{} \tilde{u}_{i}+\tilde{\nu }_{i}{\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1}\varvec{\xi }_{0}-\tilde{\nu }_{i}\\ \end{bmatrix}\succeq 0\\&\qquad \quad ~\;\left\| \varvec{\tilde{z}}_{i}\right\| _{q}\le \eta \\&\qquad \quad ~\;\tilde{\nu }_{i}\ge 0. \end{aligned} \end{aligned}$$

Hence, the claim follows. $\square$

Proof of Corollary 3.2

Note that the support set ${\Xi }=\{\varvec{\xi }\in \mathbb {R}^{m}:\varvec{C}\varvec{\xi }\le \varvec{d}\}$ with $\varvec{C}\in \mathbb {R}^{k\times m}$ and $\varvec{d}\in \mathbb {R}^{k}$ is convex. Then by Lemma A.1, we can reformulate (12b) as

$$\begin{aligned} s_{i}&\ge \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top } \varvec{Q}\varvec{\hat{y}}^{i}-\eta \left\| \varvec{\xi }-\varvec{\zeta }^{i}\right\| _{p}\right) +({\varvec{q}^{0}})^{\top }\varvec{\hat{y}}^{i}\nonumber \\&=\inf _{\left\| \varvec{\hat{z}}_{i}\right\| _{q}\le \eta }\left( \varvec{\hat{z}}_{i}^{\top }\varvec{\zeta }^{i}+ \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{Q}\varvec{\hat{y}}^{i} -\varvec{\hat{z}}_{i}^{\top }\varvec{\xi }\right) \right) +(\varvec{q}^{0})^{\top } \varvec{\hat{y}}^{i}. \end{aligned}$$

(40)

We now discuss the reformulation of the inner linear programming subproblem $\sup _{\varvec{\xi }\in {\Xi }}$ in (40). For this problem, the strong duality theory yields

$$\begin{aligned} \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{Q}\varvec{\hat{y}}^{i}- \varvec{\hat{z}}_{i}^{\top }\varvec{\xi }\right) = \left\{ \begin{aligned} \sup _{\varvec{\xi }}\quad&(\varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i})^{\top }\varvec{\xi }\\ \text{ s.t. }\quad&\varvec{C}\varvec{\xi }\le \varvec{d} \end{aligned} \right. = \left\{ \begin{aligned} \inf _{\varvec{\gamma }_{i}\ge 0}\quad&\varvec{\gamma }_{i}^{\top }\varvec{d}\\ \text{ s.t. }\quad&\varvec{C}^{\top }\varvec{\gamma }_{i}=\varvec{Q}\varvec{\hat{y}}^{i}-\varvec{\hat{z}}_{i}. \end{aligned} \right. \end{aligned}$$

(41)

By replacing the inner subproblem $\sup _{\varvec{\xi }\in {\Xi }}$ in (40) by (41), we have

$$\begin{aligned} \begin{aligned} s_{i}&\ge \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top } \varvec{Q}\varvec{\hat{y}}^{i}-\eta \left\| \varvec{\xi }-\varvec{\zeta }^{i}\right\| _{p}\right) +({\varvec{q}^{0}})^{\top }\varvec{\hat{y}}^{i}\\ \nonumber&=\inf _{\varvec{\hat{z}}_{i},\varvec{\gamma }_{i}} \left( \varvec{\hat{z}}_{i}^{\top }\varvec{\zeta }^{i}+\varvec{\gamma }_{i}^{\top } \varvec{d}\right) +({\varvec{q}^{0}})^{\top }\varvec{\hat{y}}^{i}\\ \nonumber&~~~~~\text{ s.t. }~~\;\varvec{\hat{z}}_{i}=\varvec{Q}\varvec{\hat{y}}^{i}- \varvec{C}^{\top }\varvec{\gamma }_{i}\\ \nonumber&\qquad \quad ~\;\left\| \varvec{\hat{z}}_{i}\right\| _{q}\le \eta \\ \nonumber&\qquad \quad ~\;\;\varvec{\gamma }_{i}\ge 0.\nonumber \end{aligned} \end{aligned}$$

Similarly, for the reformulation of (12c), we present the following parallel conclusion.

