American Institute of Mathematical Sciences

Advanced
2012, 2(4): 713-738. doi: 10.3934/naco.2012.2.713

Two-stage stochastic programs: Integer variables, dominance relations and PDE constraints

Rüdiger Schultz 1,

1. 

Department of Mathematics, University of Duisburg-Essen, Campus Duisburg, Lotharstr. 65, D-47048 Duisburg, Germany

Received  December 2011 Revised  November 2012 Published  November 2012

From a unified point-of-view, we present some recent developments in two-stage stochastic programming. Our discussion includes stochastic programs with integer variables, stochastic programs with dominance constraints, and PDE constrained stochastic programs.
Keywords: PDE constrained optimization., stochastic dominance, mixed-integer optimization, Stochastic integer programming.
Mathematics Subject Classification: Primary: 90C15, 90C11, 60E1.
Citation: Rüdiger Schultz. Two-stage stochastic programs: Integer variables, dominance relations and PDE constraints. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 713-738. doi: 10.3934/naco.2012.2.713
References:
[1]

B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammers, "Non-linear Parametric Optimization," Akademie-Verlag, Berlin, 1983.  Google Scholar

[2]

B. Bank and R. Mandel, "Parametric Integer Optimization," Akademie-Verlag, Berlin 1988.  Google Scholar

[3]

A. Ben-Tal, L. El-Ghaoui and A. Nemirovski, "Robust Optimization," Princeton University Press, Princeton and Oxford, 2009.  Google Scholar

[4]

J. R. Birge and F. Louveaux, "Introduction to Stochastic Programming," Springer-Verlag, New York, 1997.  Google Scholar

[5]

C. E. Blair and R. G. Jeroslow, The value function of a mixed integer program: I, Discrete Mathematics, 19 (1977), 121-138.  Google Scholar

[6]

C. C. Carøe and R. Schultz, Dual decomposition in stochastic integer programming, Operations Research Letters, 24 (1999), 37-45.  Google Scholar

[7]

M. Carrión, U. Gotzes and R. Schultz, Risk aversion for an electricity retailer with second-order stochastic dominance constraints, Computational Management Science, 6 (2009), 233-250.  Google Scholar

[8]

P. G. Ciarlet, "Mathematical Elasticity Volume I: Three-Dimensional Elasticity," Studies in Mathematics and its Applications, Vol. 20, North-Holland, 1988.  Google Scholar

[9]

, CPLEX Callable Library-9.1.3, ILOG, 2008. Available from:, , ().   Google Scholar

[10]

S. Conti, H. Held, M. Pach, M. Rumpf and R. Schultz, Shape optimization under uncertainty - a stochastic programming perspective, SIAM Journal on Optimization, 19 (2008), 1610-1632.  Google Scholar

[11]

S. Conti, H. Held, M. Pach, M. Rumpf and R. Schultz, Risk averse shape optimization, SIAM Journal on Control and Optimization, 49 (2011), 927-947.  Google Scholar

[12]

M. C. Delfour and J. P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus and Optimization," SIAM, Philadelphia, 2001.  Google Scholar

[13]

D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM Journal on Optimization, 14 (2003), 548-566.  Google Scholar

[14]

D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization with nonlinear dominance constraints, Mathematical Programming, 99 (2004), 329-350. doi: 10.1007/s10107-003-0453-z.  Google Scholar

[15]

R. Gollmer, U. Gotzes and R. Schultz, A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse, Mathematical Programming, 127 (2011), 179-190.  Google Scholar

[16]

R. Gollmer, F. Neise and R. Schultz, Stochastic programs with first-order dominance constraints induced by mixed-integer linear recourse, SIAM Journal on Optimization, 19 (2008), 552-571.  Google Scholar

[17]

U. Gotzes and F. Neise, "User's Guide to ddsip.vSD - A C Package for the Dual Decomposition of Stochastic Programs with Dominance Constraints Induced by Mixed-Integer Linear Recourse," Department of Mathematics, University of Duisburg-Essen, 2008. Google Scholar

[18]

E. Handschin, F. Neise, H. Neumann and R. Schultz, Optimal operation of dispersed generation under uncertainty using mathematical programming, International Journal of Electrical Power & Energy Systems, 28 (2006), 618-626. Google Scholar

[19]

A. Märkert and R. Gollmer, "User's Guide to ddsip - A C Package for the Dual Decomposition of Two-Stage Stochastic Programs with Mixed-Integer Recourse," Department of Mathematics, University of Duisburg-Essen, 2008; Available from: http://www.neos-server.org/neos/solvers/slp:ddsip/MPS.html. Google Scholar

[20]

A. Müller and D. Stoyan, "Comparison Methods for Stochastic Models and Risks," Wiley, Chichester, 2002.  Google Scholar

[21]

G. L. Nemhauser and L. A. Wolsey, "Integer and Combinatorial Optimization," Wiley, New York 1988.  Google Scholar

[22]

A. Prékopa, "Stochastic Programming," Kluwer, Dordrecht, 1995.  Google Scholar

[23]

