Quantitative stability of mixed-integer two-stage quadratic stochastic programs
For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is...
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| Vydané v: | Mathematical methods of operations research (Heidelberg, Germany) Ročník 75; číslo 2; s. 149 - 163 |
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| Médium: | Journal Article |
| Jazyk: | English |
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Berlin/Heidelberg
Springer-Verlag
01.04.2012
Springer Springer Nature B.V |
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| ISSN: | 1432-2994, 1432-5217 |
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| Abstract | For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs. |
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| AbstractList | For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs. For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.[PUBLICATION ABSTRACT] |
| Author | Han, Youpan Chen, Zhiping |
| Author_xml | – sequence: 1 givenname: Zhiping surname: Chen fullname: Chen, Zhiping email: zchen@mail.xjtu.edu.cn organization: Faculty of Science, Department of Scientific Computing and Applied Software, Xi’an Jiaotong University – sequence: 2 givenname: Youpan surname: Han fullname: Han, Youpan organization: Faculty of Science, Department of Scientific Computing and Applied Software, Xi’an Jiaotong University |
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| DOI | 10.1007/s00186-010-0326-1 |
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| Keywords | Stochastic quadratic programs Stability Probability metric Mixed-integer Lipschitz continuity |
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| References_xml | – reference: RömischWVigerskeSQuantitative stability of fully random mixed-integer two-stage stochastic programsOptim Lett2008237738823994681189.9011110.1007/s11590-007-0066-1 – reference: BankBGuddatJKlatteDKummerBTammerKNon-linear parametric optimization1982BerlinAkademie Verlag – reference: GranotFSkorin-KapovJSome proximity and sensitivity in quadratic integer programmingMath Program19904725926810593950714.9007310.1007/BF01580862 – reference: ChenXWomersleyRSRandom test problems and parallel methods for quadratic programs and quadratic stochastic programsOptim Method Softw20001327530617879470980.9006010.1080/10556780008805789 – reference: Römisch W (2003) Stability of stochastic programming. In: Stochastic programming, handbooks in operations research and management science, vol 10. Elsevier, Amsterdam, pp 483–554 – reference: SchultzROn structure and stability in stochastic programs with random technology matrix and complete integer recourseMath Program19957073890841.90101 – reference: Curtis EavesBOn quadratic programmingManag Sci1971171169871110.1287/mnsc.17.11.698 – reference: RockafellerRTWetsRJ-BVariational analysis1998BerlinSpringer10.1007/978-3-642-02431-3 – reference: ChoG-MLog-barrier method for two-stage quadratic stochastic programmingAppl Math Comput2005164456921299771071.6508310.1016/j.amc.2004.04.095 – reference: SenSAardalKAlgorithms for stochastic mixed-integer programming models. Chapter 9 in Discrete OptimizationHandbooks in operations research and management science, vol 122005AmsterdamElsevier515558 – reference: RachevSTRömischWQuantitative stability in stochastic programming: the methods of probability metricsMath Oper Res20022779281819391781082.9008010.1287/moor.27.4.792.304 – reference: ChenZXuZContinuity and stability of two-stage quadratic stochastic programs with mixed-integer recourseNumer Funct Anal Optim20093046247725438791176.9043610.1080/01630560902920668 – reference: Klein HaneveldWKvan der VlerkMHStochastic integer programming: general models and algorithmsAnn Oper Res199985395716855960920.9011010.1023/A:1018930113099 – reference: RockafellarRTWetsRJ-BA Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programmingMath Program Study19862863938367620599.9009010.1007/BFb0121126 – reference: TakritiSKrasenbrinkBWuLS-YA stochastic model for the unit commitment problemIEEE Trans Power Syst1996111497150810.1109/59.535691 – reference: SchultzRRates of convergence in stochastic programs with complete integer recourseSIAM J Optim199661138115214165330868.9008810.1137/S1052623494271655 – volume: 17 start-page: 698 issue: 11 year: 1971 ident: 326_CR5 publication-title: Manag Sci doi: 10.1287/mnsc.17.11.698 – volume: 85 start-page: 39 year: 1999 ident: 326_CR7 publication-title: Ann Oper Res doi: 10.1023/A:1018930113099 – volume: 27 start-page: 792 year: 2002 ident: 326_CR8 publication-title: Math Oper Res doi: 10.1287/moor.27.4.792.304 – volume: 164 start-page: 45 year: 2005 ident: 326_CR4 publication-title: Appl Math Comput doi: 10.1016/j.amc.2004.04.095 – volume: 30 start-page: 462 year: 2009 ident: 326_CR3 publication-title: Numer Funct Anal Optim doi: 10.1080/01630560902920668 – volume: 47 start-page: 259 year: 1990 ident: 326_CR6 publication-title: Math Program doi: 10.1007/BF01580862 – start-page: 515 volume-title: Handbooks in operations research and management science, vol 12 year: 2005 ident: 326_CR15 – volume: 28 start-page: 63 year: 1986 ident: 326_CR9 publication-title: Math Program Study doi: 10.1007/BFb0121126 – ident: 326_CR11 doi: 10.1016/S0927-0507(03)10008-4 – volume-title: Non-linear parametric optimization year: 1982 ident: 326_CR1 doi: 10.1007/978-3-0348-6328-5 – volume-title: Variational analysis year: 1998 ident: 326_CR10 doi: 10.1007/978-3-642-02431-3 – volume: 70 start-page: 73 year: 1995 ident: 326_CR13 publication-title: Math Program – volume: 6 start-page: 1138 year: 1996 ident: 326_CR14 publication-title: SIAM J Optim doi: 10.1137/S1052623494271655 – volume: 2 start-page: 377 year: 2008 ident: 326_CR12 publication-title: Optim Lett doi: 10.1007/s11590-007-0066-1 – volume: 11 start-page: 1497 year: 1996 ident: 326_CR16 publication-title: IEEE Trans Power Syst doi: 10.1109/59.535691 – volume: 13 start-page: 275 year: 2000 ident: 326_CR2 publication-title: Optim Method Softw doi: 10.1080/10556780008805789 |
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| Title | Quantitative stability of mixed-integer two-stage quadratic stochastic programs |
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