Decomposition and discrete approximation methods for solving two-stage distributionally robust optimization problems

Abstract

Decomposition methods have been well studied for solving two-stage and multi-stage stochastic programming problems, see Rockafellar and Wets (Math. Oper. Res. 16:119–147, 1991), Ruszczyński and Shapiro (Stochastic Programming, Handbook in OR & MS, North-Holland Publishing Company, Amsterdam, 2003) and Ruszczyński (Math. Program. 79:333–353, 1997). In this paper, we propose an algorithmic framework based on the fundamental ideas of the methods for solving two-stage minimax distributionally robust optimization (DRO) problems where the underlying random variables take a finite number of distinct values. This is achieved by introducing nonanticipativity constraints for the first stage decision variables, rearranging the minimax problem through Lagrange decomposition and applying the well-known primal-dual hybrid gradient (PDHG) method to the new minimax problem. The algorithmic framework does not depend on specific structure of the ambiguity set. To extend the algorithm to the case that the underlying random variables are continuously distributed, we propose a discretization scheme and quantify the error arising from the discretization in terms of the optimal value and the optimal solutions when the ambiguity set is constructed through generalized prior moment conditions, the Kantorovich ball and $\phi$-divergence centred at an empirical probability distribution. Some preliminary numerical tests show the proposed decomposition algorithm featured with parallel computing performs well.

Appendix

Proof of Lemma 1

The inequality holds trivially if $Q\in {\tilde{{{\mathcal {P}}}}}$. So we only consider the case when $Q\not \in {\tilde{{{\mathcal {P}}}}}$. We proceed the proof in three steps.

Step 1. By the definition of the total variation norm (see [1]), $||P||= \sup _{\Vert \phi \Vert ^*\le 1} \langle P, \phi \rangle ,$ where $\Vert \cdot \Vert ^*$ is the dual of norm $\Vert \cdot \Vert$. Moreover, by the definition of the total variation metric

$$\begin{aligned} \mathsf {dl}_{TV}(Q,\tilde{{{\mathcal {P}}}})= & {} \inf _{P\in {\tilde{{{\mathcal {P}}}}}} \mathsf {dl}_{TV}(Q,P) \\= & {} \inf _{P\in \text{ cl } \{P\in \mathscr {P}(\tilde{\varXi }): {\mathbb {E}}_P[\varPsi _E]= \mu _E, {\mathbb {E}}_P[\varPsi _I]\preceq \mu _I\}} \sup _{\Vert \phi \Vert ^*\le 1} \langle Q-P, \phi \rangle \\= & {} \sup _{\Vert \phi \Vert ^*\le 1} \min _{P\in \text{ cl } \{P\in \mathscr {P}(\tilde{\varXi }): {\mathbb {E}}_P[\varPsi _E]= \mu _E, {\mathbb {E}}_P[\varPsi _I]\preceq \mu _I\}} \langle Q-P, \phi \rangle \\= & {} \sup _{\Vert \phi \Vert ^*\le 1} \inf _{P\in \{P\in \mathscr {P}(\tilde{\varXi }): {\mathbb {E}}_P[\varPsi _E]= \mu _E, {\mathbb {E}}_P[\varPsi _I]\preceq \mu _I\}} \langle Q-P, \phi \rangle , \end{aligned}$$

where $\text{ cl }(\cdot )$ denotes closure of a set under topology of weak convergence, $\langle P, \phi \rangle :=\int _{\tilde{\varXi }}\phi (\xi )P(d\xi )$ and the exchange is justified by [7, Theorem 2] under our assumption that $\tilde{{{\mathcal {P}}}}$ is weakly compact. Note that we write $\langle P, \phi \rangle$ for ${\mathbb {E}}_P[\phi ]$ in that later on we will relax P from a probability measure to a positive measure, and we will be able to see P clearly as a variable in the moment system and the moment system is linear in P. It is easy to observe that $\mathsf {dl}_{TV}(P,{{\mathcal {P}}})\le 2$. Moreover, under the Slater type condition (16), it follows by [38, Proposition 3.4] that