$$\begin{aligned} \begin{aligned} s_{i}+\frac{\lambda \beta }{\alpha }&\ge \sup _{\varvec{\xi }\in {\Xi }} \left( (1+\frac{\lambda }{\alpha })\varvec{\xi }^{\top }\varvec{Q}\varvec{\tilde{y}}_{i}-\eta \left\| \varvec{\xi }-\varvec{\zeta }^{i}\right\| _{p}\right) +(1+\frac{\lambda }{\alpha })({\varvec{q}^{0}})^{\top } \varvec{\tilde{y}}_{i}\\ \nonumber&=\inf _{\varvec{\tilde{z}}_{i},\varvec{\mu }_{i}} \left( \varvec{\tilde{z}}_{i}^{\top }\varvec{\zeta }^{i}+\varvec{\mu }_{i}^{\top } \varvec{d}\right) +(1+\frac{\lambda }{\alpha })({\varvec{q}^{0}})^{\top }\varvec{\tilde{y}}_{i}\\ \nonumber&~~~~~\text{ s.t. }~~\;\varvec{\tilde{z}}_{i}=(1+\frac{\lambda }{\alpha }) \varvec{Q}\varvec{\tilde{y}}_{i}-\varvec{C}^{\top }\varvec{\mu }_{i}\\ \nonumber&\qquad \quad ~\;\left\| \varvec{\tilde{z}}_{i}\right\| _{q}\le \eta \\ \nonumber&\qquad \quad ~\;\;\varvec{\mu }_{i}\ge 0.\nonumber \end{aligned} \end{aligned}$$

Hence, the claim follows. $\square$

Proof of Corollary 3.3

Note that the support set ${\Xi }=\mathbb {R}^{m}$ is convex. Then by Lemma A.1, we can reformulate (12b) as

$$\begin{aligned} s_{i}&\ge \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{Q} \varvec{\hat{y}}^{i}-\eta \left\| \varvec{\xi }-\varvec{\zeta }^{i}\right\| _{p}\right) +({\varvec{q}^{0}})^{\top }\varvec{\hat{y}}^{i}\nonumber \\&=\inf _{\left\| \varvec{\hat{z}}_{i}\right\| _{q}\le \eta }\left( \varvec{\hat{z}}_{i}^{\top }\varvec{\zeta }^{i}+ \sup _{\varvec{\xi }\in {\Xi }}\left( \varvec{\xi }^{\top }\varvec{Q}\varvec{\hat{y}}^{i}- \varvec{\hat{z}}_{i}^{\top }\varvec{\xi }\right) \right) +(\varvec{q}^{0})^{\top } \varvec{\hat{y}}^{i}. \end{aligned}$$

(42)

We now discuss the reformulation of the inner unconstrained subproblem $\sup _{\varvec{\xi }\in {\Xi }}$ in (42), which implies that $\varvec{\hat{z}}_{i}=\varvec{Q}\varvec{\hat{y}}^{i}$, and then (12b) is equivalent to

$$\begin{aligned} \begin{aligned}&(\varvec{Q}\varvec{\hat{y}}^{i})^{\top }\varvec{\zeta }^{i}+ ({\varvec{q}^{0}})^{\top }\varvec{\hat{y}}^{i}\le s_{i}\\&\left\| \varvec{Q}\varvec{\hat{y}}^{i} \right\| _{q}\le \eta . \end{aligned} \end{aligned}$$

Similarly, for the reformulation of (12c), we present the following parallel conclusion.

$$\begin{aligned} \begin{aligned}&(1+\frac{\lambda }{\alpha })\left( (\varvec{Q}\varvec{\tilde{y}}^{i})^{\top } \varvec{\zeta }^{i}+({\varvec{q}^{0}})^{\top }\varvec{\tilde{y}}^{i}\right) \le s_{i}+\frac{\lambda \beta }{\alpha }\\&(1+\frac{\lambda }{\alpha })\left\| \varvec{Q}\varvec{\tilde{y}}^{i}\right\| _{q}\le \eta . \end{aligned} \end{aligned}$$

