Fully polynomial time (Σ,Π)-approximation schemes for continuous nonlinear newsvendor and continuous stochastic dynamic programs

We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone co...

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Veröffentlicht in:	Mathematical programming Jg. 195; H. 1-2; S. 183 - 242
Hauptverfasser:	Halman, Nir, Nannicini, Giacomo
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2022 Springer Springer Nature B.V
Schlagworte:	Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Cost function Full Length Paper Hardness Logistics Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Mechanical properties Numerical Analysis Polynomials Supply chains Theoretical Stochastic dynamic programming approximation sets and functions Approximation algorithms Newsvendor problem Stochastic inventory control Hardness of approximation
ISSN:	0025-5610, 1436-4646
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Abstract	We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that these problems cannot be approximated to any degree of either relative or additive error, regardless of the scheme used. To circumvent these hardness results, we generalize the concept of fully polynomial-time approximation scheme allowing arbitrarily small additive and multiplicative error at the same time, while requiring a polynomial running time in the input size and the error parameters. We develop approximation schemes of this type for the classes of problems mentioned above. In light of our hardness results, such approximation schemes are “best possible”. A computational evaluation shows the promise of this approach.
AbstractList	We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that these problems cannot be approximated to any degree of either relative or additive error, regardless of the scheme used. To circumvent these hardness results, we generalize the concept of fully polynomial-time approximation scheme allowing arbitrarily small additive and multiplicative error at the same time, while requiring a polynomial running time in the input size and the error parameters. We develop approximation schemes of this type for the classes of problems mentioned above. In light of our hardness results, such approximation schemes are “best possible”. A computational evaluation shows the promise of this approach.
Audience	Academic
Author	Nannicini, Giacomo Halman, Nir
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Keywords	Stochastic dynamic programming approximation sets and functions Approximation algorithms Newsvendor problem Stochastic inventory control Hardness of approximation
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References_xml	– reference: DworkCKenthapadiKMcSherryFMironovINaorMOur data, ourselves: privacy via distributed noise generationAdv. Cryptol. EUROCRYPT2006200648650324235601140.94336 – reference: Van HoeselAPMWagelmansCPMFully polynomial approximation schemes for single-item capacitated economic lot-sizing problemsMath. Oper. Res.200126233935718958331082.90532 – reference: PowellWBRuszczyńskiATopalogluHLearning algorithms for separable approximations of discrete stochastic optimization problemsMath. Oper. Res.200429481483621041561082.90079 – reference: ChubanovSKovalyovMYPeschEAn FPTAS for a single-item capacitated economic lot-sizing problem with monotone cost structureMath. Program.2006106245346622167901134.90003 – reference: GodfreyGAPowellWBAn adaptive, distribution-free algorithm for the newsvendor problem with censored demands, with applications to inventory and distributionManag. Sci.2001478110111121232.90053 – reference: ChowCSTsitsiklisJNThe complexity of dynamic programmingJ. Complex.19895446648810289080685.90098 – reference: LawlerELLenstraJKRinnooy KanAHGShmoysDBGravesSCRinnooy KanAHGZipkinPHSequencing and scheduling: algorithms and complexityHandbooks in Operations Research and Management Science1993AmsterdamNorth-Holland445522 – reference: Nemirovski, A.S.: Information-based Complexity of Convex Programming. Lecture Notes (1995) – reference: NascimentoJMPowellWBDynamic programming models and algorithms for the mutual fund cash balance problemManag. Sci.20105658018151232.90341 – reference: TraubJFWasilkowskiGWWozniakowskiHInformation-Based Complexity1988New YorkAcademic Press0654.94004 – reference: Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin-packing problem. 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SubjectTerms	Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Cost function Full Length Paper Hardness Logistics Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Mechanical properties Numerical Analysis Polynomials Supply chains Theoretical
Title	Fully polynomial time (Σ,Π)-approximation schemes for continuous nonlinear newsvendor and continuous stochastic dynamic programs
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