Epi-convergent discretizations of stochastic programs via integration quadratures

Epi-convergent discretizations of stochastic programs via integration quadratures

The simplest and the best-known method for numerical approximation of high-dimensional integrals is the Monte Carlo method (MC), i.e. random sampling. MC has also become the most popular method for constructing numerically solvable approximations of stochastic programs. However, certain modern integration quadratures are often superior to crude MC in high-dimensional integration, so it seems na...

Epi-convergent discretizations of multistage stochastic programs via integration quadratures

This paper presents procedures for constructing numerically solvable discretizations of multistage stochastic programs that epi-converge to the original problem as the discretizations are made finer. Epi-convergence implies, in particular, that the cluster points of the first-stage solutions of the discretized problems are optimal first-stage solutions of the original problem. The discretizatio...

Numerical study of discretizations of multistage stochastic programs

Stochastic Collocation Methods via ℓ1 Minimization Using Randomized Quadratures

In this work, we discuss the problem of approximating a multivariate function by polynomials via `1 minimization method, using a random chosen sub-grid of the corresponding tensor grid of Gaussian points. The independent variables of the function are assumed to be random variables, and thus, the framework provides a non-intrusive way to construct the generalized polynomial chaos expansions, ste...

Convergent Bounds for Stochastic Programs with Expected Value Constraints

This article elaborates a bounding approximation scheme for convex multistage stochastic programs (MSP) that constrain the conditional expectation of some decision-dependent random variables. Expected value constraints of this type are useful for modelling a decision maker’s risk preferences, but they may also arise as artefacts of stage-aggregation. It is shown that the gap between certain upp...