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

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