Stability Approximation And Decomposition In Two And Multistage Stochastic Programming

Christian Küchler studies various aspects of the stability of stochastic optimization problems as well as approximation and decomposition methods in stochastic programming. In particular, the author presents an extension of the Nested Benders decomposition algorithm related to the concept of recombining scenario trees.

The goal of the Encyclopedia of Optimization is to introduce the reader to a complete set of topics that show the spectrum of research, the richness of ideas, and the breadth of applications that has come from this field. The second edition builds on the success of the former edition with more than 150 completely new entries, designed to ensure that the reference addresses recent areas where optimization theories and techniques have advanced. Particularly heavy attention resulted in health science and transportation, with entries such as "Algorithms for Genomics", "Optimization and Radiotherapy Treatment Design", and "Crew Scheduling".

Optimization problems involving stochastic models occur in almost all areas of science and engineering, such as telecommunications, medicine, and finance. Their existence compels a need for rigorous ways of formulating, analyzing, and solving such problems. This book focuses on optimization problems involving uncertain parameters and covers the theoretical foundations and recent advances in areas where stochastic models are available. Readers will find coverage of the basic concepts of modeling these problems, including recourse actions and the nonanticipativity principle. The book also includes the theory of two-stage and multistage stochastic programming problems; the current state of the theory on chance (probabilistic) constraints, including the structure of the problems, optimality theory, and duality; and statistical inference in and risk-averse approaches to stochastic programming.

From the Preface... The preparation of this book started in 2004, when George B. Dantzig and I, following a long-standing invitation by Fred Hillier to contribute a volume to his International Series in Operations Research and Management Science, decided finally to go ahead with editing a volume on stochastic programming. The field of stochastic programming (also referred to as optimization under uncertainty or planning under uncertainty) had advanced significantly in the last two decades, both theoretically and in practice. George Dantzig and I felt that it would be valuable to showcase some of these advances and to present what one might call the state-of- the-art of the field to a broader audience. We invited researchers whom we considered to be leading experts in various specialties of the field, including a few representatives of promising developments in the making, to write a chapter for the volume. Unfortunately, to the great loss of all of us, George Dantzig passed away on May 13, 2005. Encouraged by many colleagues, I decided to continue with the book and edit it as a volume dedicated to George Dantzig. Management Science published in 2005 a special volume featuring the “Ten most Influential Papers of the first 50 Years of Management Science.” George Dantzig’s original 1955 stochastic programming paper, “Linear Programming under Uncertainty,” was featured among these ten. Hearing about this, George Dantzig suggested that his 1955 paper be the first chapter of this book. The vision expressed in that paper gives an important scientific and historical perspective to the book. Gerd Infanger

Energy is one of the world`s most challenging problems, and power systems are an important aspect of energy related issues. This handbook contains state-of-the-art contributions on power systems modeling and optimization. The book is separated into two volumes with six sections, which cover the most important areas of energy systems. The first volume covers the topics operations planning and expansion planning while the second volume focuses on transmission and distribution modeling, forecasting in energy, energy auctions and markets, as well as risk management. The contributions are authored by recognized specialists in their fields and consist in either state-of-the-art reviews or examinations of state-of-the-art developments. The articles are not purely theoretical, but instead also discuss specific applications in power systems.

Consisting of two parts, this book presents papers describing publicly available stochastic programming systems that are operational. It presents a diverse collection of application papers in areas such as production, supply chain and scheduling, gaming, environmental and pollution control, financial modeling, telecommunications, and electricity.

Motivation Stochastic Linear Programming with recourse represents one of the more widely applicable models for incorporating uncertainty within in which the SLP optimization models. There are several arenas model is appropriate, and such models have found applications in air line yield management, capacity planning, electric power generation planning, financial planning, logistics, telecommunications network planning, and many more. In some of these applications, modelers represent uncertainty in terms of only a few seenarios and formulate a large scale linear program which is then solved using LP software. However, there are many applications, such as the telecommunications planning problem discussed in this book, where a handful of seenarios do not capture variability well enough to provide a reasonable model of the actual decision-making problem. Problems of this type easily exceed the capabilities of LP software by several orders of magnitude. Their solution requires the use of algorithmic methods that exploit the structure of the SLP model in a manner that will accommodate large scale applications.

In this thesis the optimization framework of stochastic programming with recourse is considered. Emphasis is placed on programs incorporating integrality constraints, dynamic decision structures (multi-stage stochastic programs), or risk aversion requirements. In the first part, Monte Carlo approximations for two-stage stochastic programs with integrality constraints are studied. In particular, the asymptotic behavior of the optimal values is analyzed. A central limit theorem for the optimal value is proven by using empirical process theory and concepts of differentiability in infinite dimensional spaces. Such a limit theorem has formerly been known only for simpler special cases. Beside being of theoretical interest, limit theorems may be useful for getting information about the accuracy of an approximate optimal value and for determining an appropriate sample size for a practical problem. Therefore, resampling methods (bootstrap) are suitably adapted and, for illustration, applied to a test problem. For stochastic programs possibly incorporating dynamic decision structures a special strategy of risk aversion is suggested and analyzed in the second part, namely the class of polyhedral risk measures: The value of a risk functional from this class can be calculated as the optimal value of a specific stochastic program with recourse which is of particular simple nature. Polyhedral risk measures are intended for objectives of general stochastic programs. Then, the two nested stochastic programs can be unified to one stochastic program with classical linear objective. This possibility can be useful for algorithmic decomposition approaches. Polyhedral risk measures are analyzed with respect to coherence axioms from risk theory. Criteria for verifying such properties for a concrete polyhedral risk measure are deduced by means of convex duality theory. Moreover, new and known instances of polyhedral risk measures are presented and shown to satisfy these coherence axioms. Furthermore, stability statements for multi-stage stochastic programs incorporating a polyhedral risk measure in the objective are proven. These statements allow the conclusion that, for such problems, the same stability based scenario tree approximation algorithms as for non-risk-averse stochastic programs can be applied if some additional regularity requirements hold. It is shown that all the instances of polyhedral risk measures presented before satisfy these regularity requirements. Finally, the practical usefulness of polyhedral risk measures is demonstrated by a case study consisting of a stochastic programming model for medium-term optimization of electricity production and trading in a smaller power utility. Expected profit and risk in terms of a polyhedral risk measure are optimized simultaneously. The model takes into account the uncertainty of energy demands and market prices in terms of probability distributions which are approximated by a scenario tree according to the above results. The model demonstrates the possibility of integrating revenue optimization and risk management. The output of the model illustrates that the class of polyhedral risk measures is capable of reproducing different preferences for risk aversion.