Werner Römisch

Scenario reduction in stochastic programming: An approach using probability metrics

Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics. Efficient algorithms are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty. For instance, it turns out that after 50% reduction of the scenario tree the optimal reduced tree still has about 90% relative accuracy.

Scenario reduction in stochastic programming

Abstract. Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics. Efficient algorithms are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty. For instance, it turns out that after 50% reduction of the scenario tree the optimal reduced tree still has about 90% relative accuracy.

Scenario Reduction Algorithms in Stochastic Programming

We consider convex stochastic programs with an (approximate) initial probability distribution P having finite support supp P, i.e., finitely many scenarios. The behaviour of such stochastic programs is stable with respect to perturbations of P measured in terms of a Fortet-Mourier probability metric. The problem of optimal scenario reduction consists in determining a probability measure that is supported by a subset of supp P of prescribed cardinality and is closest to P in terms of such a probability metric. Two new versions of forward and backward type algorithms are presented for computing such optimally reduced probability measures approximately. Compared to earlier versions, the computational performance (accuracy, running time) of the new algorithms has been improved considerably. Numerical experience is reported for different instances of scenario trees with computable optimal lower bounds. The test examples also include a ternary scenario tree representing the weekly electrical load process in a power management model.

Scenario reduction and scenario tree construction for power management problems

Portfolio and risk management problems of power utilities may be modeled by multistage stochastic programs. These models use a set of scenarios and corresponding probabilities to model the multivariate random data process (electrical load, stream flows to hydro units, and fuel and electricity prices). For most practical problems the optimization problem that contains all possible scenarios is too large. Due to computational complexity and to time limitations this program is often approximated by a model involving a (much) smaller number of scenarios. The proposed reduction algorithms determine a subset of the initial scenario set and assign new probabilities to the preserved scenarios. The scenario tree construction algorithms successively reduce the number of nodes of a fan of individual scenarios by modifying the tree structure and by bundling similar scenarios. Numerical experience is reported for constructing scenario trees for the load and spot market prices entering a stochastic portfolio management model of a German utility.

Stochastic Lagrangian Relaxation Applied to Power Scheduling in a Hydro-Thermal System under Uncertainty

A dynamic (multi-stage) stochastic programming model for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain demand (or load) is developed. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. A stochastic Lagrangian relaxation scheme is designed by assigning (stochastic) multipliers to all constraints coupling power units. It is assumed that the stochastic load process is given (or approximated) by a finite number of realizations (scenarios) in scenario tree form. Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single (thermal or hydro) unit subproblems. The stochastic thermal and hydro subproblems are solved by a stochastic dynamic programming technique and by a specific descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the first stage (primal) decisions starting from the optimal (stochastic) multipliers is developed. Numerical results are presented for realistic data from a German power utility and for numbers of scenarios ranging from 5 to 100 and a time horizon of 168 hours. The sizes of the corresponding optimization problems go up to 200 000 binary and 350 000 continuous variables, and more than 500 000 constraints.

Scenario tree modeling for multistage stochastic programs

An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, we develop (stability) theory-based heuristics for generating scenario trees out of an initial set of scenarios. They are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps. Conditions are established implying closeness of optimal values of the original process and its tree approximation, respectively, by relying on a recent stability result in Heitsch, Römisch and Strugarek (SIAM J Optim 17:511–525, 2006) for multistage stochastic programs. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.

Stability of Stochastic Programming Problems

Abstract The behaviour of stochastic programming problems is studied in case of the underlying probability distribution being perturbed and approximated, respectively. Most of the theoretical results provide continuity properties of optimal values and solution sets relative to changes of the original probability distribution, varying in some space of probability measures equipped with some convergence and metric, respectively. We start by discussing relevant notions of convergence and distances for probability measures. Then we associate a distance with a stochastic program in a natural way and derive (quantitative) continuity properties of values and solutions by appealing to general perturbation results for optimization problems. Later we show how these results relate to stability with respect to weak convergence and how certain ideal probability metrics may be associated with more specific stochastic programs. In particular, we establish stability results for two-stage and chance constrained models. Finally, we present some consequences for the asymptotics of empirical approximations and for the construction of scenario-based approximations of stochastic programs.

Polyhedral Risk Measures in Stochastic Programming

We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties. Polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right-hand side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we propose multiperiod extensions of the Conditional-Value-at-Risk.

Scenario tree reduction for multistage stochastic programs

A framework for the reduction of scenario trees as inputs of (linear) multistage stochastic programs is provided such that optimal values and approximate solution sets remain close to each other. The argument is based on upper bounds of the Lr-distance and the filtration distance, and on quantitative stability results for multistage stochastic programs. The important difference from scenario reduction in two-stage models consists in incorporating the filtration distance. An algorithm is presented for selecting and removing nodes of a scenario tree such that a prescribed error tolerance is met. Some numerical experience is reported.

Quantitative Stability in Stochastic Programming: The Method of Probability Metrics

Quantitative stability of optimal values and solution sets to stochastic programming problems is studied when the underlying probability distribution varies in some metric space of probability measures. We give conditions that imply that a stochastic program behaves stable with respect to a minimal information (m.i.) probability metric that is naturally associated with the data of the program. Canonical metrics bounding the m.i. metric are derived for specific models, namely for linear two-stage, mixed-integer two-stage and chance-constrained models. The corresponding quantitative stability results as well as some consequences for asymptotic properties of empirical approximations extend earlier results in this direction. In particular, rates of convergence in probability are derived under metric entropy conditions. Finally, we study stability properties of stable investment portfolios having minimal risk with respect to the spectral measure and stability index of the underlying stable probability distribution.