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Twostage stochastic programs: Integer variables, dominance relations and PDE constraints
1.  Department of Mathematics, University of DuisburgEssen, Campus Duisburg, Lotharstr. 65, D47048 Duisburg, Germany 
References:
[1] 
B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammers, "Nonlinear Parametric Optimization,", AkademieVerlag, (1983). Google Scholar 
[2] 
B. Bank and R. Mandel, "Parametric Integer Optimization,", AkademieVerlag, (1988). Google Scholar 
[3] 
A. BenTal, L. ElGhaoui and A. Nemirovski, "Robust Optimization,", Princeton University Press, (2009). Google Scholar 
[4] 
J. R. Birge and F. Louveaux, "Introduction to Stochastic Programming,", SpringerVerlag, (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 secondorder stochastic dominance constraints,, Computational Management Science, 6 (2009), 233. Google Scholar 
[8] 
P. G. Ciarlet, "Mathematical Elasticity Volume I: ThreeDimensional Elasticity,", Studies in Mathematics and its Applications, (1988). Google Scholar 
[9] 
, CPLEX Callable Library9.1.3, 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/s101070030453z. Google Scholar 
[15] 
R. Gollmer, U. Gotzes and R. Schultz, A note on secondorder stochastic dominance constraints induced by mixedinteger linear recourse,, Mathematical Programming, 127 (2011), 179. Google Scholar 
[16] 
R. Gollmer, F. Neise and R. Schultz, Stochastic programs with firstorder dominance constraints induced by mixedinteger 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 MixedInteger 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 TwoStage Stochastic Programs with MixedInteger 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 MixedInteger 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,", SIAMMPS, (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, "Nonlinear Parametric Optimization,", AkademieVerlag, (1983). Google Scholar 
[2] 
B. Bank and R. Mandel, "Parametric Integer Optimization,", AkademieVerlag, (1988). Google Scholar 
[3] 
A. BenTal, L. ElGhaoui and A. Nemirovski, "Robust Optimization,", Princeton University Press, (2009). Google Scholar 
[4] 
J. R. Birge and F. Louveaux, "Introduction to Stochastic Programming,", SpringerVerlag, (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 secondorder stochastic dominance constraints,, Computational Management Science, 6 (2009), 233. Google Scholar 
[8] 
P. G. Ciarlet, "Mathematical Elasticity Volume I: ThreeDimensional Elasticity,", Studies in Mathematics and its Applications, (1988). Google Scholar 
[9] 
, CPLEX Callable Library9.1.3, 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/s101070030453z. Google Scholar 
[15] 
R. Gollmer, U. Gotzes and R. Schultz, A note on secondorder stochastic dominance constraints induced by mixedinteger linear recourse,, Mathematical Programming, 127 (2011), 179. Google Scholar 
[16] 
R. Gollmer, F. Neise and R. Schultz, Stochastic programs with firstorder dominance constraints induced by mixedinteger 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 MixedInteger 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 TwoStage Stochastic Programs with MixedInteger 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 MixedInteger 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,", SIAMMPS, (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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