This work was completed during my tenure as a scientific assistant and d- toral student at the Institute for Operations Research at the University of St. Gallen. During that time, I was involved in several industry projects in the field of power management, on the occasion of which I was repeatedly c- fronted with complex decision problems under uncertainty. Although usually hard to solve, I quickly learned to appreciate the benefit of stochastic progr- ming models and developed a strong interest in their theoretical properties. Motivated both by practical questions and theoretical concerns, I became p- ticularly interested in the art of finding tight bounds on the optimal value of a given model. The present work attempts to make a contribution to this important branch of stochastic optimization theory. In particular, it aims at extending some classical bounding methods to broader problem classes of practical relevance. This book was accepted as a doctoral thesis by the University of St. Gallen in June 2004.1 am particularly indebted to Prof. Dr. Karl Frauendorfer for - pervising my work. I am grateful for his kind support in many respects and the generous freedom I received to pursue my own ideas in research. My gratitude also goes to Prof. Dr. Georg Pflug, who agreed to co-chair the dissertation committee. With pleasure I express my appreciation for his encouragement and continuing interest in my work.

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

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.

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.

This book constitutes the refereed proceedings of the 5th International Symposium on Stochastic Algorithms, Foundations and Applications, SAGA 2009, held in Sapporo, Japan, in October 2009. The 15 revised full papers presented together with 2 invited papers were carefully reviewed and selected from 22 submissions. The papers are organized in topical sections on learning, graphs, testing, optimization and caching, as well as stochastic algorithms in bioinformatics.

This volume presents a collection of contributions dedicated to applied problems in the financial and energy sectors that have been formulated and solved in a stochastic optimization framework. The invited authors represent a group of scientists and practitioners, who cooperated in recent years to facilitate the growing penetration of stochastic programming techniques in real-world applications, inducing a significant advance over a large spectrum of complex decision problems. After the recent widespread liberalization of the energy sector in Europe and the unprecedented growth of energy prices in international commodity markets, we have witnessed a significant convergence of strategic decision problems in the energy and financial sectors. This has often resulted in common open issues and has induced a remarkable effort by the industrial and scientific communities to facilitate the adoption of advanced analytical and decision tools. The main concerns of the financial community over the last decade have suddenly penetrated the energy sector inducing a remarkable scientific and practical effort to address previously unforeseeable management problems. Stochastic Optimization Methods in Finance and Energy: New Financial Products and Energy Markets Strategies aims to include in a unified framework for the first time an extensive set of contributions related to real-world applied problems in finance and energy, leading to a common methodological approach and in many cases having similar underlying economic and financial implications. Part 1 of the book presents 6 chapters related to financial applications; Part 2 presents 7 chapters on energy applications; and Part 3 presents 5 chapters devoted to specific theoretical and computational issues.

The book covers several new research findings in the area of generalized convexity and integral inequalities. Integral inequalities using various type of generalized convex functions are applicable in many branches of mathematics such as mathematical analysis, fractional calculus, and discrete fractional calculus. The book contains integral inequalities of Hermite-Hadamard type, Hermite- Hadamard-Fejer type and majorization type for the generalized strongly convex functions. It presents Hermite-Hadamard type inequalities for functions defined on Time scales. Further, it provides the generalization and extensions of the concept of preinvexity for interval-valued functions and stochastic processes, and give Hermite-Hadamard type and Ostrowski type inequalities for these functions. These integral inequalities are utilized in numerous areas for the boundedness of generalized convex functions. Features: Covers Interval-valued calculus, Time scale calculus, Stochastic processes – all in one single book Numerous examples to validate results Provides an overview of the current state of integral inequalities and convexity for a much wider audience, including practitioners Applications of some special means of real numbers are also discussed The book is ideal for anyone teaching or attending courses in integral inequalities along with researchers in this area.