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Landelijk Netwerk Mathematische Besliskunde

# Course SP: "Stochastic Programming"

 Time: Monday 15.15 – 17.00 (March 14 - April 11, April 25 - May 23). Location: Hans Freudenthalgebouw, Room 611AB, Budapestlaan, Utrecht (De Uithof). Lecturer: Dr. W. Romeijnders (University of Groningen).

Course description:
Stochastic programming (see also http://stoprog.org) is a framework for modelling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters. When the parameters are known only within certain bounds, one approach to tackling such problems is called robust optimization. Here the goal is to find a solution which is feasible for all such data and optimal in some sense. Stochastic programming models are similar in style but take advantage of the fact that probability distributions governing the data are known or can be estimated. The goal here is to find some policy that is feasible for all (or almost all) the possible data instances and maximizes the expectation of some function of the decisions and the random variables. More generally, such models are formulated, solved analytically or numerically, and analyzed in order to provide useful information to a decision-maker.

The most widely applied and studied stochastic programming models are two-stage linear programs. Here the decision maker takes some action in the first stage, after which a random event occurs affecting the outcome of the first-stage decision. A recourse decision can then be made in the second stage that compensates for any bad effects that might have been experienced as a result of the first-stage decision. The optimal policy from such a model is a single first-stage policy and a collection of recourse decisions (a decision rule) defining which second-stage action should be taken in response to each random outcome.

The following subjects are discussed:
- Concepts and examples of stochastic programming.
- Stochastic linear programming.
- Recourse models.
- Chance constraints.
- SP calculus (e.g. convexity; approximation of distributions).
- Algorithms.
- Stochastic integer programming.
- Case study.

Prerequisites:
- Basic knowledge of probability theory: S.M. Ross, Introduction to probability models, 8th edition, Academic Press, 2003 (chapters 1-3).
- Basic knowledge of linear programming: V. Chvatal, Linear programming, Freeman, 1983.

Literature:
- W.K. Klein Haneveld, M.H. van der Vlerk, and W. Romeijnders, Stochastic Programming - Modeling Decision Problems Under Uncertainty, Graduate Texts in Operations Research, Springer, 2020. Link to book

Examination:
Take home problems, case study.