Numerical Techniques for Stochastic Optimization

This is a comprehensive and timely overview of the numerical techniques that have been developed to solve stochastic programming problems. After a brief introduction to the field, where accent is laid on modeling questions, the next few chapters lay out the challenges that must be met in this area. They also provide the background for the description of the computer implementations given in the third part of the book. Selected applications are described next. Some of these have directly motivated the development of the methods described in the earlier chapters. They include problems that come from facilities location, exploration investments, control of ecological systems, energy distribution and generation. Test problems are collected in the last chapter. This is the first book devoted to this subject. It comprehensively covers all major advances in the field (both Western and Soviet). It is only because of the recent developments in computer technology, that we have now reached a point where our computing power matches the inherent size requirements faced in this area. The book demonstrates that a large class of stochastic programming problems are now in the range of our numerical capacities

A tighter variant of Jensen's lower bound for stochastic programs and separable approximations to recourse functions

In this paper, we propose a new method to compute lower bounds on the optimal objective value of a stochastic program and show how this method can be used to construct separable approximations to the recourse functions. We show that our method yields tighter lower bounds than Jensen's lower bound and it requires a reasonable amount of computational effort even for large problems. The fundamental idea behind our method is to relax certain constraints by associating dual multipliers with them. This yields a smaller stochastic program that is easier to solve. We particularly focus on the special case where we relax all but one of the constraints. In this case, the recourse functions of the smaller stochastic program are one dimensional functions. We use these one dimensional recourse functions to construct separable approximations to the original recourse functions. Computational experiments indicate that our lower bounds can significantly improve Jensen's lower bound and our recourse function approximations can provide good solutions.Stochastic programming Lower bounds Recourse function approximation

Convex optimization methods for graphs and statistical modeling

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 209-220).An outstanding challenge in many problems throughout science and engineering is to succinctly characterize the relationships among a large number of interacting entities. Models based on graphs form one major thrust in this thesis, as graphs often provide a concise representation of the interactions among a large set of variables. A second major emphasis of this thesis are classes of structured models that satisfy certain algebraic constraints. The common theme underlying these approaches is the development of computational methods based on convex optimization, which are in turn useful in a broad array of problems in signal processing and machine learning. The specific contributions are as follows: -- We propose a convex optimization method for decomposing the sum of a sparse matrix and a low-rank matrix into the individual components. Based on new rank-sparsity uncertainty principles, we give conditions under which the convex program exactly recovers the underlying components. -- Building on the previous point, we describe a convex optimization approach to latent variable Gaussian graphical model selection. We provide theoretical guarantees of the statistical consistency of this convex program in the high-dimensional scaling regime in which the number of latent/observed variables grows with the number of samples of the observed variables. The algebraic varieties of sparse and low-rank matrices play a prominent role in this analysis. -- We present a general convex optimization formulation for linear inverse problems, in which we have limited measurements in the form of linear functionals of a signal or model of interest. When these underlying models have algebraic structure, the resulting convex programs can be solved exactly or approximately via semidefinite programming. We provide sharp estimates (based on computing certain Gaussian statistics related to the underlying model geometry) of the number of generic linear measurements required for exact and robust recovery in a variety of settings. -- We present convex graph invariants, which are invariants of a graph that are convex functions of the underlying adjacency matrix. Graph invariants characterize structural properties of a graph that do not depend on the labeling of the nodes; convex graph invariants constitute an important subclass, and they provide a systematic and unified computational framework based on convex optimization for solving a number of interesting graph problems. We emphasize a unified view of the underlying convex geometry common to these different frameworks. We describe applications of both these methods to problems in financial modeling and network analysis, and conclude with a discussion of directions for future research.by Venkat Chandrasekaran.Ph.D

Proceedings of the Third International Mobile Satellite Conference (IMSC 1993)

Satellite-based mobile communications systems provide voice and data communications to users over a vast geographic area. The users may communicate via mobile or hand-held terminals, which may also provide access to terrestrial cellular communications services. While the first and second International Mobile Satellite Conferences (IMSC) mostly concentrated on technical advances, this Third IMSC also focuses on the increasing worldwide commercial activities in Mobile Satellite Services. Because of the large service areas provided by such systems, it is important to consider political and regulatory issues in addition to technical and user requirements issues. Topics covered include: the direct broadcast of audio programming from satellites; spacecraft technology; regulatory and policy considerations; advanced system concepts and analysis; propagation; and user requirements and applications