Multi-stage scenario generation by the combined moment matching and scenario reduction method

We describe an opportunity to speed up multi-stage scenario generation and reduction using a combination of two well-known methods: the moment matching method (Høyland and Wallace, 2001) and the method for scenario reduction to approximately minimize a metric (Heitsch and Römish, 2009). Our suggestion is to combine them rather than using them in serial by making use of a stage-wise approximation to the moment matching algorithm. Computational results show that combining the methods can bring significant benefits.     

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.     

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.     

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.     

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.     

A stochastic programming approach for multi-period portfolio optimization

This paper extends previous work on the use of stochastic linear programming to solve life-cycle investment problems. We combine the feature of asset return predictability with practically relevant constraints arising in a life-cycle investment context. The objective is to maximize the expected utility of consumption over the lifetime and of bequest at the time of death of the investor. Asset returns and state variables follow a first-order vector auto-regression and the associated uncertainty is described by discrete scenario trees. To deal with the long time intervals involved in life-cycle problems we consider a few short-term decisions (to exploit any short-term return predictability), and incorporate a closed-form solution for the long, subsequent steady-state period to account for end effects.     

Scenario optimization asset and liability modelling for individual investors

We develop a scenario optimization model for asset and liability management of individual investors. The individual has a given level of initial wealth and a target goal to be reached within some time horizon. The individual must determine an asset allocation strategy so that the portfolio growth rate will be sufficient to reach the target. A scenario optimization model is formulated which maximizes the upside potential of the portfolio, with limits on the downside risk. Both upside and downside are measured vis-à-vis the goal. The stochastic behavior of asset returns is captured through bootstrap simulation, and the simulation is embedded in the model to determine the optimal portfolio. Post-optimality analysis using out-of-sample scenarios measures the probability of success of a given portfolio. It also allows us to estimate the required increase in the initial endowment so that the probability of success is improved.     

No-arbitrage conditions,scenario trees,and multi-asset financial optimization

Many numerical optimization methods use scenario trees as a discrete approximation for the true (multi-dimensional) probability distributions of the problem’s random variables. Realistic specifications in financial optimization models can lead to tree sizes that quickly become computationally intractable. In this paper we focus on the two main approaches proposed in the literature to deal with this problem: scenario reduction and state aggregation. We first state necessary conditions for the node structure of a tree to rule out arbitrage. However, currently available scenario reduction algorithms do not take these conditions explicitly into account. State aggregation excludes arbitrage opportunities by relying on the risk-neutral measure. This is, however, only appropriate for pricing purposes but not for optimization. Both limitations are illustrated by numerical examples. We conclude that neither of these methods is suitable to solve financial optimization models in asset–liability or portfolio management.     

Investment models based on clustered scenario trees

Stochastic programming is widely applied in financial decision problems. In particular, when we need to carry out the actual calculations for portfolio selection problems, we have to assign a value for each expected return and the associated conditional probability in advance. These estimated random parameters often rely on a scenario tree representing the distribution of the underlying asset returns. One of the drawbacks is that the estimated parameters may be deviated from the actual ones. Therefore, robustness is considered so as to cope with the issue of parameter inaccuracy. In view of this, we propose a clustered scenario-tree approach, which accommodates the parameter inaccuracy problem in the context of a scenario tree.     

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.     

A general framework for multistage mean-variance post-tax optimization

An investor’s decisions affect the way taxes are paid in a general portfolio investment, modifying the net redemption value and the yearly optimal portfolio distribution. We investigate the role of these decisions on multistage mean-variance portfolio allocation model. A number of risky assets grouped in wrappers with special taxation rules is integrated in a multistage financial portfolio optimization problem. The uncertainty on the returns of assets is specified as a scenario tree generated by simulation/clustering based approach. We show the impact of decisions in the yearly reallocation of the investments for three typical cases with an annual fixed withdrawal in a fixed horizon that utilizes completely the option of taper relief offered by banks in UK. Our computational framework can be used as a tool for testing decisions in this context.     

