Stochastic programming recourse models

Viac o knihe

This thesis explores the optimization framework of stochastic programming with recourse, focusing on integrality constraints, dynamic decision structures, and risk aversion. The first part examines Monte Carlo approximations for two-stage stochastic programs with integrality constraints, analyzing the asymptotic behavior of optimal values. A central limit theorem for the optimal value is established using empirical process theory and concepts of differentiability in infinite-dimensional spaces, previously known only for simpler cases. This theorem aids in assessing the accuracy of approximate optimal values and determining sample sizes for practical problems, with resampling methods like bootstrap adapted and applied to a test problem. In the second part, a strategy for risk aversion is suggested through polyhedral risk measures, which can be calculated as the optimal value of a specific stochastic program. This unification of two nested stochastic programs into one with a classical linear objective is beneficial for algorithmic decomposition approaches. The analysis of polyhedral risk measures includes coherence axioms from risk theory, with criteria for verifying these properties derived from convex duality theory. Stability statements for multi-stage stochastic programs incorporating polyhedral risk measures are proven, allowing the use of stability-based scenario tree approximation algorithms under certain regularity cond