Frederike Neise 
Risk Management in Stochastic Integer Programming 
With Application to Dispersed Power Generation

I am deeply grateful to my advisor Prof. Dr. Rüdiger Schultz for his untiring – couragement. Moreover, I would like to express my gratitude to Prof. Dr. -Ing. – mund Handschin and Dr. -Ing. Hendrik Neumann from the University of Dortmund for inspiration and support. I would like to thank PD Dr. René Henrion from the Weierstrass Institute for Applied Analysis and Stochastics in Berlin for reviewing this thesis. Cordial thanks to my colleagues at the University of Duisburg-Essen for motivating and fruitful discussions as well as a pleasurable cooperation. Contents 1 Introduction 1 1. 1 Stochastic Optimization. . . . . . . . . . . . . . . . . . . . . . . 3 1. 1. 1 The two-stage stochastic optimization problem . . . . . . 3 1. 1. 2 Expectation-based formulation. . . . . . . . . . . . . . . 5 1. 2 Content and Structure. . . . . . . . . . . . . . . . . . . . . . . . 6 2 Risk Measuresin Two-Stage Stochastic Programs 9 2. 1 Risk Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. 1. 1 Deviation measures. . . . . . . . . . . . . . . . . . . . . 10 2. 1. 2 Quantile-based risk measures . . . . . . . . . . . . . . . 11 2. 2 Mean-Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . 12 2. 2. 1 Results concerning structure and stability . . . . . . . . . 13 2. 2. 2 Deterministic equivalents. . . . . . . . . . . . . . . . . . 22 2. 2. 3 Algorithmic issues – dual decomposition method . . . . . 26 3 Stochastic Dominance Constraints 33 3. 1 Introduction to Stochastic Dominance . . . . . . . . . . . . . . . 33 3. 1. 1 Stochastic orders for the preference of higher outcomes . . 34 3. 1. 2 Stochastic orders for the preference of smaller outcomes . 38 3. 2 Stochastic Dominance Constraints . . . . . . . . . . . . . . . . . 42 3. 2. 1 First order stochastic dominanceconstraints. . . . . . . . 43 3. 2. 2 Results concerning structure and stability . . . . . . . . . 44 3. 2. 3 Deterministic equivalents. . . . . . . . . . . . . . . . . . 51 3. 2. 4 Algorithmic issues . . . . . . . . . . . . . . . . . . . . .

Table of Content

Risk Measures in Two-Stage Stochastic Programs.- Stochastic Dominance Constraints induced by Mixed-Integer Linear Recourse.- Application: Optimal Operation of a Dispersed Generation System.- Conclusion and Perspective.

About the author

Dr. Frederike Neise gained a Ph D in Mathematics from the University of Duisburg-Essen studying two-stage stochastic programming and its application to the optimal management of dispersed generation systems. She currently works as a gas market analyst with E.ON Ruhrgas AG.