Hence, the claim follows. $\square$

Proof of Proposition 5.1

Note that by using (6), the worst-case mean-CVaR problem in (28) can be rewritten as

$$\begin{aligned}&\sup _{\mathbb {P}\in \mathcal {P}}\left\{ \mathbb {E}_{\mathbb {P}} \left[ Q(\varvec{x},\varvec{\tilde{\xi }})\right] +\lambda \textrm{CVaR}_{1-\alpha } (Q(\varvec{x},\varvec{\tilde{\xi }}))\right\}&\nonumber \\&\quad =\sup _{\mathbb {P}\in \mathcal {P}}\left\{ \mathbb {E}_{\mathbb {P}} \left[ Q(\varvec{x},\varvec{\tilde{\xi }})\right] +\lambda \inf _{\beta \in \mathbb {R}}\left\{ \beta +\frac{1}{\alpha } \mathbb {E}_{\mathbb {P}}\left[ \left( Q\left( \varvec{x}, \varvec{\tilde{\xi }}\right) -\beta \right) _{+}\right] \right\} \right\} \end{aligned}$$

(43a)

$$\begin{aligned}&\quad =\sup _{\mathbb {P}\in \mathcal {P}}\inf _{\beta \in \mathbb {R}}\left\{ \lambda \beta +\mathbb {E}_{\mathbb {P}} \left[ Q(\varvec{x},\varvec{\tilde{\xi }})+\frac{\lambda }{\alpha } \left( Q\left( \varvec{x},\varvec{\tilde{\xi }}\right) -\beta \right) _{+}\right] \right\} \end{aligned}$$

(43b)

$$\begin{aligned}&\quad =\inf _{\beta \in \mathbb {R}}\left\{ \lambda \beta + \sup _{\mathbb {P}\in \mathcal {P}}\mathbb {E}_{\mathbb {P}} \left[ Q(\varvec{x},\varvec{\tilde{\xi }})+\frac{\lambda }{\alpha } \left( Q\left( \varvec{x},\varvec{\tilde{\xi }}\right) -\beta \right) _{+}\right] \right\} , \end{aligned}$$

(43c)

where the third equality is due to a stochastic saddle point theorem [33].

The subordinate worst-case expectation problem in (43c) can be expressed as

$$\begin{aligned} \sup _{\mathbb {P}\in \mathcal {M}_{+}}~&\int _{\widehat{{\Xi }}}Q(\varvec{x},\varvec{\xi }) +\frac{\lambda }{\alpha }\left( Q\left( \varvec{x},\varvec{\xi }\right) -\beta \right) _{+}\mathbb {P} \left( d \varvec{\xi }\right) \end{aligned}$$

(44a)

$$\begin{aligned} \text{ s.t. }\quad \!\!&\int _{\widehat{{\Xi }}}\mathbb {P}\left( d \varvec{\xi }\right) =1 \end{aligned}$$

(44b)

$$\begin{aligned}&\int _{\widehat{{\Xi }}}\varvec{\xi }\mathbb {P}\left( d \varvec{\xi }\right) =\varvec{\hat{\mu }} \end{aligned}$$

(44c)

$$\begin{aligned}&\int _{\widehat{{\Xi }}}\varvec{\xi }\varvec{\xi }^{\top }\mathbb {P}\left( d \varvec{\xi }\right) = \varvec{\widehat{{\Sigma }}}+\varvec{\hat{\mu }}\varvec{\hat{\mu }}^{\top }, \end{aligned}$$

(44d)

where $\mathcal {M}_{+}$ represents the one of nonnegative Borel measures on $\mathbb {R}^{mn}$.

We now assign dual variables $z_{0}\in \mathbb {R}$, $\varvec{z}\in \mathbb {R}^{mn}$, and $\varvec{Z}\in \mathbb {S}^{mn}_{+}$ with the constraints of the primal problem (44), and then obtain the following dual problem

$$\begin{aligned} \inf _{z_{0},\varvec{z},\varvec{Z}} \quad&z_{0}+\varvec{z}^{\top }\varvec{\hat{\mu }}+ \langle \varvec{Z},\varvec{\widehat{{\Sigma }}}+\varvec{\hat{\mu }} \varvec{\hat{\mu }}^{\top }\rangle \end{aligned}$$

