Evolutionary multi-stage financial scenario tree generation

Multi-stage financial decision optimization under uncertainty depends on a careful numerical approximation of the underlying stochastic process, which describes the future returns of the selected assets or asset categories. Various approaches towards an optimal generation of discrete-time, discrete-state approximations (represented as scenario trees) have been suggested in the literature. In this paper, a new evolutionary algorithm to create scenario trees for multi-stage financial optimization models will be presented. Numerical results and implementation details conclude the paper

Numerical study of discretizations of multistage stochastic programs

summary:This paper presents a numerical study of a deterministic discretization procedure for multistage stochastic programs where the underlying stochastic process has a continuous probability distribution. The discretization procedure is based on quasi-Monte Carlo techniques originally developed for numerical multivariate integration. The solutions of the discretized problems are evaluated by statistical bounds obtained from random sample average approximations and out-of-sample simulations. In the numerical tests, the optimal values of the discretizations as well as their first-stage solutions approach those of the original infinite-dimensional problem as the discretizations are made finer

Scenario Optimization for Multi-Stage Stochastic Programming Problems

The field of multi-stage stochastic programming provides a rich modelling framework to tackle a broad range of real-world decision problems. In order to numerically solve such programs - once they get reasonably large - the infinite-dimensional optimization problem has to be discretized. The stochastic optimization program generally consists of an optimization model and a stochastic model. In the multi-stage case the stochastic model is most commonly represented as a multi-variate stochastic process. The most common technique to calculate an useable discretization is to generate a scenario tree from the underlying stochastic process. In the first part of the talk we take a look at scenario optimization from the viewpoint of a decision taker, to provide rather non-technical insights into the problem. In the second part of the talk we examplify scenario tree generation by reviewing one specific algorithm based on multi-dimensional facility location applying backward stagewise clustering. An example from the area of financial engineering concludes the talk

Scenario trees and policy selection for multistage stochastic programming using machine learning

We propose a hybrid algorithmic strategy for complex stochastic optimization problems, which combines the use of scenario trees from multistage stochastic programming with machine learning techniques for learning a policy in the form of a statistical model, in the context of constrained vector-valued decisions. Such a policy allows one to run out-of-sample simulations over a large number of independent scenarios, and obtain a signal on the quality of the approximation scheme used to solve the multistage stochastic program. We propose to apply this fast simulation technique to choose the best tree from a set of scenario trees. A solution scheme is introduced, where several scenario trees with random branching structure are solved in parallel, and where the tree from which the best policy for the true problem could be learned is ultimately retained. Numerical tests show that excellent trade-offs can be achieved between run times and solution quality

On Stability of Multistage Stochastic Programs

We study quantitative stability of linear multistage stochastic programs underperturbations of the underlying stochastic processes. It is shown that the optimalvalues behave Lipschitz continuous with respect to an $L_p$-distance. Therefore, wehave to make a crucial regularity assumption on the conditional distributions, thatallows to establish continuity of the recourse function with respect to the currentstate of the stochastic process. The main stability result holds for nonanticipativediscretizations of the underlying process and thus represents a rigorous justificationof established discretization techniques

Epi-Convergent Discretization of the Generalizaed Bolza Problem in Dynamic Optimization

The paper is devoted to well-posed discrete approximations of the so-called generalized Bolza problem of minimizing variational functionals defined via extended-real-valued functions. This problem covers more conventional Bolza-type problems in the calculus of variations and optimal control of differential inclusions as well of parameterized differential equations. Our main goal is find efficient conditions ensuring an appropriate epi-convergence of discrete approximations, which plays a significant role in both the qualitative theory and numerical algorithms of optimization and optimal control. The paper seems to be the first attempt to study epi-convergent discretizations of the generalized Bolza problem; it establishes several rather general results in this direction

Multistage Stochastic Portfolio Optimisation in Deregulated Electricity Markets Using Linear Decision Rules

The deregulation of electricity markets increases the financial risk faced by retailers who procure electric energy on the spot market to meet their customers’ electricity demand. To hedge against this exposure, retailers often hold a portfolio of electricity derivative contracts. In this paper, we propose a multistage stochastic mean-variance optimisation model for the management of such a portfolio. To reduce computational complexity, we perform two approximations: stage-aggregation and linear decision rules (LDR). The LDR approach consists of restricting the set of decision rules to those affine in the history of the random parameters. When applied to mean-variance optimisation models, it leads to convex quadratic programs. Since their size grows typically only polynomially with the number of periods, they can be efficiently solved. Our numerical experiments illustrate the value of adaptivity inherent in the LDR method and its potential for enabling scalability to problems with many periods.OR in energy, electricity portfolio management, stochastic programming, risk management, linear decision rules

Dynamic Procurement of New Products with Covariate Information: The Residual Tree Method

Problem definition: We study the practice-motivated problem of dynamically procuring a new, short life-cycle product under demand uncertainty. The firm does not know the demand for the new product but has data on similar products sold in the past, including demand histories and covariate information such as product characteristics. Academic/practical relevance: The dynamic procurement problem has long attracted academic and practitioner interest, and we solve it in an innovative data-driven way with proven theoretical guarantees. This work is also the first to leverage the power of covariate data in solving this problem. Methodology:We propose a new, combined forecasting and optimization algorithm called the Residual Tree method, and analyze its performance via epi-convergence theory and computations. Our method generalizes the classical Scenario Tree method by using covariates to link historical data on similar products to construct demand forecasts for the new product. Results: We prove, under fairly mild conditions, that the Residual Tree method is asymptotically optimal as the size of the data set grows. We also numerically validate the method for problem instances derived using data from the global fashion retailer Zara. We find that ignoring covariate information leads to systematic bias in the optimal solution, translating to a 6–15% increase in the total cost for the problem instances under study. We also find that solutions based on trees using just 2–3 branches per node, which is common in the existing literature, are inadequate, resulting in 30–66% higher total costs compared with our best solution. Managerial implications: The Residual Tree is a new and generalizable approach that uses past data on similar products to manage new product inventories. We also quantify the value of covariate information and of granular demand modeling