A. Ruszczyński and A. Shapiro, "Stochastic Programming," Handbooks in Operations Research and Management Science, Elsevier, Amsterdam, 10 (2003).  Google Scholar

[24]

R. Schultz, Continuity properties of expectation functions in stochastic integer programming, Mathematics of Operations Research, 18 (1993), 578-589.  Google Scholar

[25]

R. Schultz, On structure and stability in stochastic programs with random technology matrix and complete integer recourse, Mathematical Programming, 70 (1995), 73-89.  Google Scholar

[26]

R. Schultz, Stochastic programming with integer variables, Mathematical Programming, 97 (2003), 285-309.  Google Scholar

[27]

R. Schultz and S. Tiedemann, Risk Aversion via Excess Probabilities in Stochastic Programs with Mixed-Integer Recourse, SIAM Journal on Optimization, 14 (2003), 115-138.  Google Scholar

[28]

A. Shapiro, D. Dentcheva and A. Ruszczyński, "Lectures on Stochastic Programming: Modeling and Theory," SIAM-MPS, Philadelphia, 2009.  Google Scholar

[29]

J. Sokołowski and J. P. Zolésio, "Introduction to Shape Optimization: Shape Sensitivity Analysis," Springer, 1992.  Google Scholar

show all references

References:
[1]

B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammers, "Non-linear Parametric Optimization," Akademie-Verlag, Berlin, 1983.  Google Scholar

[2]

B. Bank and R. Mandel, "Parametric Integer Optimization," Akademie-Verlag, Berlin 1988.  Google Scholar

[3]

A. Ben-Tal, L. El-Ghaoui and A. Nemirovski, "Robust Optimization," Princeton University Press, Princeton and Oxford, 2009.  Google Scholar

[4]

J. R. Birge and F. Louveaux, "Introduction to Stochastic Programming," Springer-Verlag, New York, 1997.  Google Scholar

[5]

C. E. Blair and R. G. Jeroslow, The value function of a mixed integer program: I, Discrete Mathematics, 19 (1977), 121-138.  Google Scholar

[6]

C. C. Carøe and R. Schultz, Dual decomposition in stochastic integer programming, Operations Research Letters, 24 (1999), 37-45.  Google Scholar

[7]

M. Carrión, U. Gotzes and R. Schultz, Risk aversion for an electricity retailer with second-order stochastic dominance constraints, Computational Management Science, 6 (2009), 233-250.  Google Scholar

[8]

P. G. Ciarlet, "Mathematical Elasticity Volume I: Three-Dimensional Elasticity," Studies in Mathematics and its Applications, Vol. 20, North-Holland, 1988.  Google Scholar

[9]

, CPLEX Callable Library-9.1.3, ILOG, 2008. Available from:, , ().   Google Scholar

[10]

S. Conti, H. Held, M. Pach, M. Rumpf and R. Schultz, Shape optimization under uncertainty - a stochastic programming perspective, SIAM Journal on Optimization, 19 (2008), 1610-1632.  Google Scholar

[11]

S. Conti, H. Held, M. Pach, M. Rumpf and R. Schultz, Risk averse shape optimization, SIAM Journal on Control and Optimization, 49 (2011), 927-947.  Google Scholar

[12]

M. C. Delfour and J. P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus and Optimization," SIAM, Philadelphia, 2001.  Google Scholar

[13]

D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM Journal on Optimization, 14 (2003), 548-566.  Google Scholar

[14]

D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization with nonlinear dominance constraints, Mathematical Programming, 99 (2004), 329-350. doi: 10.1007/s10107-003-0453-z.  Google Scholar

[15]

R. Gollmer, U. Gotzes and R. Schultz, A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse, Mathematical Programming, 127 (2011), 179-190.  Google Scholar

[16]

R. Gollmer, F. Neise and R. Schultz, Stochastic programs with first-order dominance constraints induced by mixed-integer linear recourse, SIAM Journal on Optimization, 19 (2008), 552-571.  Google Scholar

[17]

U. Gotzes and F. Neise, "User's Guide to ddsip.vSD - A C Package for the Dual Decomposition of Stochastic Programs with Dominance Constraints Induced by Mixed-Integer Linear Recourse," Department of Mathematics, University of Duisburg-Essen, 2008. Google Scholar

[18]

E. Handschin, F. Neise, H. Neumann and R. Schultz, Optimal operation of dispersed generation under uncertainty using mathematical programming, International Journal of Electrical Power & Energy Systems, 28 (2006), 618-626. Google Scholar

[19]

A. Märkert and R. Gollmer, "User's Guide to ddsip - A C Package for the Dual Decomposition of Two-Stage Stochastic Programs with Mixed-Integer Recourse," Department of Mathematics, University of Duisburg-Essen, 2008; Available from: http://www.neos-server.org/neos/solvers/slp:ddsip/MPS.html. Google Scholar

[20]

