An algorithm for the construction of convex hulls in simple integer recourse programming

Haneveld, Willem K. Klein; Stougie, Leen; Vlerk, van der Mh

doi:10.1007/bf02187641

Cited by 46 publications

(23 citation statements)

References 4 publications

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“…For α-approximations of expected surplus function g, the claimed error bound now follows from (9), (10), and the observation that…”

Section: Corollary 31 Assume That H Is Continuously Distributed Witmentioning

confidence: 98%

“…This loss of generality is acceptable, because it is possible to construct the convex envelope of the function Q if h is discretely distributed [10]. …”

Section: Corollary 31 Assume That H Is Continuously Distributed Witmentioning

confidence: 99%

“…The error bounds are derived in case the distributions of the random right hand sides are continuous. For the case with discretely distributed h it is possible to construct the convex envelope of the function Q , see [10].…”

Section: Notesmentioning

confidence: 99%

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Approximation in two-stage stochastic integer programming

Romeijnders

Stougie

Vlerk³

2014

Surveys in Operations Research and Management Science

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h i g h l i g h t s• We survey complexity results for stochastic (integer) programming problems.• We survey convex approximations of stochastic integer programming objectives.• We survey approximation guarantees for stochastic integer programming problems. a b s t r a c t Approximation algorithms are the prevalent solution methods in the field of stochastic programming. Problems in this field are very hard to solve. Indeed, most of the research in this field has concentrated on designing solution methods that approximate the optimal solution value. However, efficiency in the complexity theoretical sense is usually not taken into account. Quality statements mostly remain restricted to convergence to an optimal solution without accompanying implications on the running time of the algorithms for attaining more and more accurate solutions. a r t i c l e i n f oHowever, over the last thirty years also some studies on performance analysis of approximation algorithms for stochastic programming have appeared. In this direction we find both probabilistic analysis and worst-case analysis.Recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most deterministic combinatorial optimization problems. Polynomial-time approximation algorithms and their performance guarantees for stochastic linear and integer programming problems have received increasing research attention only very recently.Approximation in the traditional stochastic programming sense will not be discussed in this chapter. The reader interested in this issue is referred to surveys on stochastic programming, like the Handbook on Stochastic Programming by Ruszczyński and Shapiro (2003) or the textbooks by Birge and Louveaux (1997), Kall and Wallace (1994), Prékopa (1995), and Shapiro et al. (2009. We concentrate on the studies of approximation algorithms in relation to computational complexity theory.With this survey we intend to give a flavor of the type of results existing in the literature on approximation algorithms in two-stage stochastic integer programming rather than a complete overview of the literature on the subject. We do so by exhibiting a representative selection of results, which we present in full detail. While presenting them we do not refer to the literature; these references, together with pointers to other relevant work in this field of research, are given in an extensive notes section at the end of the survey.

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“…For α-approximations of expected surplus function g, the claimed error bound now follows from (9), (10), and the observation that…”

Section: Corollary 31 Assume That H Is Continuously Distributed Witmentioning

confidence: 98%

“…This loss of generality is acceptable, because it is possible to construct the convex envelope of the function Q if h is discretely distributed [10]. …”

Section: Corollary 31 Assume That H Is Continuously Distributed Witmentioning

confidence: 99%

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Approximation in two-stage stochastic integer programming

Romeijnders

Stougie

Vlerk³

2014

Surveys in Operations Research and Management Science

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“…Our main motivation to study this model is that it is the simplest extension of pureinteger recourse models, which we studied in a number of papers (Klein Haneveld et al 1996Louveaux and Van der Vlerk 1993;Van der Vlerk 2004). In particular, we will see that the approach which we developed to construct convex approximations for the recourse function Q in the pure-integer case, can be extended to this mixed-integer recourse model.…”

Section: Introductionmentioning

confidence: 99%

Convex approximations for a class of mixed-integer recourse models

Vlerk

2009

Ann Oper Res

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“…The L-shaped method was used to solve stochastic programs having discrete decisions in the first stage (Laporte and Louveaux [13]), This method was applied to solve a stochastic concentrator location problem (Shiina [18,19] [15] investigated the property of the problem, Klein Haneveld, Stougie, and van der Vlerk [11,12] proposed an algorithm to construct a convex envelope of the recourse function. Step…”

Section: Introductionmentioning

confidence: 99%