In this dissertation, we study several stochastic capacity expansion models in the presence of permanent, spot market, and contract capacity for acquisition. Using a scenario tree approach to handle the data uncertainty of the problems, we develop multi-stage stochastic integer programming formulations for these models. First, we study multi-period single resource stochastic capacity expansion problems, where different sources of capacity are available to the decision maker. We develop efficient algorithms that can solve these models to optimality in polynomial time. Second, we study multi-period stochastic network capacity expansion problems with different sources for capacity. The proposed models are NP-hard multi-stage stochastic integer programs and we develop an efficient, asymptotically convergent approximation algorithm to solve them. Third, we consider some decomposition algorithms to solve the proposed multi-stage stochastic network capacity expansion problem. We propose an enhanced Benders' decomposition algorithm to solve the problem, and a Benders' decomposition-based heuristic algorithm to find tight bounds for it. Finally, we extend the stochastic network capacity expansion model by imposing budget restriction on permanent capacity acquisition cost. We design a Lagrangian relaxation algorithm to solve the model, including heuristic methods to find tight upper bounds for it. / Thesis / Doctor of Philosophy (PhD)

The main objective of this thesis is to build a multi-stage stochastic pro- gram within an asset-liability management problem of a leasing company. At the beginning, the business model of such a company is introduced and the stochastic programming formulation is derived. Thereafter, three various risk constraints, namely the chance constraint, the Value-at-Risk constraint and the conditional Value-at-Risk constraint along with the second-order stochastic dominance constraint are applied to the model to control for riski- ness of the optimal strategy. Their properties and their effects on the optimal decisions are thoroughly investigated, while various risk limits are considered. In order to obtain solutions of the problems, random elements in the model formulation had to be approximated by scenarios. The Hull - White model calibrated by a newly proposed method based on maximum likelihood esti- mation has been used to generate scenarios of future interest rates. In the end, the performances of the optimal solutions of the problems for unconsid- ered and unfavourable crisis scenarios were inspected. The used methodology of such a stress test has not yet been implemented in stochastic programming problems within an asset-liability management. 1

Multi-Stage Stochastic Programming with CVaR: Modeling, Algorithms and Robustness RNDr. Václav Kozmík Abstract: We formulate a multi-stage stochastic linear program with three different risk measures based on CVaR and discuss their properties, such as time consistency. The stochastic dual dynamic programming algorithm is described and its draw- backs in the risk-averse setting are demonstrated. We present a new approach to evaluating policies in multi-stage risk-averse programs, which aims to elimi- nate the biggest drawback - lack of a reasonable upper bound estimator. Our approach is based on an importance sampling scheme, which is thoroughly ana- lyzed. A general variance reduction scheme for mean-risk sampling with CVaR is provided. In order to evaluate robustness of the presented models we extend con- tamination technique to the case of large-scale programs, where a precise solution cannot be obtained. Our computational results are based on a simple multi-stage asset allocation model and confirm usefulness of the presented procedures, as well as give additional insights into the behavior of more complex models. Keywords: Multi-stage stochastic programming, stochastic dual dynamic programming, im- portance sampling, contamination, CVaR