On level regularization with normal solutions in decomposition methods for multistage stochastic programming problems

Abstract

We consider well-known decomposition techniques for multistage stochastic programming and a new scheme based on normal solutions for stabilizing iterates during the solution process. The given algorithms combine ideas from finite perturbation of convex programs and level bundle methods to regularize the so-called forward step of these decomposition methods. Numerical experiments on a hydrothermal scheduling problem indicate that our algorithms are competitive with the state-of-the-art approaches such as multistage regularized decomposition, nested decomposition and stochastic dual dynamic programming.

Appendices

Appendix A: Proof of Lemma 1

• Item (a) Let ${{\tilde{x}}}$ be a solution of (11), then $f({{\tilde{x}}}) \le f(x(\tau ))$ (recall that both ${{\tilde{x}}}$ and $x(\tau )$ belong to X). Moreover, $f( x(\tau ))+ \frac{1}{\tau }\varphi (x(\tau )) \le f( {{\tilde{x}}})+ \frac{1}{\tau }\varphi ({{\tilde{x}}})$. These two inequalities show that $\varphi (x(\tau )) \le \varphi (\tilde{x})=\varphi ^*<\infty$.

• Item (b) The previous item ensures that $\varphi (x(\tau _k)) \le \varphi ({{\tilde{x}}})< \infty$ for all k. Since $\varphi$ is a strong convex function, its level sets are compact. As a result, $\{x(\tau _k)\}$ is a bounded sequence and, therefore, there exists a constant $L<\infty$ bounding all subgradients of $\varphi$ at $x(\tau _k)$ for all k.

• Item (c) We first prove that

  $$\begin{aligned} \varphi (x(\tau '))< \varphi (x(\tau '')) \text{ and } f(x(\tau '))<f(x(\tau '')) \hbox { for all } 0<\tau '<\tau ''. \end{aligned}$$

  (29)

It follows from optimality of $x(\tau ')$ and $x(\tau '')$ and strongly convexity of $\varphi$ that

$$\begin{aligned} f(x(\tau '))+ \varphi (x(\tau '))/\tau '< & {} f(x(\tau ''))+\varphi (x(\tau ''))/\tau ' \nonumber \\ f(x(\tau ''))+\varphi (x(\tau ''))/\tau ''< & {} f(x(\tau '))+\varphi (x(\tau '))/\tau '' \,. \end{aligned}$$

(30)

By summing these inequalities we obtain $\varphi (x(\tau '))\left( \frac{1}{\tau '} - \frac{1}{\tau ''}\right) < \varphi (x(\tau ''))\left( \frac{1}{\tau '} - \frac{1}{\tau ''}\right)$, showing that $\varphi (x(\tau ')) < \varphi (x(\tau ''))$. Inequality (30) implies $f(x(\tau '')) - f(x(\tau '))< \frac{1}{\tau ''}[\varphi (x(\tau ')) - \varphi (x(\tau ''))] < 0$, yielding $f(x(\tau '')) < f(x(\tau '))$.

Next we show that

$$\begin{aligned} x(\tau )= & {} \mathop {\mathrm{argmin}}\{ f(x)\; \text{ s.t. } x \in X,\; \varphi (x) \le \varphi (x(\tau ))\} \nonumber \\= & {} \mathop {\mathrm{argmin}}\{ \varphi (x)\; \text{ s.t. } x \in X,\; f(x) \le f(x(\tau ))\}, \quad \forall \; \tau >0. \end{aligned}$$

(31)

To this end, let $x \in X$ such that $x\ne x(\tau )$. Then $\frac{1}{\tau }[\varphi (x(\tau )) - \varphi (x)] < f(x) -f(x(\tau ))$ from the definition of $x(\tau )$. If $\varphi (x) \le \varphi (x(\tau ))$ then $f(x(\tau ))< f(x)$, showing that $x(\tau )$ is the unique solution of $\min \{ f(x)\; \text{ s.t. } x \in X,\; \varphi (x) \le \varphi (x(\tau ))\}$. Furthermore, if $f(x)\le f(x(\tau ))$, then $\varphi (x(\tau )) < \varphi (x)$, showing that $x(\tau )$ is the unique solution of $\min \{ \varphi (x)\; \text{ s.t. } x \in X,\; f(x) \le f(x(\tau ))\}$.

We are now in position to proof item (c). Recall that $\varphi (x(\tau )) \le \varphi ({{\tilde{x}}})$ from item (a). The equivalences follow from (31) $\varphi ({{\tilde{x}}})= \varphi (x(\tau )) \Leftrightarrow f({{\tilde{x}}})= f(x(\tau )) \Leftrightarrow x(\tau ) \in X^*$, and the identity $x(\tau ')=\tilde{x}$ for all $\tau ' \ge \tau$ follows from (29). $\square$

Appendix B: Detailed descriptions of the test instances

We describe the test instances used in our numerical experiments in more detail. The interconnected network structure of the hydrothermal power plants in Brazil is depicted in Fig. 6, and the detailed data on hydro and thermo plants are given in Table 4 and Table 5, respectively. In addition, we set the demand to be a constant 14,500 in each stage, and we set the unit penalty cost of not satisfying the demand to be 4500. The scenario data file is too big to be displayed here, but will be available upon request.

Table 4 Hydro plant data: the maximum allowed amount of turbined flow, the minimum/maximum level of water allowed, the initial water level, and the amount of power generated by releasing one unit of water flow in each hydro plant $h = 1,2,\ldots , 30$

Fig. 6

The interconnected network structure of the hydrothermal power plants in Brazil

Table 5 Thermo plant data: maximum allowed amount of power generated and the unit cost of thermal power generated from each thermo plant f