Introduction to stochastic programming

This book provides a complete and comprehensive reference/guide to Pyomo (Python Optimization Modeling Objects) for both beginning and advanced modelers, including students at the undergraduate and graduate levels, academic researchers, and practitioners. The text illustrates the breadth of the modeling and analysis capabilities that are supported by the software and support of complex real-world applications. Pyomo is an open source software package for formulating and solving large-scale optimization and operations research problems. The text begins with a tutorial on simple linear and integer programming models. A detailed reference of Pyomo's modeling components is illustrated with extensive examples, including a discussion of how to load data from data sources like spreadsheets and databases. Chapters describing advanced modeling capabilities for nonlinear and stochastic optimization are also included. The Pyomo software provides familiar modeling features within Python, a powerful dynamic programming language that has a very clear, readable syntax and intuitive object orientation. Pyomo includes Python classes for defining sparse sets, parameters, and variables, which can be used to formulate algebraic expressions that define objectives and constraints. Moreover, Pyomo can be used from a command-line interface and within Python's interactive command environment, which makes it easy to create Pyomo models, apply a variety of optimizers, and examine solutions. The software supports a different modeling approach than commercial AML (Algebraic Modeling Languages) tools, and is designed for flexibility, extensibility, portability, and maintainability but also maintains the central ideas in modern AMLs.

https://calhoun.nps.edu/bitstream/10945/37919/1/Dimitrov-PyomoReview.pdf

Pyomo - Optimization Modeling in Python

https://courses.nus.edu.sg/course/cheld/internet/CN5111/Reference/Retrospective%20on%20optimization.pdf

Retrospective on optimization

Encompassing all the major topics students will encounter in courses on the subject, the authors teach both the underlying mathematical foundations and how these ideas are implemented in practice. They illustrate all the concepts with both worked examples and plenty of exercises, and, in addition, provide software so that students can try out numerical methods and so hone their skills in interpreting the results. As a result, this will make an ideal textbook for all those coming to the subject for the first time. Authors' note: A problem recently found with the software is due to a bug in Formula One, the third party commercial software package that was used for the development of the interface. It occurs when the date, currency, etc. format is set to a non-United States version. Please try setting your computer date/currency option to the United States option . The new version of Formula One, when ready, will be posted on Www.

Linear Programming 1: Introduction

An inexact two-stage stochastic programming (ITSP) model is proposed for water resources management under uncertainty. The model is a hybrid of inexact optimization and two-stage stochastic programming. It can reflect not only uncertainties expressed as probability distributions but also those being available as intervals. The solution meth od for ITSP is computationally effective, which makes it applicable to practical problems. The ITSP is applied to a hypothetical case study of water resources system operation. The results indicate that reasonable solutions have been obtained. They are further analyzed and interpreted for generating decision alternatives and identifying significant factors that affect the system's performance. The information obtained through these post-optimality analyses can provide useful decision support for water managers.

An inexact two-stage stochastic programming model for water resources management under uncertainty

This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem. Many of the computational tricks developed for deterministic staircase problems are adapted to the stochastic setting and their effect on computation times is investigated. The computer code supports an arbitrary number of time periods and various types of random structures for the input data. Numerical results compare the performance of the algorithm to MINOS 5.0.

MSLiP: a computer code for the multistage stochastic linear programming problem

Data conventions for the automatic input of multiperiod stochastic linear programs are described. The input format is based on the MPSX standard and is designed to promote the efficient conversion of originally deterministic problems by introducing stochastic variants in separate files. A flexible "header" syntax generates a useful variety of stochastic dependencies. An extension using the NETGEN format is proposed for stochastic network programs.

A Standard Input Format for Multiperiod Stochastic Linear Programs

A method is described for finding logging levels to maximize harvest in a finite horizon type II model. Uncertainty is considered in the form of the risk of forest fires and other environmental hazards, which may destroy a random fraction of the existing forest. Numerical results include upper and lower bound approximations to the original problem.

Optimal harvest of a forest in the presence of uncertainty

Edmundson and Madansky have developed an upper bound for the expectation of a convex function of a multivariate random variable. The bound exists under rather general conditions. However, it is computable for n≥3 only when the random variables are independent. This paper develops a procedure for computing this bound using a linear program. We also show how to extend and sharpen the bound by utilizing it on subsets of the sample space. Moreover, the bound is applicable to arbitrary convex domains which may be unbounded. The new bounds along with Jensen’s Inequality may be applied on subsets using the procedures developed by Huang, Ziemba, and Ben-Tal to yield a procedure for obtaining upper and lower bounds on the expectation of a convex function of a multivariate random variable to an arbitrary degree of accuracy. The results are useful in a wide variety of optimization applications. Some numerical work is provided to illustrate the use of the new bounds.

A tight upper bound for the expectation of a convex function of a multivariate random variable

This article describes and compares several numerical methods for finding multivariate probabilities over a rectangle. A large computational study shows how the computation times depend on the problem dimensions, the correlation structure, the magnitude of the sought probability, and the required accuracy. No method is uniformly best for all problems and—unlike previous work—this article gives some guidelines to help establish the most promising method a priori. Numerical tests were conducted on approximately 3,000 problems generated randomly in up to 20 dimensions. Our findings indicate that direct integration methods give acceptable results for up to 12-dimensional problems, provided that the probability mass of the rectangle is not too large (less than about 0.9). For problems with small probabilities (less than 0.3) a crude Monte Carlo method gives reasonable results quickly, while bounding procedures perform best on problems with large probabilities (> 0.9). For larger problems numerical integration ...

Horand I. Gassmann

Papers

MSLiP: a computer code for the multistage stochastic linear programming problem

A Standard Input Format for Multiperiod Stochastic Linear Programs

Optimal harvest of a forest in the presence of uncertainty

A tight upper bound for the expectation of a convex function of a multivariate random variable

Computing Multivariate Normal Probabilities: A New Look