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 L _r -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.     

Adaptive discretization of convex multistage stochastic programs

We propose a new scenario tree reduction algorithm for multistage stochastic programs, which integrates the reduction of a scenario tree into the solution process of the stochastic program. This allows to construct a scenario tree that is highly adapted on the optimization problem. The algorithm starts with a rough approximation of the original tree and locally refines this approximation as long as necessary. Promising numerical results for scenario tree reductions in the settings of portfolio management and power management with uncertain load are presented.     

Generating scenario trees: A parallel integrated simulation-optimization approach

A crucial issue for addressing decision-making problems under uncertainty is the approximate representation of multivariate stochastic processes in the form of scenario tree. This paper proposes a scenario generation approach based on the idea of integrating simulation and optimization techniques. In particular, simulation is used to generate outcomes associated with the nodes of the scenario tree which, in turn, provide the input parameters for an optimization model aimed at determining the scenarios’ probabilities matching some prescribed targets. The approach relies on the moment-matching technique originally proposed in [K. Høyland, S.W. Wallace, Generating scenario trees for multistage decision problems, Manag. Sci. 47 (2001) 295-307] and further refined in [K. Høyland, M. Kaut, S.W. Wallace, A heuristic for moment-matching scenario generation, Comput. Optim. Appl. 24 (2003) 169-185]. By taking advantage of the iterative nature of our approach, a parallel implementation has been designed and extensively tested on financial data. Numerical results show the efficiency of the parallel algorithm and the improvement in accuracy and effectiveness.     

On distributionally robust multiperiod stochastic optimization

This paper considers model uncertainty for multistage stochastic programs. The data and information structure of the baseline model is a tree, on which the decision problem is defined. We consider “ambiguity neighborhoods” around this tree as alternative models which are close to the baseline model. Closeness is defined in terms of a distance for probability trees, called the nested distance. This distance is appropriate for scenario models of multistage stochastic optimization problems as was demonstrated in Pflug and Pichler (SIAM J Optim 22:1–23, 2012). The ambiguity model is formulated as a minimax problem, where the the optimal decision is to be found, which minimizes the maximal objective function within the ambiguity set. We give a setup for studying saddle point properties of the minimax problem. Moreover, we present solution algorithms for finding the minimax decisions at least asymptotically. As an example, we consider a multiperiod stochastic production/inventory control problem with weekly ordering. The stochastic scenario process is given by the random demands for two products. We determine the minimax solution and identify the worst trees within the ambiguity set. It turns out that the probability weights of the worst case trees are concentrated on few very bad scenarios.     

Airline network revenue management by multistage stochastic programming

A multistage stochastic programming approach to airline network revenue management is presented. The objective is to determine seat protection levels for all itineraries, fare classes, points of sale of the airline network and all dcps of the booking horizon such that the expected revenue is maximized. While the passenger demand and cancelation rate processes are the stochastic inputs of the model, the stochastic protection level process represents its output and allows to control the booking process. The stochastic passenger demand and cancelation rate processes are approximated by a finite number of tree structured scenarios. The scenario tree is generated from historical data using a stability-based recursive scenario reduction scheme. Numerical results for a small hub-and-spoke network are reported. This research is supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

Scenario reduction in stochastic programming with respect to discrepancy distances

Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario sets. The relevant distances of (multivariate) probability distributions for deriving quantitative stability results for such stochastic programs are ℬ-discrepancies, where the class ℬ of Borel sets depends on their structural properties. Hence, the optimal scenario reduction problem for such models is stated with respect to ℬ-discrepancies. In this paper, upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or Kolmogorov metric) in some detail and provide some numerical experience.     

