Two-Stage Stochastic Mixed Integer Linear Optimization

The primary focus of this dissertation is on optimization problems that involve uncertainty unfolding over time. In many real-world decisions, the decision-maker has to decide in the face of uncertainty. After the outcome of the uncertainty is observed, she can correct her initial decision by taking some corrective actions at a later time stage. These problems are known as stochastic optimization problems with recourse. In the case that the number of time stages is limited to two, these problems are referred to as two-stage stochastic optimization problems. We focus on this class of optimization problems in this dissertation. The optimization problem that is solved before the realization of uncertainty is called the first-stage problem and the problem solved to make a corrective action on the initial decision is called the second-stage problem. The decisions made in the second- stage are affected by both the first-stage decisions and the realization of random variables. Consequently, the two-stage problem can be viewed as a parametric optimization problem which involves the so-called value function of the second-stage problem. The value function describes the change in optimal objective value as the right-hand side is varied and understanding it is crucial to developing solution methods for two-stage optimization problems.In the first part of this dissertation, we study the value function of a MILP. We review the structural properties of the value function and its construction methods. We con- tribute by proposing a discrete representation of the MILP value function. We show that the structure of the MILP value function arises from two other optimization problems that are constructed from its discrete and continuous components. We show that our representation can explain certain structural properties of the MILP value function such as the sets over which the value function is convex. We then provide a simplification of the Jeroslow Formula obtained by applying our results. Finally, we describe a cutting plane algorithm for its construction and determine the conditions under which the pro- posed algorithm is finite.Traditionally, the solution methods developed for two-stage optimization problems consider the problem where the second-stage problem involves only continuous variables. In the recent years, however, two-stage problems with integer variables in the second- stage have been visited in several studies. These problems are important in practice and arise in several applications in supply chain, finance, forestry and disaster management, among others. The second part of this dissertation concerns the development and implementation of a solution method for the two-stage optimization problem where both the first and second stage involve mixed integer variables. We describe a generalization of the classical Benders’ method for solving mixed integer two-stage stochastic linear optimization problems. We employ the strong dual functions encoded in the branch-and-bound trees resulting from solution of the second-stage problem. We show that these can be used effectively within a Benders’ framework and describe a method for obtaining all required dual functions from a single, continuously refined branch-and-bound tree that is used to warm start the solution procedure for each subproblem.Finally, we provide details on the implementation of our proposed algorithm. The implementation allows for construction of several approximations of the value function of the second-stage problem. We use different warm-starting strategies within our proposed algorithm to solve the second-stage problems, including solving all second-stage problems with a single tree. We provide computational results on applying these strategies to the stochastic server problems (SSLP) from the stochastic integer programming test problem library (SIPLIB)