Convex Approximations for Totally Unimodular Integer Recourse Models: A Uniform Error Bound

Romeijnders, Ward; Vlerk, Maarten H. van der; Haneveld, Willem K. Klein

doi:10.1137/130945703

Cited by 18 publications

(32 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Positive results for the class of TU integer recourse models, giving a uniform bound on the approximation error Q α − Q ∞ , are reported in Romeijnders et al [2]. …”

mentioning

confidence: 83%

Erratum to: Convex approximations for complete integer recourse models

Vlerk

2013

Math. Program.

View full text Add to dashboard Cite

This note reports corrections to the results obtained by van der Vlerk [3], in which two errors were discovered by W. Romeijnders.The first problem is with Theorem 1 of this reference, which claims that the convex approximation Q α yields the convex hull of the integer recourse function Q (in the space of tender variables) in case the recourse matrix is totally unimodular. As detailed in the report [1], this result only holds under very restrictive conditions on the distribution of the random right-hand side vector ω. Essentially, these conditions are only satisfied if the components of ω are independently and uniformly distributed as in Example 2 in van der Vlerk [3]. In general, the polyhedral function Q α may not be a lower bound for Q, and the convex hull of Q may be non-polyhedral.Theorem 2 therefore incorrectly assumes that Q α is a lower bound for the integer recourse function Q (in the space of the first-stage variables). The function Q lp , obtained by relaxing the second-stage integrality conditions, is a convex lower bound of Q. It can be shown that under the stated assumptionsQ α is a lower bound for Q, then it is strictly better than Q lp in this sense. Positive results for the class of TU integer recourse models, giving a uniform bound on the approximation error Q α − Q ∞ , are reported in Romeijnders et al. [2].The online version of the original article can be found under

show abstract

“…Positive results for the class of TU integer recourse models, giving a uniform bound on the approximation error Q α − Q ∞ , are reported in Romeijnders et al [2]. …”

mentioning

confidence: 83%

Erratum to: Convex approximations for complete integer recourse models

Vlerk

2013

Math. Program.

View full text Add to dashboard Cite

show abstract

“…Recently, substantial progress has been made in deriving error bounds for convex approximations of mixed-integer recourse models with multiple non-separable recourse constraints. For example, for TU integer recourse models, Romeijnders et al [39] derive an error bound for the α-approximations from [50]. This error bound depends on the total variations of the density functions of the random right-hand side variables in the model.…”

Section: Solution Methods For Risk-neutral Mixed-integer Recourse Modelsmentioning

confidence: 99%

Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

Beesten

Romeijnders

2019

Math. Program.

View full text Add to dashboard Cite

In traditional two-stage mixed-integer recourse models, the expected value of the total costs is minimized. In order to address risk-averse attitudes of decision makers, we consider a weighted mean-risk objective instead. Conditional value-at-risk is used as our risk measure. Integrality conditions on decision variables make the model non-convex and hence, hard to solve. To tackle this problem, we derive convex approximation models and corresponding error bounds, that depend on the total variations of the density functions of the random right-hand side variables in the model. We show that the error bounds converge to zero if these total variations go to zero. In addition, for the special cases of totally unimodular and simple integer recourse models we derive sharper error bounds. Keywords Stochastic programming • Mean-risk models • Conditional value-at-risk • Mixed-integer recourse • Convex approximations Mathematics Subject Classification 90C11 • 90C15 • 90C59 Maarten H. van der Vlerk was actively involved in an early stage of this research, but passed away during the writing of this paper.

show abstract

“…The assumption in Theorem 1 that the components of ω are independently distributed is not necessary. Indeed, in Section 5 we deal with a special type of perfect dependency in the right-hand side, and in Romeijnders et al (2015Romeijnders et al ( , 2016 bounds for the dependent case are derived involving conditional density functions instead of marginal ones.…”

Section: Solution Methods For Integer Recourse Modelsmentioning

confidence: 99%