(45a)

$$\begin{aligned} \text{ s.t. }\quad&z_{0}+\varvec{z}^{\top }\varvec{\xi }+ \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle \ge Q(\varvec{x},\varvec{\xi })+\frac{\lambda }{\alpha } \left( Q\left( \varvec{x},\varvec{\xi }\right) - \beta \right) _{+}~~\forall \varvec{\xi }\in \widehat{{\Xi }} \end{aligned}$$

(45b)

$$\begin{aligned}&z_{0}\in \mathbb {R},~\varvec{z}\in \mathbb {R}^{mn},~\varvec{Z}\in \mathbb {S}^{mn}_{+}. \end{aligned}$$

(45c)

It can be shown that strong duality holds due to $\varvec{\widehat{{\Sigma }}}\succ 0$ [32]. Note that constraint (45b) can be rewritten as

$$\begin{aligned} \sup _{\varvec{\xi }\in \widehat{{\Xi }}} \left( Q(\varvec{x},\varvec{\xi })+\frac{\lambda }{\alpha } \left( Q\left( \varvec{x},\varvec{\xi }\right) - \beta \right) _{+}-z_{0}-\varvec{z}^{\top }\varvec{\xi }- \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle \right) \le 0, \end{aligned}$$

then by (29), which is further equivalent to

$$\begin{aligned} \begin{aligned}&\sup _{\varvec{\xi }\in \widehat{{\Xi }}} \left( \varvec{\xi }^{\top }\varvec{y}^{1} -z_{0}-\varvec{z}^{\top }\varvec{\xi }- \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle \right) \le 0\\&\sup _{\varvec{\xi }\in \widehat{{\Xi }}} \left( \varvec{\xi }^{\top }\varvec{y}^{2}+\frac{\lambda }{\alpha } \left( \varvec{\xi }^{\top }\varvec{y}^{2}- \beta \right) -z_{0}-\varvec{z}^{\top }\varvec{\xi }- \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle \right) \le 0. \end{aligned} \end{aligned}$$

(46)

For the optimization problem $\sup _{\varvec{\xi }\in \widehat{{\Xi }}}$ on the left-hand side of the first inequality in the constraint system (46), there exists $\varvec{\xi }=\varvec{\xi }_{0}$ such that $({\varvec{\xi }-\varvec{\xi }_{0}})^{\top }\varvec{W}^{-1}({\varvec{\xi }-\varvec{\xi }_{0}})<1$, which implies Slater’s condition is satisfied, and hence, the strong duality holds. Then we have

$$\begin{aligned}&\sup _{\varvec{\xi }\in \widehat{{\Xi }}} \left( \varvec{\xi }^{\top }\varvec{y}^{1} -z_{0}-\varvec{z}^{\top }\varvec{\xi }- \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle \right) \\&\quad =\inf _{\nu _{1}\ge 0}\sup _{\varvec{\xi }}\left( \varvec{\xi }^{\top }\varvec{y}^{1} -z_{0}-\varvec{z}^{\top }\varvec{\xi }- \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle -\nu _{1} \left( ({\varvec{\xi }-\varvec{\xi }_{0}})^{\top }\varvec{W}^{-1}({\varvec{\xi }- \varvec{\xi }_{0}})-1 \right) \right) ,\nonumber \end{aligned}$$

which can be converted into

$$\begin{aligned} \begin{aligned} \min _{\nu _{1}\ge 0,u_{1}}\quad&u_{1}\\ \text{ s.t. }\quad&\varvec{\xi }^{\top }\varvec{y}^{1} -z_{0}-\varvec{z}^{\top }\varvec{\xi }- \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle -\nu _{1} \left( ({\varvec{\xi }-\varvec{\xi }_{0}})^{\top }\varvec{W}^{-1} ({\varvec{\xi }-\varvec{\xi }_{0}})-1 \right) \le u_{1}~~\forall \varvec{\xi }\in \mathbb {R}^{mn}. \end{aligned} \end{aligned}$$

(47)

