Distances of probability measures and uniform distribution mod 1

Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming

“…Step Dealing with discrepancy distances, we meet different problems in solving the combinatorial problem (15). On the one hand, discrepancies do not allow, to the best of our understanding, a reformulation of (15) as a mixed-integer linear program that would be accessible by available solvers.…”

“…Since sometimes one may be interested in knowing the achievable minimal discrepancy, we suggest the following approach to solve (15) for moderate values of N and to reduce the time needed by complete enumeration. However, the complexity of (15) quickly increases with the dimension of the problem and real-world stochastic optimization problem are of higher dimension, in general.…”

“…The parameters used for the breadth-first search areL = min{N − 1, n + 4}, λ = 0.01, σ = 0.3 and ϑ = 50; the time needed by enumeration is significantly reduced, by up to 95% (R 2 , n = 11). With regard to the complexity of the inner problem (16), we adopted a heuristic for the outer problem (15), within that (16) has to be solved only once. More precisely, to approximate the initial measure of Example 2, we used a Quasi-Monte Carlo (QMC) approach, based on the first n points of the Halton sequence with bases 2 and 3, cf.…”

“…Consequently, this heuristic seems to be not appropriate for higher dimensional problems. We refer again to Example 2, where another heuristic is used to tackle the outer problem (15). Table 4 compares the results of a complete enumeration, the branch and bound Algorithm 4, and the Forward Selection Algorithm 2 in the case of the cell discrepancy.…”

“…Consequently, practitioners mostly have to abandon an optimal approximation and the following approach is rather of academic interest. For an approach that may be numerically more tractable for larger problems, we refer to Example 2, where we adopted a Quasi-Monte Carlo method to tackle the outer problem (15).…”