SDDP for some interstage dependent risk-averse problems and application to hydro-thermal planning

Abstract

We consider interstage dependent stochastic linear programs where both the random right-hand side and the model of the underlying stochastic process have a special structure. Namely, for equality constraints (resp. inequality constraints) the right-hand side is an affine function (resp. a given function b _t ) of the process value for the current time step t. As for m-th component of the process at time step t, it depends on previous values of the process through a function h _tm .

For this type of problem, to obtain an approximate policy under some assumptions for functions b _t and h _tm , we detail a stochastic dual dynamic programming algorithm. Our analysis includes some enhancements of this algorithm such as the definition of a state vector of minimal size, the computation of feasibility cuts without the assumption of relatively complete recourse, as well as efficient formulas for sharing optimality and feasibility cuts between nodes of the same stage. The algorithm is given for both a non-risk-averse and a risk-averse model. We finally provide preliminary results comparing the performances of the recourse functions corresponding to these two models for a real-life application.

Appendix

Proof of Theorem 2.4

By duality, $Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta_{t j} )$ may be expressed as the optimal value of the following linear program (due to Assumption (A3) the dual and the primal have the same finite optimal value):

$$ \begin{aligned} &\max_{\lambda_t, \pi_{t 1}, \pi_{t 2}, \rho_t} \; g_1( \lambda_t, \pi_{t 1}, \pi_{t 2}, \rho_t, \eta_{t j} ) \\ &C_t^{\scriptscriptstyle \top }\pi_{t 1}^{\scriptscriptstyle \top }+ A_t^{\scriptscriptstyle \top }\pi_{t 2}^{\scriptscriptstyle \top }+ { \overrightarrow{E}_{t}^i}^{\scriptscriptstyle \top }\rho_t^{\scriptscriptstyle \top }\leq\beta _t^{\scriptscriptstyle \top }\lambda_t^{\scriptscriptstyle \top }\\ &\lambda_t e =1,\qquad \rho_t e =1, \qquad \rho_t \geq0,\qquad \lambda_t \geq0,\qquad \pi _{t 2} \geq0 \end{aligned} $$

(40)

where the objective function is given by

$$ \begin{aligned} &g_1( \lambda_t, \pi_{t 1}, \pi_{t 2}, \rho_t, \eta_{t j}) \\ &\quad = \lambda_t \alpha_t + \rho_t \bigl(\overrightarrow{e}_{t}^i + \overrightarrow {{\tilde{E}}}_{t}^i( {\tilde{\varPhi}}_{t} { \xi}_{[t-1]}+{\tilde{\varPsi}}_{t} \eta_{t j} + { \tilde{\varTheta}}_{t} ) \bigr) \\ &\qquad{} + \pi_{t 2} \bigl(b_t( \varPhi_{t} { \xi}_{[t-1]}+\varPsi_{t} \eta_{t j} + \varTheta_{t} )-B_t x_{t-1} \bigr) \\ &\qquad{} + \pi_{t 1} \bigl(D_t( \varPhi_{t} { \xi }_{[t-1]}+ \varPsi_{t} \eta_{t j} + \varTheta_{t} )-E_t x_{t-1} \bigr). \end{aligned} $$

(41)

For problem (40), optimal solutions are extremal points of the feasible set. Further, the feasible set neither depends on x _t−1 nor on ξ _[t−1] and for any (x _t−1,ξ _[t−1]), row vectors $\lambda_{t}^{k j}, \pi_{t 1}^{k j}, \pi_{t 2}^{k j}, \rho_{t}^{k j}$ are extremal points of the feasible set of problem $Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta_{t j} )$ expressed as (40). It follows that $Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta_{t j})$ is bounded from below by

$$ \begin{aligned} &g_1\bigl( \lambda_t^{k j}, \pi_{t 1}^{k j}, \pi_{t 2}^{k j}, \rho_t^{k j}, \eta_{t j}\bigr) \\ &\quad = \lambda_t^{k j} \alpha_t + \rho_t^{k j} \bigl(\overrightarrow{e}_{t}^i + \overrightarrow{{\tilde{E}}}_{t}^i( {\tilde { \varPhi}}_{t} {\xi}_{[t-1]}+{\tilde{\varPsi}}_{t} \eta_{t j} + {\tilde {\varTheta }}_{t} ) \bigr) \\ &\qquad{} + \pi_{t 2}^{k j} \bigl(b_t( \varPhi_{t} {\xi}_{[t-1]}+ \varPsi_{t} \eta_{t j} + \varTheta_{t} )-B_t x_{t-1} \bigr) \\ &\qquad{} + \pi_{t 1}^{k j} \bigl(D_t( \varPhi _{t} {\xi}_{[t-1]}+ \varPsi_{t} \eta_{t j} + \varTheta_{t} )-E_t x_{t-1} \bigr) \end{aligned} $$

