Stability in multistage stochastic programming

Multistage stochastic programs are regarded as mathematical programs in a Banach spaceX of summable functions. Relying on a result for parametric programs in Banach spaces, the paper presents conditions under which linearly constrained convex multistage problems behave stably when the (input) data process is subjected to (small) perturbations. In particular, we show the persistence of optimal solutions, the local Lipschitz continuity of the optimal value and the upper semicontinuity of optimal sets with respect to the weak topology inX. The linear case with deterministic first-stage decisions is studied in more detail.

[1]  Karl Prauendorfer,et al.  Multistage stochastic programming: Error analysis for the convex case , 1994, Math. Methods Oper. Res..

[2]  B. Bank,et al.  Non-Linear Parametric Optimization , 1983 .

[3]  Werner Römisch,et al.  Stability of Solutions for Stochastic Programs with Complete Recourse , 1993, Math. Oper. Res..

[4]  Jitka Dupacová,et al.  Postoptimality for multistage stochastic linear programs , 1995, Ann. Oper. Res..

[5]  V. Zolotarev.  METRIC DISTANCES IN SPACES OF RANDOM VARIABLES AND THEIR DISTRIBUTIONS , 1976 .

[6]  Paul Olsen.  Multistage Stochastic Programming with Recourse As Mathematical Programming in an $L_p $ Space , 1976 .

[7]  Werner Römisch,et al.  Obtaining convergence rates for approximations in stochastic programming , 1987 .

[8]  M. V. F. Pereira,et al.  Multi-stage stochastic optimization applied to energy planning , 1991, Math. Program..

[9]  Sjur Didrik Flåm.  Lagrange multipliers in stochastic programming , 1992 .

[10]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[11]  Roger J.-B. Wets.  Stochastic programs with recourse: A basic theorem for multistage problems , 1972 .

[12]  Paul Olsen.  Discretizations of multistage stochastic programming problems , 1976 .

[13]  F. Hiai,et al.  Integrals, conditional expectations, and martingales of multivalued functions , 1977 .

[14]  Peter Kall,et al.  On approximations and stability in stochastic programming , 1987 .

[15]  R. Rockafellar,et al.  Nonanticipativity and L1-martingales in stochastic optimization problems , 1976 .

[16]  A. Kirsch,et al.  Continuous perturbations of infinite optimization problems , 1980 .

[17]  R. Rockafellar,et al.  The Optimal Recourse Problem in Discrete Time: $L^1 $-Multipliers for Inequality Constraints , 1978 .

[18]  Jim Freeman.  Probability Metrics and the Stability of Stochastic Models , 1991 .

[19]  I. V. Evstigneev,et al.  Measurable Selection and Dynamic Programming , 1976, Math. Oper. Res..

[20]  R. Rockafellar,et al.  Integral functionals, normal integrands and measurable selections , 1976 .

[21]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[22]  Jitka Dupa.  Postoptimality for multistage stochastic linear programs , 1995 .

[23]  S. M. Robinson.  Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems , 1976 .

[24]  S. M. Robinson,et al.  Stability in two-stage stochastic programming , 1987 .

[25]  S. M. Robinson.  Stability Theory for Systems of Inequalities. Part I: Linear Systems , 1975 .

[26]  R. Lepp.  Discrete approximation of linear two–stage stochastic programming problem , 1987 .

[27]  Jitka Dupacová,et al.  Multistage stochastic programs: The state-of-the-art and selected bibliography , 1995, Kybernetika.

[28]  J. Zowe,et al.  Regularity and stability for the mathematical programming problem in Banach spaces , 1979 .

[29]  Igor V. Evstigneev,et al.  Stochastic models of control and economic dynamics , 1987 .

[30]  J. Dupacová.  Stability and sensitivity-analysis for stochastic programming , 1991 .

[31]  Werner Römisch,et al.  Stability analysis for stochastic programs , 1991, Ann. Oper. Res..

[32]  Diethard Klatte,et al.  On Procedures for Analysing Parametric Optimization Problems , 1982 .

[33]  R. Rockafellar.  LEVEL SETS AND CONTINUITY OF CONJUGATE CONVEX FUNCTIONS , 1966 .

[34]  M. Eisner,et al.  Duality for Stochastic Programming Interpreted as L. P. in $L_p $-Space , 1975 .

[35]  M. Dempster.  On stochastic programming. II: Dynamic problems under risk , 1988 .

[36]  John R. Birge,et al.  Optimality conditions for match-up strategies in stochastic scheduling , 1992 .

[37]  Roger J.-B. Wets,et al.  On the Relation between Stochastic and Deterministic Optimization , 1975 .

[38]  R. Wets,et al.  Stochastic programming , 1989 .