Note that the semi-infinite constraint in (47) is equivalent to linear matrix inequality constraint as below:

$$\begin{aligned} \begin{aligned}&\left( \varvec{\xi }^{\top },1\right) \begin{bmatrix} \varvec{Z}+\nu _{1}\varvec{W}^{-1}&{}-\frac{1}{2}(2\nu _{1}\varvec{W}^{-1} \varvec{\xi }_{0}+\varvec{y}^{1}-\varvec{z})\\ &{}&{}\\ -\frac{1}{2}(2\nu _{1}\varvec{W}^{-1}\varvec{\xi }_{0}+ \varvec{y}^{1}-\varvec{z})^{\top }&{}~~~u_{1}+\nu _{1}{\varvec{\xi }_{0}}^{\top } \varvec{W}^{-1}\varvec{\xi }_{0}+z_{0}-\nu _{1} \end{bmatrix}\\&\quad \left( \varvec{\xi }^{\top },1\right) ^{\top }\ge 0~~\forall \varvec{\xi }\in \mathbb {R}^{mn}\\&\iff \begin{bmatrix} \varvec{Z}+\nu _{1}\varvec{W}^{-1}&{}-\frac{1}{2}(2\nu _{1}\varvec{W}^{-1}\varvec{\xi }_{0}+ \varvec{y}^{1}-\varvec{z})^{\top }\\ &{}&{}\\ -\frac{1}{2}(2\nu _{1}\varvec{W}^{-1}\varvec{\xi }_{0}+\varvec{y}^{1}- \varvec{z})^{\top }&{}~~~u_{1}+\nu _{1}{\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1}\varvec{\xi }_{0}+ z_{0}-\nu _{1} \end{bmatrix}\succeq 0,\nonumber \end{aligned} \end{aligned}$$

which in turn allows us to reformulate problem (47) as

$$\begin{aligned} \begin{aligned} \min _{\nu _{1}\ge 0,u_{1}}\quad&u_{1}\\ \nonumber \text{ s.t. }\quad&\begin{bmatrix} \varvec{Z}+\nu _{1}\varvec{W}^{-1}&{}-\frac{1}{2}(2\nu _{1}\varvec{W}^{-1} \varvec{\xi }_{0}+\varvec{y}^{1}-\varvec{z})\\ &{}&{}\\ -\frac{1}{2}(2\nu _{1}\varvec{W}^{-1} \varvec{\xi }_{0}+\varvec{y}^{1}-\varvec{z})^{\top }&{}~~~u_{1}+ \nu _{1}{\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1}\varvec{\xi }_{0}+z_{0}-\nu _{1} \end{bmatrix}\succeq 0. \end{aligned} \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \begin{aligned} 0&\ge \sup _{\varvec{\xi }\in \widehat{{\Xi }}} \left( \varvec{\xi }^{\top }\varvec{y}^{1} -z_{0}-\varvec{z}^{\top }\varvec{\xi }- \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle \right) \\ \nonumber&=\min _{\nu _{1},u_{1}}\quad u_{1}\\ \nonumber&\qquad \text{ s.t. } \begin{bmatrix} \varvec{Z}+\nu _{1}\varvec{W}^{-1}&{}-\frac{1}{2}(2\nu _{1} \varvec{W}^{-1}\varvec{\xi }_{0}+\varvec{y}^{1}-\varvec{z})\\ &{}&{}\\ -\frac{1}{2}(2\nu _{1}\varvec{W}^{-1}\varvec{\xi }_{0}+ \varvec{y}^{1}-\varvec{z})^{\top }&{}~~~u_{1}+\nu _{1}{\varvec{\xi }_{0}}^{\top } \varvec{W}^{-1}\varvec{\xi }_{0}+z_{0}-\nu _{1} \end{bmatrix}\succeq 0\\ \nonumber&\qquad \quad ~\;\nu _{1}\ge 0.\nonumber \end{aligned} \end{aligned}$$

Similarly, for the reformulation of the second inequality in the constraint system (46), we present the following parallel conclusion.