$$\begin{aligned}&\inf _{P\in \{P\in \mathscr {P}(\tilde{\varXi }): {\mathbb {E}}_P[\varPsi _E]= \mu _E, {\mathbb {E}}_P[\varPsi _I]\preceq \mu _I\}} \langle Q-P, \phi \rangle \nonumber \\&\quad =\sup _{\lambda \in \varLambda ,\lambda _0} \inf _{P\in \mathscr {M}_+(\tilde{\varXi })} \langle Q-P, \phi \rangle + \lambda \bullet (\langle P,\varPsi \rangle -\mu ) + \lambda _0(\langle P,1\rangle -1)\nonumber \\&\quad =\sup _{\lambda \in \varLambda ,\lambda _0} \inf _{P\in \mathscr {M}_+(\tilde{\varXi })} \langle Q-P, \phi -\lambda \bullet \varPsi -\lambda _0\rangle \nonumber \\&\qquad + \langle Q, \lambda \bullet \varPsi +\lambda _0\rangle -\lambda \bullet \mu -\lambda _0, \end{aligned}$$

(49)

where $\mathscr {M}_+(\tilde{\varXi })$ denotes the set of all positive measures defined on $\tilde{\varXi }$, $\varLambda :=\{(\lambda _1,\cdots ,\lambda _q): \lambda _i\succeq 0, \; \text{ for }\; i=p+1,\ldots ,q\},$ and $A \bullet B$ denotes the Frobenius product of the two matrices A and B. If there exists some $\xi _i$ such that $\phi (\xi _i) -\lambda \bullet \varPsi (\xi _i) -\lambda _0>0$, then the value of (49) is $-\infty$ because we can choose $P=\alpha \delta _{\xi _i}(\cdot )$, where $\delta _{\xi _i}(\cdot )$ denotes the Dirac probability measure at $\xi _i$, and drive $\alpha$ to $+\infty$. Thus we are left to consider that case with

$$\begin{aligned} \phi (\xi ) -\lambda \bullet \varPsi (\xi ) -\lambda _0 \le 0, \xi \in {\tilde{\varXi }}. \end{aligned}$$

Consequently we can rewrite (49) as

$$\begin{aligned}&\inf _{P\in \{P\in \mathscr {P}(\tilde{\varXi })_+: {\mathbb {E}}_P[\varPsi _E] = \mu _E, {\mathbb {E}}_P[\varPsi _I]\preceq \mu _I\}} \langle Q-P, \phi \rangle \\&\quad = \sup _{\lambda \in \varLambda ,\lambda _0} \inf _{P\in \mathscr {M}_+(\tilde{\varXi }), \xi \in \varXi _N} \langle Q-P, \phi -\lambda \bullet \varPsi -\lambda _0\rangle + \langle Q, \lambda \bullet \varPsi +\lambda _0\rangle -\lambda \bullet \mu -\lambda _0\\&\quad = \sup _{\lambda \in \varLambda ,\lambda _0}\langle Q, \phi -\lambda \bullet \varPsi -\lambda _0\rangle + \langle Q, \lambda \bullet \varPsi +\lambda _0\rangle -\lambda \bullet \mu -\lambda _0\\&\quad =\sup _{\lambda \in \varLambda ,\lambda _0} \langle Q, \phi \rangle -\lambda \bullet \mu -\lambda _0. \end{aligned}$$

The second inequality is due to the fact that the optimum is attained at $P=0$. Summarizing the discussions above, we arrive at