(42)

for j=1,…,q _t . Next, from the convexity of b _ti , we obtain

$$ b_{t i}( \varPhi_{t} {\xi}_{[t-1]}+ \varPsi_{t} \eta_{t j} + \varTheta_{t} ) \geq b_{t i} \bigl(\xi_{t j}^{k} \bigr) + s_{t i}^b \bigl( \xi_{t j}^{k} \bigr)^{\scriptscriptstyle \top }\varPhi _t \bigl(\xi_{[t-1]}- \xi_{[t-1]}^k \bigr), \quad i=1,\ldots,\ell_t, $$

(43)

and since $\pi_{t 2}^{k j} \geq0$, we have

$$ \pi_{t 2}^{k j} b_t( \varPhi_{t} {\xi}_{[t-1]}+ \varPsi_{t} \eta_{t j} + \varTheta_{t} ) \geq\pi_{t 2}^{k j} \bigl[ b_t \bigl( \xi_{t j}^{k} \bigr) + s_t^b \bigl( \xi_{t j}^{k} \bigr) \varPhi_t \bigl(\xi_{[t-1]}-\xi_{[t-1]}^k \bigr) \bigr]. $$

(44)

Plugging (44) into lower bound (42) for $Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta_{t j})$ and since $\mathcal{Q}_{t} ( x_{t-1}, \xi_{[t-1]})$ is bounded from below by $\mathbb{E}_{\eta_{t}}[Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta_{t})] = \sum_{j=1}^{q_{t}} \, p(t,j) Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta_{t j})$, we obtain a cut of the form $\theta_{t-1}^{k} + E_{t-1}^{k} x_{t-1} \geq {\tilde{E}}_{t-1}^{k} \xi_{[t-1]}+e_{t-1}^{k}$ and the result follows. □

Proof of Theorem 2.8

We show by induction, from t=T+1 down to t=2, that the announced cuts are valid and that ${\tilde{E}}_{t}^{k} \geq0$ for t=1,…,T, and k=0,1,…,iH. For t=T, we have $\mathcal{Q}_{T+1}^{i}=\mathcal{Q}_{T+1} \equiv0$. As a result, all components of $E_{T}^{k}, {\tilde{E}}_{T}^{k}$, and $e_{T}^{k}$ are null for k=0,1,…,iH. In particular, we have that ${\tilde{E}}_{t}^{k} \geq0$. This achieves the first step of the induction.

Let us now assume that for some t∈{2,…,T}, valid cuts have been built for $\mathcal{Q}_{\ell+1}, \ell=t,\ldots,T$, according to the formulas given in the theorem for $E_{\ell}^{k}, {\tilde{E}}_{\ell}^{k}$, and $e_{\ell}^{k}$, ℓ=t,…,T, k=0,1,…,iH, with all ${\tilde{E}}_{\ell}^{k} \geq0$.

We have $\mathcal{Q}_{t}(x_{t-1}, \xi_{[t-1]})$ $\geq\sum_{j=1}^{q_{t}} p(t,j) \mathcal{Q}_{t}^{i}(x_{t-1}, \xi_{[t-1]}, \eta_{t j})$ with

$$\begin{aligned} \mathcal{Q}_t^i(x_{t-1}, \xi_{[t-1]}, \eta_{t j}) \geq& g_2 \bigl( \lambda_t^{k j}, \pi_{t 2}^{k j}, \rho_t^{k j}, \eta_{t j} \bigr) \\ =& \lambda_t^{k j} \alpha_t + \rho_t^{k j} \overrightarrow{e}_{t}^i -\pi_{t 2}^{k j}B_t x_{t-1} +U_{t j}^{k} + V_{t j}^{k}+W_{t j}^{k} \end{aligned}$$

(45)

where

$$\begin{aligned} U_{t j}^{k} = & \sum_{m=1}^M \sum_{\ell=1}^{s_{t+1, m}-1} \Biggl[ \sum _{w=0}^{i H} \rho_{t}^{k j}(w) { \tilde{E}}_{t, m}^w(\ell+1) \Biggr] \xi _{t-\ell}(m), \end{aligned}$$

(46)

$$\begin{aligned} V_{t j}^{k} = & \sum_{m=1}^M \sum_{w=0}^{i H} \rho_{t}^{k j}(w) {\tilde{E}}_{t, m}^w(1) \xi_{t j}(m) , \end{aligned}$$

(47)

$$\begin{aligned} W_{t j}^{k} = & \sum_{\ell=1}^{\ell_t} \pi_{t 2}^{k j}(\ell)b_{t \ell } ( \xi_{t j} ). \end{aligned}$$

(48)

