Are quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction

Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

Quasi-Monte Carlo algorithms are studied for designing discrete approximationsof two-stage linear stochastic programs. Their integrands are piecewiselinear, but neither smooth nor lie in the function spaces considered for QMC erroranalysis. We show that under some weak geometric condition on the two-stagemodel all terms of their ANOVA decomposition, except the one of highest order,are smooth. Hence, Quasi-Monte Carlo algorithms may achieve the optimal rateof convergence $O(n^{-1+\delta}$ with $\delta \in (0,\frac{1}{2}]$ and a constant not depending on the dimension. The geometric condition is shown to be generically satisfied if the underlyingdistribution is normal. We discuss sensitivity indices, effective dimensionsand dimension reduction techniques for two-stage integrands. Numerical experimentsshow that indeed convergence rates close to the optimal rate are achievedwhen using randomly scrambled Sobol' point sets and randomly shifted latticerules accompanied with suitable dimension reduction techniques

Efficient solution selection for two-stage stochastic programs

Sampling-based stochastic programs are extensively applied in practice. However, the resulting models tend to be computationally challenging. A reasonable number of samples needs to be identified to represent the random data, and a group of approximate models can then be constructed using such a number of samples. These approximate models can produce a set of potential solutions for the original model. In this paper, we consider the problem of allocating a finite computational budget among numerous potential solutions of a two-stage linear stochastic program, which aims to identify the best solution among potential ones by conducting simulation under a given computational budget. We propose a two-stage heuristic approach to solve the computational resource allocation problem. First, we utilise a Wasserstein-based screening rule to remove potentially inferior solutions from the simulation. Next, we use a ranking and selection technique to efficiently collect performance information of the remaining solutions. The performance of our approach is demonstrated through well-known benchmark problems. Results show that our method provides good trade-offs between computational effort and solution performance

Numerical Methods for PDE Constrained Optimization with Uncertain Data

Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization. The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods

Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The twostage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives and belong to L2. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence O(n-1+delta) with 2 (0; 1 2 ] and a constant not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol’ point sets accompanied with PCA for dimension reduction