$$\begin{aligned} \mathsf {dl}_{TV}(Q,{\tilde{{{\mathcal {P}}}}})= & {} \left\{ \begin{array}{cl} \displaystyle \sup _{\Vert \phi \Vert ^*\le 1} \sup _{\lambda \in \varLambda ,\lambda _0} &{} \langle Q, \phi \rangle -\lambda \bullet \mu -\lambda _0 \\ \text{ s.t. } &{}\phi (\xi ) -\lambda \bullet \varPsi (\xi ) -\lambda _0 \le 0, \;\;\xi \in {\tilde{\varXi }} \end{array}\right. \nonumber \\= & {} \left\{ \begin{array}{cl} \displaystyle \sup _{\lambda \in \varLambda ,\lambda _0}\sup _{\Vert \phi \Vert ^*\le 1} &{} \langle Q, \phi \rangle -\lambda \bullet \mu -\lambda _0 \\ \text{ s.t. } &{}\phi (\xi ) -\lambda \bullet \varPsi (\xi ) -\lambda _0 \le 0, \;\;\xi \in {\tilde{\varXi }} \end{array}\right. \nonumber \\= & {} \left\{ \begin{array}{cl} \displaystyle \sup _{\lambda \in \varLambda ,\lambda _0} &{} \langle Q, \min \{\lambda \bullet \varPsi (\xi )+\lambda _0, 1\}\rangle -\lambda \bullet \mu -\lambda _0 \\ \text{ s.t. } &{} -1 -\lambda \bullet \varPsi (\xi ) -\lambda _0 \le 0,\;\;\text{ a.e. } \; \; \xi \in {\tilde{\varXi }}. \end{array}\right. \end{aligned}$$

(50)

The first equality is obtained by swapping the two “sup” operations because their optimal values are bounded. To see how the second equality holds, we may compare the optimal value of the second and third programs above. Let’s denote the second one by (P) and the third one by (P’). Observe that (P’) is transferred from (P) by replacing $\phi (\xi )$ in the objective by the largest possible value $\min \{\lambda \bullet \varPsi (\xi )+\lambda _0, 1\}$ and in the constraint the smallest possible value $-1$ making the feasible set largest. This means the optimal value of (P) is less or equal to that of (P’). On the other hand, for any optimal solution $(\lambda ^*, \lambda _0^*)$ of (P’), let $\phi ^*:=\min \{\lambda ^*\bullet \varPsi (\xi )+\lambda _0^*, 1\}$. Then $(\lambda ^*, \lambda _0^*, \phi ^*)$ is a feasible solution of (P) which implies in turn the optimal value of (P’) is less or equal to that of (P). Note also that the optimal value is $\mathsf {dl}_{TV}(Q,{\tilde{{{\mathcal {P}}}}})\in [0,2]$.

Step 2. We show that the optimization problem at the right hand side of (50) has a bounded optimal solution. Let

$$\begin{aligned} {{\mathcal {F}}}:=\{(\lambda ,\lambda _0) \in \varLambda \times \mathbb {R}: -1 -\lambda \bullet \varPsi (\xi ) -\lambda _0 \le 0,\;\; \forall \xi \in {\tilde{\varXi }}\} \end{aligned}$$

denote the feasible set of (50) and

$$\begin{aligned} {{\mathcal {C}}}:=\{(\lambda ,\lambda _0) \in \varLambda \times \mathbb {R}: \Vert \lambda \Vert +|\lambda _0|=1, -\lambda \bullet \varPsi (\xi ) -\lambda _0 \le 0,\;\; \forall \xi \in {\tilde{\varXi }}\}. \end{aligned}$$

(51)

Then

$$\begin{aligned} {{\mathcal {F}}}=\{(0_q,-1)+t{{\mathcal {C}}}: t\in [0_q,+\infty )\}. \end{aligned}$$

To see this, let $(\lambda , \lambda _0)\in {{\mathcal {F}}}$ and $t=\Vert \lambda \Vert +|\lambda _0+1|$. If $t=0$, then $(0_q,-1)\in \{(0_q,-1)+t{{\mathcal {C}}}: t\in [0,+\infty )\}$. Consider the case that $t\ne 0$. Then it is easy to verify that $(\lambda , \lambda _0+1)/t\in {{\mathcal {C}}}$ and hence $(\lambda , \lambda _0)\in (0_q,-1)+t{{\mathcal {C}}}$. This shows ${{\mathcal {F}}}\subset \{(0_q,-1)+t{{\mathcal {C}}}: t\in [0,+\infty )\}$. The converse inclusion is obvious.