Scenario tree generation approaches using K-means and LP moment matching methods

We consider in this paper the efficient ways to generate multi-stage scenario trees. A general modified K-means clustering method is first presented to generate the scenario tree with a general structure. This method takes the time dependency of the simulated path into account. Based on the traditional and modified K-means analyses, the moment matching of multi-stage scenario trees is described as a linear programming (LP) problem. By simultaneously utilizing simulation, clustering, non-linear time series and moment matching skills, a sequential generation method and another new hybrid approach which can generate the whole multi-stage tree right off are proposed. The advantages of these new methods are: the vector autoregressive and multivariate generalized autoregressive conditional heteroscedasticity (VAR-MGARCH) model is adopted to properly reflect the inter-stage dependency and the time-varying volatilities of the data process, the LP-based moment matching technique ensures that the scenario tree generation problem can be solved more efficiently and the tree scale can be further controlled, and in the meanwhile, the statistical properties of the random data process are maintained properly. What is more important, our new LP methods can guarantee at least two branches are derived from each non-leaf node and thus overcome the drawback in relevant papers. We carry out a series of numerical experiments and apply the scenario tree generation methods to a portfolio management problem, which demonstrate the practicality, efficiency and advantages of our new approaches over other models or methods.     

Solving Sequences of Refined Multistage Stochastic Linear Programs

Multistage stochastic programs with continuous underlying distributions involve the obstacle of high-dimensional integrals where the integrands' values again are given by solutions of stochastic programs. A common solution technique consists of discretizing the support of the original distributions leading to scenario trees and corresponding LPs which are – up to a certain size – easy to solve. In order to improve the accuracy of approximation, successive refinements of the support result in rapidly expanding scenario trees and associated LPs. Hence, the solvability of the multistage stochastic program is limited by the numerical solvability of sequences of such expanding LPs. This work describes an algorithmic technique for solving the large-scale LP of refinement ν based on the solutions at the previous ν?1 refinements. Numerical results are presented for practical problem statements within financial applications demonstrating significant speedup (depending on the size of the LP instances).     

Barycentric scenario trees in convex multistage stochastic programming

This work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support. Based on earlier results, sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem. Error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation. Convergence of the approximate solutions is proven under the stated assumptions. Preliminary computational results are outlined. This work has been supported by Schweizerischen Nationalfonds Grant Nr. 21-39 575.93.     

A mixed integer programming model for multistage mean–variance post-tax optimization

In this paper, we introduce a mixed integer stochastic programming approach to mean–variance post-tax portfolio management. This approach takes into account of risk in a multistage setting and allows general withdrawals from original capital. The uncertainty on asset returns is specified as a scenario tree. The risk across scenarios is addressed using the probabilistic approach of classical stochastic programming. The tax rules are used with stochastic linear and mixed integer quadratic programming models to compute an overall tax and return-risk efficient multistage portfolio. The incorporation of the risk term in the model provides robustness and leads to diversification over wrappers and assets within each wrapper. General withdrawals and risk aversion have an impact on the distribution of assets among wrappers. Computational results are presented using a study with different scenario trees in order to show the performance of these models.     

Scenario formulation in an algebraic modelling language

Algebraic modelling languages have simplified management of many types of large linear programs but have not specifically supported stochastic modelling. This paper considers modelling language support for multistage stochastic linear recourse problems with finite distributions. We describe basic language requirements for formulation of finite event trees in algebraic modelling languages and show representative problems in AMPL using three commonly used scenario types.     

Applications of stochastic programming: Achievements and questions

When solving a decision problem under uncertainty via stochastic programming it is essential to choose or to build a suitable stochastic programming model taking into account the nature of the real-life problem, character of input data, availability of software and computer technology. Besides a brief review of history and achievements of stochastic programming, selected modeling issues concerning applications of multistage stochastic programs with recourse (the choice of the horizon, stages, methods for generating scenario trees, etc.) will be discussed.     

Testing the structure of multistage stochastic programs

A fixed topology of stages and/or a fixed branching scheme are common assumptions for applications and numerical solution of scenario based multistage stochastic programs. Using contamination technique to test this structure, we extend the results of Dupačová (Contamination for multistage stochastic programs. In: Hušková M, Janžura M (eds) Prague stochastics. Matfyzpress, Praha, pp 91–101, 2006a) to stochastic programs with multistage polyhedral risk objectives. The ideas are exemplified by bond portfolio management problems and complemented by illustrative numerical results.     

