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, (1983).   Google Scholar

[2]

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

[3]

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

[4]

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

[6]

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

[8]

P. G. Ciarlet, "Mathematical Elasticity Volume I: Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, (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.   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.   Google Scholar

[12]

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

[13]

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

[14]

D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization with nonlinear dominance constraints,, Mathematical Programming, 99 (2004), 329.  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.   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.   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, (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.   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, (2008).   Google Scholar

[20]

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

[21]

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

[22]

A. Prékopa, "Stochastic Programming,", Kluwer, (1995).   Google Scholar

[23]

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

[24]

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

[26]

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

[28]

A. Shapiro, D. Dentcheva and A. Ruszczyński, "Lectures on Stochastic Programming: Modeling and Theory,", SIAM-MPS, (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, (1983).   Google Scholar

[2]

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

[3]

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

[4]

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

[6]

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

[8]

P. G. Ciarlet, "Mathematical Elasticity Volume I: Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, (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.   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.   Google Scholar

[12]

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

[13]

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

[14]

D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization with nonlinear dominance constraints,, Mathematical Programming, 99 (2004), 329.  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.   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.   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, (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.   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, (2008).   Google Scholar

[20]

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

[21]

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

[22]

A. Prékopa, "Stochastic Programming,", Kluwer, (1995).   Google Scholar

[23]

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

[24]

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

[26]

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

[28]

A. Shapiro, D. Dentcheva and A. Ruszczyński, "Lectures on Stochastic Programming: Modeling and Theory,", SIAM-MPS, (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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