Note that ${{\mathcal {C}}}\ne \emptyset$. Otherwise we would have

$$\begin{aligned}&{\left\{ \begin{array}{cl} \displaystyle {\sup _{\lambda \in \varLambda ,\lambda _0}} &{} \langle Q, \min \{\lambda \bullet \varPsi +\lambda _0, 1\}\rangle -\lambda \bullet \mu -\lambda _0\\ \text{ s.t. } &{} -1 -\lambda \bullet \varPsi (\xi ) -\lambda _0 \le 0,\;\;\text{ a.e. } \; \; \xi \in {\tilde{\varXi }} \end{array}\right. }\\&\quad \le \sup _{(\lambda , \lambda _0)= (0,-1)} \langle Q, \lambda \bullet \varPsi \rangle -\lambda \bullet \mu =0, \end{aligned}$$

which implies $P\in \tilde{{{\mathcal {P}}}}$, a contradiction. In what follows, we consider the case when ${{\mathcal {C}}}\ne \emptyset$. From (51) we immediately have

$$\begin{aligned} -\lambda \bullet \langle P, \varPsi \rangle -\lambda _0 \langle P, 1\rangle \le 0, \forall (\lambda ,\lambda _0)\in {{\mathcal {C}}}, P\in \mathscr {M}_+(\tilde{\varXi }). \end{aligned}$$

(52)

On the other hand, by Assumption 1 and (17),

$$\begin{aligned} (0,0_{q}) \in \text{ int } \; [ (\langle P, 1 \rangle -1, \langle P, \varPsi (\xi )\rangle -\mu , -\{0\}\times \{0_p\} \times {{\mathcal {K}}}_-^{q-p}: P\in \mathscr {M}_+(\tilde{\varXi })] \end{aligned}$$

and there exists a closed neighborhood of $(0,0_{q})$ with radius $\epsilon _0$, denoted by $\epsilon _0{{\mathcal {B}}}$, such that

$$\begin{aligned} \epsilon _0{{\mathcal {B}}}\subset \text{ int } \; [ (\langle P, 1 \rangle -1, \langle P, \varPsi (\xi )\rangle -\mu ) - \{0\}\times \{0_p\} \times \mathcal{K}_-^{q-p}: P\in \mathscr {M}_+(\tilde{\varXi })]. \end{aligned}$$

Let $({\tilde{\lambda }}, {\tilde{\lambda }}_0)\in {{\mathcal {C}}}$. Then there exists $\tilde{w}\in \epsilon _0{{\mathcal {B}}}$ (depending on $({\tilde{\lambda }}, {\tilde{\lambda }}_0)$) such that $\langle {\tilde{w}}, (\tilde{\lambda },{\tilde{\lambda }}_0)\rangle \le -\epsilon _0$. In other words, there exist $\tilde{P}\in \mathscr {M}_+(\tilde{\varXi })$ and $\eta \in {{\mathcal {K}}}_-^{q-p}$ such that

$$\begin{aligned} {\tilde{\lambda }}_0 (\langle \tilde{P}, 1 \rangle -1)+ {\tilde{\lambda }} \bullet (\langle \tilde{P}, \varPsi (\xi )\rangle -\mu ) -\eta \bullet {\tilde{\lambda }}_I \le -\epsilon _0. \end{aligned}$$

(53)

Since $-\eta \bullet {\tilde{\lambda }}_I\ge 0$, we deduce from (52) and (53) that $- \tilde{\lambda }_0- {\tilde{\lambda }} \bullet \mu \le -\epsilon _0.$ The inequality holds for every $({\tilde{\lambda }}, {\tilde{\lambda }}_0)\in {{\mathcal {C}}}$. Note that for any $(\lambda ,\lambda _0)\in {{\mathcal {F}}}$, we may write it in the form

$$\begin{aligned} (\lambda ,\lambda _0) = ({\hat{\lambda }},{\hat{\lambda }}_0) + t(\tilde{\lambda },\tilde{\lambda }_0), \end{aligned}$$

