Problem-based optimal scenario generation and reduction in stochastic programming

Abstract

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem.

Appendix

We consider semi-infinite programs of the form

$$\begin{aligned} P[V]\qquad \min \{g_{0}(u):u\in U,\,g_{j}(u,v)\le 0,\,j=1,\ldots ,p,\, \forall v\in V\}, \end{aligned}$$

where $U\subset \mathbb {R}^{m}$ is closed, $V\subset \mathbb {R}^{k}$ is compact and the functions $g_{0}:U\rightarrow \mathbb {R}$, $g_{j}:U\times V\rightarrow \mathbb {R}$, $j=1,\ldots ,p$, are continuous. Let $V_{i}$, $i\in \mathbb {N}_{0}$, be an increasing sequence of finite subsets of V such that $\lim _{i\rightarrow \infty }\sup _{v\in V}\min _{v_{i}\in V_{i}}\Vert v-v_{i}\Vert =0$.

• Discretization algorithm:

• Step 0: Set $i=0$, $D_{0}=V_{0}$.

• Step 1: Find a solution $u_{i}$ of $P[D_{i}]$.

• Step 2: Find a solution $v_{i}$ of $\max _{v\in V_{i+1}} \max _{j=1,\ldots ,p}g_{j}(u_{i},v)$.

• Step 3: If $\gamma _{i}=\max _{j=1,\ldots ,p}g_{j}(u_{i},v_{i})>0$, then select a set $D_{i+1}$ such that

  $$\begin{aligned} D_{i}\cup \{v_{i}\}\subseteq D_{i+1}\subseteq V_{i+1}. \end{aligned}$$

• Step 4: If $\gamma _{i}\le 0$ then stop.

• Step 5: Set $i=i+1$ and go to Step 1.

If the feasible set F[V] of P[V] is nonempty and the level set $\{u\in F[V_{0}]:g_{0}(u)\le g_{0}(u_{0})\}$ is bounded for some $u_{0}\in F[V]$, the infima of $P[D_{i}]$ converge to the infimum of P[V] and the sequence $(u_{i})$ has an accumulation point which solves P[V]. For a proof of this result we refer to [39, Theorem 2.1] and for further information and discussion to [40].