Two-stage stochastic programming models are considered as attractive tools for making optimal decisions under uncertainty. Traditionally, optimality is formalized by applying statistical parameters such as the expectation or the conditional value at risk to the distributions of objective values. Uwe Gotzes analyzes an approach to account for risk aversion in two-stage models based upon partial orders on the set of real random variables. These stochastic orders enable the incorporation of the characteristics of whole distributions into the decision process. The profit or cost distributions must pass a benchmark test with a given acceptable distribution. Thus, additional objectives can be optimized. For this new class of stochastic optimization problems, results on structure and stability are proven and a tailored algorithm to tackle large problem instances is developed. The implications of the modelling background and numerical results from the application of the proposed algorithm are demonstrated with case studies from energy trading.

Optimization problems whose constraints involve partial differential equations (PDEs) are relevant in many areas of technical, industrial, and economic app- cations. At the same time, they pose challenging mathematical research problems in numerical analysis and optimization. The present text is among the ?rst in the research literature addressing stochastic uncertainty in the context of PDE constrained optimization. The focus is on shape optimization for elastic bodies under stochastic loading. Analogies to ?nite dim- sional two-stage stochastic programming drive the treatment, with shapes taking the role of nonanticipative decisions.The main results concern level set-based s- chastic shape optimization with gradient methods involving shape and topological derivatives. The special structure of the elasticity PDE enables the numerical - lution of stochastic shape optimization problems with an arbitrary number of s- narios without increasing the computational effort signi?cantly. Both risk neutral and risk averse models are investigated. This monograph is based on a doctoral dissertation prepared during 2004-2008 at the Chair of Discrete Mathematics and Optimization in the Department of Ma- ematics of the University of Duisburg-Essen. The work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the Priority Program “Optimi- tion with Partial Differential Equations”. Rüdiger Schultz Acknowledgments I owe a great deal to my supervisors, colleagues, and friends who have always supported, encouraged, andenlightenedmethroughtheirownresearch, comments, and questions.

Stochastic programming provides a framework for modelling, analyzing, and solving optimization problems with some parameters being not known up to a probability distribution. Such problems arise in a variety of applications, such as inventory control, financial planning and portfolio optimization, airline revenue management, scheduling and operation of power systems, and supply chain management. 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 approach combines the concept of cut sharing with a specific aggregation procedure and prevents an exponentially growing number of subproblem evaluations. Convergence results and numerical properties are discussed.

Optimization problems involving uncertain data arise in many areas of industrial and economic applications. Stochastic programming provides a useful framework for modeling and solving optimization problems for which a probability distribution of the unknown parameters is available.Motivated by practical optimization problems occurring in energy systems with regenerative energy supply, Debora Mahlke formulates and analyzes multistage stochastic mixed-integer models. For their solution, the author proposes a novel decomposition approach which relies on the concept of splitting the underlying scenario tree into subtrees. Based on the formulated models from energy production, the algorithm is computationally investigated and the numerical results are discussed.

Two-stage stochastic programming models are considered as attractive tools for making optimal decisions under uncertainty. Traditionally, optimality is formalized by applying statistical parameters such as the expectation or the conditional value at risk to the distributions of objective values. Uwe Gotzes analyzes an approach to account for risk aversion in two-stage models based upon partial orders on the set of real random variables. These stochastic orders enable the incorporation of the characteristics of whole distributions into the decision process. The profit or cost distributions must pass a benchmark test with a given acceptable distribution. Thus, additional objectives can be optimized. For this new class of stochastic optimization problems, results on structure and stability are proven and a tailored algorithm to tackle large problem instances is developed. The implications of the modelling background and numerical results from the application of the proposed algorithm are demonstrated with case studies from energy trading.

Optimization problems whose constraints involve partial differential equations (PDEs) are relevant in many areas of technical, industrial, and economic app- cations. At the same time, they pose challenging mathematical research problems in numerical analysis and optimization. The present text is among the ?rst in the research literature addressing stochastic uncertainty in the context of PDE constrained optimization. The focus is on shape optimization for elastic bodies under stochastic loading. Analogies to ?nite dim- sional two-stage stochastic programming drive the treatment, with shapes taking the role of nonanticipative decisions.The main results concern level set-based s- chastic shape optimization with gradient methods involving shape and topological derivatives. The special structure of the elasticity PDE enables the numerical - lution of stochastic shape optimization problems with an arbitrary number of s- narios without increasing the computational effort signi?cantly. Both risk neutral and risk averse models are investigated. This monograph is based on a doctoral dissertation prepared during 2004-2008 at the Chair of Discrete Mathematics and Optimization in the Department of Ma- ematics of the University of Duisburg-Essen. The work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the Priority Program “Optimi- tion with Partial Differential Equations”. Rüdiger Schultz Acknowledgments I owe a great deal to my supervisors, colleagues, and friends who have always supported, encouraged, andenlightenedmethroughtheirownresearch, comments, and questions.

Stochastic programming provides a framework for modelling, analyzing, and solving optimization problems with some parameters being not known up to a probability distribution. Such problems arise in a variety of applications, such as inventory control, financial planning and portfolio optimization, airline revenue management, scheduling and operation of power systems, and supply chain management. 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 approach combines the concept of cut sharing with a specific aggregation procedure and prevents an exponentially growing number of subproblem evaluations. Convergence results and numerical properties are discussed.

Optimization problems involving uncertain data arise in many areas of industrial and economic applications. Stochastic programming provides a useful framework for modeling and solving optimization problems for which a probability distribution of the unknown parameters is available.Motivated by practical optimization problems occurring in energy systems with regenerative energy supply, Debora Mahlke formulates and analyzes multistage stochastic mixed-integer models. For their solution, the author proposes a novel decomposition approach which relies on the concept of splitting the underlying scenario tree into subtrees. Based on the formulated models from energy production, the algorithm is computationally investigated and the numerical results are discussed.