Solving stochastic complementarity problems in energy market modeling using scenario reduction

In this paper, we analyze market equilibrium models with random aspects that lead to stochastic complementarity problems. While the models presented depict energy markets, the results are believed to be applicable to more general stochastic complementarity problems. The contribution is the development of new heuristic, scenario reduction approaches that iteratively work towards solving the full, extensive form, stochastic market model. The methods are tested on three representative models and supporting numerical results are provided as well as derived mathematical bounds.     

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.     

Optimizing fuzzy portfolio selection problems by parametric quadratic programming

This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are obtained by equivalent value (EV) reduction methods. For common type-2 triangular and trapezoidal fuzzy variables, their reduced fuzzy variables are studied in the current development. The parametric possibility distributions of reduced fuzzy variables are first derived, then the second moment formulas for the reduced fuzzy variables are established. Taking the second moment as a new risk measure, the reward-risk and risk-reward models are developed to optimize fuzzy portfolio selection problems. The mathematical properties of the proposed optimization models are analyzed, including the analytical representations for the second moments of linear combinations of reduced fuzzy variables as well as the convexity of second moments with respect to decision vectors. On the basis of the analytical representations for the second moments, the reward-risk and risk-reward models can be turned into their equivalent parametric quadratic convex programming problems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling ideas and the efficiency of solution method.     

A scenario decomposition algorithm for 0–1 stochastic programs

We propose a scenario decomposition algorithm for stochastic 0–1 programs. The algorithm recovers an optimal solution by iteratively exploring and cutting-off candidate solutions obtained from solving scenario subproblems. The scheme is applicable to quite general problem structures and can be implemented in a distributed framework. Illustrative computational results on standard two-stage stochastic integer programming and nonlinear stochastic integer programming test problems are presented.     

A note on scenario reduction for two-stage stochastic programs

We extend earlier work on scenario reduction by relying directly on Fortet-Mourier metrics instead of using upper bounds given in terms of mass transportation problems. The importance of Fortet-Mourier metrics for quantitative stability of two-stage models is reviewed and some numerical results are also provided.     

Moment matching method for numerical solution of MTL equations in interconnect analysis

Multiconductor transmission line (MTL) analysis is a popular technique for evaluating high-speed electrical interconnects. Typically, MTLs are modeled in the Laplace domain and similarity transformations are used to decouple the MTL equations. For high-speed systems, however, direct solution of the MTL equations at a large number of frequencies is computationally very expensive. Recent studies have employed moment matching techniques to approximate the solution for the MTL equations and improve the computational efficiency. In this study, a generalization of the method of characteristics is further studied for solving the MTL equations for lossy transmission lines. An efficient recursive solution for generating the moments of eigenvalues and eigenvectors is presented. Numerical results of this moment matching technique agree with the direct solution methods up to 10GHz.     

Behaviour of queueing approximations based on sample moments

We investigate moment–based queueing approximations in the presence of sampling error. Let L be the steady–state mean number in the system for a GI/M/1 queue. We focus on the estimation of L under the assumption that only sample moments of the interarrival–time distribution are known. A simulation experiment is carried out for several interarrival–time distributions. For each case, sample moments from the interarrival–time distribution are matched to an approximating phase–type distribution and the corresponding estimate L is obtained. We show that the sampling error in the moments induces bias as well as variability in L. Based on our simulation experiment, we suggest matching only two moments when the sample coefficient of variation is low or when sample size is low; otherwise, matching three moments is preferable.     

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.     

The sample average approximation method for empty container repositioning with uncertainties

One of the challenges faced by liner operators today is to effectively operate empty containers in order to meet demand and to reduce inefficiency in an uncertain environment. To incorporate uncertainties in the operations model, we formulate a two-stage stochastic programming model with random demand, supply, ship weight capacity, and ship space capacity. The objective of this model is to minimize the expected operational cost for Empty Container Repositioning (ECR). To solve the stochastic programs with a prohibitively large number of scenarios, the Sample Average Approximation (SAA) method is applied to approximate the expected cost function. To solve the SAA problem, we consider applying the scenario aggregation by combining the approximate solution of the individual scenario problem. Two heuristic algorithms based on the progressive hedging strategy are applied to solve the SAA problem. Numerical experiments are provided to show the good performance of the scenario-based method for the ECR problem with uncertainties.