Improving the performance of the stochastic dual dynamic programming algorithm using Chebyshev centers

Abstract

In hydro predominant systems, the long-term hydrothermal scheduling problem (LTHS) is formulated as a multistage stochastic programming model. A classical optimization technique to obtain an operational policy is the stochastic dual dynamic programming (SDDP), which employs a forward step, for generating trial state variables, and a backward step to construct Benders-like cuts. To assess the quality of the obtained policy (the cuts obtained over the iterations), a confidence interval is computed on the optimality gap. As the SDDP is a cutting-plane based method, it exhibits slow convergence in large-scale problems. To improve computational efficiency, we explore different regions in which the cuts are usually constructed by the classical algorithm. For that, the cuts in the forward step are translated using ideas related to the definition of Chebyshev centers of certain polyhedrons. Essentially, the cuts are lifted by a parameter that vanishes along with iterations without harming the convergence analysis. The proposed technique is assessed on an instance of the Brazilian LTHS problem with individualized monthly decisions per plant, indicating a higher policy quality in comparison with the classical approach, since it computes lower optimality gaps throughout the iterative process.

Appendix

In this section, we present the proof of Proposition 1. For this case, it is convenient to alternatively rewrite (2) as:

$$\begin{aligned} &\mathop {\hbox{min} }\limits_{{x_{t} \ge 0,s_{t} }} \, s_{t} \hfill \\ &{\text{s}} . {\text{t}} .\;A_{t} x_{t} = b_{t} - B_{t} \bar{x}_{t - 1} , & (c_{t} + \upbeta_{t + 1}^{j} )^{{ \top }} x_{t} + \upalpha_{t + 1}^{j} \le s_{t} ,\;\forall \;j \in J_{t + 1} . \hfill \\ \end{aligned}$$

(20)

Let (x ^K_t , s ^K_t ) be an optimal solution for this LP (K standing for Kelley iterates). Note that $s_{t}^{K} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Q} (\bar{x}_{t - 1} ,\upxi_{t} )$, which is the optimal value of (2). If $s_{t}^{K} < \bar{z}(\upxi_{[t]} )$, it is consistent to remark that (x_t, s_t, σ_t, w_t) = (x ^K_t , s ^K_t , 0, 0) is feasible for (5). On the other hand, (x_t, s_t, σ_t, w_t) = (x ^K_t , s ^K_t , $\bar{z}(\upxi_{[t]} ) - s_{t}^{K}$, 0) is feasible for (5) if $s_{t}^{K} \ge \bar{z}(\upxi_{[t]} )$. The second constraint in (5) yields $\bar{z}(\upxi_{[t]} ) \ge \upsigma_{t} + s_{t} \ge \upsigma_{t} + s_{t}^{K}$, showing that the objective function σ_t of (5) is bounded from above by $\bar{z}(\upxi_{[t]} ) - s_{t}^{K}$. As a result, LP (5) has an optimal solution whenever $\bar{z}(\upxi_{[t]} )$ is finite, provided (2) has a solution. This proves item (1) of Preposition 1. Suppose $\bar{z}(\upxi_{[t]} ) \le s_{t}^{K} ( = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Q} (\bar{x}_{t - 1} ,\upxi_{t} ))$ and let (x ^C_t , s ^C_t , σ ^C_t , w ^C_t ) (C standing for Chebyshev) be a solution of (5), ensured to exist by item (1). It follows from the constraints of (5), the nonnegativity of w_t and the definition of (x ^K_t , s ^K_t ) that

$$\begin{aligned} s_{t}^{C} & \ge \max_{{j \in J_{t + 1} }} \left\{ {(c_{t} + \upbeta_{t + 1}^{j} )^{{ \top }} x_{t} + \upalpha_{t + 1}^{j} + w_{t} \left\| {(1,c_{t} + \upbeta_{t + 1}^{j} )} \right\|_{*} } \right\} \\ & \ge \max_{{j \in J_{t + 1} }} \left\{ {(c_{t} + \upbeta_{t + 1}^{j} )^{{ \top }} x_{t} + \upalpha_{t + 1}^{j} } \right\} \\ & \ge \max_{{j \in J_{t + 1} }} \left\{ {(c_{t} + \upbeta_{t + 1}^{j} )^{{ \top }} x_{t}^{K} + \upalpha_{t + 1}^{j} } \right\} = s_{t}^{K} . \\ \end{aligned}$$

(21)

