Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 90C15, 90C11, 60E15.

 Citation:

•  [1] B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammers, "Non-linear Parametric Optimization," Akademie-Verlag, Berlin, 1983. [2] B. Bank and R. Mandel, "Parametric Integer Optimization," Akademie-Verlag, Berlin 1988. [3] A. Ben-Tal, L. El-Ghaoui and A. Nemirovski, "Robust Optimization," Princeton University Press, Princeton and Oxford, 2009. [4] J. R. Birge and F. Louveaux, "Introduction to Stochastic Programming," Springer-Verlag, New York, 1997. [5] C. E. Blair and R. G. Jeroslow, The value function of a mixed integer program: I, Discrete Mathematics, 19 (1977), 121-138. [6] C. C. Carøe and R. Schultz, Dual decomposition in stochastic integer programming, Operations Research Letters, 24 (1999), 37-45. [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. [8] P. G. Ciarlet, "Mathematical Elasticity Volume I: Three-Dimensional Elasticity," Studies in Mathematics and its Applications, Vol. 20, North-Holland, 1988. [9] CPLEX Callable Library-9.1.3, ILOG, 2008. Available from: http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/. [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. [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. [12] M. C. Delfour and J. P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus and Optimization," SIAM, Philadelphia, 2001. [13] D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM Journal on Optimization, 14 (2003), 548-566. [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. [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. [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. [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. [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. [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. [20] A. Müller and D. Stoyan, "Comparison Methods for Stochastic Models and Risks," Wiley, Chichester, 2002. [21] G. L. Nemhauser and L. A. Wolsey, "Integer and Combinatorial Optimization," Wiley, New York 1988. [22] A. Prékopa, "Stochastic Programming," Kluwer, Dordrecht, 1995. [23] A. Ruszczyński and A. Shapiro, "Stochastic Programming," Handbooks in Operations Research and Management Science, Elsevier, Amsterdam, 10 (2003). [24] R. Schultz, Continuity properties of expectation functions in stochastic integer programming, Mathematics of Operations Research, 18 (1993), 578-589. [25] R. Schultz, On structure and stability in stochastic programs with random technology matrix and complete integer recourse, Mathematical Programming, 70 (1995), 73-89. [26] R. Schultz, Stochastic programming with integer variables, Mathematical Programming, 97 (2003), 285-309. [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. [28] A. Shapiro, D. Dentcheva and A. Ruszczyński, "Lectures on Stochastic Programming: Modeling and Theory," SIAM-MPS, Philadelphia, 2009. [29] J. Sokołowski and J. P. Zolésio, "Introduction to Shape Optimization: Shape Sensitivity Analysis," Springer, 1992.