Study on Optimality Conditions in Stochastic Linear Programming

Abstract

In the rapidly changing world of today, people have to make decisions under some degree of uncertainty. At the same time, the development of computing technologies enables people to take uncertain factors into considerations while making their decisions.Stochastic programming techniques have been widely applied in finance engineering, supply chain management, logistics, transportation, etc. Such applications often involve a large, possibly infinite, set of scenarios. Hence the resulting programstend to be large in scale.The need to solve large scale programs calls for a combination of mathematical programming techniques and sample-based approximation. When using sample-based approximations, it is important to determine the extent to which the resulting solutions are dependent on thespecific sample used. This dissertation research focuses on computational evaluation of the solutions from sample-based two-stage/multistage stochastic linear programming algorithms, with a focus on the effectiveness of optimality tests and the quality ofa proposed solution.In the first part of this dissertation, two alternative approaches of optimality tests of sample-based solutions, adaptive and non-adaptive sampling methods, are examined and computationally compared. The results of the computational experiment are in favor of the adaptive methods. In the second part of this dissertation, statistically motivated bound-based solution validation techniques in multistage linear stochastic programs are studied both theoretically and computationally. Different approaches of representations of the nonanticipativity constraints are studied. Bounds are established through manipulations of the nonanticipativity constraints.