Some Unconventional Stochastic Programs

Stochastic programming is a mathematical optimization model for decision making when the uncertainty is characterized by random events. This thesis is concerned with some stochastic programs that deviate from the conventional modeling or assumptions. We first study the stochastic programs without relatively complete recourse. For a very long time, the relatively complete recourse condition is a key assumption in analyzing solution approaches such as the sample average approximation method or the stochastic approximation algorithm. Nevertheless, this assumption fails for many real-world problems, e.g., linear regression problems with data-dependent constraints. Without the condition, the solutions generated may be infeasible. For the class of problems having chain-constrained domain, we derive probability bounds on the feasibility of the sample average approximation solutions. The result is then strengthened when convexity is involved. The second topic we investigate is the multistage stochastic programs with optimal stopping. The optimal stopping problem has a long history and finds interesting applications in house selling, option pricing, etc. It turns out that the idea of stopping time can be incorporated in the framework of multistage stochastic programming quite naturally. We provide a uniform treatment of the time consistency of the solutions when we consider the multistage problems with optimal stopping in the risk averse setting.Ph.D