Therefore, it follows from the second constraint of (5) that $\bar{z}(\upxi_{[t]} ) - s_{t}^{C} \le s_{t}^{K} - s_{t}^{C} \le 0$. As shown in the proof of item (1), (x ^K_t , s ^K_t , $\bar{z}(\upxi_{[t]} ) - s_{t}^{K}$, 0) is feasible for (5). Consequently, $\bar{z}(\upxi_{[t]} ) - s_{t}^{K} \le \upsigma_{t}^{C} \le \bar{z}(\upxi_{[t]} ) - s_{t}^{c}$, revealing [through (21)] that s ^K_t  = s ^C_t . This equality inserted in (5) gives w ^C_t  = 0. Thus, we have shown that (x ^C_t , s ^C_t , σ ^C_t , w ^C_t ) = (x ^C_t , s ^K_t , $\bar{z}(\upxi_{[t]} ) - s_{t}^{K}$, 0), therefore

$$\begin{aligned} s_{t}^{K} = s_{t}^{C} & \ge \max_{{j \in J_{t + 1} }} \left\{ {(c_{t} + \upbeta_{t + 1}^{j} )^{{ \top }} x_{t}^{C} + \upalpha_{t + 1}^{j} } \right\} \\ & \ge \max_{{j \in J_{t + 1} }} \left\{ {(c_{t} + \upbeta_{t + 1}^{j} )^{{ \top }} x_{t}^{K} + \upalpha_{t + 1}^{j} } \right\} = s_{t}^{K} {\mkern 1mu} , \\ \end{aligned}$$

(22)

Equation (22) shows that (x ^C_t , s ^C_t ) also solves (20), which implies that x ^C_t is a Kelley iterate in this case. This completes the proof of item (2). Thereafter, item (3) can be demonstrated by simply disclosing that σ ^C_t  = w ^C_t whenever $\bar{z}(\upxi_{[t]} ) > s_{t}^{K} ( = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Q} (\bar{x}_{t - 1} ,\upxi_{t} ))$. Suppose, for contradiction purpose, that σ ^C_t  < w ^C_t . In that case, the point (x_t, s_t, σ_t, w_t) = (x ^C_t , s ^K_t , σ ^C_t , σ ^C_t ) is feasible for (5) and satisfies $v = \mathop {\hbox{max} }\limits_{{j \in J_{t + 1} }} \left\{ {(c_{t} + \upbeta_{t + 1}^{j} )^{{ \top }} x_{t}^{C} + \upalpha_{t + 1}^{j} + \upsigma_{t}^{C} \left\| {(1,c_{t} + \upbeta_{t + 1}^{j} )} \right\|_{*} } \right\} < s_{t}^{C}$. It follows from the above inequality and the second constraint of (5) that $\upsigma_{t}^{v} > \bar{z}(\upxi_{[t]} ) - s_{t}^{C} \ge \upsigma_{t}^{C}$, being $\upsigma_{t}^{v} = \bar{z}(\upxi_{[t]} ) - v$. As a result, the point (x_t, s_t, σ_t, w_t) = (x ^C_t , v, σ ^v_t , σ ^C_t ) is feasible for (5) and provides an objective value strictly greater than σ ^C_t , hereby contradicting the optimality of (x ^C_t , s ^K_t , σ ^C_t , w ^C_t ). Accordingly, σ ^C_t  = w ^C_t and the first part of item (3) holds true from the Chebyshev centers definition, in Sect. 3. Let’s now focus on item (4). Since the second constraint in (5) is active at any of its solutions (x ^C_t , s ^C_t , σ ^C_t , w ^C_t ) (otherwise, the optimal value of (5) could be improved by increasing σ_t), we can replace the variable σ_t by $\bar{z}(\upxi_{[t]} ) - s_{t}$ and remove it from (5):

$$\left\{ \begin{aligned} \mathop {\hbox{max} }\limits_{{x_{t} ,s_{t} ,w_{t} }} & \bar{z}(\upxi_{[t]} ) - s_{t} \hfill \\ {\text{s}} . {\text{t}} .& A_{t} x_{t} = b_{t} - B_{t} \bar{x}_{t - 1} \hfill \\ & (c_{t} + \upbeta_{t + 1}^{j} )^{{ \top }} x_{t} + \upalpha_{t + 1}^{j} + w_{t} \left\| {(1,c_{t} + \upbeta_{t + 1}^{j} )} \right\|_{*} \le s_{t} ,\;j \in J_{t + 1} \hfill \\ & \bar{z}(\upxi_{[t]} ) - s_{t} \le w_{t} ,x_{t} \ge 0,{\mkern 1mu} w_{t} \ge 0,{\mkern 1mu} s_{t} \in \Re {\mkern 1mu} . \hfill \\ \end{aligned} \right.$$

(23)

Formulation (6) comes from the above LP by recalling that $\bar{z}(\upxi_{[t]} )$ is a constant, max -s_t = -min s_t and by setting $s_{t} = c_{t}^{{ \top }} x_{t} + r_{t + 1}$.