Julia L. Higle

An Introductory Tutorial on Stochastic Linear Programming Models

Linear programming is a fundamental planning tool. It is often difficult to precisely estimate or forecast certain critical data elements of the linear program. In such cases, it is necessary to address the impact of uncertainty during the planning process. We discuss a variety of LP-based models that can be used for planning under uncertainty. In all cases, we begin with a deterministic LP model and show how it can be adapted to include the impact of uncertainty. We present models that range from simple recourse policies to more general two-stage and multistage SLP formulations. We also include a discussion of probabilistic constraints. We illustrate the various models using examples taken from the literature. The examples involve models developed for airline yield management, telecommunications, flood control, and production planning.

Capacity expansion under stochastic demands

We consider the problem of optimally meeting a stochastically growing demand for capacity over an infinite horizon. Under the assumption that demand for product follows either a nonlinear Brownian motion or a non-Markovian birth and death process, we show that this stochastic problem can be transformed into an equivalent deterministic problem. Consistent with earlier work by A. Manne, the equivalent problem is formed by replacing the stochastic demand by its deterministic trend and discounting all costs by a new interest rate that is smaller than the original, in approximate proportion to the uncertainty in the demand.

The C 3 Theorem and a D 2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification

This paper considers the two-stage stochastic integer programming problem, with an emphasis on instances in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C3 (Common Cut Coefficients) Theorem. Based on the C3 Theorem, we develop a decomposition algorithm which we refer to as Disjunctive Decomposition (D2). In this new class of algorithms, we work with master and subproblems that result from convexifications of two coupled disjunctive programs. We show that when the second stage consists of 0-1 MILP problems, we can obtain accurate second stage objective function estimates after finitely many steps. This result implies the convergence of the D2 algorithm.

Sensitivity Analysis and Uncertainty in Linear Programming

Linear programming (LP) is one of the great successes to emerge from operations research and management science. It is well developed and widely used. LP problems in practice are often based on numerical data that represent rough approximations of quantities that are inherently difficult to estimate. Because of this, most LP-based studies include a postoptimality investigation of how a change in the data changes the solution. Researchers routinely undertake this type of sensitivity analysis (SA), and most commercial packages for solving linear programs include the results of such an analysis as part of the standard output report. SA has shortcomings that run contrary to conventional wisdom. Alternate models address these shortcomings.

Statistical verification of optimality conditions for stochastic programs with recourse

Statistically motivated algorithms for the solution of stochastic programming problems typically suffer from their inability to recognize optimality of a given solution algorithmically. Thus, the quality of solutions provided by such methods is difficult to ascertain. In this paper, we develop methods for verification of optimality conditions within the framework of Stochastic Decomposition (SD) algorithms for two stage linear programs with recourse. Consistent with the stochastic nature of an SD algorithm, we provide termination criteria that are based on statistical verification of traditional (deterministic) optimality conditions. We propose the use of “bootstrap methods” to confirm the satisfaction of generalized Kuhn-Tucker conditions and conditions based on Lagrange duality. These methods are illustrated in the context of a power generation planning model, and the results are encouraging.

Duality and statistical tests of optimality for two stage stochastic programs

We present alternative methods for verifying the quality of a proposed solution to a two stage stochastic program with recourse. Our methods revolve around implications of a dual problem in which dual multipliers on the nonanticipativity constraints play a critical role. Using randomly sampled observations of the stochastic elements, we introduce notions of statistical dual feasibility and sampled error bounds. Additionally, we use the nonanticipativity multipliers to develop connections to reduced gradient methods. Finally, we propose a statistical test based on directional derivatives. We illustrate the applicability of these tests via some examples.

Conditional Stochastic Decomposition: An Algorithmic Interface for Optimization and Simulation

Simulation and optimization are among the most commonly used elements in the OR toolkit. Often times, some of the data elements used to define an optimization problem are best described by random variables, yielding a stochastic program. If the distributions of the random variables cannot be specified precisely, one may have to resort to simulation to obtain observations of these random variables. In this paper, we present conditional stochastic decomposition (CSD), a method that may be construed as providing an algorithmic interface between simulation and optimization for the solution of stochastic linear programs with recourse. Derived from the concept of the stochastic decomposition of such problems, CSD uses randomly generated observations with a Renders decomposition of the problem. In this paper, our method is analytically verified and graphically illustrated. In addition, CSD is used to solve several test problems that have appeared in the literature. Our computational experience suggests that CSD ...

A Summary and Illustration of Disjunctive Decomposition with Set Convexification

In this paper we review the Disjunctive Decomposition (D2) algorithm for two-stage Stochastic Mixed Integer Programs (SMIP). This novel method uses principles of disjunctive programming to develop cutting-plane-based approximations of the feasible set of the second stage problem. At the core of this approach is the Common Cut Coefficient Theorem, which provides a mechanism for transforming cuts derived for one outcome of the second stage problem into cuts that are valid for other outcomes. An illustrative application of the D2 method to the solution of a small SMIP illustrative example is provided.

Statistical approximations forstochastic linear programming problems

Sampling and decomposition constitute two of the most successful approaches foraddressing large‐scale problems arising in statistics and optimization, respectively. In recentyears, these two approaches have been combined for the solution of large‐scale stochasticlinear programming problems. This paper presents the algorithmic motivation for suchmethods, as well as a broad overview of issues in algorithm design. We discuss both basicschemes as well as computational enhancements and stopping rules. We also introduce ageneralization of current algorithms to handle problems with random recourse.

Inexact subgradient methods with applications in stochastic programming

In many instances, the exact evaluation of an objective function and its subgradients can be computationally demanding. By way of example, we cite problems that arise within the context of stochastic optimization, where the objective function is typically defined via multi-dimensional integration. In this paper, we address the solution of such optimization problems by exploring the use of successive approximation schemes within subgradient optimization methods. We refer to this new class of methods as inexact subgradient algorithms. With relatively mild conditions imposed on the approximations, we show that the inexact subgradient algorithms inherit properties associated with their traditional (i.e., exact) counterparts. Within the context of stochastic optimization, the conditions that we impose allow a relaxation of requirements traditionally imposed on steplengths in stochastic quasi-gradient methods. Additionally, we study methods in which steplengths may be defined adaptively, in a manner that reflects the improvement in the objective function approximations as the iterations proceed. We illustrate the applicability of our approach by proposing an inexact subgradient optimization method for the solution of stochastic linear programs.