A. Müller and D. Stoyan, "Comparison Methods for Stochastic Models and Risks," Wiley, Chichester, 2002.  Google Scholar

[21]

G. L. Nemhauser and L. A. Wolsey, "Integer and Combinatorial Optimization," Wiley, New York 1988.  Google Scholar

[22]

A. Prékopa, "Stochastic Programming," Kluwer, Dordrecht, 1995.  Google Scholar

[23]

A. Ruszczyński and A. Shapiro, "Stochastic Programming," Handbooks in Operations Research and Management Science, Elsevier, Amsterdam, 10 (2003).  Google Scholar

[24]

R. Schultz, Continuity properties of expectation functions in stochastic integer programming, Mathematics of Operations Research, 18 (1993), 578-589.  Google Scholar

[25]

R. Schultz, On structure and stability in stochastic programs with random technology matrix and complete integer recourse, Mathematical Programming, 70 (1995), 73-89.  Google Scholar

[26]

R. Schultz, Stochastic programming with integer variables, Mathematical Programming, 97 (2003), 285-309.  Google Scholar

[27]

R. Schultz and S. Tiedemann, Risk Aversion via Excess Probabilities in Stochastic Programs with Mixed-Integer Recourse, SIAM Journal on Optimization, 14 (2003), 115-138.  Google Scholar

[28]

A. Shapiro, D. Dentcheva and A. Ruszczyński, "Lectures on Stochastic Programming: Modeling and Theory," SIAM-MPS, Philadelphia, 2009.  Google Scholar

[29]

J. Sokołowski and J. P. Zolésio, "Introduction to Shape Optimization: Shape Sensitivity Analysis," Springer, 1992.  Google Scholar

[1]

René Henrion, Christian Küchler, Werner Römisch. Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 363-384. doi: 10.3934/jimo.2008.4.363
[2]

Zhimin Liu, Shaojian Qu, Hassan Raza, Zhong Wu, Deqiang Qu, Jianhui Du. Two-stage mean-risk stochastic mixed integer optimization model for location-allocation problems under uncertain environment. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2783-2804. doi: 10.3934/jimo.2020094
[3]

Elham Mardaneh, Ryan Loxton, Qun Lin, Phil Schmidli. A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1601-1623. doi: 10.3934/jimo.2017009
[4]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
[5]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[6]

Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353
[7]

Ye Tian, Cheng Lu. Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1027-1039. doi: 10.3934/jimo.2011.7.1027
[8]

Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557
[9]

Meng Xue, Yun Shi, Hailin Sun. Portfolio optimization with relaxation of stochastic second order dominance constraints via conditional value at risk. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2581-2602. doi: 10.3934/jimo.2019071
[10]

Louis Caccetta, Syarifah Z. Nordin. Mixed integer programming model for scheduling in unrelated parallel processor system with priority consideration. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 115-132. doi: 10.3934/naco.2014.4.115
[11]

Edward S. Canepa, Alexandre M. Bayen, Christian G. Claudel. Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming. Networks & Heterogeneous Media, 2013, 8 (3) : 783-802. doi: 10.3934/nhm.2013.8.783
[12]

Mahdi Roozbeh, Saman Babaie–Kafaki, Zohre Aminifard. Two penalized mixed–integer nonlinear programming approaches to tackle multicollinearity and outliers effects in linear regression models. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3475-3491. doi: 10.3934/jimo.2020128
[13]

Fanwen Meng, Kiok Liang Teow, Kelvin Wee Sheng Teo, Chee Kheong Ooi, Seow Yian Tay. Predicting 72-hour reattendance in emergency departments using discriminant analysis via mixed integer programming with electronic medical records. Journal of Industrial & Management Optimization, 2019, 15 (2) : 947-962. doi: 10.3934/jimo.2018079
[14]

Wan Nor Ashikin Wan Ahmad Fatthi, Adibah Shuib, Rosma Mohd Dom. A mixed integer programming model for solving real-time truck-to-door assignment and scheduling problem at cross docking warehouse. Journal of Industrial & Management Optimization, 2016, 12 (2) : 431-447. doi: 10.3934/jimo.2016.12.431
[15]

Yu Chen, Yonggang Li, Bei Sun, Chunhua Yang, Hongqiu Zhu. Multi-objective chance-constrained blending optimization of zinc smelter under stochastic uncertainty. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021169
[16]

Zhenhuan Yang, Wei Shen, Yiming Ying, Xiaoming Yuan. Stochastic AUC optimization with general loss. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4191-4212. doi: 10.3934/cpaa.2020188
[17]

Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2675-2701. doi: 10.3934/jimo.2019075
[18]

Zhenbo Wang, Shu-Cherng Fang, David Y. Gao, Wenxun Xing. Global extremal conditions for multi-integer quadratic programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 213-225. doi: 10.3934/jimo.2008.4.213
[19]

Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67
[20]

Mohamed A. Tawhid, Ahmed F. Ali. A simplex grey wolf optimizer for solving integer programming and minimax problems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 301-323. doi: 10.3934/naco.2017020

 Impact Factor: 

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences