A central limit theorem for the optimal value is proven by using empirical process theory and concepts of differentiability in infinite dimensional spaces. Such a limit theorem has formerly been known only for simpler special cases. Beside being of theoretical interest, limit theorems may be useful for getting information about the accuracy of an approximate optimal value and for determining an appropriate sample size for a practical problem.
Therefore, resampling methods (bootstrap) are suitably adapted and, for illustration, applied to a test problem. For stochastic programs possibly incorporating dynamic decision structures a special strategy of risk aversion is suggested and analyzed in the second part, namely the class of polyhedral risk measures: The value of a risk functional from this class can be calculated as the optimal value of a specific stochastic program with recourse which is of particular simple nature. Polyhedral risk measures are intended for objectives of general stochastic programs. Then, the two nested stochastic programs can be unified to one stochastic program with classical linear objective. This possibility can be useful for algorithmic decomposition approaches. Polyhedral risk measures are analyzed with respect to coherence axioms from risk theory. Criteria for verifying such properties for a concrete polyhedral risk measure are deduced by means of convex duality theory. Moreover, new and known instances of polyhedral risk measures are presented and shown to satisfy these coherence axioms. Furthermore, stability statements for multi-stage stochastic programs incorporating a polyhedral risk measure in the objective are proven. These statements allow the conclusion that, for such problems, the same stability based scenario tree approximation algorithms as for non-risk-averse stochastic programs can be applied if some additional regularity requirements hold.
It is shown that all the instances of polyhedral risk measures presented before satisfy these regularity requirements. Finally, the practical usefulness of polyhedral risk measures is demonstrated by a case study consisting of a stochastic programming model for medium-term optimization of electricity production and trading in a smaller power utility. Expected profit and risk in terms of a polyhedral risk measure are optimized simultaneously. The model takes into account the uncertainty of energy demands and market prices in terms of probability distributions which are approximated by a scenario tree according to the above results. The model demonstrates the possibility of integrating revenue optimization and risk management. The output of the model illustrates that the class of polyhedral risk measures is capable of reproducing different preferences for risk aversion.
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