Scenarios for Multistage Stochastic Programs

A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative scenarios (or a representative scenario tree) relative to the problem being modeled.     

Monotonic bounds in multistage mixed-integer stochastic programming

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.     

Scenario tree generation and multi-asset financial optimization problems

We compare two popular scenario tree generation methods in the context of financial optimization: moment matching and scenario reduction. Using a simple problem with a known analytic solution, moment matching–when ensuring absence of arbitrage–replicates this solution precisely. On the other hand, even if the scenario trees generated by scenario reduction are arbitrage-free, the solutions are biased and highly variable. These results hold for correlated and uncorrelated asset returns, as well as for normal and non-normal returns.     

On capacity expansion planning under strategic and operational uncertainties based on stochastic dominance risk averse management

A new scheme for dealing with uncertainty in scenario trees is presented for dynamic mixed 0–1 optimization problems with strategic and operational stochastic parameters. Let us generically name this type of problems as capacity expansion planning (CEP) in a given system, e.g., supply chain, production, rapid transit network, energy generation and transmission network, etc. The strategic scenario tree is usually a multistage one, and the replicas of the strategic nodes root structures in the form of either a special scenario graph or a two-stage scenario tree, depending on the type of operational activity in the system. Those operational scenario structures impact in the constraints of the model and, thus, in the decomposition methodology for solving usually large-scale problems. This work presents the modeling framework for some of the risk neutral and risk averse measures to consider for CEP problem solving. Two types of risk averse measures are considered. The first one is a time-inconsistent mixture of the chance-constrained and second-order stochastic dominance (SSD) functionals of the value of a given set of functions up to the strategic nodes in selected stages along the time horizon, The second type is a strategic node-based time-consistent SSD functional for the set of operational scenarios in the strategic nodes at selected stages. A specialization of the nested stochastic decomposition methodology for that problem solving is outlined. Its advantages and drawbacks as well as the framework for some schemes to, at least, partially avoid those drawbacks are also presented.     

Practical arbitrage‐free scenario tree reduction methods and their applications in financial optimization

We construct an arbitrage‐free scenario tree reduction model, from which some arbitrage‐free scenario tree reduction algorithms are designed. They ensure that the reduced scenario trees are arbitrage free. Numerical results show the practicality and efficiency of the proposed algorithms. Results for multistage portfolio selection problems demonstrate the necessity and importance for guaranteeing that the reduced scenario trees are arbitrage free, as well as the practicality of the proposed arbitrage‐free scenario tree reduction algorithms for financial optimization.     

Stability Properties of a Bond Portfolio Management Problem

The bond portfolio management problem is formulated as a multiperiod two-stage or multistage stochastic program based on interest rate scenarios. These scenarios depend on the available market data, on the applied estimation and sampling techniques, etc., and are used to evaluate coefficients of the resulting large scale mathematical program. The aim of the contribution is to analyze stability and sensitivity of this program on small changes of the coefficients – the (scenario dependent) values of future interest rates and prices. We shall prove that under sensible assumptions, the scenario subproblems are stable linear programs and that also the optimal first-stage decisions and the optimal value of the considered stochastic program possess acceptable continuity properties.     

BFC-MSMIP: an exact branch-and-fix coordination approach for solving multistage stochastic mixed 0–1 problems

We present an exact algorithmic framework, so-called BFC-MSMIP, for optimizing multistage stochastic mixed 0–1 problems with complete recourse. The uncertainty is represented by using a scenario tree and lies anywhere in the model. The problem is modeled by a splitting variable representation of the Deterministic Equivalent Model of the stochastic problem, where the 0–1 variables and the continuous variables appear at any stage. The approach uses the Twin Node Family concept within the algorithmic framework, so-called Branch-and-Fix Coordination, for satisfying the nonanticipativity constraints in the 0–1 variables. Some blocks of additional strategies are used in order to show the performance of the proposed approach. The blocks are related to the scenario clustering, the starting branching and the branching order strategies, among others. Some computational experience is reported. It shows that the new approach obtains the optimal solution in all instances under consideration.