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Problem-based optimal scenario generation and reduction in stochastic programming
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