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 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

Guaranteed Bounds for General Nondiscrete Multistage Risk-Averse Stochastic Optimization Programs

In general, multistage stochastic optimization problems are formulated on the basis of continuous distributions describing the uncertainty. Such “infinite” problems are practically impossible to solve as they are formulated, and finite tree approximations of the underlying stochastic processes are used as proxies. In this paper, we demonstrate how one can find guaranteed bounds, i.e., finite tree models, for which the optimal values give upper and lower bounds for the optimal value of the original infinite problem. Typically, there is a gap between the two bounds. However, this gap can be made arbitrarily small by making the approximating trees bushier. We consider approximations in the first-order stochastic sense, in the convex-order sense, and based on subgradient approximations. Their use is shown in a multistage risk-averse production problem

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

Successive discretization procedures for stochastic programming with recourse

Includes bibliographical references (leaves [30]-[31]).by Randall Hiller

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

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