Let us first bound from below $W_{t j}^{k}$. Using the convexity of b _tℓ , we obtain

$$b_{t \ell}( \xi_{t j} ) \geq b_{t \ell} \bigl( \xi_{t j}^{k} \bigr) + s_{t \ell}^b \bigl( \xi_{t j}^{k} \bigr)^{\scriptscriptstyle \top }\bigl( \xi_{t j} - \xi_{t j}^{k} \bigr) $$

for every ℓ=1,…,ℓ _t . Using these inequalities and the fact that $\pi_{t 2}^{k j} \geq0$, we have

$$ W_{t j}^{k} = \pi_{t 2}^{k j} b_{t}( \xi_{t j} ) \geq\pi_{t 2}^{k j} b_{t} \bigl( \xi_{t j}^{k} \bigr) + \pi_{t 2}^{k j} s_{t}^b \bigl( \xi_{t j}^{k} \bigr) \bigl( \xi _{t j} - \xi_{t j}^{k} \bigr). $$

(49)

Similarly, using the convexity of h _tm , we have

$$\begin{aligned} \xi_{t j}(m)-\xi_{t j}^{k}(m) =& h_{t m}\bigl(\xi_{t-1:t-p_{t}(m)}(m),\eta_{t j}(m) \bigr)-h_{t m}\bigl(\xi_{t-1:t-p_{t}(m)}^k(m), \eta_{t j}(m)\bigr) \\ \geq&\sum_{w=1}^{p_t(m)} s_{t m}^h \bigl(\xi _{t-1:t-p_{t}(m)}^k(m),\eta_{t j}(m) \bigr) (w) \bigl( \xi_{t-w}(m) \\ &{}- \xi _{t-w}^k(m) \bigr) \end{aligned}$$

(50)

for every m=1,…,M. Using Assumption (A4), each component of each subgradient of b _tℓ is nonnegative. As a result, all elements in matrix $s_{t}^{b}( \xi_{t j}^{k})$ are nonnegative. Using this observation and relations (49) and (50), we obtain for $W_{t j}^{k}$ the lower bound $\pi_{t 2}^{k j} b_{t}( \xi _{t j}^{k})$ plus

$$\begin{aligned} &\sum_{m=1}^M \sum _{w=1}^{p_t(m)} \Biggl[ \sum _{\ell=1}^{\ell_t} \pi_{t 2}^{k j}(\ell) s_{t \ell}^b \bigl( \xi_{t j}^{k} \bigr) (m) s_{t m}^h \bigl(\xi _{t-1:t-p_{t}(m)}^k(m), \eta_{t j}(m) \bigr) (w) \Biggr] \\ &\quad {} \times \bigl(\xi _{t-w}(m)-\xi _{t-w}^k(m) \bigr). \end{aligned}$$

(51)

Let us now bound from below $V_{t j}^{k}$. Using relation (50) and the nonnegativeness of ${\tilde{E}}_{t, m}^{w}(1)$ (induction hypothesis) and of $\rho_{t}^{k j}(w)$, we obtain for $V_{t j}^{k}$ the lower bound

$$\begin{aligned} &\sum_{m=1}^M \sum_{w=0}^{i H} \rho_{t}^{k j}(w) {\tilde{E}}_{t, m}^w(1) \displaystyle{\sum _{u=1}^{p_t(m)}} s_{t m}^h \bigl( \xi _{t-1:t-p_{t}(m)}^k(m),\eta_{t j}(m) \bigr) (u) \bigl(\xi _{t-u}(m)-\xi _{t-u}^k(m) \bigr) \\ &\quad{} + \sum_{m=1}^M \sum _{w=0}^{i H} \rho_{t}^{k j}(w) { \tilde{E}}_{t, m}^w(1) \xi_{t j}^{k}(m). \end{aligned}$$

(52)

Plugging into (45) relation (46) as well as lower bounds (51) and (52) for respectively $W_{t j}^{k}-\pi_{t 2}^{k j} b_{t}( \xi_{t j}^{k})$ and $V_{t j}^{k}$, we obtain for $\mathcal{Q}_{t}$ a cut of form $-E_{t-1}^{k} x_{t-1}+{\tilde{E}}_{t-1}^{k} \xi_{[t]}+e_{t-1}^{k}$ with the desired values of $E_{t-1}^{k}, {\tilde{E}}_{t-1}^{k}$, and $e_{t-1}^{k}$. If remains to check that all components of ${\tilde{E}}_{t-1}^{k}$ are nonnegative. We had already observed that for all functions b _ti , all components of all subgradients are nonnegative, due to Assumption (A4). The same remark holds for functions h _tm , due to Assumption (A6). By induction hypothesis, all coefficients $({\tilde{E}}_{t m}^{j}(\ell))_{j, \ell, m}$ are nonnegative. Using the nonnegativity of these coefficients, as well as the nonnegativity of row vectors $\rho _{t}^{k j}$ and $\pi_{t 2}^{k j}$, together with the formula for ${\tilde{E}}_{t-1}^{k}$, we obtain that ${\tilde{E}}_{t-1}^{k} \geq0,\;k=0,1,\ldots,iH$. □