where $({\hat{\lambda }},{\hat{\lambda }}_0) =(0_q, -1)$, $(\tilde{\lambda },\tilde{\lambda }_0)\in {{\mathcal {C}}}$ and $t\ge 0$. Observe that $\langle Q, \min \{\lambda \bullet \varPsi +\lambda _0, 1\}\rangle \in [-1,1]$ and

$$\begin{aligned} -\lambda \bullet \mu -\lambda _0 = -\mu \bullet {\hat{\lambda }} -{\hat{\lambda }}_0 - t(\mu \bullet \tilde{\lambda }+\tilde{\lambda }_0) \le -\mu \bullet {\hat{\lambda }} -{\hat{\lambda }}_0 - t \epsilon _0=1-t \epsilon _0. \end{aligned}$$

Note that the optimal value of the optimization problem at the right hand side of (50) is positive, which implies for any optimal solution $(\lambda ^*, \lambda _0^*) = (0_q, -1) + (\bar{\lambda }^*, \bar{\lambda }_0^*)$ with $(\bar{\lambda }^*, \bar{\lambda }_0^*)\in t^*{{\mathcal {C}}}$ we must have $1-\mu \bullet 0_q -(-1) - t^* \epsilon _0 = 2-t^*\epsilon _0>0$ or equivalently $t^*< \frac{2}{\epsilon _0}.$ Let $t_1= \frac{2}{\epsilon _0}$ and ${{\mathcal {F}}}_1:={{\mathcal {F}}}_0 + \{t{{\mathcal {C}}}: t_1\ge t\ge 0\}$. Based on the discussions above, we conclude that the optimization problem at the right hand side of (50) has an optimal solution in ${{\mathcal {F}}}_1$.

Step 3. Let $C_1:=\max _{(\lambda , \lambda _0)\in {{\mathcal {F}}}_1}\Vert \lambda \Vert$. Then

$$\begin{aligned} d_{TV}(Q,{{\mathcal {P}}})= & {} \sup _{(\lambda , \lambda _0)\in {{\mathcal {F}}}_1} \langle Q, \min \{\lambda \bullet \varPsi (\xi (\omega ))+\lambda _0, 1\}\rangle -\lambda \bullet \mu -\lambda _0\\\le & {} \sup _{(\lambda , \lambda _0)\in {{\mathcal {F}}}_1} \langle Q, \lambda \bullet \varPsi (\xi (\omega ))+\lambda _0\rangle -\lambda \bullet \mu -\lambda _0\\= & {} \sup _{(\lambda , \lambda _0)\in {{\mathcal {F}}}_1}{\lambda }\bullet ({\mathbb {E}}_Q[\varPsi (\xi )]-\mu )\\\le & {} \sup _{(\lambda , \lambda _0)\in {{\mathcal {F}}}_1} \; \sum _{i=1}^p \lambda _i({\mathbb {E}}_Q[\varPsi _i(\xi )]-\mu _i) +\sum _{i=p+1}^q \lambda _i({\mathbb {E}}_Q[\varPsi _i(\xi )]-\mu _i)\\\le & {} \sup _{(\lambda , \lambda _0)\in {{\mathcal {F}}}_1} \; \sum _{i=1}^p |\lambda _i||{\mathbb {E}}_Q[\varPsi _i(\xi )]-\mu _i| +\sum _{i=p+1}^q \lambda _i({\mathbb {E}}_Q[\varPsi _i(\xi )]-\mu _i)_+\\\le & {} C_1 ( \Vert ({\mathbb {E}}_Q[\varPsi _E(\xi )]-\mu _E)\Vert + \Vert ({\mathbb {E}}_Q[\varPsi _I(\xi )]-\mu _I)_+\Vert ), \end{aligned}$$

where $C_1=\frac{2}{\epsilon _0}+1$. The inequality also holds for the case when ${{\mathcal {C}}}=\emptyset$ because ${{\mathcal {F}}}_0\subset {{\mathcal {F}}}_1$. Note that the above result holds for all ${\tilde{{{\mathcal {P}}}}}$ when $\bar{\varXi }\subset {\tilde{\varXi }}\subset \varXi$, which include both continuous and discrete support set case, the proof is complete. $\square$