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The paper addresses the unit commitment problem in power plant operation planning. For a real power system comprising coal and gas fired thermal as well as pumped storage hydro plants a large-scale mixed integer optimization model for unit commitment is developed. Then primal and dual approaches to solving the optimization problem are presented and results of test runs are reported.

{\begin{footnotesize} This thesis is concerned with structural properties and the stability behaviour of two-stage stochastic programs. Chapter~1 gives an introduction into stochastic programming and a summary of the main results of the thesis. In Chapter~2 we present easily verifiable sufficient conditions for the strong convexity of the expected-recourse function in a stochastic program with linear complete recourse. Different levels of randomness in the data are considered. We start with models where only the right-hand side of the constraints is random and extend these results to the situation where also the technology matrix contains random entries. The statements on strong convexity imply new stability estimates for sets of optimal solutions when perturbing the underlying probability measure. We work out Hölder estimates (in terms of the $\mbox{L}_1$-Wasserstein distance) for optimal solution sets to linear recourse models with random technology matrix. In Chapter~3 ({\it joint work with Werner Römisch, Berlin}) we are aiming at the Lipschitz stability of optimal solution sets to linear recourse models with random right-hand side. To this end , we first adapt the distance notion for the underlying probability measures to the structure of the model and derive a Lipschitz estimate for optimal solutions based on that distance. Here, the strong convexity established in Chapter~2 turns out as an essential assumption. For applications, however, a Lipschitz estimate with respect to a more accesssible probability distance is desirable. Structural properties of the expected-recourse function finally permit such an estimate in terms of the Kolmogorov-Smirnov distance of linear transforms of the underlying measures. The general analysis is specified to estimation via empirical measures. We obtain a law of iterated logarithm, a large deviation estimate and an estimate for the asymptotic distribution of optimal solution sets. Chapters~4 and~5 deal with two-stage linear stochastic programs where integrality constraints occur in the second stage. In Chapter~4 we study basic continuity properties of the expected-recourse function for models with random right-hand side and random technology matrix. The joint continuity with respect to the decision variable and the underlying probability measure leads to qualitative statements on the stability of local optimal values and local optimal solutions. In Chapter~5 we demonstrate that a variational distance of probability measures based on a suitable Vapnik-\v{C}ervonenkis class of Borel sets leads to convergence rates of the Hölder type for the expected recourse as a function of the underlying probability measure. The rates carry over to the convergence of local optimal values. As an application we again consider estimation via empirical measures. Beside qualitative asymptotic results for optimal values and optimal solutions we obtain a law of iterated logarithm for optimal values. \end{footnotesize}}

Integer stochastic linear programming is considered from the viewpoint of discontinuous optimization. After reviewing solution approaches via mollifier subgradients and decomposition we outline how to base a solution method on efficient pointwise calculation of the objective employing computer algebra.

Strong Convexity in Stochastic Programs with Complete Recourse II: Partially Random Right-Hand Side
(1995)

We establish a verifiable sufficient condition for strong convexity of the expected recourse as a function of the tender variable in a two-stage stochastic program with linear recourse. Generalizing a former result where all components of the second-stage right-hand side vector were random we treat the case where only a subvector of the right-hand side is random. As prerequisite, a refined analysis of the polyhedral complex of lineality regions of the second-stage value function is carried out. The sufficient condition for strong convexity allows to widen the class of recourse models for which certain quantitative results on stability and asymptotic convergence of optimal solutions are valid.

In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Gröbner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the integer expected recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest possible improvements to economize on the number of function evaluations needed.

Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.

A Two-Stage Stochastic Program for Unit Commitment Under Uncertainty in a Hydro-Thermal Power System
(1998)

We develop a two-stage stochastic programming model with integer first-stage and mixed-integer recourse for solving the unit commitment problem in power generation in the presence of uncertainty of load profiles. The solution methodology rests on a novel scenario decomposition method for stochastic integer programming. This method combines Lagrangian relaxation of non-anticipativity constraints with branch-and-bound. It can be seen as a decomposition algorithm for large-scale mixed-integer linear programs with block-angular structure. With realistic data from a German utility we validate our model and carry out test runs. Sizes of these problems go up to 20.000 integer and 150.000 continuous variables together with up to 180.000 constraints.

We present an algorithm for solving stochastic integer programming problems with recourse, based on a dual decomposition scheme and Lagrangian relaxation. The approach can be applied to multi-stage problems with mixed-integer variables in each time stage. %We outline a branch-and-bound algorithm for obtaining primal feasible and %possibly optimal solutions. Numerical experience is presented for some two-stage test problems.

Preprocessing in two-stage stochastic programming is considered from the viewpoint of Fourier-Motzkin elimination. Although of exponential complexity in general, Fourier-Motzkin elimination is shown to provide valuable insights into specific topics such as solving integer recourse stochastic programs or verifying stability conditions. Test runs with the computer code PORTA [1994] are reported.

Integrals of optimal values of random optimization problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure. Under fairly broad conditions, it is proved that uniform convergence of empirical approximations of the right hand sides of the constraints implies uniform convergence of the optimal values in the linear and convex case.