Proof of Proposition 2.9

We show the cuts are valid when there are both equality and inequality constraints. A similar proof can be done when there are not equality constraints. Let k∈{(i−1)H+1,…,iH} and j∈{1,…,q _t }. If $\eta_{t j} \in\varOmega_{t}^{k}$ then $Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta _{t j} )$ is bounded from below by $g_{1}( \lambda_{t}^{k j}, \pi_{t 1}^{k j}, \pi_{t 2}^{k j}, \rho _{t}^{k j}, \eta_{t j} )$ where the expression of g ₁ is given by (40). Next, for every j such that $\eta_{t j} \notin\varOmega_{t}^{k}$, since all $(\lambda_{t}, \pi_{t 1}, \pi_{t 2}, \rho_{t}) \in\mathcal{M}_{t}^{i}$ belong to the feasible set of problem $Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta_{t j} )$, we have $Q_{t}^{i} ( x_{t-1}, \xi_{[t-1]}, \eta_{t j} ) \geq g_{1}( \lambda_{t}, \pi _{t 1}, \pi_{t 2}, \rho_{t}, \eta_{t j} )$ for every $(\lambda_{t}, \pi_{t 1}, \pi_{t 2}, \rho_{t}) \in\mathcal {M}_{t}^{i}$. As a result,

$$\begin{aligned} Q_t^i ( x_{t-1}, \xi_{[t-1]}, \eta_{t j} ) \geq& \max_{(\lambda_t, \pi_{t 1}, \pi_{t 2}, \rho_t) \in\mathcal {M}_t^i} \; g_1( \lambda_t, \pi_{t 1}, \pi_{t 2}, \rho_t, \eta_{t j} ) \\ = & g_1\bigl( \lambda_t^{k j}, \pi_{t 1}^{k j}, \pi_{t 2}^{k j}, \rho_t^{k j}, \eta_{t j} \bigr) \quad \mbox{ using } \mbox{(25)}. \end{aligned}$$

We then conclude as in the proof of Theorem 2.4. □

Proof of Theorem 2.10

Let x _t−1 be a feasible state at the end of time step t−1 at a given node of this time step with history ξ _[t−1]. Since for one of the son nodes, the realization of η _t is η _tj , the optimal value of (32) is 0. As a result, the optimal value of the dual of (32) is 0. This dual problem can be written

$$ \begin{cases} \max_{\pi_{t 1}, \pi_{t 2}, \sigma_t } \;f( \pi_{t 1}, \pi_{t 2}, \sigma_t )\\ C_t^{\scriptscriptstyle \top }\pi_{t 1}^{\scriptscriptstyle \top }+ A_t^{\scriptscriptstyle \top }\pi_{t 2}^{\scriptscriptstyle \top }+ {\overrightarrow{F}}_{t}^{\scriptscriptstyle \top }\sigma_t^{\scriptscriptstyle \top }\leq0\\ -e \leq\pi_{t 1}^{\scriptscriptstyle \top }\leq e, \qquad 0 \leq\pi_{t 2}^{\scriptscriptstyle \top }\leq e, \qquad 0 \leq\sigma_t^{\scriptscriptstyle \top }\leq e, \end{cases} $$

(53)

where the objective function f is given by

$$\begin{aligned} &\pi_{t 1} \bigl[ D_t( \varPhi_t \xi_{[t-1]}+\varPsi_t \eta_{t j} + \varTheta_t ) -E_t x_{t-1} \bigr]+ \pi_{t 2} \bigl[b_t( \varPhi_t \xi_{[t-1]} \\ &\quad {}+\varPsi _t \eta _{t j} + \varTheta_t )-B_t x_{t-1} \bigr] \\ &\quad {}+ \sigma_t \bigl[ \overrightarrow{{\tilde{F}}}_{t}( {\tilde{\varPhi}}_t \xi _{[t-1]}+ {\tilde{\varPsi}}_t \eta_{t j} + {\tilde{\varTheta}}_t ) + \overrightarrow{f}_{t} \bigr]. \end{aligned}$$

For this dual problem, since the optimal value is 0 and since $(\pi_{t 1}^{k j}, \pi_{t 2}^{k j}, \sigma_{t}^{k j})$ is feasible, we obtain

$$0 \geq f \bigl( \pi_{t 1}^{k j}, \pi_{t 2}^{k j}, \sigma_t^{k j} \bigr). $$

We conclude using (43) and (44). □

Proof of Proposition 2.11

We follow the proofs of Theorems 2.8 and 2.10. □

Proof of Theorem 3.1

The proof is similar to the proof of Theorem 2.4. □