$$\begin{aligned} \begin{aligned} \frac{\lambda \beta }{\alpha }&\ge \sup _{\varvec{\xi }\in \widehat{{\Xi }}} \left( (1+\frac{\lambda }{\alpha })\varvec{\xi }^{\top }\varvec{y}^{2} -z_{0}-\varvec{z}^{\top }\varvec{\xi }- \langle \varvec{Z},\varvec{\xi }\varvec{\xi }^{\top }\rangle \right) \\ \nonumber&=\min _{\nu _{2},u_{2}}\quad u_{2}\\ \nonumber&\qquad \text{ s.t. } \begin{bmatrix} \varvec{Z}+\nu _{2}\varvec{W}^{-1}&{}-\frac{1}{2}(2\nu _{2} \varvec{W}^{-1}\varvec{\xi }_{0}+(1+\frac{\lambda }{\alpha })\varvec{y}^{2}- \varvec{z})\\ &{}&{}\\ -\frac{1}{2}(2\nu _{2}\varvec{W}^{-1}\varvec{\xi }_{0}+(1+\frac{\lambda }{\alpha }) \varvec{y}^{2}-\varvec{z})^{\top }&{}~~~u_{2}+\nu _{2} {\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1}\varvec{\xi }_{0}+z_{0}-\nu _{2} \end{bmatrix}\succeq 0\\ \nonumber&\qquad \quad ~\;\nu _{2}\ge 0.\nonumber \end{aligned} \end{aligned}$$

In conclusion, problem (28) with moment-based ambiguity set (31) is equivalent to

$$\begin{aligned} \min _{\varvec{x},\varvec{y}^{1},\varvec{y}^{2},\varvec{z},\varvec{Z},z_{0},\beta ,\nu _{1},\nu _{2},u_{1},u_{2}} \quad&\left\{ \varvec{c}^{\top }\varvec{x}+\lambda \beta +z_{0}+\varvec{z}^{\top }\varvec{\hat{\mu }}+ \langle \varvec{Z},\varvec{\widehat{{\Sigma }}}+ \varvec{\hat{\mu }}\varvec{\hat{\mu }}^{\top }\rangle \right\} \\&\text{ s.t. }~~~~~~\quad \begin{bmatrix} \varvec{Z}+\nu _{1}\varvec{W}^{-1}&{}-\frac{1}{2}(2\nu _{1} \varvec{W}^{-1}\varvec{\xi }_{0}+\varvec{y}^{1}-\varvec{z})\\ &{}&{}\\ -\frac{1}{2}(2\nu _{1}\varvec{W}^{-1} \varvec{\xi }_{0}+\varvec{y}^{1}-\varvec{z})^{\top }&{}~~~u_{1}+\nu _{1}{\varvec{\xi }_{0}}^{\top } \varvec{W}^{-1}\varvec{\xi }_{0}+z_{0}-\nu _{1} \end{bmatrix}\succeq 0 \\&\begin{bmatrix} \varvec{Z}+\nu _{2}\varvec{W}^{-1}&{}-\frac{1}{2}(2\nu _{2}\varvec{W}^{-1} \varvec{\xi }_{0}+(1+\frac{\lambda }{\alpha })\varvec{y}^{2}- \varvec{z})\\ &{}&{}\\ -\frac{1}{2}(2\nu _{2}\varvec{W}^{-1}\varvec{\xi }_{0}+ (1+\frac{\lambda }{\alpha })\varvec{y}^{2}-\varvec{z})^{\top }&{}~~~u_{2}+\nu _{2} {\varvec{\xi }_{0}}^{\top }\varvec{W}^{-1}\varvec{\xi }_{0}+z_{0}-\nu _{2} \end{bmatrix}\succeq 0 \\&\sum _{j=1}^{n}y^{1}_{rj}=x_{r},~\sum _{j=1}^{n}y^{2}_{rj}=x_{r}~~\forall r\in [m]\\&\sum _{r=1}^{m}y^{1}_{rj}=a_{j},~\sum _{r=1}^{m}y^{2}_{rj}=a_{j}~~\forall j\in [n]\\&u_{1}\le 0,u_{2}\le \frac{\lambda \beta }{\alpha }\\&\varvec{x}\ge 0,~\varvec{y}^{1}\ge 0,~\varvec{y}^{2}\ge 0, ~\nu _{1}\ge 0,~\nu _{2}\ge 0,~\varvec{Z}\in \mathbb {S}^{mn}_{+}. \end{aligned}$$

This completes the proof. $\square$