Quasi-Monte Carlo methods for two-stage stochastic mixed-integer programs

We consider randomized QMC methods for approximating the expected recourse in two-stage stochastic optimization problems containing mixed-integer decisions in the second stage. It is known that the second-stage optimal value function is piecewise linear-quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets. This structure is exploited to provide conditions implying … Read more

Optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic optimization problems. Earlier approaches for optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based on distances containing minimal information, i.e., on data appearing … Read more

SDDP for multistage stochastic linear programs based on spectral risk measures

We consider risk-averse formulations of multistage stochastic linear programs. For these formulations, based on convex combinations of spectral risk measures, risk-averse dynamic programming equations can be written. As a result, the Stochastic Dual Dynamic Programming (SDDP) algorithm can be used to obtain approximations of the corresponding risk-averse recourse functions. This allows us to define a … Read more

Quantitative Stability Analysis of Stochastic Generalized Equations

We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem and a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with … Read more

Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures

We define a risk averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it. The approach is based on dynamic programming equations written for a risk averse formulation of the problem. This formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures. Dual representations of … Read more